The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

THE MA YER-VIETORIS PRINCIPLE F OR GR OTHENDIECK-WITT GROUPS OF SCHEMES MARCO SCHLICHTING Abstract. W e prov e lo calization and Zariski-May er-Vietoris for hi gher Gro- thendiec k-Witt groups, alias hermitian K -groups, of sche mes admitting a n ample f ami ly of line-bundles. No assumption on the c haracteristic is needed, and our sc hemes can b e singular. Along the wa y , w e prov e additivity , ﬁbration and approximation theorems for the herm i tian K -theory of exact categories with weak equiv alences and duality . Contents 1. Int ro duction 1 2. The Gr othendiec k-Witt space 3 3. Additivit y theorems 12 4. Change of weak equiv a lences and Coﬁna lit y 19 5. Approximation, Change of exac t structure and Resolution 27 6. F rom exact catego ries to chain complexes 34 7. DG-Algebras on ringed spaces and dualities 40 8. Higher Grothendieck-Witt gro ups of schemes 44 9. Lo calization a nd Za riski-excision in p ositive degr ees 46 10. Extension to negative Grothendieck-Witt gr oups 58 References 65 1. Introduction A classical in v aria n t of a scheme X is its Gro thendiec k-Witt group GW 0 ( X ) of sy mmetric bilinear spaces over X . According t o Knebusch [Kne77, § 4], this is the ab elian group generated by iso metry classes [ V , ϕ ] of vector bundles V ov er X equipp ed with a non-singular symmetr ic bilinear form ϕ : V ⊗ O X V → O X mo dulo the relations [( V , ϕ ) ⊥ ( V ′ , ϕ ′ )] = [ V , ϕ ] + [ V ′ , ϕ ′ ] and [ M , ϕ ] = [ H ( N ) ] for ev ery metabo lic space ( M , ϕ ) with La grangian subbundle N = N ⊥ ⊂ M and asso ciated h yp erbo lic space H ( N ). Grothendieck-Witt gr oups natura lly o ccur in A 1 -homotopy theory [Mor04] a nd are to oriented Chow groups what alge braic K - theory is to o rdinary Chow g roups, see [B M00], [FS07], [Ho r08]. Using a hermitian v ersion of Quillen’s Q -construction, we have deﬁned in [Sc h08] the higher Grothendieck-Witt groups GW i ( X ), i ∈ N , o f a scheme X , gene ralizing 1991 Mathematics Subje ct Classiﬁc ati on. 19G38 (19E08, 19G12, 11E81, 14C35). Key wor ds and phr ases. Grothendiec k-Witt groups, May er-Vietoris, S • -construction. The author was partially supp orted by NSF gran t D M S-0604583 . 1 2 MARC O SCHLICHTING the group GW 0 ( X ). The purp ose of this article is to prove the following May er- Vietoris principle for op en cov ers. 1.1. Theorem. L et X = U ∪ V b e a scheme with an ample family of line-bund les (e.g., quasi-pr oje ctive over an aﬃne scheme, or r e gular sep ar ate d no etherian) which is c over e d by t wo op en quasi-c omp act subschemes U, V ⊂ X . Then ther e is a long exact se quenc e, i ∈ Z , · · · GW i +1 ( U ∩ V ) → GW i ( X ) → GW i ( U ) ⊕ GW i ( V ) → GW i ( U ∩ V ) → GW i − 1 ( X ) · · · This is a sp ecial case of our theorem 10.13 which also includes versions of theorem 1.1 for skew-symmetric forms, for for ms with co eﬃcien ts in line-bundles other tha n O X and for certa in non-commutativ e schemes. Note that we don’t need the common assumption 1 2 ∈ Γ( X , O X ), and X can be singular! Theorem 1.1 is a co nsequence of tw o theorems, “ Loca lization” and “Zar iski- excision”. T o explain the implication, let X be a scheme, L a line bundle o n X , n ∈ Z a n integer, and Z ⊂ X a closed subscheme w ith op en complement U . With this set of data, w e asso ciate in deﬁnition 8.2 a topologic al s pace GW n ( X on Z , L ) which, for Z = X , n = 0 a nd L = O X , yields the Grothendieck-Witt space GW ( X ) int ro duced in [Sch08] whose homoto p y gro ups are the highe r Gro thendiec k-Witt groups GW i ( X ) in theorem 1.1, s ee coro llary 8.5. The space GW n ( X on Z , L ) is the Gro thendiec k-Witt space (as deﬁned in 2.1 1 ) of a n exact ca tegory with weak equiv alences a nd dualit y , namely , the exact category o f bounded chain co mplexes of vector bundles on X which a re (cohomologica lly) supported in Z , eq uipped with the set of quasi-isomorphisms a s weak equiv alences a nd duality E 7→ H om ( E , L [ n ]), where L [ n ] denotes the complex which is L in degr ee − n . If Z = X , we write GW n ( X, L ) for GW n ( X on Z , L ). The non-negative par t of theorem 1.1 is a consequence of the following tw o theorems (proved in theor ems 9.2 and 9 .3). They are extended to ne gativ e Gro thendiec k-Witt groups in § 10 (theorems 10.11 and 10.12). 1.2. Theorem (Lo caliza tion). L et X b e a scheme with an ample family of line- bund les, let U ⊂ X b e a quasi-c omp act op en subscheme with close d c omplement Z = X − U . L et L b e a line bund le on X , and n ∈ Z an inte ger. Then ther e is a homotopy ﬁbr ation GW n ( X on Z , L ) − → GW n ( X, L ) − → GW n ( U, j ∗ L ) . 1.3. Theorem (Za riski excisio n). L et j : U ⊂ X b e qu asi-c omp act op en subscheme of a scheme X which has an ample family of line-bund les. L et Z ⊂ X b e a close d subset su ch that Z ⊂ U . Then r estriction of ve ctor-bund les induc es a homotopy e quivalenc e for al l n ∈ Z and al l line bund les L on X GW n ( X on Z, L ) ∼ − → GW n ( U on Z, j ∗ L ) . Theorems 1.1 – 1.3 have well-known analogs in a lgebraic K -theo ry prov ed by Thomason in [TT90] based on the work of W aldhausen [W al85] a nd Gr othendiec k et al. [SGA6]. In fact, our theorems 9.2, 10.11, 9 .3, 10.12 and 10.13 – sp ecial case s THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 3 of which are theo rems 1.2, 1.3 and 1 .1 – are generaliza tions o f the co rresp onding the- orems in Thomason’s work. More r ecen tly , Ba lmer [Bal01] a nd Ho rn bo stel [Hor05] prov ed r esults reminisc en t of o ur theorems 1.1 – 1 .3. Both need X to be regular no e- therian and separa ted and they need 2 to be a unit in the ring of r egular functions on X . Balmer w orks with (triangula r) Witt-gr oups ins tead of Grothendieck-Witt groups, and Hor n bos tel works with Karoubi’s hermitian K -groups of rings extended to regular separated schemes using Joua nolou’s device of replacing such a scheme by an aﬃne vector-bundle torso r. Neither Balmer ’s nor Hornbostel’s metho ds c an be genera lized to cov er our the- orems 1.2, 1 .3 and 1.1. This is b ecause the assumption 1 2 ∈ Γ( X , O X ) is ubiquito us in their w ork, the analog of theorem 1.1 for Balmer’s triangula r Witt gro ups fails to hold for singular quasi-pr o jective sc hemes (see [Sch b] for a co un ter example even with 1 2 ∈ Γ( X, O X )), Hornbos tel imp oses homotopy inv a riance which do esn’t hold for s ingular schemes, and his pro of uses Karo ubi’s fundamental theo rem [Ka r80 ] which fails to hold for higher Grothendieck-Witt g roups when 1 2 / ∈ Γ( X , O X ) (see [Scha ] for a counter example). Instead, we generalize Tho mason’s work [TT90]. His pro ofs of the K -theory ana logs of theorems 1.2, 1.3 a nd 1 .1 are base d o n a ﬁbration theorem o f W aldhausen [W al85, 1.6.4] and o n “inv ariance of K - theory under der iv ed equiv alences ” [TT9 0, Theorem 1 .9.8] which itself is a co nsequence of W aldhaus en’s approximation theorem [W al8 5, 1.6 .7]. W e prov e in theorem 4.2 the analog of W aldhausen’s ﬁbration theorem for hig her Grothendiec k-Witt groups. Its pro of, how ever, is not a formal cons equence o f “a dditivit y” (prov ed for hig her Grothendieck-Witt gr oups in § 3), contrary to the K -theor y situation. Our pro of relies o n the author’s cone construction in [Sch 08]. “Inv ar iance under deriv ed equiv- alences” as well a s the naive gener alization of W aldhausen’s approximation theorem fail to hold for higher Grothendieck-Witt gro ups when “2 is no t a unit” (see [Scha] for a coun ter ex ample). W e pro ve in theorems 5.1 and 5.1 1 versions of W aldhausen’s approximation theorem for hig her Grothendieck-Witt g roups. Tho ugh not as gen- eral as one might wish, they are enough to show theorems 1.2, 1.3 and 1.1 and their generaliza tions in § 9 a nd § 10. Prerequisites. The ar ticle can b e rea d independently from [TT90] and [W al85], though, o f co urse, muc h of our inspiration der iv es from thes e tw o pap ers. The reader is a dvised to have some background in homotopy theory in the form of [GJ99, I- IV] a nd in the theory of triangulated catego ries in the for m of [K el96], [Nee92], [Nee96, § 1-2]. Also , we will freque n tly use r esults fro m [Sch08]. Ac knowledgmen ts. Part of the results of this article were obtained and written down while I was v isiting I.H.E.S. in Paris and Ma x-Planck-Institut in Bonn. I would like to express my gra titude for their hospita lit y . 2. The Grothendieck-Witt sp ace In this section w e in tr oduce the Grothendieck-Witt g roup and the Gr othendiec k- Witt space of an ex act c ategory with weak equiv alence s a nd duality (Deﬁnitions 2.4 and 2.11), and we show in prop osition 2 .20 that the Grothendieck-Witt space of an exact ca tegory with duality deﬁned here is equiv alen t to the one deﬁned in [Sch08]. W e star t with recalling deﬁnitions from [Sc h0 8]. Note that our terminology (for “ category with dua lit y”, “dua lit y preserving functor”) s ometimes diﬀers from standard terminology as in [Sch85], [Knu91]. 4 MARC O SCHLICHTING 2.1. Categories with duali t y , C h and form functors. A c ate gory with duality is a triple ( C , ∗ , η ) with C a categ ory , ∗ : C op → C a functor, η : 1 → ∗∗ a na tural transformatio n, calle d double dual identiﬁc ation , such that 1 A ∗ = η ∗ A ◦ η A ∗ for all ob jects A in C . If η is a na tural iso morphism, we say that the duality is str ong . In case η is the identit y (in whic h case ∗∗ = i d ), we call the duality st rict . A symmetric form in a category with dualit y ( C , ∗ , η ) is a pair ( X , ϕ ) where ϕ : X → X ∗ is a morphism in C satisfying ϕ ∗ η X = ϕ . A map o f symmetric forms ( X, ϕ ) → ( Y , ψ ) is a map f : X → Y in C s uc h that ϕ = f ∗ ◦ ψ ◦ f . Comp osition of such maps is c ompositio n in C . F or a c ategory with dualit y ( C , ∗ , η ), we deno te by C h the c ate gory of symmetric forms in C . It has ob jects the symmetric forms in C a nd maps the maps b etw een symmetric forms. A form fu n ctor fro m a categ ory with duality ( A , ∗ , α ) to a nother such ca tegory ( B , ∗ , β ) is a pair ( F, ϕ ) with F : A → B a functor a nd ϕ : F ∗ → ∗ F a natural trans- formation, called duality c omp atibility morphism , such that ϕ ∗ A β F A = ϕ A ∗ F ( α A ) for every ob ject A of A . Ther e is an ev iden t deﬁnition of co mposition o f form func- tors, see [Sch08, 3.2]. The categor y F un( A , B ) of functors A → B is a category with duality , where the dual F ♯ of a functor F is ∗ F ∗ , and double dual identiﬁcation η F : F → F ♯♯ at an ob ject A of A is the map β F ( A ∗∗ ) ◦ F ( α A ) = F ( α A ) ∗∗ ◦ β F A . T o give a form functor ( F , ϕ ) is the same as to g iv e a symmetric form ( F, ˆ ϕ ) in the category with duality F un( A , B ) in view of the formulas ϕ A = F ( α A ) ∗ ◦ ˆ ϕ A ∗ and ˆ ϕ A = ϕ A ∗ ◦ F ( α A ). A natura l transfo rmation ( F , ϕ ) → ( G, ψ ) of for m functors is a map ( F , ˆ ϕ ) → ( G, ˆ ψ ) of symmetric forms in F un( A , B ). A duality pr eserving functor b et ween catego ries with duality ( A , ∗ , α ) and ( B , ∗ , β ) is a functor F : A → B which commutes with dualities a nd double dual identiﬁ- cations, that is, we hav e F ∗ = ∗ F and F ( α ) = β F . In this case, ( F, id ) is a fo rm functor. W e will co nsider duality preser ving functors F a s fo rm functor s ( F , id ). Note that our use of the phra se “dualit y preserving functor” may diﬀer from its use by other authors! Recall that an exact c ate gory is an additive category E equipped with a fa mily of sequences of ma ps in E , called c onﬂations (or admissible short exact se quen c es , or s imply exact se quenc es ), X i → Y p → Z satisfying a list of axioms, see [Qui73], [Kel96, § 4], [Sch 08, 2 .1]. The ma p i in an exa ct sequence is ca lled inﬂation (or admissible monomor phism) and may b e depicted a s ֌ , and the map p is called deﬂation (or admissible epimorphism) and may be depicted as ։ in diagra ms. Unless other wise stated, a ll exact categor ies in this a rticle will b e (essentially) sma ll. 2.2. Exact categories wi th w eak equiv alences. An exact c ate gory with we ak e quivalenc es is a pair ( E , w ) with E an exac t categor y and w ⊂ Mor E a set of morphisms, c alled weak equiv alences , which contains all ident ity morphisms, is closed under isomorphisms, retracts, push-o uts alo ng inﬂations, pull-backs alo ng deﬂations, comp osition and the 2 out of three prop ert y for comp osition (if 2 o f the 3 maps among a , b , ab ar e in w then so is the third). A weak equiv alence is usually depicted as ∼ → in diagrams . A functor F : A → B b et w een e xact categor ies with weak equiv alences ( A , w ) and ( B , w ) is called exact if it sends conﬂatio ns to conﬂations and weak equiv alenc es to weak equiv alences. THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 5 F or an exact categor y with w eak equiv alences ( E , w ), we will write w E for the sub category of w eak equiv alences in E . Its ob jects are the ob jects o f E and its maps the maps in w . Also, we will regar d a n e xact categ ory E (without sp ecifying w eak equiv alences ) a s a n ex act category with weak equiv alences ( E , i ) where i is the set of is omorphisms in E . 2.3. Exact categories with w eak equiv alences and duality . An exact c ate gory with we ak e quivalenc es and duality is a quadr uple ( E , w , ∗ , η ) with ( E , w ) an exa ct category with weak equiv a lences and ( E , ∗ , η ) a category with duality such that ∗ : ( E op , w ) → ( E , w ) is an exact functor (in particular , ∗ ( w ) ⊂ w ) and η : id → ∗ ∗ is a natural weak equiv alence, that is, η X ∈ w for all ob jects X in E . W e may simply say E or ( E , w ) is a n exact category with weak equiv alences and dualit y if the remaining data a re under stoo d. Note that if E is an e xact category with weak equiv alences and duality , the category w E of weak equiv a lences in E is a category with duality . A symmetric form ( X , ϕ ) in ( E , w , ∗ , η ) is ca lled non-singular if ϕ is a weak equiv alence. In this case, we call the pair ( X, ϕ ) a symmetric sp ac e in ( E , w , ∗ , η ). A form functor ( F , ϕ ) : ( A , w , ∗ , η ) → ( B , w, ∗ , η ) is c alled ex act if F : ( A , w ) → ( B , w ) is exact. It is called non-singu lar , if the dualit y co mpatibilit y mor phism ϕ : F ∗ → ∗ F is a natura l weak equiv alence. An exact c ate gory with duality is an exact ca tegory with weak equiv alences and duality wher e the set of w eak equiv alences is the set of isomorphisms. In particular, the double dual identiﬁcation has to b e a natural iso morphism. 2.4. Deﬁnition. The Gr othendie ck-Witt gr oup GW 0 ( E , w, ∗ , η ) of an exact categor y with weak equiv alences a nd duality ( E , w , ∗ , η ) is the free ab elian gr oup generated by isomor phism classes [ X, ϕ ] of s ymmetric spaces ( X , ϕ ) in ( E , w , ∗ , η ), sub ject to the following relations (a) [ X , ϕ ] + [ Y , ψ ] = [ X ⊕ Y , ϕ ⊕ ψ ] (b) if g : X → Y is a weak equiv alence, then [ Y , ψ ] = [ X , g ∗ ψ g ], a nd (c) if ( E • , ϕ • ) is a symmetric space in the categ ory of exac t sequences in E , that is, a map E • : ϕ • ≀   E − 1 / / i / / ϕ − 1 ≀   E 0 p / / / / ϕ 0 ≀   E 1 ϕ 1 ≀   E ∗ • : E ∗ 1 / / p ∗ / / E ∗ 0 i ∗ / / / / E ∗ − 1 of ex act sequence s with ( ϕ − 1 , ϕ 0 , ϕ 1 ) = ( ϕ ∗ 1 η , ϕ ∗ 0 η , ϕ ∗ − 1 η ) a weak e quiv a - lence, then [ E 0 , ϕ 0 ] = h E − 1 ⊕ E 1 ,  0 ϕ 1 ϕ − 1 0 i . 2.5. Remark. If in deﬁnitio n 2.4, the set of weak equiv a lences is the set of isomor- phisms, then we reco ver the classica l Gro thendiec k-Witt group of an exact category with duality , see for instance [Sch08, 2.9]. In this case, rela tion (c) s a ys that the class [ E 0 , ϕ 0 ] o f a metab olic space ( E 0 , ϕ 0 ) with Lag rangian i : E − 1 ֌ E 0 is equiv- alent in the Gr othendiec k-Witt group to the cla ss of the h yp erbolic spa ce H ( E − 1 ) of the Lagrangian E − 1 . In particular, if E is the category V ect ( X ) of v e ctor b undles 6 MARC O SCHLICHTING on X , ∗ is the duality functor E 7→ H om ( E , O X ) and η is the usual canonical dou- ble dual identiﬁcation, the group GW 0 ( E , i, ∗ , η ) is Knebusch’s Grothendieck-Witt group GW 0 ( X ) of a scheme X , deno ted L ( X ) in [Kne77]. 2.6. Deﬁnition. The Witt gro up W 0 ( E , w, ∗ , η ) of an exact categor y with weak equiv alences a nd duality ( E , w , ∗ , η ) is the free ab elian gr oup generated by isomor phism classes [ X, ϕ ] of s ymmetric spaces ( X , ϕ ) in ( E , w , ∗ , η ), sub ject to the relations 2.4 (a), (b) and (c’) if ( E • , ϕ • ) is a symmetric space in the category of exact sequences in ( E , w, ∗ , η ), then [ E 0 , ϕ 0 ] = 0. The her mitian S • -construction of [SY96], [HS04, 1 .5], which gives rise to the Grothendieck-Witt space to be deﬁned in 2.11, is the edgewise sub division of W ald- hausen’s S • -construction [W al85]. W e review the relev ant deﬁnitions and start with the edgewis e s ubdivision of a simplicia l ob ject, see [W al85, 1.9 App endix], [Seg73, Appendix 1]. 2.7. Edgewise sub division. Let ∆ be the c ategory with ob jects [ n ] the tota lly ordered sets [ n ] = { 0 < 1 < ... < n } , n ∈ N , and morphisms the o rder pr eserving maps. Let n be the totally orde red set n = { n ′ < ( n − 1) ′ < ... < 0 ′ < 0 < ... < n } . It is (uniquely) iso morphic to [2 n + 1 ]. The assignment T : [ n ] 7→ n deﬁnes a functor ∆ → ∆ where a ma p θ : [ n ] → [ m ] go es to the map T ( θ ) : n → m : p 7→ θ ( p ) , p ′ 7→ θ ( p ) ′ . F or a simplicial ob ject X • , the e dge-wise sub division X e • of X • is the simplicia l o b ject X • ◦ T . The inclusion [ n ] ֒ → n : i 7→ i deﬁnes a map X e → X of s implicial ob jects. It is known [Seg73, App endix 1] that for a simplicial set X • , the top ological realization of X • and of its edg e-wise sub division X e • are homeomorphic. W e need the following (well-kno wn) v ariant. 2.8. Lemma. F or any simplicial set X • , the map X e • → X • is a homotopy e quiv- alenc e. Pr o of. Let X • be a s implicial set. F or a sma ll catego ry C , write X C for the set Hom( N ∗ C , X • ) of simplicia l maps fro m the nerve N ∗ C o f C to X • . Note that X e is the simplicia l set [ n ] 7→ X n . W e deﬁne bisimplicial sets X e •• and X •• by the formulas X e m,n = X m × [ n ] and X m,n = X [ m ] × [ n ] . Co nsider the following diagra m of bisimplicial sets X e 0 • ∼ / / ≀   X e ••   X e • 0 ∼ o o   X 0 • ∼ / / X •• X • 0 ∼ o o in which the horizo n tal maps a re the ca nonical inclusions of ho rizontally resp ec- tively vertically consta n t bisimplicial se ts, and the vertical ma ps ar e induced by the inclusions [ m ] ⊂ m . Once we show that all a rrows lab eled ∼ → ar e homo top y equiv alences , w e are done, b ecause the right vertical map can b e identiﬁed with X e • → X • . THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 7 The left vertical map X e 0 • → X 0 • is a ho motop y equiv alence since it can be ident iﬁed with the map X I • → X • which is ev aluation at 0, where I = N ∗ 0 ∼ = N ∗ [1] is the s tandard simplicial interv al. In order to see that the uppe r r igh t hor izon tal map X e • 0 → X e •• is a homotopy equiv alence, it s uﬃces to prov e that fo r every n , the map X e • 0 → X e • n is a homoto p y eq uiv a lence of simplicial sets. Since the map [0] → [ n ] : 0 7→ 0 induces a retra ction X e • n → X e • 0 , w e hav e to s ho w tha t the comp osition X e • n → X e • 0 → X e • n is homo topic to the identit y . The (unique) natural tra nsformation from the cons tan t functor [ n ] → [ n ] : i 7→ 0 to the identit y functor [ n ] → [ n ] deﬁnes a functor h : [1] × [ n ] → [ n ] such that the restr ictions to { i } × [ n ] → [ n ], i = 0 , 1 are the constant resp ectively the ident ity functor. W e hav e a map o f simplicia l sets (1) I e × X e • n → X e • n which in degree m sends the pair ( ξ , f ) ∈ I m × X m × [ n ] to the compo sition N ∗ ( m × [ n ]) (1 ,ξ ) × 1 − → N ∗ ( m × [1] × [ n ]) (1 × h ) − → N ∗ ( m × [ n ]) f − → X • Since the two p oints { 0 , 1 } = { 0 , 1 } e ⊂ I e are path connected in I e , the map (1) deﬁnes the desired homotopy . The other hor izon tal homoto p y equiv alences in the diagram are similar, and we omit the details .  Now we reca ll W aldhausen’s S • -construction [W a l85, § 1 .3]. 2.9. W aldhausen’s S • -construction. Let Ar [ n ] deno te the categ ory who se ob- jects a re the arrows of [ n ] = { 0 < 1 < ... < n } a nd whose morphisms a re the commutativ e s quares in [ n ]. F or an exa ct categor y with weak equiv alences ( E , w ), W aldhausen deﬁnes S n E ⊂ F un(Ar[ n ] , E ) as the full sub category of the catego ry F un(Ar[ n ] , E ) of functors A : Ar[ n ] → E : ( p ≤ q ) 7→ A p,q for which A p,p = 0 and A p,q ֌ A p,r ։ A q,r is a conﬂation whenever p ≤ q ≤ r , p, q , r ∈ [ n ]. The categ ory S n E is an exact categ ory with w eak equiv a lences where a sequence A → B → C of functor s Ar[ n ] → E in S n E is ex act if A p,q ֌ B p,q ։ C p,q is exa ct in E , and a map A → B of functors in S n E is a weak equiv a lence if A p,q → B p,q is a weak e quiv alence in E for all p ≤ q ∈ [ n ]. The cosimplicial ca tegory n 7→ Ar[ n ] makes the a ssignmen t n 7→ S n E into a simplicial ex act ca tegory with weak e quiv a lences. According to [W al85], [T T90], the K -theory spa ce K ( E , w ) of a n exa ct categor y with weak equiv a lences ( E , w ) is the space K ( E , w ) = Ω | wS • E | . 2.10. The hermiti an S • -construction. The categor y [ n ] has a unique str ucture of a category wit h strict dualit y [ n ] op → [ n ] : i 7→ n − i . This induces a s trict dualit y on the category Ar[ n ] of arrows in [ n ]. F o r an exact categ ory with w eak equiv a lences and duality ( E , w, ∗ , η ), the categ ory F un (Ar[ n ] , E ) is there fore a categor y with duality (see 2.1). This duality pr eserves the s ubcatego ry S n E ⊂ F un (Ar[ n ] , E ), and makes S n E into a n exact c ategory with weak equiv a lences and dua lit y . It turns out that the simplicial structure maps of n 7→ S n E are not compatible with dualities. How ever, its edgewis e sub division S e • E : n 7→ S e n E = S 2 n +1 E 8 MARC O SCHLICHTING is a simplicial exa ct category with weak equiv alences a nd dualit y; the simplicial structure maps b eing duality preserving . Considering S e n E as a full sub category of F un(Ar( n ) , E ), the dual A ∗ of an ob ject A : Ar( n ) → E satisﬁes ( A ∗ ) p,q = A ∗ q ′ ,p ′ for p ≤ q ∈ n where p ′′ trickily deno tes p . The double dual identiﬁcation A → A ∗∗ at ( p ≤ q ) is η A p,q . 2.11. Deﬁnition. Let ( E , w , ∗ , η ) be an exa ct category with weak equiv alences and duality . By 2.10, the assignment n 7→ S e n E deﬁnes a simplicia l exact ca tegory S e • E with weak equiv alences and duality . The sub categories of w eak equiv alences deﬁne a simplicial categor y with duality n 7→ w S e n E . T aking asso ciated catego ries of symmetric forms (see 2.1), we obtain a simplicial category ( w S e • E ) h . The comp osition ( w S e • E ) h → w S e • E → wS • E of s implicial catego ries, in which the ﬁrst arr o w is the forgetful functor ( X , ϕ ) 7→ X , and the s econd is the canonica l map X e • → X • of s implicial ob jects (see 2.7 ), yields a map o f classifying spa ces (2) | ( wS e • E ) h | → | wS • E | whose homotopy ﬁbr e (with resp ect to a z ero o b ject of E as ba se p oint of wS • E ) is deﬁned to be the Gr othendie ck-Witt sp ac e GW ( E , w , ∗ , η ) of ( E , w, ∗ , η ). If ( ∗ , η ) a re understo o d, we may simply write GW ( E , w ) instea d of GW ( E , w , ∗ , η ). W e deﬁne the higher Gr othendie ck-Witt gr oups of ( E , w , ∗ , η ) as the homotopy groups GW i ( E , w, ∗ , η ) = π i GW ( E , w , ∗ , η ) , i ≥ 1 , and show in prop osition 3.8 b elo w that π 0 GW ( E , w , ∗ , η ) ∼ = GW 0 ( E , w, ∗ , η ), so that our deﬁnition here extends that in 2.4. 2.12. Remark. Orthog onal sum makes the spa ces ( wS e • E ) h and GW ( E , w , ∗ , η ) int o commutativ e H -spaces. Since the commutativ e mo noid of connected comp o- nent s o f these spaces are gr oups (see prop osition 3.8 and remark 3 .9 b elow), b oth spaces a re a ctually commutative H -g roups. 2.13. F unctorialit y. A non-singular exact fo rm functor ( F, ϕ ) : ( A , w, ∗ , η ) → ( B , w , ∗ , η ) b etw een exact ca tegories with weak equiv a lences and duality induces maps ( F, ϕ ) : ( wS e • A ) h → ( w S e • B ) h : ( A, α ) 7→ ( F A , ϕ A F ( α )) , and F : wS • A → wS • B : A 7→ F A of simplicia l categ ories compatible with comp osition of for m functors . T a king ho- motopy ﬁbr es of ( w S e • ) h → w S • , we obtain an induced map GW ( F, ϕ ) : GW ( A , w, ∗ , η ) → GW ( B , w, ∗ , η ) of a sso ciated Grothendieck-Witt spaces . F o r the next lemma, recall that a na tural transformatio n of form functors ( F , ϕ ) → ( G, ψ ) is a map of asso ciated symmetric forms in F un( A , B ). It is a natura l weak equiv alence if F A → GA is a weak equiv alence for all ob jects A of A . THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 9 2.14. Lemma. L et ( F , ϕ ) ∼ → ( G, ψ ) b e a n atur al we ak e quivalenc e of non- singular exact form functors ( A , w, ∗ , η ) → ( B , w , ∗ , η ) b etwe en exact c ate gories with we ak e quivalenc es and duality. Then, on asso ciate d Gr othendie ck-Witt sp ac es, ( F , ϕ ) and ( G, ψ ) induc e homotopic maps GW ( A , w , ∗ , η ) → GW ( B , w, ∗ , η ) . Pr o of. The natur al weak equiv alence ( F , ϕ ) ∼ → ( G, ψ ) induces natura l tr ansforma- tions o f functors ( w S e n A ) h → ( wS e n B ) h and w S n A → wS n B . These natural trans- formations deﬁne functors [1] × ( w S e n A ) h → ( w S e n B ) h and [1] × wS n A → w S n B whose r estrictions to 0 , 1 ∈ [1] are the tw o given functors . They are compatible with the simplicial structur e and induce, after topo logical realiz ation, the ho mo- topy b et w een GW ( F, ϕ ) a nd GW ( G, ψ ).  Next, we will as socia te to every exact categ ory with w eak equiv ale nces ( E , w ) a category with weak equiv alence s a nd dua lit y ( HE , w ) such that the Grothendieck- Witt space of ( H E , w ) is e quiv alent to the K -theory space of ( E , w ). In this se nse, Grothendieck-Witt theory is a g eneralization of alg ebraic K -theor y . 2.15. Hyp erb olic categories. Let C be a catego ry . Its hyper bolic catego ry is the category with strict dua lit y H C = ( C × C op , ∗ ) wher e ( X , Y ) ∗ = ( Y , X ). F or any category with duality A there is a functor A h → A : ( X , ϕ ) 7→ X that “for gets the forms”. W e deﬁne the functor ( HC ) h → C as the comp osition of the functor ( HC ) h → HC and the pro jection HC = C × C op → C onto the ﬁrst factor . 2.16. Lemma. F or any smal l c ate gory C , the functor ( H C ) h → C is a homotopy e quivalenc e. Pr o of. The categor y ( HC ) h of symmetric forms in HC is isomor phic t o the category whose ob jects are ma ps f : X → Y in C a nd wher e a map from f to f ′ : X ′ → Y ′ is a pair of maps a : X → X ′ , b : Y ′ → Y in C such that f = bf ′ a . Compo sition is comp osition in C of the a ’s and b ’s . The functor ( HC ) h → C in the lemma sends the ob ject f : X → Y to X a nd the map ( a, b ) to a . W rite F for this functor , and let A b e an ob ject of the targ et ca tegory C . W e will show that the comma categories ( A ↓ F ) a re contractible. By Q uillen’s theore m A [Qui73, § 1], this implies the lemma. The ca tegory ( A ↓ F ) has ob jects sequences A x → X f → Y of maps in C . A morphism from ( x, f ) to A x ′ → X ′ f ′ → Y ′ is a pair a : X → X ′ , b : Y ′ → Y of maps in C such that x ′ = ax and f = bf ′ a . In pa rticular, the catego ry ( A ↓ F ) is non- empt y as (1 A , 1 A ) is one of its o b jects. Let C 0 ⊂ ( A ↓ F ) b e the full sub catego ry of o b jects ( x, f ) with x = 1 A . The inclusion ha s a r igh t adjoint ( A ↓ F ) → C 0 : ( x, f ) 7→ (1 A , f x ) with counit o f adjunction (1 A , f x ) → ( x, f ) given by the pair of maps x : A → X and 1 : Y → Y . It follows tha t the inclusion C 0 ⊂ ( A ↓ F ) is a ho motop y equiv a lence. Since the ca tegory C 0 has a terminal ob ject, namely (1 A , 1 A ), the categor ies C 0 and ( A ↓ F ) are contractible.  If ( E , w ) is an ex act ca tegory with weak equiv alences, we make HE into an exact category with weak equiv alences and (strict) duality by declar ing a map ( a, b ) : ( X , Y ) → ( X ′ , Y ′ ) in HE to be a weak equiv alence if a : X → X ′ and b : Y ′ → Y are weak equiv alences in E . Note that w HE = H w E as c ategories with strict duality . 10 MARC O SCHLICHTING 2.17. Prop osition. L et ( E , w ) b e an exact c ate gory with we ak e quivalenc es, then ther e is a n atur al homotop y e quivalenc e GW ( HE , w ) ≃ K ( E , w ) . Pr o of. Consider the commutativ e diagr am of simplicial catego ries ( wS e • HE ) h = / /   ( H w S e • E ) h   ∼ ' ' O O O O O O O O O O O O wS e • HE = / / ≀   wS e • E × ( w S e • E ) op ≀   p 1 / / wS e • E ≀   wS • HE = / / wS • E × ( w S • E ) op p 1 / / wS • E where the upp er vertical maps are the functors that “ forget the for ms”. The lower vertical maps are induced by the inclusion [ n ] ֒ → n and are thus homo top y equiv a- lences, by lemma 2.8. The dia gonal map is a homoto p y equiv alence by lemma 2.16. By the “o ctahedron axiom” for homotopy ﬁbres applied to the upper r igh t triangle, it follows that the homotopy ﬁbre of the comp osition of the left vertical ma ps is equiv alent to the lo op space of the ﬁbre o f p 1 : wS • E × ( w S • E ) op → w S • E which is the lo op o f the ﬁbre of ( w S • E ) op → (p oin t) which is K ( E , w ).  2.18. Remark. Let ( E , w, ∗ , η ) b e an exact category with weak e quiv a lences and duality . The functor F : ( E , w , ∗ , η ) → ( HE , w ) : X 7→ ( X , X ∗ ) together with the dualit y compatibility mor phism (1 , η X ) : ( X ∗ , X ∗∗ ) → ( X ∗ , X ) is called for getful form functor . It is a no n-singular exact form functor b etw ee n exact categor ies with weak equiv alences and dua lit y . The map ( w S e • E ) h → w S • E deﬁning the Grothendieck-Witt spa ce factors as ( wS e • E ) h F → ( w S e • HE ) h ∼ → w S • E , where the second map is the homotopy equiv a lence in the diagra m of the pro of of prop osition 2 .17 (going right, diag onally and down). It follows that the Grothendieck- Witt space GW ( E , w, ∗ , η ) is naturally homotopy equiv alent to the homotopy ﬁbre of ( wS e • E ) h F → ( w S e • HE ) h . W e ﬁnish the sec tion with a comparison result b et ween the deﬁnition o f the Grothendieck-Witt space of an exact categor y with dualit y ( E , i, ∗ , η ) in terms o f the her mitian S • -construction and the deﬁnition given in [Sc h08, Deﬁnition 4.6] in terms of the hermitian Q -constr uction. W e recall the relev an t deﬁnitions. THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 11 2.19. The hermiti an Q -construction. Reca ll from [Q ui73 ] that for an exact category E its Q -construction is the ca tegory with o b jects the ob jects of E and maps X → Y equiv alence clas ses of diagr ams (3) X p և U i ֌ Y with p a deﬂa tion and i an inﬂatio n. The datum ( U, i, p ) is equiv a len t to ( U ′ , i ′ , p ′ ) if there is an iso morphism g : U → U ′ in E such that p = p ′ ◦ g and i = i ′ ◦ g . The comp osition in Q E of maps X → Y a nd Y → Z represented by the data ( U, i, p ) and ( V , j, q ) is given by the datum ( U × Y V , p ¯ q, j ¯ i ) where ¯ q : U × Y V → U and ¯ i : U × Y V → V a re the canonica l pro jections to U and V , resp ectively . F or an exact catego ry with duality ( E , ∗ , η ), the hermitian Q -construction is the categor y Q h ( E , ∗ , η ) with o b jects the s ymmetric spa ces ( X , ϕ ) in E . A map ( X, ϕ ) → ( Y , ψ ) is a ma p X → Y in Quille n’s Q -construction, that is , a n e quiv alence class of dia grams (3), such that the square o f maps p , i , p ∗ ϕ and i ∗ ψ is comm uta tiv e and bicartesia n. Compo sition of maps is as in Quillen’s Q -construction. F or mor e details, we refer the reader to [Sch08, 4.2 ] and the refer ences in [Sch08, Remar k 4.3]. In [Sch08, Deﬁnition 4.6], we deﬁned the Grothendieck-Witt space of ( E , ∗ , η ) as the homotop y ﬁbre of the forgetful functor Q h E → Q E : ( X , ϕ ) 7→ X . The following prop osition reco nciles this deﬁnition with the one given in 2.11. This allows us to freely use the results prov ed in [Sch08]. 2.20. Prop osition. F or an exact c ate gory with duality ( E , ∗ , η ) , ther e ar e natu r al homotopy e qu ival enc e b etwe en | Q h E | and | ( i S e • E ) h | and b etwe en the homotopy ﬁbr e of the for getful functor | Q h E | → | Q E | and the Gr othendie ck-Witt sp ac e as deﬁne d in 2.11. Pr o of. F or the ﬁrst homotopy equiv alence, the pro of is the same a s in [W al85, 1.9 Appendix]. In detail, let i Q h • E b e the simplicia l categor y which in degr ee n is the category iQ h n E whose ob jects ar e sequences X 0 → X 1 → ... → X n of maps in Q h E and a map in iQ h n E is an isomor phism of suc h sequences. As n v a ries, iQ h n E deﬁnes a simplicial categ ory wher e face and deg eneracy maps ar e deﬁned as in the usua l ne rv e construction. The ner v e of iQ h • E as a bisimplicial set is isomorphic to the nerve of the simplicia l category which in degree m are the sequences X 0 → X 1 → ... → X m of isomor phisms in Q h E and where maps are maps of sequenc es in Q h E (which are no t neces sarily isomo rphisms). The latter simplicial categor y is degree-wise equiv alent to Q h E (via the embedding of Q h E as the cons tan t sequences). Thus the latter simplicia l c ategory (and ther fore also iQ h • E ) is ho motop y equiv alen t to Q h E . Every ob ject ( A p,q ) p ≤ q ∈ n in ( S e n E ) h deﬁnes a string of maps A 0 ′ , 0 → A 1 ′ , 1 → ... → A n ′ ,n in Q h E . This deﬁnes a map ( iS e • E ) h → i Q h • E of simplicia l categories which is degree-wise an equiv alence. Therefore, this map deﬁnes a homotopy equiv alence o n top ological r ealizations. 12 MARC O SCHLICHTING F or the second homotopy equiv a lence, consider the c omm utativ e diagra m of top ological s paces | ( iS e • E ) h |   | ( iS e • E ) h | 1 o o ∼ / /   | iQ h • E |   | Q h E | ∼ o o   | iS • E | | iS e • E | ∼ o o ∼ / / | iQ • E | | Q E | ∼ o o in which the low er right tw o ho rizontal maps are deﬁned in a similar wa y as their hermitian ana logs ab ov e them (see [W a l85, 1.9 App endix]), the three right vertical maps are “forgetful” functor s and all maps labele d ∼ → ar e homotopy equiv alences. It follo ws that the homotopy ﬁbr e of the ﬁrst v ertical ma p is equiv alent to the homotopy ﬁbre of the last vertical map.  3. Additivity theorems Additivit y theorems are fundamental in a lgebraic K -theo ry . They imply , for instance, W aldhausen’s ﬁbra tion theorem [W a l85, 1.6.4] whic h is the ba sis fo r the K -theor y v e rsion of theor em 1.2. In this sectio n, we prov e the analo gs of additivity for higher Grothendiec k-Witt theo ry . In order to form ulate them, we reca ll the concept of “admissible short complexes” from [Sch08, § 7]. 3.1. Admissibl e short complexes and the i r hom ology. Le t ( E , w , ∗ , η ) b e an exact categ ory with weak equiv alences and dualit y . A short c omplex in E is a complex (4) A • : 0 → A 1 d 1 → A 0 d 0 → A − 1 → 0 , ( d 0 ◦ d 1 = 0) in E concentrated in degr ees − 1 , 0 , 1. It is admissible if d 1 and d 0 are inﬂa tion and deﬂation, resp ectively , and the map A 1 → k er( d 0 ) (or e quiv a len tly coker( d 1 ) → A − 1 ) is an inﬂation (deﬂation). A sequence A • → B • → C • of admiss ible sho rt complexes is exact if A i → B i → C i is exact in E ; a map A • → B • is a w eak equiv alence if A i → B i is a weak equiv alence in E , i = − 1 , 0 , 1. The dual o f the complex (4) is the (admissible short) co mplex A ∗ − 1 d ∗ 0 → A ∗ 0 d ∗ 1 → A ∗ 1 , and the do uble dual identiﬁcation η A • : A • → ( A • ) ∗∗ is η A i in degr ee i = − 1 , 0 , 1. W e denote b y ( s Cx( E ) , w , ∗ , η ) the ex act categor y with weak equiv alences and du- ality of admissible short complexes in E . If ( A • , α • ) is a symmetric form in s Cx( E ), w e write H 0 ( A • , α • ) for its zeroth homology symmetric form. Its underlying ob ject is H 0 ( A • ) = ker( d 0 ) / im ( d 1 ), and it is eq uipped with the form ¯ α w hic h is the unique symmetric form such that ¯ α | ker( d 0 ) = α | ker( d 0 ) . This makes H 0 : s Cx( E ) → E into a non-singular e xact form functor for every exact ca tegory with weak equiv alences and dua lit y E . F o r mo re details, we refer the reader to [Sch08, § 7]. In the special case of an e xact category wit h dualit y (where all weak equiv alences are isomo rphisms), the following tw o theorems were prov ed in [Sch08, Theor ems 7.1, 7.4]. The K -theory version is due to W a ldhausen in [W al85, The orem 1.4.2 and prop osition 1.3.2], in view of the equiv alence S e 1 E → s Cx( E ) : A 7→ ( A 1 ′ 0 ′ → A 1 ′ 1 → A 01 ) of exact catego ries with weak equiv a lences a nd duality . THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 13 3.2. Theorem (Additivity for short complexes). L et ( E , w, ∗ , η ) b e an ex act c ate- gory with we ak e quivalenc es and duality. Then the non-singular ex act form functor H 0 : s Cx( E ) → E to gether with the exact fun ctor ev 1 : s Cx( E ) → E : A • 7→ A 1 induc e homotopy e quivalenc es ( H 0 , ev 1 ) : ( wS e • s Cx E ) h ∼ − → ( wS e • E ) h × wS • E GW ( s Cx E , w , ∗ , η ) ∼ − → GW ( E , w , ∗ , η ) × K ( E , w ) . F or the next theorem, recall that a for m functor ( A , ∗ , η ) → ( B ∗ , η ) betw een categorie s with duality is nothing else than a symmetric for m ( F, ϕ ) in the category with duality of functors (F un ( A , B ) , ♯, η ), see 2.1. 3.3. Theorem (Additivit y for functors). Le t ( A , w , ∗ , η ) and ( B , w , ∗ , η ) b e exact c ate gories with we ak e quivalenc es and duality. Given a n on-singular exact form functor ( F • , ϕ • ) : ( A , w, ∗ , η ) → s Cx( B , w, ∗ , η ) , that is, a c ommutative diagr am of exact funct ors F i : ( A , w ) → ( B , w ) F • : ϕ • ≀   F 1 / / d 1 / / ϕ 1 ≀   F 0 d 0 / / / / ϕ 0 ≀   F − 1 ϕ − 1 ≀   F ♯ • : F ♯ − 1 / / d ♯ 0 / / F ♯ 0 d ♯ 1 / / / / F ♯ 1 with d 0 d 1 = 0 , F 1 ֌ k er d 0 an inﬂation, and ( ϕ 1 , ϕ 0 , ϕ − 1 ) = ( ϕ ♯ − 1 η , ϕ ♯ 0 η , ϕ ♯ 1 η ) . Then t he two non-singular ex act form functors (5) ( F 0 , ϕ 0 ) and H 0 ( F • , ϕ • ) ⊥  F 1 ⊕ F − 1 ,  0 ϕ − 1 ϕ 1 0  induc e homotopic maps on Gr othendie ck-Witt sp ac es GW ( A , w , ∗ , η ) → GW ( B , w , ∗ , η ) . W e will reduce the pro ofs o f the additivity theorems 3.2 a nd 3 .3 to the ca se of exa ct categ ories with dualities dealt with in [Sch08 , § 7]. This will b e do ne with the help of the s implicial r esolution lemma 3.7 b elow. F or that, we need to replace an exact ca tegory with weak equiv a lences and dualit y (wher e the double dual identiﬁcation is a natur al w eak equiv alence) by one w ith a str ong duality (where the double dual identiﬁcation is a natural isomo rphism) without c hanging its her mitian K -theory . This will b e done in the strictiﬁcation lemma 3.4 b elow. W e introduce nota tion for lemma 3.4. Let E xW e Du be the ca tegory of sma ll exact categories with w eak equiv alences and duality; and non-s ingular exact form functors as maps. Recall that a ca tegory with dualit y ( A , ∗ , η ) has a strict duality if η = id (and in particula r, ∗ ∗ = id ). Let E xW eD u str be the category of small exact categories with weak equiv a lences and strict dualit y; and dualit y pr eserving functors as maps. W r ite la x : E xW eD u str ⊂ E xW eD u for the natura l inclusion. 14 MARC O SCHLICHTING 3.4. Lemma (Strictiﬁca tion le mma). Ther e is a ( s trictiﬁc ation) functor str : E xW eD u → E xW eD u str : ( A , w , ∗ , η ) 7→ ( A str w , w , ♯, id ) and natura l tra nsformations (Σ , σ ) : id → lax ◦ str and (Λ , λ ) : lax ◦ str → id such that the c omp ositions (Σ , σ ) ◦ (Λ , λ ) and (Λ , λ ) ◦ (Σ , σ ) ar e we akly e qu ival ent to the identity form fun ctor. In p articular, for any exact c ate gory with we ak e quiv- alenc es and du ality ( A , w , ∗ , α ) , we have a homotopy e quivalenc e of Gr othendie ck- Witt s p ac es GW (Σ , σ ) : GW ( A , w , ∗ , α ) ∼ − → GW ( A str w , w , ♯, id ) . Pr o of. The construction of the strictiﬁca tion functor is as follows. Let ( A , w , ∗ , α ) be a n exac t categor y with weak equiv alences and dualit y . The ob jects of A str w are triples ( A, B , f ) with A , B ob jects of A and f : A ∼ → B ∗ a w eak equiv alence in A . A morphism from ( A 0 , B 0 , f 0 ) to ( A 1 , B 1 , f 1 ) is a pair ( a, b ) of morphisms a : A 0 → A 1 and b : B 1 → B 0 in A such that f 1 a = b ∗ f 0 . Comp osition is comp osition of the a ’s and b ’s in A . A map ( a, b ) is a weak equiv alence if a and b are weak equiv alences in A . A sequence ( A 0 , B 0 , f 0 ) → ( A 1 , B 1 , f 1 ) → ( A 2 , B 2 , f 2 ) is exa ct if A 0 → A 1 → A 2 and B 2 → B 1 → B 0 are exa ct in A . The duality ♯ : ( A str w ) op → A str w is deﬁned by ( A, B , f : A → B ∗ ) ♯ = ( B , A, f ∗ α B ) on ob jects, and b y ( a, b ) ♯ = ( b, a ) on morphisms. The double dual ident iﬁcation is the identit y na tural trans formation. The categor y thus constructed A str w = ( A str w , w , ♯, id ) is an exact ca tegory with weak equiv alences and strict duality . W e may write A str , or A str w for ( A str w , w , ♯, id ) if the remaining da ta are under stoo d. If ( F , ϕ ) : ( A, w, ∗ , α ) → ( B , v , ∗ , β ) is a no n- singular exact fo rm functor , its image under the strictiﬁcation fun ctor is the functor F str : A str w → B str v given by F str : ( A, B , f ) 7→ ( F A, F B , ϕ B F ( f )) on ob jects, a nd by ( a, b ) 7→ ( F a, F b ) on morphisms. One checks that F str preserves comp osition and commut es with dualities. The natural transfor mations (Σ , σ ) : i d → lax ◦ str and (Λ , λ ) : lax ◦ str → id a re deﬁned a s follows. The functor Σ : A → A str w sends an ob ject A to ( A, A ∗ , α A ) and a morphism a to ( a, a ∗ ). The dua lit y compatibilit y morphism σ : Σ( A ∗ ) → Σ( A ) ♯ is the map (1 , α A ) : ( A ∗ , A ∗∗ , α A ∗ ) → ( A ∗ , A, 1 ). The functor Λ : A str → A sends the ob ject ( A, B , f ) to A and a mo rphism ( a, b ) to a , and is eq uipped with the duality compatibility mor phism λ : Λ[( A, B , f ) ♯ ] → [Λ( A, B , f )] ∗ the map f ∗ α B : B → A ∗ . The comp osition (Σ , σ ) ◦ (Λ , λ ) is the functor sending ( A, B , f ) to ( A, A ∗ , α A ), and the morphism ( a, b ) to ( a, a ∗ ). It is equipped with the duality co mpatibilit y mor- phism ( f ∗ α B , f ) : ( B , B ∗ , α B ) → ( A ∗ , A, 1 ). There is a natural weak equiv ale nce of form functor s (Σ , σ ) ◦ (Λ , λ ) ∼ → id given by the ma p (1 , f ∗ α B ) : ( A, A ∗ , α A ) → ( A, B , f ). Regarding the o ther comp osition, we hav e (Λ , λ ) ◦ (Σ , σ ) = id . By lemma 2.14, the for m functor (Σ , σ ) : A → A str w induces a homo top y equiv a - lence of Grothendieck-Witt spaces with ho motop y inv erse (Λ , λ ).  3.5. Remark. Here is a s ligh t ge neralization of lemma 3.4. If v is another set of weak equiv ale nces for ( A , ∗ , α ) such that w ⊂ v , then ( A str w , v , ♯, id ) is also an exact category with weak equiv a lences and strict duality . The form functor s (Λ , λ ) : ( A str w , v ) → ( A , v ) and (Σ , σ ) : ( A , v ) → ( A str w , v ) ar e still exact and non-singular with co mpositions tha t a re naturally weakly e quiv alent to the identit y functors . I n THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 15 particular, we hav e a homoto p y equiv a lence GW (Σ , σ ) : GW ( A , v , ∗ , α ) ∼ − → GW ( A str w , v , ♯, id ) . The next le mma will sometimes allow us to r eplace an exact c ategory with weak equiv alences and duality by a simplicial exact c ategory with dua lit y (where weak equiv alences a nd double dual identiﬁcation ar e isomo rphisms). 3.6. Notation for l emma 3.7. Let ( E , w , ∗ , η ) b e a n exact c ategory with weak equiv alences and duality , and let D b e an ar bitrary (small) ca tegory . Reca ll fro m 2.1 that the category F un ( C , E ) of functors D → E is a ca tegory with dualit y . It is an exact categor y with weak equiv a lences and duality if we decla re maps F → G (s equences F − 1 → F 0 → F 1 ) of functors D → E to b e a weak equiv alence (conﬂation) if F ( A ) → G ( A ) is a weak equiv alence ( F − 1 ( A ) → F 0 ( A ) → F 1 ( A ) is a co nﬂation) in E for a ll ob jects A o f D . W e wr ite F un w ( D , E ) ⊂ F un( D , E ) for the full sub categor y of those functors F : D → C for which the image F ( d ) of a ll maps d of D are weak equiv alences in E : F ( d ) ∈ w E . The ca tegory F un w ( D , E ) inherits from F un( D , E ) the structure of an exac t categor y with weak equiv alences and duality . In par ticular, for n ∈ N and ( E , w, ∗ , η ) an exact catego ry with weak equiv alences and str ong dua lit y , the ca tegory F un w ( n , E ) is an exact ca tegory with weak equiv alences and s trong duality . It has o b jects string s of weak equiv alence s and maps commutativ e diagr ams in E . V arying n , the ca tegories F un w ( n, E ) deﬁne a simplicial exact categ ory with w eak equiv alences and strong dua lit y . Recall tha t the symbol i stands for the set of isomor phisms in a categ ory . 3.7. Lemma (Simplicial resolution lemma). L et ( E , w, ∗ , η ) b e an exact c ate gory with we ak e quivalenc es and str ong duality. Then ther e ar e homotopy e quivalenc es ( wS e • E ) h ≃ | n 7→ ( iS e • F un w ( n , E )) h | , GW ( E , w ) ≃ | n 7→ GW (F un w ( n , E ) , i ) | which ar e functorial for exact form functors ( F , ϕ ) for which ϕ is an isomorphism. Pr o of. W e star t with so me gener al rema rks. Let C b e a catego ry , and r ecall tha t i stands for the set of isomo rphisms in C . Since C = F un i ([0] , C ), inclusio n of degree - zero simplices yie lds a map of simplicia l catego ries C → ( n 7→ F un i ([ n ] , C )) which is degree-wise an equiv alence of categories , and th us induces a ho motop y equiv alenc e after top ological r ealization. Using the equa lit y of bisimplicial s ets p, q 7→ N p F un i ([ q ] , C ) = N q i F un([ p ] , C ) , where N • stands for the nerve of a ca tegory , we obtain a homotopy equiv a lence |C | ∼ → | n 7→ i F un([ n ] , C ) | . Since the top ological realiza tions of p 7→ i F un([ p ] , C ) a nd of p 7→ i F un([ p ] op , C ) are isomor phic, we hav e a ho motop y equiv alence (6) C ∼ → | n 7→ i F un([ n ] op , C ) | . The ho motop y equiv alence is natural in the ca tegory C . Let ( C , ∗ , η ) b e a categ ory with strong dua lit y . There is an equiv alence of cate- gories i F un([ n ] op , C h ) → ( i F un( n , C )) h which sends an ob ject ( X n , ϕ n ) f n → ( X n − 1 , ϕ n − 1 ) f n − 1 → · · · f 1 → ( X 0 , ϕ 0 ) 16 MARC O SCHLICHTING of the left ha nd ca tegory to the ob ject X n f n → X n − 1 f n − 1 → · · · f 1 → X 0 ϕ 0 → X ∗ 0 f ∗ 1 → X ∗ 1 f ∗ 2 → · · · f ∗ n → X ∗ n equipp ed with the form ( η X n , ..., η X 0 , 1 , ..., 1). A ma p ( g n , ..., g 0 ) (which is an iso - morphism co mpatible with forms) is sent to the map ( g n , ..., g 0 , ( g ∗ 0 ) − 1 , ..., ( g ∗ n ) − 1 ). The equiv alence is functorial in [ n ] ∈ ∆ and thu s induces a homotopy e quiv a lence after top ologica l rea lization. T ogether with (6), we o btain a ho motop y equiv alence of to polog ical spaces (7) |C h | ∼ − → | n 7→ ( i F un( n , C )) h | which is natural for categor ies with strong dualit y ( C , ∗ , η ) and form functors ( F , ϕ ) betw een them for which ϕ is an iso morphism. F or an exac t ca tegory with weak equiv alences and s trong duality ( E , w , ∗ , η ), we apply the general homotopy equiv a lence (7) to the form functor w S e p E → w S e p HE induced by the forg etful form functor E → H E (see remark 2.18). V ary ing p , we obtain a map of homotopy equiv alences after to polog ical realiza tion ( wS e • E ) h ∼ / /   | n 7→ ( iS e • F un w ( n, E )) h |   ( wS e • HE ) h ∼ / / | n 7→ ( iS e • H F un w ( n, E )) h | . The top ro w gives the ﬁrst homotopy equiv a lence of t he lemma. By r emark 2.18, the left vertical homotopy ﬁbre o f the diagram is GW ( E , w , ∗ , η ). In view of Bousﬁeld- F riedlander ’s theorem [BF78, B4], [GJ9 9, Theor em IV 4.9], the homotopy ﬁbre of the right vertical map is the simplicial rea lization of the degr ee-wise ho motop y ﬁbres. By rema rk 2.18, this is | n 7→ GW (F un w ( n , E ) , i, ∗ , η ) | .  Before proving the additivity theorems, we g iv e a ﬁrst a pplication of the s im- plicial res olution lemma and show that π 0 GW ( E , w, ∗ , η ) is the Grothendieck-Witt group as deﬁned in 2.4. 3.8. Prop osition (Pres en tation of GW 0 ). L et ( E , w, ∗ , η ) b e an exact c ate gory with we ak e quivalenc es and duality. Ther e is a natura l isomorphism GW 0 ( E , w, ∗ , η ) ∼ = π 0 GW ( E , w, ∗ , η ) . Pr o of. In view of r elation 2.4 (b) a nd lemma 2 .14, weakly equiv alent non-s ingular exact form functors ( F , ϕ ) ∼ − → ( G, ψ ) induce the same map on GW 0 and on π 0 GW . Therefore, w e c an r eplace ( E , w ) by its strictiﬁcation ( E str w , w ) from lemma 3.7 whic h has a strong (in fact s trict) duality . So w e ca n ass ume the duality o n ( E , w ) to be strong which will allo w us to use the simplicial resolution lemma 3.7. F or a bisimplicial set X •• , there is a co-eq ualizer diagram (8) π 0 | X 1 • | d 1 / / d 0 / / π 0 | X 0 • | / / π 0 | X •• | . THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 17 This is well known and follows from an examination o f the usual skeletal ﬁltr ation of | X •• | = | n 7→ X n,n | = | n 7→ X n • | – which the rea der ca n ﬁnd in [GJ99, Diagram IV.1 (1 .6)], for ins tance – using the fact that the functor π 0 : ∆ op Sets → Sets preserves push-out dia grams a s it is left adjoint to the inclusion functor Sets → ∆ op Sets. By the simplicial res olution lemma 3.7 and the fa ct (proven in [Sch08, Prop osition 4.14 ] tog ether with prop osition 2.2 0 ) that the prop osition holds when the set o f w eak equiv alences is the set of iso morphisms, w e deduce that π 0 GW ( E , w ) is the co-equalizer of the diagra m GW 0   F un w (1 , E )   d 1 / / d 0 / / GW 0   F un w (0 , E )   of Gr othendiec k-Witt gr oups of exact catego ries with dualit y . It suﬃces therefor e to display GW 0 ( E , w, ∗ , η ) as the co-equa lizer o f the same diagra m. Ev aluation at the ob ject 0 ′ of 0 deﬁnes a non-singula r exact form functor F : (F un w (0 , E ) , i ) → ( E , w ) which s ends the ob ject f : X 0 ′ → X 0 to F ( f ) = X 0 ′ and has duality c ompatibilit y morphism F ( f ∗ ) → F ( f ) ∗ the map f ∗ : X ∗ 0 → X ∗ 0 ′ . The form functor induces a map GW 0 (F un w (0 , E ) , i ) → GW 0 ( E , w ) which eq ualizes d 0 and d 1 in view of the relation 2.4 (b). W e therefore o btain a map from the co -equalizer of the diagr am to GW 0 ( E , w ). T o cons truct its in verse, consider the map from the free ab elian group g enerated by isomo rphism class es [ X , ϕ ] o f symmetr ic spaces in ( E , w ) to GW 0 (F un w (0 , E ) , i ) sending ( X , ϕ ) to ϕ : X → X ∗ equipp ed with the non-singular form ( η X , 1). This ma p is s urjectiv e and factors through r elations 2.4 (a ) and (c). The map induces a surjective map to the co- equalizer which factors through relatio n 2.4 (b) and th us induces a surjective map fr om GW 0 ( E , w, ∗ , η ) to the co-equaliz er. Since compo sition with the map from the co-equalizer to GW 0 ( E , w, ∗ , η ) is the ident ity , the claim fo llo ws.  3.9. Remark. Using the co-equa lizer diagra m (8), one can show directly the is o- morphism π 0 | ( wS e • E ) h | ∼ = W 0 ( E , w, ∗ , η ) witho ut the need of the simplicia l resolu- tion le mma. Pr o of of t he or em 3.2. In view of the str ictiﬁcation lemma 3 .4 we can assume the duality on E to b e str ong. B y the s implicial resolutio n lemma 3.7 the pro of r educes further to the case of an ex act c ategory with d uality (in whic h all w eak equiv ale nces are iso morphisms). This ca se w as prov ed in [Sch08, theorems 7.4, 7.10] in view of prop osition 2.2 0.  3.10. Pr o of of the or em 3.3. Theo rem 3.3 is a formal co nsequence o f theorem 3.2. Let ( E , w, ∗ , η ) b e an exact categ ory with weak equiv ale nces and dualit y . Con- sider the form functor s s Cx( E ) → HE : E • 7→ ( E 1 , E ∗ − 1 ) and HE → s Cx( E ) : ( E 1 , E − 1 ) 7→ ( E 1 → E 1 ⊕ E ∗ − 1 → E ∗ − 1 ) with duality compatibility morphisms (1 , η ) : ( E ∗ − 1 , E ∗∗ 1 ) → ( E ∗ − 1 , E 1 ) and ( η ,  0 1 η 0  , 1) : ( E − 1 → E − 1 ⊕ E ∗ 1 → E ∗ 1 ) → ( E ∗∗ − 1 → E ∗ 1 ⊕ E ∗∗ − 1 → E ∗ 1 ). T oge ther with the form functor H 0 : s Cx( E ) → E and the duality preserving functor E → s Cx( E ) : E 7→ (0 → E → 0) they deﬁne non-singular exact form functors E × HE → s Cx( E ) → E × HE whose c ompositio n is weakly equiv alent to the identit y functor. By the additiv- it y theorem for sho rt co mplexes (theorem 3.2), the seco nd form functor induces a 18 MARC O SCHLICHTING homotopy equiv a lence in hermitian K -theory . It follows that the t wo for m func- tors induce in v erse ho motop y equiv a lences on Gro thendiec k-Witt spaces a nd o n hermitian S • constructions. Therefore, the co mpositions A ( F • ,ϕ • ) − → s Cx( B ) ev 0 − → B and A ( F • ,ϕ • ) − → s Cx( B ) → B × H B → s Cx( B ) ev 0 − → B induce homotopic ma ps in hermitian K -theory . These co mpositions are (weakly equiv alent to) the ma ps in (5).  3.11. Remark. Iterated a pplication of theorem 3.3 implies homotopy equiv alences ( wS e • S e n E ) h ∼ − →            ( wS e • E ) h × k Y p =1 wS • E , n = 2 k + 1 k Y p =1 wS • E , n = 2 k . This allows us to identify ( wS e • S e • E ) h with the Bar constr uction of the H -gro up wS • E acting on ( w S e • E ) h and leads to a homotopy ﬁbra tion ( wS e • E ) h → ( w S e • S e • E ) h → w S • S • E in whic h the ﬁr st map is “inclusio n of degree zero simplices” and the se cond ma p is the “ forgetful ma p” ( E , ϕ ) 7→ E followed b y the canonica l homotopy equiv alenc e X e • → X • . In particular , the iterated hermitia n S • -construction ( wS e • S e • E ) h is not a delo oping of ( w S e • E ) h contrary to the K -theory situation, co mpare [W al85, prop osition 1.5 .3 and rema rk thereafter]. 3.12. Remark. Deﬁne the Witt-the ory sp ac e W ( E , w , ∗ , η ) as the colimit of the top row in the sequence of homotopy ﬁbra tions GW ( E , w ) / / ( wS e • E ) h / /   ( wS e • S e • E ) h / /   ( wS e • S e • S e • E ) h / /   · · · wS • E wS • S • E wS • S • S • E Since the space s in the se cond r o w g et highe r and higher connected, we see that π 0 W ( E , w , ∗ , η ) = W 0 ( E , w, ∗ , η ) and π 1 W ( E , i, ∗ , η ) = W f or m ( E , ∗ , η ), where the group W f or m ( E , ∗ , η ) is the Witt-group of formations in ( E , ∗ , η ), that is , the co k- ernel of the hyperb olic ma p K 0 ( E ) → GW f or m ( E ) : [ X ] 7→ [ H X, X , X ∗ ] to the Grothendieck-Witt gro up o f for mations GW f or m ( E ) deﬁned in [Sc h08, 4.11]. If E is a Z [ 1 2 ]-linear category a nd “complicia l”, we s ho w in [Scha ] that the Grothendieck-Witt s pace W ( E , w , ∗ , η ) is the inﬁnite lo op space a sso ciated with (t he ( − 1)-connected cov er of ) Ranicki’s L -theory sp ectrum, and its homotopy groups π i W ( E , w , ∗ , η ) = W − i ( w − 1 E , ∗ , η ) , i ≥ 0 , are Ba lmer’s Witt groups W − i ( w − 1 E , ∗ , η ) of the triangulated categ ory with dualit y ( w − 1 E , ∗ , η ). At this p oint, I don’t k no w how to calc ulate π n W ( E , i, ∗ , η ), n ≥ 2, for (complicial) ( E , w, ∗ , η ) when E is not Z [ 1 2 ]-linear. THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 19 4. Change of weak equiv al ences a nd Cofinality In this section we prove in theorems 4.2 and 4.10 the hig her Gro thendiec k-Witt theory analogs of W aldha usen’s ﬁbration theorem [W a l85, 1.6.4] and of Thomason’s coﬁnality theo rem [TT90, 1 .10.1]. W aldhausen’s K -theory v ersion of theorem 4.2 b elow n eeds a “cy linder functor”. The purp ose of the next deﬁnition is to deﬁne the higher Grothendieck-Witt theory analog. W e ﬁrst ﬁx some notation. F or an ex act catego ry with w eak equiv alences ( E , w ), write E w ⊂ E for the full sub category of w -acyclic ob jects, that is, those ob jects E o f E fo r which the unique map 0 → E is a weak equiv alence. The categ ory E w is closed under extensions in E , and thus inherits an exact structure from E such that the inclus ion E w ⊂ E is fully exa ct. 4.1. Deﬁnition. Let ( E , w , ∗ , η ) be an exact categor y with weak equiv alences and duality . A symmetric c one o n ( E , w , ∗ , η ) is given by the following data: (a) exact functors P : E → E w , and C : E → E w , (b) a natura l deﬂation p E : P E ։ E and a natural inﬂation i E : E ֌ C E , (c) a natural map γ E : P ( E ∗ ) → ( C E ) ∗ such that i ∗ E γ E = p E ∗ for all ob jects E of E . It is conv enien t to deﬁne ¯ γ E : C ( E ∗ ) → ( P E ) ∗ by ¯ γ E = P ( η E ) ∗ ◦ γ ∗ E ∗ ◦ η C ( E ∗ ) . Then p ∗ E = ¯ γ E ◦ i E ∗ , and γ E = C ( η E ) ∗ ◦ ¯ γ ∗ E ∗ ◦ η P ( E ∗ ) . In other words, the s equence P → id → C deﬁnes an exa ct form functor from E to the catego ry of se quences E − 1 ։ E 0 ֌ E 1 in E with duality co mpatibilit y map ( γ , 1 , ¯ γ ). F or examples of s ymmetric cones , see 7.5. The pro of of the nex t theor em will o ccupy most o f this section. 4.2. Theorem (Change of w eak equiv alences). L et ( E , w , ∗ , η ) b e an exact c ate gory with we ak e quivalenc es and duality which has a symmetric c one. L et v b e another set of we ak e quivalenc es in E c ontaining w and which is close d under the duality. Then the duality ( ∗ , η ) on E makes ( E v , w ) , ( E v , v ) , ( E , v ) into exact c ate gories with we ak e quivalenc es and duality su ch that the c ommutative squar e of duality pr eserving inclusions ( E v , w ) / /   ( E v , v )   ( E , w ) / / ( E , v ) induc es a homotopy c artesian squ ar e of asso ciate d Gr othendie ck-Witt sp ac es. Mor e- over, the upp er right c orner has c ontr actible Gr othendie ck-Witt sp ac e. 4.3. Remark. A sq uare of homotopy c omm utativ e H -g roups (such as Gr othen- dieck-Witt spaces) is homo top y car tesian if a nd only if the ma p betw een, say , horizontal homotopy ﬁbres is a homotopy equiv alence, and the map of ab elian groups be t w een ho rizont al co k ernels o f π 0 ’s is a monomorphism. 4.4. Remark. In theorem 4.2, the map GW 0 ( E , w, ∗ , η ) → GW 0 ( E , v , ∗ , η ) is no t surjective, in gener al, contrary to the K -theor y situa tion in [W al85, Fibration the- orem 1 .6.4]. 20 MARC O SCHLICHTING W e will reduce the pro of of theorem 4.2 to idemp otent complete exact categ ories with weak equiv alences and dualit y . W e recall the re lev ant deﬁnitions. 4.5. Idemp oten t completion. Recall tha t the idemp otent c ompletion ˜ E o f an exact categ ory E has ob jects pa irs ( A, p ) with p = p 2 : A → A a n idemp oten t in E . A map ( A, p ) → ( B , q ) is a map f : A → B in E s uc h that f = f p = qf . Comp osition is compo sition of maps in E . The idempotent co mpletion ˜ E has a canonical str ucture of a n exact categor y suc h that the inclus ion E ⊂ ˜ E : A 7→ ( A, 1) is fully exact (see [TT90, The orem A.9.1], wher e “idemp oten t completion” is called “Karo ubianisation”). Any duality ( ∗ , η ) on E extends to a dualit y ( A, p ) ∗ = ( A ∗ , p ∗ ) on ˜ E w ith double dual identiﬁcation η A ◦ p : ( A, p ) → ( A, p ) ∗∗ . If ( E , w , ∗ , η ) is an exact categ ory with dua lit y , call a map in the idemp oten t completion ˜ E we ak e quivalenc e if it is a re tract of a w eak equiv alence in E . Then ( ˜ E , w, ∗ , η ) is an exact catego ry with weak equiv a lences and dua lit y . Note that the natural inclusion f E w ⊂ ( ˜ E ) w is an equiv alence o f catego ries if ( E , w , ∗ , η ) ha s a (symmetr ic) c one. This is b ecause for an ob ject X in ( ˜ E ) w , the weak equiv ale nce 0 → X in ˜ E is, by deﬁnition, a retr act of a weak equiv alence f : Y → Z in E , a nd, by functoriality , also a retract of 0 → C ( f ), where C ( f ) is the push-out of i Y : Y → C Y a long f . Since C ( f ) is in E w , the ob ject X is (isomorphic to a n ob ject) in f E w . 4.6. Lemma. L et ( E , w , ∗ , η ) b e an exact c ate gory with we ak e quivalenc es and str ong duality which has a symmetric c one. Then the c ommut ative diagr am of exact c ate- gories with duality ( E w , i ) / /   ( ˜ E w , i )   ( E , i ) / / ( ˜ E , i ) induc es a homotopy c artesian squar e of Gr othendie ck-Witt sp ac es. Pr o of. By the coﬁnality theo rem in [Sch08, 5.5], the horizo n tal homo top y ﬁbres o f asso ciated Grothendiec k-Witt spaces are con tractible. Ther efore, it suﬃces to sho w that the map GW 0 ( ˜ E w ) /GW 0 ( E w ) → GW 0 ( ˜ E ) /GW 0 ( E ) betw een the cokernels o f horizontal π 0 ’s is injective. F o r an exact ca tegory with dualit y A , the quotient GW 0 ( ˜ A ) /GW 0 ( A ) is the ab elian monoid of isometry cla sses of symmetric spa ces in ˜ A mo dulo the submono id of symmetric spaces in A [Sch08 , 5.2]. In particular , a symmetric space in ˜ A yields the zero class in GW 0 ( ˜ A ) /GW 0 ( A ) if and only if it is stably in A . Let ( A, α ) be a symmetric space in ˜ E w whose clas s in GW 0 ( ˜ E ) /GW 0 ( E ) is zero. Then there ar e s ymmetric spaces ( X , ϕ ) and ( Y , ψ ) in E and a n is ometry ( A, α ) ⊥ ( Y , ψ ) ∼ = ( X, ϕ ). In pa rticular, there is a n exact sequence 0 → A → X f → Y → 0 in ˜ E . The push-out C ( f ) of f along the E - inﬂation X ֌ C X is in E . In the exact sequence 0 → A → C X → C ( f ) → 0, we hav e A a nd C X in ˜ E w , so that C ( f ) is also in ˜ E w , hence C ( f ) ∈ ˜ E w ∩ E = E w . Cho ose ¯ A ∈ ˜ E w such that A ⊕ ¯ A ∈ E w . Then ¯ A ⊕ C X is in E w as it is an extension of A ⊕ ¯ A and C ( f ), b oth b eing in E w . It follows that the symmetric spaces ( A, α ) ⊕ H ( ¯ A ⊕ C X ) and H ( ¯ A ⊕ C X ) lie in E w , where H E = ( E ⊕ E ∗ ,  0 1 η 0  ) is the hyperb olic space ass ocia ted with E . This implies tha t ( A, α ) is trivia l in GW 0 ( ˜ E w ) /GW 0 ( E w ).  THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 21 F or an e xact category with weak equiv alenc es and dualit y ( E , w , ∗ , η ), the cate- gory Mo r E = F un([1] , E ) of morphisms in E is an exac t category with weak equiv a- lences and duality suc h that the fully exact inclus ion E ⊂ Mor E : E 7→ i d E , induced by the unique map [1] → [0], is duality preserving. The inclusion facto rs throug h the fully exa ct sub category Mor w E = F un w ([1] , E ) ⊂ Mor E of weak equiv alences in E a nd deﬁnes a dualit y pr eserving functor I : E → Mor w E . Note tha t (Mor w E ) w = Mor w ( E w ) = Mor( E w ). The following prop osition is the key to proving Theo rem 4.2. 4.7. Prop osition. L et ( E , w , ∗ , η ) b e an ex act c ate gory with we ak e quivalenc es and str ong duality. Assum e that ( E , w, ∗ , η ) has a symmetric c one. Then the c ommuta- tive diagr am (9) E w / /   Mor w E w   E I / / Mor w E of duality pr eserving inclusions of exact c ate gories with duality ( al l we ak e quiv- alenc es b eing isomorphisms) induc es a homotopy c artesian squar e of asso ciate d Gr othendie ck-Witt sp ac es. The pro of uses the co ne c ategory construction of [Sch08, § 9]. W e r ecall the relev a n t deﬁnitions and facts. 4.8. Cone exact categories. Let A ⊂ U b e a duality preser ving fully exact in- clusion of idempotent complete ex act categor ies with duality ( ∗ , η ). In [Sch08, § 9], we c onstructed a duality preserving fully e xact inclusion Γ : U ⊂ C ( U , A ) of exact categorie s with duality , dep ending functorially on the pair A ⊂ U , such that the duality preserving commutativ e sq uare (10) A Γ / /   C ( A , A )   U Γ / / C ( U , A ) of e xact categor ies with duality induces a homotopy cartesia n s quare of asso ciated Grothendieck-Witt spaces, and the upp er right corner C ( A , A ) (also wr itten as C ( A )) has co n tractible Grothendieck-Witt space [Sch08, theor em 9.9]. W e rec all the de ﬁnition o f the c one c ate gory C ( U , A ), details ca n b e found in [Sch08, § 9 .1-9.3]. One ﬁr st constructs a category C 0 ( U , A ), a lo calization of which is C ( U , A ). O b jects of C 0 ( U , A ) ar e commutativ e diag rams in U (11) U 0 / / ∼ / /   U 1 / / ∼ / /   U 2 / / ∼ / /   U 3 / / ∼ / /   · · · U 0 U 1 ∼ o o o o U 2 ∼ o o o o U 3 ∼ o o o o · · · ∼ o o o o such that the maps U i ∼ ֌ U i +1 and U i +1 ∼ ։ U i , i ∈ N are inﬂations with cokernel in A and deﬂations with kernel in A , respectively . Moreover, there has to be an integer 22 MARC O SCHLICHTING d s uc h that for every i ≥ j , the map U j → U i + d is an inﬂation with cok ernel in A and the map U i + d → U j is a deﬂa tion with k ernel in A . If the maps in diagr am (11) are understo o d, we may abbreviate the diagram a s ( U • → U • ). Maps in C 0 ( U , A ) are natural tra nsformations of diagrams. A sequence of diagrams in C 0 ( U , A ) is exact if at ea c h U i , U j sp ot it is a n ex act seq uence in U . T he dua l of the diag ram (11) is obtained by applying the duality to the diag ram: ( U • → U • ) ∗ = (( U • ) ∗ → ( U • ) ∗ ). F or each diagr am (11), for getting the upper left corner U 0 gives us a new ob ject ( U • → U • ) [1] = ( U • +1 → U • ) and a ca nonical map ( U • → U • ) → ( U • +1 → U • ). Similarly , forgetting the low er left c orner U 0 deﬁnes a new ob ject ( U • → U • ) [1] = ( U • → U • +1 ) and a c anonical map ( U • → U • +1 ) → ( U • → U • ). Finally , the category C ( U , A ) is the lo calization of C 0 ( U , A ) with r espect to the tw o type s of canonical maps just deﬁned. A sequence in C ( U , A ) is a conﬂation if and only if it is isomo rphic in C ( U , A ) to the imag e under the lo caliza tion functor C 0 ( U , A ) → C ( U , A ) of a conﬂation in C 0 ( U , A ). There is a fully exact duality pr eserving inclusio n Γ : U ⊂ C ( U , A ) which sends an ob ject U of U to the constant dia gram U / / = / / id   U / / = / / id   U / / = / / id   U / / = / / id   · · · U U = o o o o U = o o o o U = o o o o · · · = o o o o Pr o of of pr op osition 4.7. By Lemma 4.6, we c an (and will) assume E to b e idem- po ten t complete. Then all categ ories in diag ram (9) ar e idempo ten t complete. W e will write C ( E , w ) and C ( E w ) instead o f the categories C ( E , E w ) and C ( E w , E w ) of 4.8, and we will write ( F , ϕ ) ∼ ( G, ψ ) if the tw o non-singula r exact form functors ( F, ϕ ), ( G, ψ ) induce homo topic maps on Gro thendiec k-Witt spaces. The strategy of pro of is as follows. W e will ex tend diagram (9) to a commutativ e diagra m o f exact categor ies with duality and non-singular exact form functors (12) E w / /   Mor w E w   / / C ( E w )   / / C (Mor w E w )   E I / / Mor w E ( F, ϕ ) / / C ( E , w ) C ( I ) / / C (Mor w E , w ) where the right hand squar e is obtained from (9) by functor ialit y of the cone cate- gory construction in 4.8. W e will show: ( † ) ( GW applied to) the comp ositions ( F, ϕ ) ◦ ( I , i d ) a nd ( C ( I ) , id ) ◦ ( F, ϕ ) are homotopic to the consta n t diag ram inclusions (Γ , i d ) o f 4.8 in such a wa y that the ho motopies r estricted to E w and Mor w E w hav e image s in the Grothendieck-Witt spa ces of C ( E w ) and C (Mo r w E w ), resp ectively . Assuming ( † ), the outer diagra m o f the left t wo squa res and the outer dia gram of the right t wo squa res induce homoto p y car tesian diagrams of Grothendieck-Witt spaces, b y [Sch08, theorem 9.9]. By remark 4.3, this implies the claim of propositio n 4.7. THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 23 T o cons truct dia gram (1 2), we will deﬁne the non-singula r exact form functor ( F, ϕ ) : Mo r w E → C ( E , w ) as the comp osition of a non-sing ular exa ct form func- tor ( F 0 , ϕ ) : Mo r w E → C 0 ( E , w ) and the lo calization functor C 0 ( E , w ) → C ( E , w ) such that its restr iction to Mor E w has image in C 0 ( E w ). Recall that ( E , w , ∗ , η ) is assumed to hav e a symmetric cone (see deﬁnition 4.1), wher e i : id ֌ C and p : P ։ id denote the natural inﬂation and deﬂation which ar e part of the struc- ture o f a symmetric cone. The functor ( F , ϕ ) (or ra ther ( F 0 , ϕ )) sends an ob ject g : X → Y of Mor w E to the ob ject F ( g ) given by the diag ram (13) X / / / / g   X ⊕ P Y ( g p i 0 )   / / / / X ⊕ P Y ⊕ C X „ g p 0 i 0 1 0 1 0 «   / / / / X ⊕ P Y ⊕ C X ⊕ P Y   / / / / · · · Y Y ⊕ C X o o o o Y ⊕ C X ⊕ P Y o o o o Y ⊕ C X ⊕ P Y ⊕ C X o o o o · · · o o o o of C 0 ( E , w ). In the notation ( U • → U • ) of 4 .8 cor respo nding to diagram (11), the ob ject F ( g ) is given by U n = X ⊕ P Y ⊕ C X ⊕ P Y ⊕ C X ⊕ · · · ( n + 1 s ummands), U n = Y ⊕ C X ⊕ P Y ⊕ C X ⊕ P Y ⊕ · · · ( n + 1 summands). The maps U n → U n +1 and U n +1 → U n are the canonical inclusions into the ﬁrst n + 1 summands and the canonica l pro jections onto the ﬁrst n + 1 facto rs. The maps U n → U n are given by the matrix ( u r s ) 0 ≤ r,s ≤ n =     g p 0 0 0 ··· i 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 . . . 0 . . .     with u r s = 0 unless r = s = 0 o r | r − s | = 1, a nd u 0 , 0 = g , u 0 , 1 = p Y , u 1 , 0 = i X , u r,r +1 = 1, u r +1 ,r = 1 for r ≥ 1 . The construction of diagr am (13) is functoria l in g , so that F : Mo r w E → C ( E , w ) is indee d a functor. The duality compatibility map ϕ g : F ( g ∗ ) → ( F g ) ∗ for g : X → Y is the identit y on X ∗ and Y ∗ , it is γ X on the summands P ( X ∗ ) and ¯ γ Y on C ( Y ∗ ). It is clear that F s ends the sub catego ry Mor w E w to the full sub category C ( E w ) of C ( E , w ). This deﬁnes diagr am (12). W e are left with proving ( † ). Since U 0 = X is the initial ob ject of dia gram (13), it deﬁnes a map j = j g : X = U 0 → F ( g ), where U 0 (and X ) is considered an ob ject of C ( E , w ) via the constant dia gram embedding Γ : E → C ( E , w ). The map j : X → F ( g ) is an inﬂation in C ( E , w ) with cokernel in C ( E w ) b ecause j : X → ( U • → U • +1 ) is a n inﬂa tion in C 0 ( E , w ) with co k ernel in C 0 ( E w ). V arying g , the map j g deﬁnes a natural transforma tion j : Γ → F . Similarly , U 0 is the ﬁnal ob ject of diagram (13) and thus deﬁnes a (functorial) map q = q g : F ( g ) → U 0 = Y with kernel in C ( E w ). W e hav e g = q j . Let ˆ ϕ : F → F ♯ = ∗ F ∗ b e the symmetric for m on the functor F asso ciated with the duality compatibility map ϕ (se e 2.1). The form ˆ ϕ : F → F ♯ ﬁts into a commutativ e diagram in C 0 ( E , w ) (14) X j g / / η X   F ( g ) ˆ ϕ g   q g / / Y η Y   X ∗∗ q ∗ g ∗ / / F ( g ∗ ) ∗ j ∗ g ∗ / / Y ∗∗ . 24 MARC O SCHLICHTING W rite ( F I , ϕ F I ) for the composition ( F , ϕ ) ◦ ( I , id ) o f form funct ors. The natural transformatio n j ab o ve makes the ca nonical inclusion (Γ , i d ) : E → C ( E , w ) into an admissible subfunctor j : Γ ⊂ F I o f F I . Commutativit y o f diagram (14) for g = id X implies that η = j ♯ ˆ ϕ F I j , that is , j deﬁnes a map (Γ , η ) → ( F I , ˆ ϕ F I ) of symmetric spaces ass ocia ted with the form functors (Γ , id ) a nd ( F I , ϕ F I ). Since the maps id and ϕ F I are isomorphisms, the symmetric space ( F I , ˆ ϕ F I ) decompose s in F un ( E , C ( E , w )) as (Γ , η ) ⊥ ( A, ˆ ϕ A ) with ( A, ˆ ϕ A ) the orthog onal co mplemen t o f (Γ , η ) in ( F I , ˆ ϕ F I ). As mentioned ab ov e , the cokernel of j : X → F I ( X ) = F (1 X ) is in C ( E w ). Therefo re, the form functor ( A, ϕ A ) factors thro ugh the ca tegory C ( E w ) (whose hermitian K -theory space is contractible [Sch08, co rollary 9.8]), so that ( A, ϕ A ) ∼ 0. Hence, ( F I , ϕ F I ) ∼ = (Γ , id ) ⊥ ( A, ϕ A ) ∼ (Γ , id ) . By construction, the homotopy restricted to E w has image in the Gro thendiec k-Witt space of C ( E w ). This s ho ws the ﬁrst half of the claim ( † ). F or the second ha lf, write ( I F, ϕ I F ) and ( I F 0 , ϕ I F 0 ) for the comp ositions o f form functors ( C ( I ) , id ) ◦ ( F, ϕ ) and ( C ( I ) , id ) ◦ ( F 0 , ϕ ), a nd no te that ( I F, ϕ I F ) is just the compo sition of ( I F 0 , ϕ I F 0 ) with the lo caliza tion functor C 0 (Mor w E , w ) → C (Mor w E , w ). There is a n obvious is omorphism of exact categories with dua lit y Mor w C 0 ( E , w ) ∼ = C 0 (Mor w E , w ) suc h that the comp osition I F 0 sends the ob ject ( g : X → Y ) ∈ Mor w E to id F ( g ) : F ( g ) → F ( g ), and the duality compatibility morphism ϕ I F 0 bec omes ( ϕ g , ϕ g ) : id F ( g ∗ ) → id F ( g ) ∗ . Consider the functor ial bicartesian square ( X j → F ( g )) ( j, 1) / / (1 ,q )   ( F ( g ) 1 → F ( g )) (1 ,q )   ( X g → Y ) ( j, 1) / / ( F ( g ) q → Y ) in Mor w C 0 ( E , w ). The total complex o f the squa re (considered as a bicomplex) is a conﬂation in Mor w C 0 ( E , w ) = C 0 (Mor w , w ). It is therefore a lso a conﬂa tion in C (Mor w , w ), hence the squa re is also bicartesian in C (Mor w , w ). In C (Mor w , w ), the horizontal maps in the square a re inﬂations with cokernel in C (Mor w E w ) since ( j, 1) is iso morphic to the C 0 (Mor w , w )-inﬂation ( j, 1) [1] which ha s cokernel in C 0 (Mor w E w ). Similarly , the vertical maps in the squa re ar e deﬂations in C (Mor w , w ) with kernel in C (Mo r w E w ) since (1 , q ) is iso morphic to the C 0 (Mor w , w )-deﬂation (1 , q ) [1] with kernel in C 0 (Mor w E w ). Commutativit y of dia gram (14) implies that the for m ˆ ϕ I F 0 on the upper rig h t corner I F 0 of the square extends to a for m o n the whole bicartesian squar e such that its restr iction to the low er left corner is the co nstan t diagram inclusio n (Γ , η ) : Mor w E → C 0 (Mor w E , w ). It follows that ker(1 , q ) ⊂ I F is a totally isotropic subfunctor of ( I F , ϕ I F ) with induced form on ( X g → Y ) = ker (1 , q ) ⊥ / ker(1 , q ) isometric to the cons tan t diagra m inclusion (Γ , id ) : Mor w E → C (Mor w E , w ). By the additivity theorem [Sch08, theorem 7 .1] (or its genera lization in theo- rem 3.3), the for m functors (Γ , id ) ⊥ H ker (1 , q ) and ( I F , ϕ I F ) induce homo topic maps on Gr othendiec k-Witt s paces. Since H ker(1 , q ) has ima ge in C (Mor w E w ) whose Grothendieck-Witt space is co n tractible, w e hav e ( I F , ϕ I F ) ∼ (Γ , id ) ⊥ H ker(1 , q ) ∼ (Γ , id ). The ho motopies r estricted to Mor w E w hav e image in the Grothendieck-Witt spac e of C (Mor w E w ). This is clear for the second homo top y , and for the ﬁrst, it follows from the pro of of additivity in 3.1 0.  THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 25 Next, we prov e a v ariant o f the Cha nge-of-weak-equiv alence theorem. 4.9. Prop osition. L et ( E , w , ∗ , η ) b e an ex act c ate gory with we ak e quivalenc es and str ong duality which has a symmetr ic c one. T hen t he fol lowing c ommut ative di- agr am of ex act c ate gories with we ak e quivalenc es and duality induc es a homotopy c artesian squar e of Gr othendie ck-Witt sp ac es with c ontr actible upp er right c orner (15) ( E w , i ) / /   ( E w , w )   ( E , i ) / / ( E , w ) . Pr o of. F rom lemma 2 .14 it is clear that GW ( E w , w ) is contractible since 0 → id is a natur al weak equiv alence in ( E w , w ). Consider the commutativ e diagr am of (sim- plicial) exact categorie s with dualities (all weak equiv alences be ing isomo rphisms) E w / /   F un w (0 , E w )   / /   n 7→ F un w ( n, E w )     E / / F un w (0 , E ) / /   n 7→ F un w ( n, E )   in whic h the left hand squar e can b e identiﬁed with the squar e o f pro position 4.7 and induces ther efore a homotopy cartesian s quare o f Grothendieck-Witt spac es. On Grothendieck-Witt spa ces, the right vertical map c an be identiﬁed with the map ( E w , w ) → ( E , w ) in view of the simplicial reso lution lemma 3.7. The right hand squar e is the inclusion of degr ee zer o simplices. The pro of of prop osition 4.9 is th us reduced to showing that the right hand squar e of the diagram induces a homotopy cartesian square of Grothendieck-Witt spaces. Let F un 1 w ( n , E ) ⊂ F un w ( n, E ) be the full sub categor y o f those fun ctors A : n → E for which A p → A q is an inﬂation, and A q ′ → A p ′ is a deﬂation, 0 ≤ p ≦ q ≤ n . It inherits the structur e o f an exa ct ca tegory with duality from F un w ( n , E ). F urther, let F un 0 w ( n , E ) be the categ ory which is equiv alent to F un 1 w ( n, E ) but where an ob ject is an ob ject A of F un 1 w ( n , E ) to gether with a choice of subquo tien ts A p,q = A q / A p = coker ( A p ∼ ֌ A q ) ∈ E w and induced maps A p,q → A p,q , and together with a choice of kernels A q ′ ,p ′ = ker( A q ′ ∼ ։ A p ′ ) ∈ E w for 0 ≤ p ≤ q ≤ n . The category F un 0 w ( n , E ) is a n exact category with dua lit y such that the forgetful functor F un 0 w ( n , E ) → F un 1 w ( n, E ) is an equiv alence of ex act categor ies with dualit y . W e hav e an ex act functor F un 0 w ( n , E ) → S n E w : A 7→ ( A p,q ) 0 ≤ p ≤ q ≤ n . By the additivity theorem [Sch08, theo rem 7.1] (or 3.3), this functor induces a ma p which is pa rt of a split homotopy ﬁbration GW (F un w (0 , E )) → GW (F un 0 w ( n, E )) → K ( S n E w ) . 26 MARC O SCHLICHTING The same argument applies to ( E w , w ) instead of ( E , w ) so that, v arying n , we obtain a map of homotopy ﬁbratio ns after top ological realiza tion GW F un w (0 , E w )   / / | n 7→ GW F un 0 w ( n, E w ) |   / / | n 7→ K ( S n E w ) | GW F un w (0 , E ) / / | n 7→ GW F un 0 w ( n, E ) | / / | n 7→ K ( S n E w ) | which s ho ws that the left square is homotopy cartesia n. Since F un 0 w → F un 1 w is an equiv alenc e of exact categories with dualit y , the propo - sition follows once w e show that the inclusion I : F un 1 w ( n , A ) ⊂ F un w ( n, A ) induces a homotopy equiv alence on Gro thendiec k-Witt spaces for A = E , E w . W e illustrate the argument for A = E and n = 1. The general case is m utatis m utandis the same. W e deﬁne tw o functors F , G : F un w (1 , E ) → F un 1 w (1 , E ). The functor F sends E 1 ′ ∼ → E 0 ′ ∼ → E 0 ∼ → E 1 to E 1 ′ ⊕ P E 0 ′ ∼ ։ E 0 ′ ∼ → E 0 ∼ ֌ E 1 ⊕ C E 0 , the functor G sends the same ob ject to P E 0 ′ ∼ ։ 0 ∼ → 0 ∼ ֌ C E 0 . Bo th functors F a nd G are equippe d with cano nical duality compatibility morphis ms, induced by γ and ¯ γ fr om deﬁnition 4.1, such that F and G are non- singular exa ct form functors. By the additivity theorem, w e hav e I F ∼ i d ⊥ I G a nd F I ∼ id ⊥ GI , so that GW ( F ) − GW ( G ) deﬁnes an inv erse o f GW ( I ), up to homotopy .  Pr o of of the or em 4.2. By lemma 2.14, GW ( E w , w ) is contractible. Let A = E str w be the s trictiﬁcation of E from lemma 3.4, and recall that it ha s a str ong dualit y . Consider the co mm utativ e diagram of exa ct categ ories with weak equiv a lences a nd duality ( A w , i ) / /   ( A w , w )   ( A v , i ) / /   ( A v , w ) / /   ( A v , v )   ( A , i ) / / ( A , w ) / / ( A , v ) . By the strictiﬁcation lemma 3.4 and lemma 2.14, the square in theore m 4.2 is equiv alent to the low er r igh t sq uare in the diag ram. B y prop osition 4.9, the upp er square and the outer diag ram of the left t w o s quares a re ho motop y ca rtesian in hermitian K - theory . Since the le ft vertical maps a re surjective on GW 0 (beca use A = E str w ), it follows that the lower left square is homotopy cartesia n in hermitian K -theor y , by rema rk 4.3. Again, b y pro positio n 4.9, the outer diagram of the tw o low er squa res induces a homoto p y car tesian square of Gr othendiec k-Witt spaces. T ogether with the facts that t he lower left square is homotopy cartes ian in hermitian K -theor y and that the low er le ft vertical map is surjective on GW 0 , this implies that the low er r igh t squa re induces a homotopy ca rtesian s quare of Grothendieck- Witt spaces.  4.10. Theorem (Coﬁnality). L et ( E , w , ∗ , η ) b e an exact c ate gory with we ak e quiv- alenc es and duality which has a symmetric c one. Le t A ⊂ K 0 ( E , w ) b e a su b gr oup THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 27 close d u nder the duality actio n on K 0 ( E , w ) , and let E A ⊂ E b e the ful l su b c ate- gory of those obje cts whose class in K 0 ( E , w ) b elongs to A . Then t he c ate gory E A inherits t he structur e of an exact c ate gory with we ak e quivalenc es and duality fr om ( E , w, ∗ , η ) , and the induc e d map on Gr othendie ck Witt sp ac es GW ( E A , w , ∗ , η ) − → GW ( E , w, ∗ , η ) is an isomorphism on π i , i ≥ 1 , and a monomorphi sm on π 0 . Pr o of. Let U = E str w , and c onsider the the diagram o f exact ca tegories with weak equiv alences a nd duality ( U w A , i ) / / ( U A , i )   / / ( U A , w )   ( U w , i ) / / ( U , i ) / / ( U , w ) . On Grothendieck-Witt spaces, the r igh t vertical map can b e identiﬁed (up to ho- motopy) with the ma p in the theorem, by lemmas 3.4 and 2.1 4. The rows ar e homotopy ﬁbratio ns, by prop osition 4.9, and the r igh t horizontal maps are surjec- tive on GW 0 (as U = E str w ). It follows that the r igh t squa re induces a homotopy cartesian square o f Gro thendiec k-Witt space s. Since U A ⊂ U is a co ﬁnal inclusion of exact catego ries with duality , the coﬁnality theorem of [Sch08, co rollary 5.5 ] shows that t he homotop y ﬁbre of the Gr othendiec k-Witt spaces of the middle vertical map is contractible. As the rig h t s quare is homotopy cartes ian in hermitian K -theory , the same is true for the rig h t vertical map.  5. Appro xima tion, Change of exact structure and Resolution In this section w e prov e in theorems 5 .1 a nd 5.11 v ar ian ts o f W aldha usen’s ap- proximation theor em [W al85, Theorem 1 .6.7] which hold for higher Grothendieck- Witt groups. W e expla in tw o immediate consequences , one concerning the condi- tions under which a c hange o f exact structur e has no eﬀect o n Grothendieck-Witt groups (lemma 5 .3, co mpare [TT90, Theor em 1.9.2]) and the other co ncerning a n analog of Q uillen’s r esolution theorem (lemma 5.6, co mpare [Q ui73, § 4 Corollar y 1]). 5.1. Theorem (Approximation I). L et ( F, ϕ ) : A → B b e a non-singular exact form functor b etwe en exact c ate gories with we ak e quivalenc es and duality. Assume the fol lowing. (a) Every map in A c an b e written as t he c omp osition of an inﬂation fol lowe d by a we ak e quivalenc e. (b) A map in A is a we ak e quivalenc e iﬀ its image in B is a we ak e quivalenc e. (c) F or every map f : F A → E in B , ther e is a map a : A → B in A and a we ak e quivalenc e g : F B ∼ → E in B such that f = g ◦ F a : F A ∀ f / / _ _ _ F a   E ∃ F B ∼ ; ; x x x x x (d) The duality c omp atibility morphism ϕ A : F ( A ∗ ) → F ( A ) ∗ is an isomor- phism for every A in A . 28 MARC O SCHLICHTING (e) F or every map f : F A → F B in B , ther e is an ex act functor L : A → A , a natura l we ak e quivalenc e λ : L ∼ → id A and a map a : LA → B in A su ch that F a = f ◦ F λ A : ∃ F LA ∼ / / F a 5 5 F A ∀ f / / _ _ _ F B . (f ) F or every map a : A → B in A such that F a = 0 in B , ther e is an ex act functor L : A → A and a n atur al we ak e qu ival enc e λ : L ∼ → id A such that a ◦ λ A = 0 in A : ∃ LA ∼ / / 0 6 6 A ∀ a / / B , F a = 0 . Then ( F , ϕ ) induc es homotopy e quivalenc es ( wS e • A ) h ∼ − → ( wS e • B ) h and GW ( A , w , ∗ , η ) ∼ − → GW ( B , w, ∗ , η ) . Pr o of. F or the purp ose of the pro of, we c all lattic e a pair ( L, λ ) with L : A → A an e xact functor and λ : L ∼ → id A a na tural weak equiv a lence. Lattices form a n asso ciative monoid under co mposition ( L 2 , λ 2 ) ◦ ( L 1 , λ 1 ) := ( L 2 L 1 , λ 2 ◦ L 2 ( λ 1 )) . Since λ 2 ◦ L 2 ( λ 1 ) = λ 1 ◦ λ 2 ,L 1 (as λ 2 is a natural transformation), lattices b ehav e like a multiplicativ e set in a commutativ e ring. Mo re precisely , comp osition of lattices allows us to generaliz e prop erties (e) and (f ) to ﬁnite families of maps: (e’) for any ﬁnite set of ma ps f i : F A i → F B i in B , i = 1 , ..., n , there are a lattice ( L, λ ) and maps a i : LA i → B i such that F a i = f i ◦ F λ A i , i = 1 , ..., n , and (f ’) for any ﬁnite s et o f maps a i : A i → B i in A suc h that F a i = 0 in B , i = 1 , ..., n , there is a lattice ( L, λ ) such that a i λ A i = 0 in A , i = 1 , ..., n . W e will refer to (e’) and (f ’) as “clea ring denominator s”, in ana logy with the lo- calization of a commutativ e ring with resp ect to a multiplicativ e subset. “Clearing denominators” together with (b) and (d) implies that we can lift non-degener ate symmetric forms from B to A in the following se nse: ( † ) F or any non-degenerate symmetric form ( F A, α ) ∈ ( w B ) h on the image F A of an ob ject A o f A , there is a la ttice ( L, λ ) and a non-degenera te symmetric form ( LA, β ) ∈ ( w A ) h on LA suc h that F ( λ A ) is a map of symmetric s paces F ( λ A ) : F ( LA, β ) ∼ → ( F A, α ). The pro of is the same as the classical pro of which s ho ws that a no n-degenerate symmetric for m ov er the fra ction ﬁeld o f a Dedekind domain can b e lifted to a (usual) lattice in the ring. In detail, the map ϕ − 1 A α : F A → F ( A ∗ ) lifts to a map a 1 : L 1 A → A ∗ such that ϕ A F a 1 = αF λ 1 ,A for s ome la ttice ( L 1 , λ 1 ). The ma p a : λ ∗ 1 ,A a 1 : L 1 A → ( L 1 A ) ∗ is a weak eq uiv a lence but not necessar ily symmetric. How ever, the diﬀerence δ = a − a ∗ η L 1 A satisﬁes F δ = 0 , so that there is a second lat- tice ( L 2 , λ 2 ) such that δ λ 2 ,L 1 A = 0. Then β = λ ∗ 2 ,L 1 A aλ 2 ,L 1 A : L 2 L 1 A → ( L 2 L 1 A ) ∗ is a non- degenerate symmetric form on L A with ( L, λ ) = ( L 2 , λ 2 ) ◦ ( L 1 , λ 1 ), and F ( λ A ) deﬁnes a map of symmetric space s F ( L A, β ) = ( F LA, ϕ LA F β ) ∼ → ( F A, α ). THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 29 Apart from “clearing denominators” , the proof of the ﬁrst homotop y equiv alence in the theorem pro ceeds now as the pro of of [Sch06, theor em 1 0] which was based on the pro of o f [W al8 5, theo rem 1 .6.7]. W e ﬁr st note that under the as sumptions of the theorem, the non-sing ular exa ct form functor s S n ( F, ϕ ) : S n A → S n B a lso satisfy (a) - (f ), n ∈ N . F or (a) - (c), this is in [W al85, lemma 1 .6.6] (using the fact that in the pr esence of (a), the map a : A → B in (c) can b e replaced by an inﬂation), (d) extends by functoriality , and the extensio n of (e), (f ) to S n easily follows by induction on n by succes siv ely clea ring denominator s. In order to sho w th at ( w S e • A ) h → ( w S e • B ) h is a homotop y equiv a lence, it suﬃces to prove that ( wS e n A ) h → ( w S e n B ) h is a homotopy equiv alence for every n ∈ N , which, by the argument of the previo us pa ragraph, only needs to b e check ed fo r n = 0, that is, it is suﬃcient to pr o ve that F : ( w A ) h → ( w B ) h is a homo top y equiv alence. The last claim will follow from Quillen’s theor em A once we show that for every o b ject X = ( X , ψ ) of ( w B ) h , the co mma categor y ( F ↓ X ) is non-empty and co n tractible. By (a) with A = 0, there is an ob ject B of A and a w eak equiv alence F B ∼ → X , hence a map ( F B , ψ | F B ) → ( X, ψ ) in ( w B ) h . By ( † ) ab ov e, there is a s ymmetric space ( C , γ ) in A and a map F ( C , γ ) → ( F B , ψ | F B ) in ( w B ) h . Hence, the categ ory ( F ↓ X ) is non-empty . In or der to show that ( F ↓ X ) is con tractible it s uﬃces to show that ev e ry functor P → ( F ↓ X ) from a ﬁnite p oset P to the comma category ( F ↓ X ) is homotopic to a constant map (see for instance [Sch06, Lemma 14]). Such a functor is given by a triple ( A , α, f ) where A is a functor A : P → w A : i 7→ A i , ( i ≤ j ) 7→ a j,i together with a co llection α of non-degener ate symmetr ic forms α i : A i → A ∗ i in A , i ∈ P , such tha t α i = α j | A i whenever i ≤ j in P , a nd f is a collection of compatible maps of symmetric s paces f i : F ( A i , α i ) → ( X, ψ ) in ( w B ) h such that f i = f j F a j,i whenever i ≤ j ∈ P . It is co n v enient to consider f as a map F ( A , α ) → ( X , ψ ) of functors P → ( w B ) h , wher e ob jects in ( w B ) h (or in A , B , ( w A ) h ) such a s ( X , ψ ) a re interpreted as constant P -diagr ams. By [Sch06, Lemma 13], there is a map b : B → A of functors P → w A such that b i : B i → A i is a weak equiv alence, i ∈ P , and the P -diagram B : P → A ha s a colimit in A s uc h that F ( B ) → F (colim P B ) repr esen ts the colimit of F ( B ) in B (the diagram B is a coﬁbra n t replacement of A in a suitable coﬁbratio n structure on the categ ory of functor s P → A , see [Sch06, Appendix A.2], co ﬁbran t ob jects ha ve colimits, and F , b eing an exa ct functor, preserves coﬁbra n t o b jects and their colimits). By (c), the natural map F (colim P B ) = colim P F ( B ) → X induced by b ◦ f factors as F (colim P B ) → F ( C ) c → X where the ﬁrst map is in the image of F and the second ma p is a weak equiv alence in B . Let g : B → C be the comp osition B → colim P B → C which, by (b), is a weak equiv alence since b , f , and c are. The nu ll-homotopy to b e c onstructed ca n b e rea d o ﬀ the following diag ram (16) L ′ LB , β β = α | L ′ LB = γ | L ′ LB λ ′ / / LB λ / / Lg   B b / / g   A, α f      LC, γ λ / / C c / / _ _ _ X , ψ 30 MARC O SCHLICHTING of which w e have co nstructed the right hand square, so far. In the dia gram, a dashed arrow A 99K X stands for a n a rrow F A → X in B , a nd so lid ar rows are arrows in A . By ( † ), there is a lattice ( L, λ ) a nd a non-degenera te s ymmetric form ( LC, γ ) ∈ ( w A ) h such that F λ C : F ( LC, γ ) → ( F C , ψ | F C ) deﬁnes a map in ( w B ) h . The restr ictions of γ a nd α i to LB i may not c oincide, but their images under F co incide since in B , b oth are (up to c ompositio n with the iso morphism ϕ B i ) the r estriction o f ψ to F LB i . Clea ring denominators, we ca n ﬁnd a lattice ( L ′ , λ ′ ) such that γ | L ′ LB i = α i | L ′ LB i =: β i for a ll i ∈ P simultaneously . The outer par t of the diag ram inv o lving ( L ′ LB , β ), ( A, α ), ( LC , γ ), ( X, ψ ) a nd the maps betw e en them, deﬁnes a homo top y bet ween ( A, α, f ) and the constant functor ( LC, γ , cF λ ) : P → ( F ↓ X ) via the functor ( L ′ LB , β , f F ( b λλ ′ )) : P → ( F ↓ X ). This ﬁnis hes the pro of o f the ﬁrst homo top y equiv a lence in the theo rem. The same pro of (for getting forms ), or [W a l85, 1.6.7], implies that wS • A → wS • B is a homotopy equiv alence. Therefore, the map GW ( A , w, ∗ , η ) → GW ( B , w , ∗ , η ) is a lso a ho motop y equiv alence.  5.2. Change of exact structure. Let ( A , w , ∗ , η ) b e an additive c ategory with weak equiv alences a nd duality , and ass ume that A can b e equipp ed with tw o exact structures E 1 and E 2 one smaller than the other, E 1 ⊂ E 2 , so that the iden tit y functor ( A , E 1 ) → ( A , E 2 ) is exact. Assume further more that the duality functor ∗ : A op → A is exac t for b oth exact structur es E 2 and E 2 , so that the identit y deﬁnes a dua lit y preserving exact functor (17) ( A , E 1 , w , ∗ , η ) → ( A , E 2 , w , ∗ , η ) . The following is an immediate consequence of theo rem 5.1. 5.3. Lemma. In the situation of 5.2, if every map in A c an b e written as the c om- p osition of an inﬂation in E 1 fol lowe d by a we ak e quivalenc e, then the map (17) in- duc es an e quivalenc e of hermitian S • -c onst ructions and of asso ciate d Gr othendie ck- Witt s p ac es.  The purp ose of the next lemma (lemma 5.5 be lo w) is to simplify s ome of the hypothesis of theorem 5.1 provided the exact categories with weak equiv a lences in the theorem are “categor ies o f co mplexes”. 5.4. Deﬁnition. W e call an e xact catego ry with w eak equiv alences ( C , w ) a c at- e gory of c omplexes (with underly ing additive catego ry C 0 ) if C ⊂ Ch C 0 is a full additive sub category of the category Ch C 0 of chain complexes C 0 such that (a) C is closed under degr ee-wise split extensions in Ch C 0 , (b) degree-w ise split exact sequences ar e exact in C , (c) w ith a complex A in C its usual cone C A (that is, the cone on the identit y map of A ) and all its shifts A [ i ], i ∈ Z ar e in C , and (d) the set of weak eq uiv a lences w contains at leas t all us ual homotopy eq uiv- alences b et w een complexes in C . 5.5. Lemma. L et ( A , w ) and ( B , w ) b e c ate gories of c omplexes with asso ciate d ad- ditive c ate gories A 0 and B 0 , and let F : ( A , w ) → ( B , w ) b e an exact functor which is the induc e d functor on chain c omplexes of an additive functor A 0 → B 0 . Then c ondition 5.1(a) holds and c ondition 5.1 ( c) is implie d by c onditions 5.1 (b), (e) and t he fol lowing c ondition. THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 31 (c’) F or every obje ct E of B ther e is an obje ct A of A and a we ak e quivalenc e F A ∼ − → E . Pr o of. Every map f : A → B in A can b e written as the comp osition A / / “ f i ” / / B ⊕ C A ( 1 0 ) ∼ / / B , where i : A ֌ C A is the canonical inclusion of A into its co ne C A . This shows that co ndition 5 .1 (a) holds. T o prove condition 5.1 (c) as suming conditions 5.1 (b), (e) and (c’) ar e satisﬁed, let f : F A → E b e a map in B with A in A . By (c’), there is a weak equiv ale nce s : F ( B ′ ) → E in B w ith B ′ in A . Let M b e the pull-back of f : F A → E alo ng the degree-wis e split sur jection ( s, p ) : F ( B ′ ) ⊕ P E → E wher e p : P E → E is the canonical deg ree-wise split surjection from the contractible complex P E = C E [ − 1] to E , so that we have a homotopy c omm utative dia gram M / /   F ( A ) f   F ( B ′ ) ∼ s / / E . Since F ( B ′ ) ⊕ P E → E is a deﬂation a nd a weak equiv alence, its pull-back, the map M → F A , is a deﬂation and a weak equiv alence, to o. By condition (c’), there is a weak equiv alence F ( A ′ ) → M with A ′ in A , and by co ndition 5.1 (e) we can assume the co mpositio ns F ( A ′ ) → F A a nd F ( A ′ ) → F ( B ′ ) to b e the images F α and F β o f maps α : A ′ → A and β : A ′ → B in A . The res ulting s quare in volving F ( A ′ ), F ( B ′ ), F A a nd E ho motop y commutes, and the map α : A ′ → A is a weak equiv alence, b y 5.1 (b). Replacing B ′ with B ′ ⊕ C A ′ , we obtain a co mm utativ e diagram F ( A ′ ) / / F α ∼ / / “ F β F i ”   F ( A ) f   F ( B ′ ) ⊕ F ( C A ′ ) ( s g ) / / E , where i : A ′ ֌ C A ′ is the cano nical inclusion o f A ′ int o its c one, g : F ( C A ′ ) = C F ( A ′ ) → E a map s uc h that g ◦ F i is the n ull-homotopic map f ◦ F α − s ◦ F β : F ( A ′ ) → E . Let B b e the pus h-out of α : A ′ → A along the degree- wise split inclusion A ′ ֌ B ′ ⊕ C A ′ and call a : A → B and b : B ′ ⊕ C A ′ → B the induced maps. Since α : A ′ → A is a weak equiv alence, so is b . The functor F pres erves push-outs along inﬂations so that we obtain an induced map t : F ( B ) → E which is a weak equiv alence since F ( b ) a nd ( s, g ) ar e. By cons truction, we hav e t ◦ F a = f .  Next, we prov e an a nalog of Quillen’s r esolution theorem [Qui7 3, § 4]. 5.6. Lemma (Resolution). L et ( B , w, ∗ , η ) b e an exact c ate gory with we ak e quiva- lenc es and duality such that ( B , w ) is a c ate gory of c omplexes. L et A ⊂ B b e a ful l sub c ate gory close d under the duality, de gr e e-wise split extensions and under t aking c ones and shifts in B . Restricting ( w , ∗ , η ) to A makes A into an exact c ate gory 32 MARC O SCHLICHTING with we ak e quivalenc es and duality. A ssume that t he fol lowing r esolution c ondition holds: F or every obje ct E of B , ther e is an obje ct A of A and a we ak e quivalenc e A ∼ − → E . Then t he duality pr eserving inclusion A ⊂ B induc es homotopy e quivalenc es ( wS e • A ) h ∼ − → ( wS e • B ) h and GW ( A , w , ∗ , η ) ∼ − → GW ( B , w, ∗ , η ) . 5.7. Pr o of. This follows from theorem 5.1 in view of lemma 5.5 and the fact that conditions 5 .1 (e) and (g) trivia lly hold since A ⊂ B is fully faithful, and condition (d) holds since A → B is duality preserving .  W e ﬁnish the section with theorem 5.1 1 below. It is a v a rian t of theor em 5 .1 when conditions 5.1 (e) and (f ) can not be achiev ed in a functoria l wa y but the form functor is a lo calization by a calculus of rig h t fractions . 5.8. Deﬁnition. Call a functor F : A → B b et w een small categor ies a lo c aliza - tion by a c alculus of right fr actions if the fo llo wing three conditions hold (compa re theorem 5.1 (e), (f )). (a) The functor F : A → B is es sen tially surjective. (b) F or every map f : F ( A ) → F ( B ) b et w een the imag es of ob jects A , B of A , there are maps s : A ′ → A and g : A ′ → B in A with F ( s ) a n iso morphism in B and f ◦ F ( s ) = F ( g ). (c) F or a n y tw o maps a, b : A → B in A s uc h that F ( a ) = F ( b ) there is a map s : A ′ → A such that F ( s ) is an isomor phism in B and as = bs . 5.9. Remark. A functor F : A → B be t w een sma ll categor ies is a lo caliza tion b y a calculus of rig h t fractions if and only if the set Σ o f maps f in A such that F ( f ) is an iso morphism in B satisﬁes a calculus o f rig h t fractions (the dua l of [GZ67, § I.2.2]) and the induced functor A [Σ − 1 ] → B is an equiv alence of ca tegories. 5.10. Con text for theorem 5. 11. Cons ider an a dditiv e functor A → B b etw een additive categor ies. It induces an exact functor F : Ch b A → Ch b B b etw ee n the asso ciated exact ca tegories of b ounded chain complexes where we call a sequence of chain complexes exa ct if it is degree-wise split exact. W e a ssume that F is part of a n exact form functor ( F, ϕ ) : (Ch b A , w , ∗ , η ) → (Ch b B , w , ∗ , η ) betw een exa ct catego ries with weak equiv alences and dua lit y such that the duality compatibility map F ∗ → ∗ F is a natura l isomor phism. 5.11. Theorem (Approximation I I). I f in the situation 5.10 ab ove, a map in Ch b A is a we ak e quivalenc e iﬀ its image in Ch b B is a we ak e quivalenc e, and if the functor A → B is a lo c alization by a c alculus of right fr actions, then ( F , ϕ ) induc es homotopy e quivalenc es (18) ( wS e • Ch b A ) h ∼ − → ( w S e • Ch b B ) h and GW (Ch b A , w , ∗ , η ) ∼ − → GW (Ch b B , w , ∗ , η ) . THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 33 The pro of of theor em 5.11 is a co nsequence of the fo llo wing tw o lemma s. 5.12. Lemma. L et F : A → B b e a lo c alization by a c alculus of right fr actions. Then t he fol lowing holds. (a) F or every inte ger n ≥ 0 , the induc e d functor F un([ n ] , A ) → F un([ n ] , B ) on diagr am c ate gories is a lo c alization by a c alculus of right fr actions. (b) If ( F, ϕ ) : ( A , ∗ , η ) → ( B , ∗ , η ) is a form fun ctor b etwe en c ate gories with duality such t hat the duality c omp atibility map ϕ : F ∗ → ∗ F is a n atu r al isomorphi sm, then the induc e d functor A h → B h on asso ciate d c ate gories of symmetric forms is a lo c alization by a c alculus of right fr actions. (c) I f F i s an additive functor b etwe en additive c ate gories, then the induc e d functors Ch b A → Ch b B and S n A → S n B ar e lo c alizatio ns by a c alculus of right fr actions. Pr o of. The pro of is a n exer cise in clear ing denominator s, and we omit the details.  5.13. Lemma. L et F : A → B b e a functor b etwe en smal l c ate gories A and B which is a lo c alization by a c alculus of right fr actions. Then F induc es a homotopy e quivalenc e on classifying sp ac es |A| ∼ − → |B | . Pr o of. F or a categ ory C , and a sub category w C , recall that the categor y F un w ([ n ] , C ) is the full sub categor y o f the category F un([ n ] , C ) o f functors [ n ] → C which hav e image in w C . Maps ar e natural tr ansformations of functors [ n ] → C . There are homotopy equiv alences of top ologica l realizations of simplicial catego ries (a v ar ian t of which already app eared in the pro of of lemma 3.7) (19) |C | ∼ → | n 7→ F un w ([ n ] , C ) | ≃ | n 7→ w F un([ n ] , C ) | which is functorial in the pair ( C , w C ). In the ﬁrst map, the category C is considered as a constant simplicial ca tegory , and the functor C → F un w ([ n ] , C ) sends a n ob ject C ∈ C to the string consis ting of only identit y maps on C . This functor is a homotopy equiv alence with in verse the functor F un w ([ n ] , C ) → C which sends a string of maps C 0 → · · · → C n to C 0 . The comp osition of the tw o functor s is the ident ity in one dir ection, and in the other, it is ho motopic to the iden tit y , where the homotopy is given by the natura l transfor mation from C 0 1 → C 0 1 → · · · C 0 to C 0 → C 1 → · · · C n induced by the structur e maps of the la st str ing. Therefore, the ﬁrs t map in (19) is a homotopy eq uiv a lence. The se cond map in (19) is in fact a homeo morphism as it is the realiz ation in tw o diﬀerent orders of the sa me bisimplicial set. In order to prov e the lemma, let σ A ⊂ A b e the sub categor y o f A whos e maps are the maps which are s en t to iso morphisms in B . By the natural homo top y equiv alences in (19), the map |A| → |B | in the lemma is equiv alent to the map | n 7→ σ F un([ n ] , A ) | → | n 7→ i F un([ n ] , B ) | which is the rea lization o f a map o f simplicial ca tegories, so that it suﬃces to show tha t for each int eger n ≥ 0, the functor σ F un([ n ] , A ) → i F un([ n ] , B ) is a homotopy equiv alence. By lemma 5.1 2 (a), this functor is a lo calization by a calculus of frac tions, so that we are re duced to proving that G : σ A → i B is a homotopy equiv alence whenever A → B is 34 MARC O SCHLICHTING a lo caliza tion by a c alculus o f right fractions. In this case, the comma category ( G ↓ B ) is left ﬁltering for e v ery ob ject B of B , hence contractible. By theorem A of Q uillen, the functor σ A → i B is a homotopy equiv alence.  5.14. Pr o of of the or em 5.11. W e only pr o ve the ﬁrst homotopy equiv alence in the theorem, the second homotopy equiv alence follows from this and the homoto p y equiv alence w S • Ch b A → wS • Ch b B which is prov ed in the same way (forgetting forms). A map of simplicia l categor ies which is degree-wise a homotopy equiv alence in- duces a homotopy equiv a lence after top ological rea lization. Therefor e, it suﬃces to show that for every n ≥ 0, the for m functor ( F , ϕ ) induces a homotopy equiv alence (20) ( wS e n Ch b A ) h − → ( wS e n Ch b B ) h . By le mma 5.12 (c) and in v iew o f the isomorphism S n Ch b E = Ch b S n E of exact categorie s applied to E = A , B , we are reduced to showing that the ma p (20) is a ho motop y equiv alence for n = 0 . The functor Ch b A → Ch b B is a lo calization by a calculus of r igh t fra ctions, by 5.12 (c). The assumption tha t Ch b A → Ch b B preserves and detec ts weak equiv alences, implies that, o n sub categories o f weak equiv alences , the functor w Ch b A → w Ch b B is also a lo calization by a calculus of right fractions. By 5 .12 (b) , the induced functor on categ ories of sy mmetric forms (Ch b A ) h → (Ch b B ) h , which is the map (20) in degree n = 0, is a lo calization by a calculus of right fractions and therefore a homo top y eq uiv a lence, by lemma 5 .13.  6. From exact ca tegories to chain complexes The purpose of this section is to prov e p rop osition 6.5 which allows us the r eplace the Grothendieck-Witt space of an e xact ca tegory w ith duality by the Grothendiec k- Witt space of the asso ciated categor y of b ounded chain co mplexes. 6.1. Chain complexes and dualiti es. Let ( E , ∗ , η ) b e an exact catego ry with duality , that is, an ex act ca tegory with weak equiv alences and dua lit y wher e all weak equiv alences are isomor phisms. Let Ch b ( E ) b e the categ ory of b ounded chain complexes ( E , d ) : · · · → E n − 1 d n − 1 → E n d n → E n +1 → · · · , d n d n − 1 = 0 , in E . A sequence of chain complexes ( E ′ , d ) → ( E , d ) → ( E ′′ , d ) is ex act if it is degree-wise exact in E , that is, if the sequence E ′ n ֌ E n ։ E ′′ n is exact for all n . Call a c hain complex ( E , d ) in E strictly acyclic if every diﬀerential d n is the comp osition E n ։ im d n ֌ E n +1 of a deﬂation fo llo wed by an inﬂa tion, and the sequences im d n − 1 ֌ E n ։ im d n are exact in E . A chain complex is c alled acyclic if it is homotopy equiv alent to a strictly a cyclic chain c omplex. A map o f chain complexes is a quasi-isomorphism if its cone is acyclic . W rite quis for the set of quasi-isomo rphisms, then the tr iple (Ch b ( E ) , quis) is a n exact categ ory with weak e quiv a lences. THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 35 F or n ∈ Z , the duality ( ∗ , η ) induces a (naive) duality ( ∗ n , η n ) on Ch b E which on ob jects ( E , d ) and o n chain maps f : ( E , d ) → ( E ′ , d ) is given by the for m ulas ( E ∗ n ) i = ( E − i − n ) ∗ , ( f ∗ n ) i = ( f − i − n ) ∗ , ( d ∗ n ) i = ( d − i − 1 − n ) ∗ , ( η n E ) i = ( − 1) n ( n − 1) 2 η E i . With these deﬁnitions, w e have an exact catego ry with weak equiv a lences and duality (Ch b ( E ) , quis , ∗ n , η n ) . If n = 0 we may simply wr ite ( ∗ , η ) for ( ∗ 0 , η 0 ). 6.2. Remark. The functor T : Ch b E → Ch b E given by the formula ( T E ) i = E i +1 , T ( f ) i = f i +1 , ( d T E ) i = d i +1 deﬁnes a dua lit y preserving isomor phism o f ex act categ ories with duality T : (Ch b E , ∗ n , η n ) ∼ = (Ch b E , ∗ n +2 , − η n +2 ) . 6.3. Remark. There is ano ther (more natural) s ign choice for deﬁning induced dualities ( ♯ n , can n ) on Ch b E co ming from the in ternal hom o f chain co mplexes, compare 7.4. They ar e given by the for m ulas ( E ♯ n ) i = ( E − i − n ) ∗ , ( f ♯ n ) i = ( f − i − n ) ∗ , ( d ♯ n ) i = ( − 1 ) i +1 ( d − i − 1 − n ) ∗ , (ca n n E ) i = ( − 1 ) i ( i + n ) η E i . The identit y functor on Ch b E tog ether with the duality compatibility isomo rphism ε n E : E ∗ n → E ♯ n which in degr ee i is ( ε n E ) i = ( − 1) i ( i +1) 2 id E ∗ − i − n : ( E ∗ n ) i → ( E ♯ n ) i deﬁnes an isomor phism o f exa ct catego ries with duality ( id, ε n ) : (Ch b E , ∗ n , η n ) ∼ = − → (Ch b E , ♯ n , can n ) . F or the purp ose of proving prop osition 6.5 below, the duality ( ∗ , η ) is conv enien t. F or most other purpo ses, the dualit y ( ♯, ca n) is more natural. It is the latter dualit y , which we will us e fr om § 8 o n. In any case, b oth give rise to isomorphic exact categorie s with duality and th us hav e isomor phic Grothendieck-Witt spa ces. 6.4. Exercise. Show that the exact categor ies with weak equiv alences and duality (Ch b E , ∗ n , η n ) and (Ch b E , ♯ n , can n ) hav e symmetric cones in the sens e of deﬁnition 4.1 (hint: s ee 7.5). F or an ex act categor y with duality ( E , ∗ , η ), inclusion a s complexes concentrated in deg ree 0 , deﬁnes a duality pres erving exa ct functor (21) ( E , i, ∗ , η ) → (Ch b E , quis , ∗ , η ) . The following pr opo sition generalize s [TT90, Theorem 1.11 .7], see also re mark 6 .8. 36 MARC O SCHLICHTING 6.5. Prop osition. F or an exact c ate gory with duality ( E , ∗ , η ) , the functor (21) induc es a homotopy e quivalenc e of Gr othendie ck-Witt sp ac es GW ( E , ∗ , η ) ∼ − → GW (Ch b E , quis , ∗ , η ) . W e will reduce the proof of the pr opo sition to “se mi-idempotent complete” exa ct categorie s. This has the adv antage that for such categorie s, every acyclic complex is s trictly acy clic. Here are the relev an t deﬁnitio ns a nd facts. 6.6. Semi-idem p oten t compl etions. Call an ex act categ ory E semi-idemp otent c omplete if any map p : A → B which has a section s : B → A , pi = 1, is a deﬂation in E . A semi-idemp oten t complete exact categ ory ha s the following prop erty: a n y map B → C for which there is a map A → B such that the comp osition A → C is a deﬂation, is itself a deﬂation. This is b ecause a semi-idemp otent co mplete exa ct category satisﬁes Thomaso n’s axiom [TT90, A.5.1] so that the sta ndard em bedding of E into the ca tegory of left exact functor s E op → h ab i into the ca tegory of ab elian groups is closed under k ernels of s urjections [TT90, Theorem A.7.1 and Propo sition A.7.16 (b)]. F o r a s emi-idempotent complete exa ct catego ry , every a cyclic co mplex is s trictly acy clic. The semi-idemp oten t completion of a n exact category E is the full s ubcatego ry e E semi ⊂ e E o f the idemp otent completion e E o f E of those ob jects which are stably in E . Clear ly , e E semi is semi-idemp oten t complete, and the map K 0 ( E ) → K 0 ( e E semi ) is a n isomo rphism. If ( E , ∗ , η ) is an exact categor y with duality , then ( e E semi , ∗ , η ) is an exact category with duality such that the fully exa ct inclusion E ⊂ e E semi is duality preserving. If ( A , w , ∗ , η ) is an exa ct categor y with weak equiv alences and duality , then ( e A semi , w , ∗ , η ) inherits the structure of a n ex act categ ory with w eak equiv alence and duality from ( e A , w , ∗ , η ), s ee 4 .5, so that the inclusion A ⊂ e A semi is dua lit y pre serving. 6.7. Lemma. L et ( A , w, ∗ , η ) b e an exact c ate gory with we ak e quivalenc es and str ong duality, then the inclusion A ⊂ e A semi induc es a homotopy e quivalenc e of Gr othendie ck- Witt s p ac es GW ( A , w , ∗ , η ) → GW ( e A semi , w , ∗ , η ) . Pr o of. F or an exact category with duality ( E , ∗ , η ), the map G W i ( E ) → GW i ( e E semi ) is an isomor phism for i ≥ 1 and it is injective for i = 0, by coﬁnality prov ed in [Sch08]. The map GW 0 ( E ) → GW 0 ( e E semi ) is a lso surjective, hence a n iso morphism, since for every s ymmetric s pace ( X, ϕ ) in e E semi , we can ﬁnd an A in E such that X ⊕ A is in E , and thus, [ X , ϕ ] = [( X, ϕ ) ⊥ H A ] − [ H A ] is in the image of the map. Therefore, lemma 6.7 holds whe n w is the s et of isomor phisms. The lemma now follows from this case and the simplicial reso lution lemma 3.7 since F un w ( n , e A semi ) is the semi-idemp otent co mpletion of F un w ( n, A ) (for a string X n ′ → · · · → X n of weak eq uiv a lences in e A semi , there is a n ob ject A o f A such that X i ⊕ A is in A for all i ∈ n , so that ( X n ′ → · · · → X n ) ⊕ ( A 1 → · · · 1 → A ) is a string of weak equiv alences in A ).  THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 37 Pr o of of pr op osition 6.5. In v iew of lemma 6.7, we can assume E to b e s emi- idempo ten t co mplete, so that acyclic complexes are str ictly acyclic. Moreov er, by the strictiﬁcation lemma 3.4, we can ass ume E to hav e a s trict duality , s ince an exact categ ory with duality is equiv alen t to its strictiﬁcation. Let Ac b E ⊂ Ch b E be the full sub category of acyclic chain complexes. It inherits the str ucture o f an exact catego ry with w eak equiv alences and strict duality fro m Ch b E . Consider the commutativ e diagra m of exact categor ies with weak equiv alences a nd strict duality 0 / /   (Ac b E , i )   / / (Ac b E , quis)   ( E , i ) / / (Ch b E , i ) / / (Ch b E , quis) . W e will show that (a) the left square induces a homo top y cartesia n square of Gr othendiec k-Witt spaces, and that (b) the map GW 0 ( E , i ) → GW 0 (Ch b E , quis) is surjective. By pr opos ition 4 .9, the right hand squa re induces a homotopy car tesian square of Grothendieck-Witt s paces, so that by (a) the same is true for the o uter square. T ogether with (b), this implies the pr opos ition. W e prove (a). F or n ≥ 0, le t C h b [ − n,n ] E ⊂ C h b E and Ac b [ − n,n ] E ⊂ Ac b E b e the full sub categories of those chain complexes which are concentrated in degrees [ − n, n ]. They inherit a structur e of exact ca tegories with duality . Note that the inclusion Ac b [ − n,n ] E ⊂ C h b [ − n,n ] E is 0 ⊂ E for n = 0. The natura l inclus ions induce a commut ative diagr am of exact categorie s with duality (22) 0 / /   Ac b [ − n,n ] E / /   Ac b [ − n − 1 ,n +1] E   E / / C h b [ − n,n ] E / / C h b [ − n − 1 ,n +1] E . W e will show that the right-hand square induces a homotopy cartesian s quare o f Grothendieck-Witt spaces. Then, by inductio n, the o uter squar e induces a homo- topy cartesia n squa re, to o. T aking the colimit ov er n o f the Gro thendiec k-Witt spaces of the outer square s yields the des ired homo top y cartes ian squa re, s ince the Grothendieck-Witt space functor GW co mm utes with ﬁltered co limits. Consider the following for m functors b etw een exa ct categ ories with duality (equipp ed with the obvious duality compatibility maps): 38 MARC O SCHLICHTING HE × Ac b [ − n,n ] E s → Ac b [ − n − 1 ,n +1] E : ( A, B ) , C 7→ A (1 0) − → A ⊕ C − n → C − n +1 → · · · → C n − 1 → C n ⊕ B ∗ (0 1) − → B ∗ C h b [ − n − 1 ,n +1] E ρ → HE × C h b [ − n,n ] E : C 7→ ( C − n − 1 , ( C n +1 ) ∗ ) , C − n → · · · → C n HE × Ac b [ − n,n ] E → HE × C h b [ − n,n ] E : ( A, B ) , C 7→ ( A, B ) , A ⊕ C − n → C − n +1 → · · · → C n − 1 → C n ⊕ B ∗ . These functors, together with the natural inclusio ns, ﬁt in to a c omm utativ e diagr am of e xact categ ories with duality Ac b [ − n,n ] E / / ( 0 1 ) ( ( P P P P P P P P P P P P   C h b [ − n,n ] E ( 0 1 ) v v m m m m m m m m m m m m   HE × Ac b [ − n,n ] E / / s v v n n n n n n n n n n n n HE × C h b [ − n,n ] E Ac b [ − n − 1 ,n +1] E / / C h b [ − n − 1 ,n +1] E . ρ h h Q Q Q Q Q Q Q Q Q Q Q Q The upper square induces a homotopy car tesian squar e of Gro thendiec k-Witt spa ces. By additivity (theorem 3.3 or [Sch08, 7.1 ]), the diagonal maps s a nd ρ in the low er square induce homoto p y eq uiv a lences o f Grothendieck-Witt spaces (details b elow). Therefore, the outer squar e induces a homotopy cartesia n square as well. T o see that the functors s and ρ induce homotopy equiv alences, consider the following functors of categories with duality Ac b [ − n − 1 ,n +1] E r → HE × Ac b [ − n,n ] E : C 7→ ( C − n − 1 , ( C n +1 ) ∗ ) , C − n /C − n − 1 → C − n +1 → · · · → C n − 1 → ker( C n → C n +1 ) HE × C h b [ − n,n ] E σ → C h b [ − n − 1 ,n +1] E : ( A, B ) , C 7→ A 0 → C − n → C − n +1 → · · · → C n − 1 → C n 0 → B ∗ . W e hav e ρσ = id . The identit y functor id on C h b [ − n − 1 ,n +1] has a totally isotropic subfunctor G ⊂ id given by G     0 / /   · · · / / 0 / /   C n +1 1   id C − n − 1 / / · · · / / C n / / C n +1 . By a dditivit y [Sc h08, 7.1], the iden tit y functor id a nd σ ρ = G ⊥ /G ⊕ H G induce homotopic ma ps on Grothendieck-Witt spac es, th us ρ induces a homo top y equiv a - lence. Similarly , we have rs = id , and the iden tity functor id on Ac b [ − n − 1 ,n +1] E has THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 39 a totally isotropic subfunctor F ⊂ i d given by F     C − n − 1 1 / / 1   C − n − 1     / / 0   / / · · · / / 0   id C − n − 1 / / / / C − n / / C − n +1 / / · · · / / / / C n +1 . Again, by a dditivit y [Sch08], the identit y functor id on Ac b [ − n − 1 ,n +1] E a nd sr = F ⊥ /F ⊕ H F induce homoto pic maps on Gro thendiec k-Witt spaces. It follows that s induces a homotopy eq uiv a lence. This ﬁnishes the pr o ve o f (a). W e are left with pr o ving (b). W e will show that (c) a symmetric space ( A, α ), where A is s upported in [ − n, n ], n ≥ 1, equals [ A, α ] = [ B , β ] + [ H ( C )] in the Gr othendiec k-Witt g roup GW 0 (Ch b E , quis), where B is suppo rted in [ − n + 1 , n − 1]. By induction, [ A, α ] is then a sum of hyperb olic ob jects plus a symmetric space suppo rted in degree 0. Since the latter tw o kinds of symmetric spaces are obviously in the image of GW 0 ( E , i ) → GW 0 (Ch b E , quis), this prov es ( b ). T o show (c), let n ≥ 1 and let ( A, α ) b e a symmetric space s upported in [ − n , n ]. Since the cone of α is a cyclic (hence strictly acyclic, by semi-idemp otent completeness of E ), the map  d − n α − n  : A − n → A − n +1 ⊕ A ∗ n is an inﬂa tion. Deﬁne a complex ˜ A , also supp orted in [ − n, n ], b y A − n “ d − n α − n ” ֌ A − n +1 ⊕ A ∗ n ( d 0 ) − → A − n +2 d → · · · d → A n − 2 ( d 0 ) − → A n − 1 ⊕ A n ( d n − 1 1 ) ։ A n . The complex ˜ A is equipp ed with a non-singula r s ymmetric form ˜ α which is α i in degree i ex cept in degre es i = − n + 1 , n − 1 where it is α i ⊕ 1 . W e hav e a symmetric space in the category of admissible shor t complexes in Ch b E A ∗ n [ n − 1] ֌ ˜ A ։ A n [ − n + 1] with form (1 , ˜ α, η ), where for an ob ject E of E , w e denote by E [ i ] the complex which is E in degree − i and 0 elsewhere. The maps A ∗ n ֌ ˜ A − n +1 = A − n +1 ⊕ A ∗ n and A n − 1 ⊕ A n = ˜ A n − 1 → A n are the canonical inclusions and p ro jections, resp ectively . Since ( A, α ) is the zero-th homology o f this admissible short complex equipped with its for m, we hav e [ ˜ A, ˜ α ] = [ A, α ] + [ H ( A n [ − n + 1])] in GW 0 (Ch b E , quis). There is another s ymmetric space in the ca tegory o f admissible s hort complexes in Ch b E C A − n [ n − 1] ֌ ˜ A ։ C A n [ n ] with non-singular from ( α − n , ˜ α, α n ), where for an ob ject E of E , we write C E [ i ] for the complex E 1 → E placed in degrees i and i − 1. The maps C A − n [ n − 1 ] ֌ ˜ A and ˜ A ։ C A n [ n ] are the unique maps which are the iden tit y in deg ree − n and n , resp ectively . Since C A − n [ n − 1] and C A n [ n ] ar e acyclic, the form o n the admissible short complex is non-singula r and its zero- th homology , whic h is concentrated in degrees [ − n + 1 , n − 1 ], has the same clas s in GW 0 (Ch b E , quis) as ( ˜ A, ˜ α ).  6.8. Remark. W e can equip (Ch b E , quis , ∗ , η ) with t wo exact structures, the o ne deﬁned in 6.1, and the degr ee-wise split ex act s tructure. By lemma 5.3, the t wo yield homotopy equiv a len t Gro thendiec k-Witt space s. 40 MARC O SCHLICHTING 7. DG-Algebras on ringed sp aces and dualities In this section we recall basic deﬁnitions and facts ab out diﬀerential gr aded algebras and mo dules ov e r them. B esides ﬁxing terminology , the main po in t here is the c onstruction of the canonical symmetric cone in 7.5, and the int erpretatio n of c ertain for m functors a s symmetric forms in dg bimo dule categ ories, see 7.7. 7.1. DG κ -mo dules. Le t κ b e a co mm utativ e ring. Unless o therwise indicated, mo dules will alwa ys mean left module, tensor pr oduct ⊗ will b e tensor pro duct ⊗ κ ov er κ , and homomorphis m sets H om ( M , N ) betw een κ -mo dules means set of κ -linear homomorphisms, and is itself a κ -mo dule. Rec all that a diﬀeren tial graded κ -mo dule M is a g raded κ -mo dule L n ∈ Z M n together with a κ -linear map d : M n → M n +1 , n ∈ Z , ca lled diﬀerential of M , satisfying d ◦ d = 0. In other words, M is a ch ain complex of κ -mo dules. A map o f dg κ -mo dules is a map of graded κ -mo dules commuting w ith the diﬀerentials. F or tw o dg κ -modules M , N , the tensor pro duct dg κ -mo dule M ⊗ N and the homomorphism dg κ -mo dule [ M , N ] are deﬁned by the usual formulas ( M ⊗ N ) n = L i + j M i ⊗ N j d ( x ⊗ y ) = dx ⊗ y + ( − 1) | x | x ⊗ dy [ M , N ] n = Q j − i = n H om ( M i , N j ) d ( f ) = d N ◦ f − ( − 1) | f | f ◦ d M . 7.2. DGAs and mo dules o v er them. A diﬀerential gr aded κ -alg ebra (dga) is a dg κ -mo dule A equipp ed with dg κ -mo dule maps · : A ⊗ A → A and κ → A , called multiplication and unit, making the usual asso ciativity and unit dia grams commute [Mac71, dia grams (1), (2), p. 16 6]. In other words, A is an a ssoc iativ e graded κ - algebra with multiplication sa tisfying d ( a · b ) = ( da ) · b + ( − 1) | a | a · ( db ). F or dg algebra s A and B , we denote by A -Mo d - B the categor y o f left A and right B -mo dules. Its ob jects are the dg κ -mo dules M equipped with dg κ -maps A ⊗ M → M and M ⊗ B → M , called multiplication, b oth of which a re asso ciative and unital, and further more, ( am ) b = a ( mb ) for a ll a ∈ A , m ∈ M and b ∈ B . W e also denote b y A -Mo d = A -Mo d - κ , Mo d - A = κ -Mo d - A , A -Bimo d = A -Mo d - A the categor ies of dg left A -mo dules, r igh t A -mo dules, and of dg A -bimo dules. Let A , B , C b e dg algebra s. Recall that for a right B mo dule M and left B - mo dule N , tensor pro duct M ⊗ B N over o f M and N ov er B is the dg κ - module which is the co-equalize r M ⊗ B ⊗ N 1 ⊗ µ / / µ ⊗ 1 / / M ⊗ N / / M ⊗ B N in t he category of dg κ -mo dules, where µ stands for the m ultiplications M ⊗ B → M and B ⊗ N → N . F or tw o dg right C -mo dules M and N , the dg κ -mo dule [ M , N ] C of r igh t C - module mor phisms is the equa lizer in the c ategory of dg κ -mo dules [ M , N ] C / / [ M , N ] [ µ, 1] / / [1 ,µ ] ◦ (? ⊗ 1 C ) / / [ M ⊗ C, N ] , where (? ⊗ 1 C ) : [ M , N ] → [ M ⊗ C , N ⊗ C ] is the dg κ -mo dule map f 7→ f ⊗ 1 C deﬁned b y ( f ⊗ 1 C )( x ⊗ c ) = f ( x ) ⊗ c for x ∈ M and c ∈ C . Similarly , one can deﬁne the dg κ -mo dule of le ft C -mo dule morphisms C [ M , N ] for t wo dg left C -mo dules. THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 41 T ensor pro duct ⊗ B and right C -mo dule morphisms [ , ] C deﬁne functor s ⊗ B : A -Mo d - B × B - Mod - C − → A -Mo d - C : M , N 7→ M ⊗ B N [ , ] C : ( B -Mod - C ) op × A -Mo d - C − → A -Mo d - B : M , N 7→ [ M , N ] C . 7.3. DGAs wi th inv oluti on. F o r a dg κ -mo dule M , we denote b y M op the op- po site dg κ -mo dule whic h, as a dg κ - module, is simply M itself. T o av oid confusion we may so metimes write x op , y op , etc, for the elemen ts in M op corres ponding to x, y ∈ M , so that d ( x op ) = ( dx ) op , for instance, denotes a n equatio n in M op as opp osed to in M . F or a dg κ -algebra A , its opp osite dga A op has underlying dg κ -mo dule the opp osite mo dule of A and mult iplication x op y op = ( − 1) | x || y | ( y x ) op . A dg algebr a with involution is a dg κ -algebra A together with an iso morphism A → A op : a 7→ ¯ a of dgas s atisfying ¯ ¯ a = a for all a ∈ A . The ba se commutativ e ring κ is always considered as a dga with trivia l inv olution κ → κ op : x 7→ x . Let A, B b e dg as with involution. F or a left A , r igh t B -mo dule M ∈ A -Mo d - B its opp osite mo dule M op is a left B op and right A op -mo dule which we consider as a left B and right A -mo dule via the isomor phisms A → A op , B → B op , that is, m op · a = ( − 1) | a || m | (¯ a · m ) op , b · m op = ( − 1) | b || m | ( m · ¯ b ) op for a ∈ A , b ∈ B , m ∈ M . F or dgas with inv o lution A , B , C and M ∈ A -Mo d - B , N ∈ B -Mo d - C , the com- m utativity isomor phism of the tensor pro duct deﬁnes an A -Mo d - C -isomorphism c : M ⊗ B N ∼ = − → ( N op ⊗ B M op ) op : x ⊗ y 7→ ( − 1) | x || y | ( y op ⊗ x op ) op . This can b e iterated to obtain for rings with inv olution A , B , C , D and M ∈ A -Mo d - B , N ∈ B -Mo d - C , P ∈ C -Mo d - D an isomorphism c 3 in A -Mo d - D de- ﬁned by ( M ⊗ B N ⊗ C P ) c 3 − → ( P op ⊗ C N op ⊗ B M op ) op x ⊗ y ⊗ z 7→ ( − 1) | x || y | + | x || z | + | y || z | ( z op ⊗ y op ⊗ x op ) op . 7.4. DG-mo dules and duali ties. L et A b e a dga with inv olution, and let I be an A -bimo dule equipp ed with an A bimodule isomorphism i : I → I op such that i op ◦ i = i d , for insta nce A itself with i ( x ) = ¯ x . W e call the pair ( I , i ) a duality c o eﬃcient for the ca tegory A -Mo d of dg A -mo dules, a s it deﬁnes a duality ♯ ( I ,i ) : ( A -Mo d) op → A - Mod by M ♯ ( I ,i ) = [ M op , I ] A . The canonical double dual identiﬁcation can ( I ,i ) ,M : M → M ♯ ( I ,i ) ♯ ( I ,i ) is the left A -mo dule map g iv en by the for m ula can ( I ,i ) ,M ( x )( f op ) = ( − 1) | x || f | i ( f ( x op )) for f ∈ [ M op , I ] A and x ∈ M . It is a straight forward to check the identit y can ♯ ( I ,i ) ,M can ( I ,i ) ,M ♯ = 1 M ♯ , so that the triple ( A -Mo d , ♯ ( I ,i ) , can ( I ,i ) ) is a category with duality . In this pa per, for the dualit y ♯ ( I ,i ) , the double dual identiﬁcation will always be the natural map can ( I ,i ) , so tha t we will wr ite ( A -Mo d , ♯ ( I ,i ) ) for the category with duality ( A -Mo d , ♯ ( I ,i ) , can ( I , i )), the double dual ident iﬁca- tion b eing ca n ( I ,i ) . If i : I → I op is understo o d, we may wr ite ♯ I instead of ♯ ( I ,i ) . 42 MARC O SCHLICHTING T o give a symmetric form ϕ : M → [ M op , I ] A in ( A -Mo d , ♯ I ) is the same as to give a n A -bimo dule map ˆ ϕ : M ⊗ M op → I such that the diagr am M ⊗ M op ˆ ϕ / / c   I i   ( M ⊗ M op ) op ˆ ϕ op / / I op commutes. The bijectio n is given by the iden tit y ˆ ϕ ( x ⊗ y op ) = ϕ ( x )( y op ). 7.5. The canonical symmetric cone. Let C = κ · 1 C ⊕ κ · ǫ b e the dg κ -mo dule whose underlying κ -mo dule is free of r ank 2 with basis 1 C and ǫ in deg rees 0 and − 1 , res pectively , and diﬀerential dǫ = 1 C . In fact, C is a commutativ e dga with unit 1 C and uniq ue multiplication. Let A b e a dga and M a (left) dg A - mo dule. W e wr ite C : A -Mo d → A -Mo d for the functor M 7→ M ⊗ C , and i M for the natura l inclusion M → C M = M ⊗ C : x 7→ x ⊗ 1 C . Similar ly , we write P : A -Mo d → A -Mo d for the functor M ⊗ [ C op , κ ] and p M for the natural surjection P M = M ⊗ [ C op , κ ] → M : m ⊗ g 7→ m · g (1 op C ). If A is a dg a with inv olution, and ( I , i ) a duality co eﬃcient for A -Mo d, we deﬁne a na tural tra nsformation γ M : [ M op , I ] A ⊗ [ C op , κ ] − → [( M ⊗ C ) op , I ] A by the formula γ M ( f ⊗ g )(( x ⊗ a ) op ) = ( − 1) | a || x | f ( x op ) · g ( a op ). One chec ks the equality i ♯ I M ◦ γ M = p M ♯ I . Therefore, an exa ct category with weak equiv alences and duality ( E , w , ∗ , η ) which admits a fully faithful and duality preser ving embedding into ( A -Mo d , ♯ I ) has a symmetric c one in the sense of 4.1 provided the functors C and P restrict to exact endofunctors o f E , the natura l maps i M and p M are inﬂation and deﬂation, and the ob jects C M and P M are w -acyclic for all M ∈ E . 7.6. Symmetric forms in bi mo dule categories and their tensor pro duct. Let A and B be dgas with inv olution, and let ( I , i ), ( J, j ) b e duality co eﬃcients for A -Mo d and B -Mo d, resp ectively . A s ymm et ric form in A -Mod - B , with re sp e ct t o the duality c o eﬃcients ( I , i ) and ( J, j ), is a pair ( M , ϕ ) wher e M ∈ A -Mo d - B is a left A and rig h t B -mo dule, a nd ϕ : M ⊗ B J ⊗ B M op → I is an A -bimo dule map making the dia gram of A -bimo dule maps (23) M ⊗ B J ⊗ B M op ϕ / / c 3 ◦ (1 ⊗ j ⊗ 1)   I i   ( M ⊗ B J ⊗ B M op ) op ϕ op / / I op commute. Isometrie s and o rthogonal sums o f symmetric for ms in A -Mo d - B are deﬁned in the ob vious wa y . T ensor pr oduct of symmetric forms is de ﬁned a s follows. Let A , B , C b e dg algebras with inv olution, and let ( I , i ), ( J, j ), ( K, k ) b e duality co eﬃcients for A -Mo d, B -Mo d and C -Mo d, r espectively . F urther, let M ∈ A -Mo d - B and N ∈ B -Mo d - C be equipp ed with s ymmetric forms given by the A - and B -bimo dule maps ϕ : M ⊗ B J ⊗ B M op → I and ψ : N ⊗ C K ⊗ C N op → J (making diagram (23) and its analog for ψ commute). The tensor pro duct ( M , ϕ ) ⊗ B ( N , ψ ) THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 43 of the symmetric for ms ( M , ϕ ) a nd ( N , ψ ) has M ⊗ B N as under lying left A and right C -mo dule, and is equipp ed with the symmetric form which is the A -bimo dule map ( M ⊗ B N ) ⊗ C K ⊗ C ( M ⊗ B N ) op c → M ⊗ B N ⊗ C K ⊗ C N op ⊗ B M op ψ → M ⊗ B J ⊗ B M op ϕ → I . 7.7. F o rm functors as tenso r pro duct w i th symmetric fo rm s. Let A and B be dga s with inv olution, and let ( I , i ), ( J, j ) be dualit y co eﬃcients for A -Mo d and B -Mo d, resp ectively . W e want to think of (certain) for m functors ( B -Mo d , ♯ J ) → ( A -Mo d , ♯ I ) as tensor pro duct with symmetric forms in A -Mo d - B . F or that, let ( M , ϕ ) b e a symmetric for m in A - Mod - B . It deﬁnes a form functor ( M , ϕ ) ⊗ B ? : ( B - Mod , ♯ J ) ( F, Φ) − → ( A - Mod , ♯ I ) , where F ( P ) = M ⊗ B P a nd the duality co mpatibilit y map is the left A -mo dule homomorphism M ⊗ B [ P op , J ] B Φ P − → [( M ⊗ B P ) op , I ] A deﬁned by Φ( x ⊗ f )(( y ⊗ t ) op ) = ( − 1) | y || t | ϕ ( x ⊗ f ( t op ) ⊗ y op ) for x, y ∈ M , f ∈ [ P op , J ] B , a nd t ∈ P . 7.8. Basic prop erties of ( M , ϕ ) ⊗ B ? . Let A , B , C be dg algebras with involution, and let ( I , i ), ( J, j ), ( K , k ) b e duality co eﬃcients for A -Mo d, B -Mo d and C -Mod, resp ectively . F urther, let ( M , ϕ ), ( M ′ , ϕ ′ ) b e symmetric for ms in A -Mo d - B and ( N , ψ ) a symmetric form in B -Mo d - C . F o rm functors induced by tensor pro duct with symmetric forms hav e the following ele men tary pro perties. (a) T ensor pro duct ( A, µ I ) ⊗ A ? with the symmetric form µ I : A ⊗ A I ⊗ A A op → I : a ⊗ t ⊗ b 7→ a · t · ¯ b on the A -bimo dule A induces the identit y form functor on ( A, ♯ I ). (b) An isometry ( M , ϕ ) ∼ = ( M ′ , ϕ ′ ) b et w een symmetric forms in A -Mo d - B deﬁnes an isometry of a sso ciated form functors ( M , ϕ ) ⊗ B ? ∼ = ( M ′ , ϕ ′ ) ⊗ B ? : ( B -Mo d , ♯ J ) → ( A -Mo d , ♯ I ) . (c) O rthogonal sum ( M , ϕ ) ⊥ ( M ′ , ϕ ′ ) of symmetric forms in A -Mo d - B cor re- sp onds to o rthogonal sum o f as socia ted fo rm functors: ( M , ϕ ) ⊗ B ? ⊥ ( M ′ , ϕ ′ ) ⊗ B ? ∼ = [( M , ϕ ) ⊥ ( M ′ , ϕ ′ )] ⊗ B ? (d) T ensor pr oduct of s ymmetric for ms co rresp onds to co mposition of asso ciated form functors: [( M , ϕ ) ⊗ B ( N , ψ )] ⊗ C ? ∼ = [( M , ϕ ) ⊗ B ?] ◦ [( N , ψ ) ⊗ C ?] These prop erties follow dire ctly fr om the deﬁnitions, a nd we omit the details . 44 MARC O SCHLICHTING 7.9. T ens or pro duct of dgas with i n volution. F or tw o dgas A , V , the tensor pro duct dg κ -mo dule AV = A ⊗ κ V is a dga with m ultiplication ( a ⊗ v ) · ( b ⊗ w ) = ( − 1) | b || v | ( a · b ) ⊗ ( v · w ). If A , V are dgas with in volution, then the tensor pro duct dga AV is a dga with inv olution AV → ( AV ) op : a ⊗ v 7→ ( ¯ a ⊗ ¯ v ) op . F urthermore , if ( I , i ) and ( U, u ) are duality co eﬃcien ts for A -Mo d and V -Mo d, then I U = I ⊗ U is an AV -bimo dule with left multiplication ( a ⊗ v ) · ( t ⊗ x ) = ( − 1) | v || t | a · t ⊗ v · x and r igh t m ultiplication ( t ⊗ x ) · ( a ⊗ v ) = ( − 1) | a || x | t · a ⊗ x · v , for a ∈ A , v ∈ V , t ∈ I , x ∈ U , and the AV -bimo dule map iu = i ⊗ u : I ⊗ U → ( I ⊗ U ) op : t ⊗ x 7→ ( i ( t ) ⊗ u ( x )) op makes the pair ( I U, i u ) into a duality co eﬃcien t for AV -Mo d. If B is ano ther dga with inv olution, and ( J, j ) is a duality coeﬃcient for B -Mo d, then, with the sa me formulas as in 7.6, 7.7, an y symmetric fr om ( M , ϕ ) in A -Mo d - B with resp ect to the dualit y co eﬃcien ts ( I , i ) a nd ( J, j ) deﬁnes a form functor ( M , ϕ ) ⊗ B ? : ( B V -Mo d , ♯ ( J U,ju ) ) → ( AV -Mo d , ♯ ( I U,iu ) ) satisfying the prop erties in 7.8. 7.10. Extension to ringe d spaces. Let ( X , O X ) b e a ringe d spac e with O X a sheaf of co mm utativ e rings o n a top olog ical space X . Replacing in 7.1 - 7.9 the ground r ing κ with the s heaf of comm uta tiv e rings O X , all deﬁnitions and prop erties from 7.1 - 7.9 extend to mo dules over diﬀerential grade d sheaves of O X -algebra s. Deﬁnitions a re extended b y a pplying the deﬁnitions o f 7.1 - 7.9 to sections o v er op en subsets of X . F o r instance, let A be a sheaf of dg O X -algebra s with inv olution, ( I , i ) a duality co eﬃcien t for A -Mo d, and P a sheaf of left dg A -mo dules. The ca nonical double dual identiﬁcation can : P → [[ P op , I ] op A , I ] A is deﬁned by sending a s ection x ∈ P ( U ), U ⊂ X , to the map of sheaves of dg mo dules can( x ) : ([ P op , I ] op A ) | U → I | U deﬁned on V ⊂ U b y can( x )( f op ) = ( − 1) | x || f | i ( f ( x op | V )) for f ∈ [ P op , I ] A ( V ). 8. Higher Grothendieck-Witt groups of schemes Let X b e a sc heme, A X be a quasi- coherent sheaf of O X -algebra s with in volution, L a line bundle on X , Z ⊂ X a clos ed subscheme and n ∈ Z an integer. The pur pose of this section is to introduce the Grothendieck-Witt s pace GW n ( A X on Z , L ) o f symmetric spa ces ov er A X with coeﬃcients in the n -th shifted line bundle L [ n ] and supp ort in Z . W e w ork in this generality in o rder to b e able to extent the lo calization and excision theor ems of § 9 to nega tiv e deg rees. Recall tha t, unless otherwise indica ted, “mo dule” will alwa ys mean “ left mo d- ule”. In what follows, we will denote by ⊗ the tensor pr oduct ⊗ O X of O X -mo dules. 8.1. V e ctor bundles and strictly p erfect complexes . Let A X be a quas i- coherent sheaf of O X -algebra s with inv olution. The categ ory of quasi-coher en t left A X -mo dules (dg-modules co ncen trated in degree 0) is a fully exact ab elian sub category o f the ab elian categor y of left A X -mo dules. W e denote by V ect( A X ) the full sub categor y of A X vector bundles , that is, of those quas i-coherent left A X -mo dules F for which F ( U ) is a ﬁnitely gener ated pr o jective A X ( U )-mo dule for ev ery aﬃne U ⊂ X . As usual, the last condition only needs to b e chec ked for those U running through a choice o f an a ﬃne op en cov er of X . The ca tegory of A X vector bundles inherits the notion o f exa ct sequences from the category of a ll (quasi-coher en t) A X -mo dules. Note that A X vector bundles need not b e lo cally THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 45 free, s ince A X,x may not b e comm utative nor a lo cal ring fo r x ∈ X . In case A X = O X , the category V ect( X ) is the usual exact ca tegory of vector-bundles on X . A strictly p erfe ct co mplex of A X -mo dules is a dg le ft A X -mo dule M such that M n = 0 f or all but ﬁ nitely man y n ∈ Z and M n is an A X vector bundle for all n ∈ Z . Denote by sPerf ( A X ) the category of strictly pe rfect complexes of A X -mo dules, in o der words, the c ategory o f b ounded chain complexes of A X vector bundles. Let L b e a line bundle on X . Then A X L [ n ] = A X ⊗ L [ n ] is a dg A X -bimo dule via the multiplication deﬁned on sections by a ( x ⊗ l ) b = ( − 1) | b || l | axb ⊗ l for a, b, x ∈ A X and l ∈ L [ n ]. W e equip A X L [ n ] with a dg A X -bimo dule is omorphism i : A X L [ n ] → ( A X L [ n ]) op : a ⊗ l 7→ ¯ a ⊗ l satisfying i op ◦ i = 1, so that ( A X L [ n ] , i ) is a duality co eﬃcient for A X -Mo d. If ε ∈ { +1 , − 1 } , then ( A X L [ n ] , ε i ) is a lso a dualit y co eﬃcien t for A X -Mo d. In the notation o f 7 .9, the duality co eﬃcien t ( A X L [ n ] , ε i ) is the tensor pro duct of t he duality coeﬃcient ( A X , µ ) for A X -Mo d and the dualit y co eﬃcien t ( L [ n ] , ε ) fo r O X -Mo d. F or a strictly p erfect co mplex of A X -mo dules M , the left dg A X -mo dule M ♯ n εL = [ M op , A X L [ n ] ] A X is also strictly p erfect, the functor M 7→ M ♯ n εL is exac t and pr eserves quasi-isomor- phisms. More o ver, the double dual identiﬁcation can ( A X L [ n ] ,εi ) deﬁned in 7 .4 is an isomorphism. There fore, the tr iple (sPerf ( A X ) , quis , ♯ n εL ) deﬁnes an exact category with weak e quiv a lences and duality , the double dual ident iﬁcation b eing under stoo d as can ( A X L [ n ] ,εi ) . If n = 0 (or ε = 1, or L = O X ), we may omit the la bel corresp onding to n (or ε , or L , re spectively), so that ( A X -Mo d , ♯ n ) means ( A X -Mo d , ♯ n 1 ,O X ), for instance. By restrictio n of str ucture, we hav e an exact categ ory with duality (V ect( A X ) , ♯ εL ) . Let Z ⊂ X be a clo sed subscheme with op en complemen t U = X − Z . A strictly per fect co mplex M o f A X -mo dules has c ohomolo gic al supp ort in Z if the restriction M | U of M to U is acyclic. W e wr ite sPerf ( A X on Z ) for the categ ory of strictly p erfect c omplexes o f A X -mo dules which hav e coho mological supp ort in Z . By restrictio n of structure, we have exa ct categ ories with weak equiv alences and duality (24) ( sPerf ( A X on Z ) , quis , ♯ n εL , ) . 8.2. Deﬁnition. Let X b e a scheme, A X be a quasi-co heren t sheaf of O X -algebra s with in v olution, L a line bundle on X , Z ⊂ X a closed subsc heme, n ∈ Z an in teger, and ε ∈ { +1 , − 1 } . The Grothendieck-Witt spa ce ε GW n ( A X on Z, L ) of ε -symmetric spaces ov er A X with co eﬃcien ts in the n -th shifted line bundle L [ n ] and (cohomologica l) supp ort in Z is the Gro thendiec k-Witt space o f the ex- act categ ory with weak equiv alences and duality (24). If Z = X (or ε = 1, or L = O X , or n = 0), we may omit the la bel cor resp onding to Z ( ε , L , n , resp ec- tively). F or instance, the space GW ( A X , L ) denotes the Grothendieck-Witt s pace 1 GW 0 ( A X on X , L ). 46 MARC O SCHLICHTING 8.3. Remark. By 7.5, the e xact ca tegory with weak equiv alences and duality (24) has a symmetric cone in the sense of 4.1. In the following prop osition, we write µ : O X ⊗ O X → O X for the multiplication in O X . 8.4. Prop osition. T ensor pr o duct with the ( − 1) -symmetric sp ac e ( O X [1] , µ ) in- duc es an e quivalenc e of exact c ate gories with we ak e quivalenc es and duality ( sPerf ( A X on Z ) , quis , ♯ n εL , ) ∼ = ( sPerf ( A X on Z ) , q uis , ♯ n +2 − εL , ) . In p articular, we have homotopy e qu ivale nc es of Gr othendie ck-Witt sp ac es ( O X [1] , µ ) ⊗ ? : ε GW n ( A X on Z, L ) ≃ − ε GW n +2 ( A X on Z , L ) and ( O X [2] , µ ) ⊗ ? : ε GW n ( A X on Z, L ) ≃ ε GW n +4 ( A X on Z , L ) . Pr o of. The pair ( O X [1] , µ ) deﬁnes a symmetric space in O X -Mo d with r espect to the duality co eﬃcient ( O X [2] , − 1). T ensor pro duct ( O X [1] , µ ) ⊗ O X ? deﬁnes a form functor ( A X -Mo d , ♯ n εL ) → ( A X -Mo d , ♯ n +2 − εL ) as explained in 7.7 – 7.1 0. Since ( O X [1] , µ ) ⊗ O X ( O X [ − 1] , − µ ) and ( O X [ − 1] , − µ ) ⊗ O X ( O X [1] , µ ) ar e isometr ic to ( O X , µ ) which induces the identit y for m functor, the e quiv alence o f catego ries with duality and the ﬁrst ho motop y eq uiv a lence follow. The second map of spaces is a homotopy equiv ale nce with in verse giv en b y the tensor pr oduct with the symmetric space ( O X [ − 2] , µ ).  8.5. Corollary. F or n ∈ Z , ther e ar e functorial homotopy e quivalenc es GW 4 n ( A X , L ) ≃ GW (V ect ( A X ) , ♯ L , can L ) , and GW 4 n +2 ( A X , L ) ≃ GW (V ect( A X ) , ♯ L , − can L ) . wher e the Gr othendie ck-W itt sp ac es on t he right hand side ar e the ones asso ciate d with the exact c ate gories with duality (V ect( A X ) , ♯ L , ± can L ) as deﬁne d in [Sch08, 4.6] . Pr o of. The ho motop y equiv a lences follow fr om prop osition 6.5, rema rk 6.3, and prop osition 8.4.  9. Localiza tion and Zariski-excision in positive degrees 9.1. Sc he mes with an ample family of line bun dl es. A sc heme X has an ample family of line bund les if there is a ﬁnite se t L 1 , ..., L n of line bundles with globa l sections s i ∈ Γ( X , L i ) suc h that the non-v anishing lo ci X s i = { x ∈ X | s i ( x ) 6 = 0 } form an op en a ﬃne cov er of X , se e [TT9 0, Deﬁnition 2 .1], [SGA6, I I 2.2.4]. Recall that if f ∈ Γ( X , L ) is a g lobal section of a line bundle L on a scheme X , then the op en inclusion X f ⊂ X is a n aﬃne map (as can b e seen b y choosing an o pen aﬃne cov er of X trivializing the line bundle L ). As a spec ial ca se, X f is aﬃne whenever X is a ﬃne. Thus, for the aﬃne c o ver a bov e X = S X s i , all ﬁnite int ersections of the X s i ’s a re aﬃne. In particula r, a scheme with an ample family of line bundles is quas i-compact (as a ﬁnite union of a ﬃne, hence quas i-compact, subschemes) and it is quas i-separated. Reca ll that the latter means that the in- tersection o f any tw o qua si-compact op en s ubsets is quasi- compact (a conditio n THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 47 which only needs to b e chec ked for the pair-wise intersections U i ∩ U j of a cov er of X = S i U i by quasi- compact op en subs ets U i ; in our case, we can take U i = X s i ). F or a scheme X with an a mple family of line bundles, there is a s et L i , i ∈ I , of line bundles o n X with global sections s i ∈ Γ( X , L i ) such that the op en s ubsets X s i , i ∈ I , form an op en aﬃne basis for the top ology of X [TT90, 2 .1.1 (b)]. F or examples of schemes with an ample family of line bundles, see [TT90, 2.1 .2]. An y quas i-compact op en o r closed subscheme of a scheme with a n ample family of line bundles has itself an ample family of line bun dles. Any sc heme quasi-pro jective ov er a n a ﬃne scheme, and any separated r egular no etherian scheme has a n ample family o f line bundles. The main purp ose o f this s ection is to pr o ve the following t wo theo rems. 9.2. Theorem (Lo caliza tion). L et X b e a scheme with an ample family of line- bund les, let Z ⊂ X b e a close d subscheme with quasi-c omp act op en c omplement j : U ⊂ X , and let L a line bund le on X . L et A X b e a quasi-c oher ent she af of O X -algebr as with involution. Then for every n ∈ Z ther e is a homotopy ﬁbr ation of Gr othendie ck-Witt sp ac es GW n ( A X on Z , L ) − → GW n ( A X , L ) − → GW n ( A U , j ∗ L ) . 9.3. Theorem (Zariski-ex cision). L et X b e a scheme with an ample family of line- bund les, let Z ⊂ X b e a close d su bscheme with quasi-c omp act op en c omplement, let L b e a line bund le on X and let A X b e a quasi-c oher ent s he af of O X -algebr as with involution. Then for every n ∈ Z and every quasi-c omp act op en subscheme j : V ⊂ X c ontaining Z , r estriction of ve ct or-bu nd les induc es a homotopy e quivalenc e GW n ( A X on Z, L ) ∼ − → GW n ( A V on Z , j ∗ L ) . The pro ofs of theorems 9.2 and 9.3 will o ccupy the r est of this sectio n. First, w e int ro duce so me termino logy . F or an op en s ubsc heme U ⊂ X , ca ll a map M → N of left dg A X -mo dules a U -isomor phism ( U - quasi-isomorphism) if its restriction M | U → N | U to U is a n isomorphism (quasi-iso morphism). A left dg A X -mo dule M is c alled U -acyclic if its restrictio n M | U to U is acyclic. 9.4. Notation. In 9.5, 9.6, 9.7 b elow, we consider the following ob jects: (a) a scheme X which is quasi-compac t a nd quasi-s eparated, (b) a ﬁnite set of line bundles L i , i = 1 , ..., n together with globa l s ections s i ∈ Γ( X , L i ), (c) the union U = S n i =1 X s i of the non-v anishing lo ci X s i of the s i ’s, denoting j : U ⊂ X the corr esponding op en immers ion, and (d) a quas i-coherent sheaf of O X -algebra s A X . 9.5. T runcated Koszul comp l exes. In the situation of 9.4, the g lobal sections s i deﬁne maps s i : O X → L i of line-bundles whose O X -duals are denoted b y s − 1 i : L − 1 i → O X . W e co nsider the maps s − 1 i as (cohomologic ally g raded) chain- complexes with O X placed in degree 0. F or a n l -tuple n = ( n 1 , ..., n l ) of nega tiv e 48 MARC O SCHLICHTING int egers, the Koszul complex (25) l O i =1 ( L n i i s n i → O X ) is acyclic over U , since the map s n i = ( s − 1 i ) ⊗| n i | : L n i i → O X is an X s i -isomorphis m, so that the Koszul complex (2 5) is acyclic (even contractible) over ea c h X s i . Let K ( s n ) denote the bounded complex whose non-zero part (whic h w e place in degrees − l + 1 , ..., 0) is the Koszul complex (25) in degrees ≤ − 1. The last diﬀerential d − 1 of the Koszul complex deﬁnes a map K ( s n ) = " l O i =1 ( L n i i s n i → O X ) # ≤− 1 [ − 1] ε − → O X of strictly p erfect complexe s of O X -mo dules which is a U -quasi-is omorphism since its cone, the Koszul complex, is U -a cyclic. F o r a left dg A X -mo dule M , we wr ite ε M for the tensor pro duct map ε M = 1 M ⊗ ε : M ⊗ K ( s n ) → M ⊗ O X ∼ = M . The following lemma is a consequence o f the w ell-known tec hniques of extending a section of a quas i-coherent s heaf from an op en subset c ut out by a diviso r to the sc heme itse lf [EGAI, Th´ eor` eme 9.3.1 ]. It is implicit in the pro of of [TT90, Prop osition 5 .4.2]. 9.6. Lemma. In the situation 9.4, let M b e a c omplex of qu asi-c oher ent left A X - mo dules and let A b e a strictly p erfe ct c omplex of A X -mo dules. Then the fol lowing holds. (a) F or every map f : j ∗ A → j ∗ M of left dg A U -mo dules b etwe en the r e- strictions of A and M to U , ther e is an l -tuple of ne gative inte gers n = ( n 1 , ..., n l ) , and a m ap ˜ f : A ⊗ K ( s n ) → M of left dg A X -mo dules such that f ◦ j ∗ ( ε A ) = j ∗ ( ˜ f ) . (b) F or every map f : A → M of left dg A X -mo dules such t hat j ∗ ( f ) = 0 , ther e is an l -tuple of ne gative inte gers n = ( n 1 , ..., n l ) such t hat f ◦ ε A = 0 . Pr o of. F or any c omplex of O X -mo dules K , to give a ma p A ⊗ K → M of chain complexes of A X -mo dules is the same as to give a map K → A X [ A, M ] o f chain complexes of O X -mo dules, by adjointn ess of the tensor pro duct A ⊗ ? : O X -Mo d → A X -Mo d and the left A X -mo dule map functor A X [ A, ] : A X -Mo d → O X -Mo d. Note that if A is strictly per fect, the complex A X [ A, M ] is a c omplex of quasi- coherent O X -mo dules and the natural ma p A X [ A, j ∗ j ∗ M ] → j ∗ j ∗ A X [ A, M ] is a n isomorphism. The adjunction a llo ws us to reduce the pr oof of the lemma to A X = O X and A = O X (concentrated in degr ee 0). Every map O X → M of chain complex es of O X -mo dules fac tors through the sub complex Z 0 M ⊂ M of M w hic h is the complex ker( d 0 : M 0 → M 1 ) concen trated in degree 0. By adjunction, every map O X → j ∗ j ∗ M factor s as O X → j ∗ Z 0 j ∗ M = j ∗ j ∗ Z 0 M → j ∗ j ∗ M . This allows us to further reduce the pr oof to M a complex with M i = 0 , i 6 = 0. In this c ase, the pro of for l = 1 is c lassical, s ee [EGAI, Th´ eor` eme 9 .3.1], [TT9 0, Lemma 5.4 .1], so that the lemma is proved in ca se l = 1. Before we treat the case l > 1 (a nd A X = O X ; A = O X , M c oncen trated in degree 0), we prov e the following. THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 49 ( † ) F or every map A → G of complexes of quasi-cohe ren t O X -mo dules with A strictly p erfect and j ∗ G cont ractible, there is an l -tuple of nega tiv e integers ( n 1 , ..., n l ) and a commutativ e dia gram of complexes of O X -mo dules A 1 A ⊗ ι / / ( ( P P P P P P P P P P P P P P P P P A ⊗ h N l i ( L n i i s n i → O X ) i   G, where ι is the ca nonical em b edding o f O X (concentrated in de gree 0) in to the Koszul complex. It suﬃces to pr o ve ( † ) for l = 1, since the genera l case is a rep eated application of the ca se l = 1 . F o r the pro of of ( † ) with l = 1, we can assume A = O X , a s ab o ve. The comp osition O X → G → j ∗ j ∗ G has contractible target and there fore factors throug h the cone ( O X → O X ) of O X . By the case l = 1 of the lemma (prov ed a bov e), there is an n < 0 such that the comp osition ( L n → L n ) s n → ( O X → O X ) → j ∗ j ∗ G lifts to G . The tw o maps (0 → L n ) s n → (0 → O X ) → G and (0 → L n ) → ( L n → L n ) → G may no t be the same, but their comp ositions with G → j ∗ j ∗ G are, so that, again b y case l = 1 of the lemma, precompos ing b oth maps with L n + t s t → L n makes the tw o maps with tar get G equal. Replacing n with n + t , we can a ssume that the tw o ma ps (0 → L n ) → G ab ov e coincide. Then we o btain a ma p from the push-o ut ( L n s n → O X ) of ( L n → L n ) ← (0 → L n ) s n → (0 → O X ) to G . This proves ( † ) for l = 1, hence for a ll l . F or part (a) of the gener al ca se o f lemma 9.6 (and A X = O X ; A = O X , M concentrated in degree 0), we apply ( † ) to the ma p of chain-complexes o f O X - mo dules (0 → O X ) → ( M → j ∗ j ∗ M ), and obtain a factor ization of that map through the Koszul complex N i ( L n i i s n i → O X ) for some l - tuple of neg ativ e integers ( n 1 , ..., n l ). The canonica l m ap fro m the stupid truncation in degree s ≤ − 1 (shifted by 1 degree) to its degree 0 part K ( s n ) = h N l i =1 ( L n i i s n i → O X ) i ≤− 1 [ − 1]   d − 1 / / O X   M = [ M → j ∗ j ∗ M ] ≤− 1 [ − 1] d − 1 / / j ∗ j ∗ M yields (a). The g eneral ca se of part (b) is a rep eated applica tion of the case l = 1.  F or an ex act c ategory with weak e quiv alences ( C , w ), we write D ( C , w ) for its derived ca tegory , that is, the c ategory C [ w − 1 ] obtained from C b y formally in verting the ar rows in w . If ( C , w ) is a category o f complex es in the sense of deﬁnition 5.4, its derived category D ( C , w ) is a triang ulated categ ory . In this case, it can a lso be obtained as the lo calization by a calculus of fr actions of the homotopy categ ory K ( C ) of C which is the fac tor category of C mo dulo the ideal of maps which are homotopic to zero. 50 MARC O SCHLICHTING 9.7. Lemma. In the situation 9.4, let A ⊂ sP erf ( A X ) b e a ful l su b c ate gory of the c ate gory of strictly p erfe ct A X -mo dules su ch that the inclusion A ⊂ sPerf ( A X ) is close d under de gr e e-wise split extensions, usual shifts and c ones. A s sume further- mor e that for al l A ∈ A , k ≤ 0 and i = 1 , ...n , we have A ⊗ L k i ∈ A . Then for every U -quasi-isomorphism M → A of c omplexes of qu asi-c oher ent A X -mo dules with A ∈ A , t her e is a U -quasi-isomorphism B → M of c omplexes of A X -mo dules with B ∈ A : B ∃ U − quis / / M ∀ U − quis / / A. In p articular, the inclusion A ⊂ sPerf ( A X ) induc es a ful ly faithful triangle fun ct or D ( A , U - quis) ⊂ D ( sPerf ( A X ) , U - quis) . Pr o of. W e ﬁrst prove the following statement. ( † ) Let s ∈ Γ( X , L ) be a global s ection of a line-bundle L such that X s is aﬃne. Then for every X s -quasi-iso morphism N → E of c omplexes of quasi- coherent A X -mo dules with E strictly p erfect on X , there is an X s -quasi- isomorphism E ⊗ L − k → N fo r so me integer k > 0. W rite j : X s ⊂ X for the o pen inclusion. Since X s is aﬃne, we have an equiv- alence of catego ries b et w een quasi-coher en t A X s -mo dules a nd A X ( X s )-mo dules under which the map j ∗ N → j ∗ E be comes a quasi-is omorphism of c omplexes of A X ( X s )-mo dules with j ∗ E a b ounded complex of pro jectives. Such a map always has a section up to homotopy f : j ∗ E → j ∗ N which is then a quasi-isomor phism. By 9.6 wit h l = 1, there is a ma p of complexe s ˜ f : E ⊗ L k → N such that j ∗ ˜ f = f · s k , for some k < 0. In pa rticular, ˜ f is a U -quasi-is omorphism. Now we prov e the lemma by induction on n . F or n = 1, this is ( † ). Let U 0 = S n − 1 i =1 X s i . By our induction hypo thesis, there is a U 0 -quasi-iso morphism B 0 → M with B 0 ∈ A . L et M 0 and A 0 be the cones of the maps B 0 → M a nd B 0 → A , that is, the push-out of these maps a long the canonica l (degree-wise split) injection of B 0 int o its cone C B 0 . W e obtain a commut ative (in fact bicartesian) diagram inv o lving M , M 0 , A and A 0 with M → M 0 and A → A 0 degree-wise split injective. F actor the map A → A 0 as in the diagr am M / / / /   M 1 / / / /   M 0   A / / / / A ⊕ P A 0 / / / / A 0 with A ⊕ P A 0 → A 0 degree-wise split sur jectiv e and P A 0 = C A 0 [ − 1] ∈ Ch b A contractible. Then M → M 0 factors thro ugh the pull-back M 1 of M 0 → A 0 along the surjectio n A ⊕ P A 0 → A . The map M → M 1 is deg ree-wise split injective (as M → M 0 is), a nd has cokernel the contractible complex P A 0 . It follows that M → M 1 is a ho motop y eq uiv a lence, and we can c ho ose a homoto p y inv erse M 1 → M . By co nstruction, A 0 and M 0 are acyclic o ver U 0 and A 0 ∈ A . Mor eov er , the map M 0 → A 0 is an X s n -quasi-iso morphism. By ( † ), there is a n X s n -quasi-iso morphism A 0 ⊗ L − k → M 0 for some k > 0 . The complex A 0 ⊗ L − k is U 0 -acyclic since A 0 is, THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 51 so that the map A 0 ⊗ L − k → M 0 is in fact a U - quasi-isomorphism. Let B be the pull-back of A 0 ⊗ L − k → M 0 along the surjection M 1 → M 0 . The resulting map B → M 1 is a U -qua si-isomorphism. Moreover B is an o b ject of A since B is also t he pull-back of A 0 ⊗ L − k → A 0 along the (degree-wise split) surjection A ⊕ P A 0 → A 0 . Comp osing the map B → M 1 with the homotopy inv erse M 1 → M of M → M 1 yields the desired U -qua si-isomorphism B → M .  9.8. The derived category of quasi-coheren t A X -mo dules. Recall that a Grothendieck ab elian categ ory is an a belian categor y A in which all set-index ed direct sums e xist, ﬁltered co limits are exa ct, and A ha s a set of gener ators. W e remind the reader that a se t I of ob jects o f A ge nerates the ab elian ca tegory A if for every ob ject E ∈ A , there is a s urjection L A i ։ E fro m a set indexed direct sum o f (p ossibly rep eated) ob jects A i ∈ I to E . If X is a scheme with an a mple family of line-bundles, and A X a quas i-coherent O X -algebra , then the catego ry Qcoh( A X ) of quasi-coher en t A X -mo dules is an ab elian categ ory with g enerating s et the set A X ⊗ L k i , i = 1 , ..., n , k ≤ 0, where L 1 , ..., L n is a set of line bundles on X with g lobal sections s i ∈ Γ( X , L i ) such that the non-v a nishing lo ci X s i are aﬃne and cov er X . F or a Grothendieck ab elian catego ry A , write D ( A ) for the unbounded der iv ed category of A , that is, the tria ngulated category D (Ch A , quis). This category has small homomo rphism sets, by [W ei94, remark 10.4 .5]. Copr oducts o f com- plexes are also copr oducts in D A , so that the tr iangulated category D A has all set- indexed copro ducts. This a pplies in particular to the unbounded de riv ed category D Qcoh( A X ) of quasi-co heren t A X mo dules. If Z ⊂ X is a closed s ubset with quasi- compact op en complement U = X − Z , w e write D Z Qcoh( A X ) ⊂ D Q coh( A X ) for the full triangulated sub categor y of those complexes E of quasi-coher en t A X - mo dules whose re striction E | U to U are acy clic. 9.9. Lemma. L et A b e a Gr othendie ck ab elian c ate gory with gener ating set of ob- je cts I . Then, an obje ct E of the triangulate d c ate gory D A is zer o iﬀ every map A [ j ] → E in D A is the zer o map for A ∈ I and j ∈ Z . Pr o of. Let E be an ob ject of the derived category D A of A such that every map A [ j ] → E in D A is the zero map for A ∈ I and j ∈ Z . W e can choose a surjectio n L J A j ։ k er( d 0 ) in A with A j ob jects in the genera ting s et I . The inclusion of complexes ker( d 0 ) → E yields a map of complexes L J A j → k er( d k ) → E whic h induces a surjective map L J A j ։ ker( d 0 ) ։ H 0 E on c ohomology . Since every map L J A j → E is zero in D A , the induced surjective map L J A j ։ H 0 E is the zero map, he nce H 0 E = 0. The same a rgument a pplied to E [ k ] instead o f to E shows that H k E = 0 for all k ∈ Z , so that E is qua si-isomorphic to the zero complex.  Next, w e r ecall the concept of compactly generated tria ngulated categor ies due to Neema n [Nee9 2] in the form o f [Nee96]. 9.10. Compactly generated triangulated categories . Let T b e a triangu- lated ca tegory in which (all set-indexed) co pro ducts exists. An ob ject A of T is ca lled c omp act [Nee96, Deﬁnition 1.6] if the natural map L j ∈ J H om ( A, M j ) → H om ( A, L j ∈ J M j ) is a n is omorphism fo r any set M j , j ∈ J , of o b jects in T . The 52 MARC O SCHLICHTING full sub categor y T c ⊂ T of compact ob jects is closed under shifts and cones and th us is a triangulated sub categor y . A triangulated ca tegory T is c omp actly gener ate d [Nee9 6, Deﬁnition 1 .7] if T contains all set-indexed direct s ums, and if there is a set I o f compact o b jects in T such that an ob ject M o f T is the zero ob ject iﬀ all maps A → M ar e the z ero map fo r A ∈ I . A set I o f compact ob jects in a co mpactly genera ted tr iangulated catego ry T is called a gener ating set [Nee96, Deﬁnition 1.7] if I is closed under shifts and if an ob ject M of T is the zero ob ject iﬀ all maps A → M are the zero ma p for A ∈ I . The following theorem is due to Neeman [Nee9 6, Theo rem 2.1 ]. 9.11. Theorem (Neeman). (a) L et T b e a c omp actly gener ate d triangulate d with gener ating set of obje cts I . Then the ful l triangulate d sub c ate gory T c of c omp act obje cts in T is the smal lest idemp otent c omplete triangulate d sub c ate gory of T c ont aining I . (b) L et R b e a c omp actly gener ate d triangulate d c ate gory, S ⊂ R c a set of c omp act obje cts close d un der taking shifts. L et S ⊂ R b e the smal lest ful l triangulate d su b c ate gory close d un der formation of c opr o ducts in R which c ontains S . Then S and R / S ar e c omp actly gener ate d triangulate d c ate- gories with gener ating set S and the image of R c in R / S . Mor e over, the functor R c / S c → R / S induc es an e quivalenc e b etwe en the idemp otent c om- pletion of R c / S c and t he c ate gory of c omp act obje cts in R / S . (c) Le t S → R b e a triangle fu nctor b etwe en c omp actly gener ate d triangulate d c ate gories which pr eserves c opr o duct and c omp act obje cts. Then S → R is an e quivalenc e iﬀ the functor S c → R c on c omp act obje cts is an e quivalenc e. The following tw o pro positions are essen tially due to Thomason [TT9 0]. W e include the proo fs here b ecause only the commutativ e situation is considered in [TT90], a nd we need the e xplicite versions b elow. F or an exac t catego ry E , write D b ( E ) f or the triangulated categ ory D (Ch b E , quis). Recall that a fully faithful functor A → B o f a dditiv e catego ries is called c oﬁnal if every ob ject of B is a direct facto r of an ob ject of A . 9.12. Prop osition. L et X b e a quasi-c omp act and qu asi-sep ar ate d scheme which is the un ion X = S n i =1 X s i of op en aﬃne non-vanishing lo ci X s i of glob al se ct ions s i ∈ Γ( X , L i ) of line-bund les L i , i = 1 , ..., n . L et A X b e a quasi-c oher ent O X - algebr a. Then the triangulate d c ate gory D Qco h( A X ) is c omp actly gener ate d by t he set of obje ct s A X ⊗ L k i [ j ] for k ≤ 0 , i = 1 , ..., n and j ∈ Z . Mor e over, the inclusion V ect( A X ) ⊂ Qcoh( A X ) induc es a ful ly faithful triangle functor D b V ect( A X ) ⊂ D Qco h ( A X ) which identiﬁes, u p to e quivalenc e, the c ate- gory D b V ect( A X ) with the ful l triangulate d sub c ate gory D c Qcoh( A X ) of c omp act obje cts in D Qcoh( A X ) . Pr o of. In the tria ngulated ca tegory D Qcoh( A X ), every s trictly p erfect complex o f A X mo dules is a compact ob ject. T o see this, note that for an A X vector bundle A and a se t M j , j ∈ J , of quasi-coher en t A X -mo dules the ca nonical map of sheav es of homomorphisms L j H om A X ( A, M j ) → H om A X ( A, L j M j ) is an isomorphism since this can be chec ked o n an aﬃne open cover of X where the statement is clear. T a king global se ctions, we o btain an isomorphism L j H om A X ( A, M j ) ∼ = → THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 53 H om A X ( A, L M j ) . This isomorphism extends to an iso morphism of homo mor- phism sets of c omplexes o f A X -mo dules for A a str ictly per fect complex and M a n arbitrar y complex of q uasi-coherent A X -mo dules. F or s uc h complex es, the isomor - phism induces an isomorphism (26) M j H om K Qcoh( A X ) ( A, M j ) ∼ = → H om K Qcoh( A X ) ( A, M j M j ) of homo morphism se ts in the homoto p y categ ory K Qcoh( A X ) of chain complexes of qua si-coherent A X -mo dules. It follows fro m lemma 9.7 with U = X and A = sPerf ( A X ) that maps in D Qcoh( A X ) from a s trictly p erfect complex A to an arbitrar y co mplex M of quas i-coherent A X -mo dules can be computed as the ﬁltered colimit colim B ∼ → A H om K Qcoh( A X ) ( B , M ) ∼ = − → H om D Qc oh( A X ) ( A, M ) of homomorphism sets in K Q coh( A X ) where the indexing category is the left ﬁl- tering category of homotopy clas ses of quasi-isomo rphisms B ∼ → A of strictly pe r- fect complex es with target A . T ak ing the co limit ov er this ﬁltering catego ry of the isomor phism (26) yields the isomorphis m which pr o ves that A is compa ct in D Qcoh( A X ). Since the set A X ⊗ L k i , i = 1 , ..., n , k ≤ 0 is a se t of g enerators fo r the Grothendieck a belian categ ory Qcoh( A X ) all of which are compact in the derived category D Qcoh( A X ), lemma 9.9 shows that D Q coh( A X ) is a compactly gener- ated tr iangulated c ategory with generating set A X ⊗ L k i [ j ], i = 1 , ..., n , k ≤ 0, j ∈ Z . The inclusio n V ect( A X ) ⊂ Qcoh( A X ) of vector bundles int o all quasi-coher en t A X -mo dules induces a triangle functor D b V ect( A X ) → D Qcoh( A X ) which is fully faithful, by the existence of an ample family of line bundles and the crite- rion in [Kel9 6, 12.1]. Since the exact categ ory V e ct( A X ) is idemp otent complete, its b ounded der iv ed catego ry D b V ect( A X ) is also idempo ten t complete [BS01, Theorem 2.8]. By Neeman’s theorem 9.11 (a), the inclusion D b V ect( A X ) → D c Qcoh( A X ) is an equiv a lence.  9.13. Reminde r on Rj ∗ . Let X be a scheme wit h an ample f amily of line bund les, and let j : U ֒ → X b e an op en immers ion from a quasi-co mpact op en subs et U to X . W e recall one p ossible construction of the right-deriv ed fu nctor Rj ∗ : D Qcoh( U ) → D Qcoh( X ) of j ∗ : Qco h( U ) → Q coh( X ). F or that, cho ose a ﬁnite co ver U = { U 0 , ..., U n } of U such that the inclusion of all ﬁnite int ersections U i 0 ∩ ... ∩ U i k ⊂ X int o X ar e aﬃne maps, i 0 , ..., i k ∈ { 0 , ..., n } . F or instance, we ca n take as U an op en cov er of U by a ﬁnite num ber of non-v anishing lo ci X s i asso ciated with a set o f line bundles L i on X and global sections s i ∈ Γ( X , L i ), i = 0 , ..., n . F or a k + 1-tuple i = ( i 0 , ..., i k ), set U i = U i 0 ∩ ... ∩ U i k and write write j i : U i ⊂ U for the corres ponding o pen immersio n. F or a quas i-coherent A U mo dule F , consider the sheaﬁﬁed ˇ Cech complex ˇ C ( U , F ) asso ciated with this cov er ing. In degr ee k it is the qua si-coherent A X -mo dule ˇ C ( U , F ) k = M i j i , ∗ j ∗ i F where the indexing set is taken ov er all k + 1-tuples i = ( i 0 , ..., i k ) with 0 ≤ i 0 < · · · < i k ≤ n . The diﬀerential d k : ˇ C ( U , F ) k → ˇ C ( U , F ) k +1 for the co mponent 54 MARC O SCHLICHTING i = ( i 0 , ..., i k +1 ) is given by the formula ( d k ( x )) i = k +1 X l =0 ( − 1) l j i , ∗ j ∗ i x ( i 0 ,..., ˆ i l ,...,i k +1 ) . Note tha t the co mplex ˇ C ( U , F ) is concentrated in degrees 0 , ..., n . The units of a djunction F → j i ∗ j ∗ i F deﬁne a ma p F → ˇ C ( U , F ) 0 = L n i =0 j i ∗ j ∗ i F int o the degree zero part of the ˇ Cech c omplex with d 0 ( F ) = 0, and thus a map of complexes of qua si-coherent A X -mo dules F → ˇ C ( U , F ) . This map is a quasi- isomorphism for any quasi-coherent A X -mo dule F as can b e chec k ed by restricting the map to the o pen subsets U i of the cov er o f U . Since, by a ssumption, for every k + 1-tuple, i = ( i 0 , ..., i k ), the op en inclusion j ◦ j i : U i ⊂ X is an aﬃne map, the functor j ∗ ˇ C ( U ) : Qcoh( A X ) → Ch Qcoh( A X ) : F 7→ j ∗ ˇ C ( U , F ) is exact. T ak ing total complexes, this functor extends to a functor on all co mplexes j ∗ T ot ˇ C ( U ) : Ch Qco h( A X ) → Ch Q coh( A X ) : F 7→ j ∗ T ot ˇ C ( U , F ) = T ot j ∗ ˇ C ( U , F ) which preserves qua si-isomorphisms as it is exa ct and sends acyclics to acy clics. This functor is equipped with a natural q uasi-isomorphism (27) F ∼ − → T ot ˇ C ( U , F ) . Therefore, it represents the right derived functor R j ∗ of j ∗ , that is, Rj ∗ = j ∗ T ot ˇ C ( U ) : D Qcoh( A U ) → D Qco h( A X ) . 9.14. Lemma. L et X b e a scheme with an ample family of line bu nd les, j : U ⊂ X a quasi-c omp act op en subscheme, Z ⊂ X a close d subset with qu asi-c omp act op en c omplement X − Z such that Z ⊂ U , then we have an e quivalenc e of triangulate d c ate gories j ∗ : D Z Qcoh( A X ) ≃ − → D Z Qcoh( A U ) with inverse the functor Rj ∗ . Pr o of. W e ﬁr st check that Rj ∗ preserves cohomo logical supp ort. Denote by j U : U − Z ⊂ U , j X : X − Z ⊂ X and j Z : U − Z ⊂ X − Z the corr esponding o pen immersions, and note that the canonical m ap j ∗ X j ∗ M → j Z ∗ j ∗ U M is an isomorphism for every quasi-coherent A U -mo dule M . By the existence of an a mple f amily of line bundles on X , we can choose a ﬁnite op en cov er U = { U 0 , ..., U n } o f U such that all inclusions U i 0 ∩ ... ∩ U i k ⊂ X are aﬃne maps. F or a complex F o f quasi-coher en t A U -mo dules, we have j ∗ X Rj ∗ F = j ∗ X j ∗ T ot ˇ C ( U , F ) = j Z ∗ j ∗ U T ot ˇ C ( U , F ) = j Z ∗ T ot ˇ C ( U − Z , j ∗ U F ) where U − Z is the co v er { U 0 − Z , ..., U n − Z } of U − Z . As pull-backs of aﬃne maps, all inclusions ( U i 0 − Z ) ∩ ... ∩ ( U i k − Z ) ⊂ X − Z are also a ﬃne maps, so that the functor j Z ∗ T ot ˇ C ( U − Z ) repr esen ts R j Z ∗ , and we obtain a natural isomor phism of functor s j ∗ X ◦ Rj ∗ ∼ = − → Rj Z ∗ ◦ j ∗ U . In pa rticular, Rj ∗ sends D Z Qcoh( A U ) int o D Z Qcoh( A X ). F or the pro of of the lemma, note t hat the comp osition j ∗ Rj ∗ = j ∗ j ∗ T ot ˇ C ( U ) = T ot ˇ C ( U ) is naturally qua si-isomorphic to the identit y functor via the map (27). F urthermor e, the unit of adjunction F → R j ∗ ◦ j ∗ ( F ), which is adjoint to the map (27) applied to j ∗ F , is a quasi-is omorphism for F ∈ D Z Qcoh( A X ) since b oth THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 55 complexes hav e cohomolog ical suppo rt in Z , so that we can ch eck this prop erty by restricting the map to U , wher e it is the quas i-isomorphism (27).  9.15. Prop osition. L et X b e a scheme with an ample family of line-bund les. L et Z ⊂ X b e a close d subscheme with quasi-c omp act op en c omplement X − Z . L et j : U ⊂ X b e a quasi-c omp act op en su bscheme. L et A X b e a quasi-c oher en t s he af of O X -algebr as. Then the fol lowing hold. (a) R est riction of ve ctor bund les induc es a ful ly faithful triangle functor j ∗ : D sPerf ( A X on Z, U - quis) ֒ → D sPerf ( A U on Z ∩ U, U - quis) . (b) The triangulate d c ate gory D Z Qcoh( A X ) is c omp actly gener ate d and t he tri- angle fu n ctor D sPerf ( A X on Z , quis) → D Z Qcoh( A X ) induc es an e quiv- alenc e of D sPerf ( A X on Z , quis) with the ful l triangulate d sub c ate gory of c omp act obje cts in D Z Qcoh( A X ) . (c) I f Z ⊂ U , then r estriction of ve ctor bund les induc es an e quivalenc e of t ri- angulate d c ate gories j ∗ : D sPerf ( A X on Z , quis) ≃ → D sPerf ( A U on Z , quis) . (d) The triangle functor in (a) is c oﬁnal. Pr o of. The functor in (a) is clearly c onserv ative. It is full by the fo llo wing ar gumen t. Let A and B b e strictly p erfect complexes of A X -mo dules with supp ort in Z , and let j ∗ A ∼ ← E → j ∗ B b e a diagr am in sP erf ( A U on U ∩ Z ) r epresenting a map f : j ∗ A → j ∗ B in D c ( A U on Z ∩ U, U - quis) where E ∼ → j ∗ A is a U - quasi- isomorphism. Let M b e the pull-back of j ∗ E → j ∗ j ∗ A and the U -isomo rphism A → j ∗ j ∗ A . The induced maps M → j ∗ E and M → A a re U -isomorphism and U -quas i-isomorphism, respectively . By lemma 9.7 with A = sPerf ( A X on Z ), ther e is a A 0 ∈ A and a U -qua si-isomorphism A 0 → M . By lemma 9.6, there is a map A 0 ⊗ K ( s n ) → B such that the tw o ma ps A 0 ⊗ K ( s n ) → A 0 → M → j ∗ E → j ∗ j ∗ B and A 0 ⊗ K ( s n ) → B → j ∗ j ∗ B co incide. If follows that the ma p f : j ∗ A → j ∗ B is the image of the map in D c ( A X on Z , U - quis) whic h is represented by the diagram A ∼ ← A 0 ⊗ K ( s n ) → B . Therefor e, the functor in (a) is full. An y conser v a tiv e a nd full triangle functor is faithful, hence the tr iangle functor in (a) is fully fa ithful. It follows fro m prop osition 9 .12 and Neeman’s theor em 9.11 (a) that the functor in (a) is c oﬁnal for Z = X since in this cas e b oth c ategories contain as a coﬁnal sub category the triangula ted catego ry generated by A X ⊗ L wher e L runs throug h the line bundles on X . This shows part (d) when Z = X . In orde r to prove (b), write R for the compactly g enerated triangulated ca te- gory D Q coh( A X ) with categor y o f compact ob jects R c = D sPerf ( A X , quis), see 9.12. Let S ⊂ R b e the full triangulated sub category clo sed under the formatio n of co products in R which is generated b y the set S = sPerf ( A X on Z , quis) ⊂ R c of c ompact ob jects. By par t (d) for Z = X prov ed ab ov e and prop osition 9.12, we hav e a co ﬁnal inclusio n R c / S c → D c Qcoh( A U ). By Neeman’s theorem 9.1 1 (b) and (c) , this implies that the functor R / S → D Qcoh( A U ) is an equiv a lence. In particular S is the kernel category of the functor D Qcoh( A X ) → D Qcoh( A U ), so that S = D Z Qcoh( A U ) is compactly genera ted by D s P erf ( A X on Z , quis). Since D sPerf ( A X , quis) = D c Qcoh( A X ) is ide mpotent complete, its epaisse sub category D sPerf ( A X on Z , quis) is also idempo ten t complete, so that we have the identiﬁ- cation D sPerf ( A X on Z , quis) = D c Z Qcoh( A X ), by Neeman’s theorem 9.1 1 (a). 56 MARC O SCHLICHTING In vie w o f (b), the map in (c) is the r estriction to compact ob jects of the equiv- alence of lemma 9.14. It is therefore a lso a n equiv a lence. F or the pro of of (d), w e simplify notation by writing D c ( A X on Z , w ) for the triangulated catego ry D sPerf ( A X on Z, w ). Let V = U ∪ ( X − Z ) and c onsider the commutativ e diagra m of triangulated catego ries D c ( A X on Z , V - quis) / /   D c ( A X , V - quis) / /   D c ( A X , X − Z - quis)   D c ( A V on Z ∩ U, V - quis) / / D c ( A V , V - quis) / / D c ( A V , X − Z - quis) in which all vertical functors ar e fully faithful, by (a), and the middle and the r igh t vertical functors ar e coﬁnal since all four categ ories hav e as co ﬁnal subca tegory the triangulated ca tegory generated by A X ⊗ L , where L runs through the line-bundles on X , by pro positio n 9.12. The refore, the right t wo vertical functors are equiv a- lences a fter idemp otent co mpletion. Since the t wo left triangulated categor ies are the “k ernel categories” of the tw o righ t ho rizont al functors , and this proper t y is pre- served under idemp oten t c ompletion, the left vertical functor is also an eq uiv a lence after idempo ten t c ompletion. Thus, the left vertical functor is coﬁnal. The functor D c ( A X on Z , U - quis) ֒ → D c ( A U on Z ∩ U, U - quis) in (a) ca n be ident iﬁed with the left vertical functor in the diagram since U -q uasi-isomorphisms are V - quasi-isomorphis ms for complexes of A X -mo dules cohomologically supported in Z , and sinc e the functor D c ( A V on Z ∩ U, V - quis) → D c ( A U on Z ∩ U, U - quis) is a n equiv alence, by (b).  9.16. Corollary. Le t X b e a scheme which has an ample family of line-bund les, let Z ⊂ X b e a close d su bset with qu asi-c omp act op en c omplement X − Z , and let j : U ⊂ X b e a quasi-c omp act op en subscheme. L et M b e a quasi-c oher ent A X -mo dule such that j ∗ M is strictly p erfe ct on U and has c ohomolo gic al supp ort in Z ∩ U . If the class [ j ∗ M ] ∈ K 0 ( A U on Z ∩ U ) is in the image of the map K 0 ( A X on Z ) → K 0 ( A U on Z ∩ U ) , then ther e is a U -quasi-isomorphism A ∃ U − quis / / M with A a strictly p erfe ct c omplex of A X -mo dules which has c ohomol o gic al supp ort in Z . Pr o of. W e star t with a standa rd fact a bout K 0 of triangula ted ca tegories. Let T 0 ⊂ T 1 be a (fully faithful and) coﬁnal functor betw een triangula ted categor ies. Then an o b ject T of T 1 is isomo rphic to an ob ject of T 0 if and only if its class [ T ] ∈ K 0 ( T 1 ) is in the image of K 0 ( T 0 ) → K 0 ( T 1 ). This is be cause the cokernel of K 0 ( T 0 ) → K 0 ( T 1 ) can b e iden tiﬁed with the quotien t monoid of the abe lian monoid of isomor phism clas ses o f ob jects in T 1 under direct sum mo dulo the submonoid o f isomorphism class es of ob jects in T 0 , so that an ob ject of T 1 deﬁnes the zero class in the cokernel if and only if it is stably in T 0 . But for triang ulated c ategories, an ob ject is stably in T 0 iﬀ it is isomorphic to an ob ject in T 0 . F or the pr oo f of the corollar y , we apply this argument to the inclusion in pr opo- sition 9.1 5 (a) which is coﬁnal, by 9.15 (d) . W e see that j ∗ M is isomo rphic in D c ( A U on Z ∩ U, U - quis) to an ob ject j ∗ B , where B is a per fect co mplex of A X - mo dules with cohomological suppor t in Z . I t follo ws that there is a zig -zag of THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 57 U -quas i-isomorphisms j ∗ M ∼ ← F ∼ → j ∗ B . Let P b e the pull-back of j ∗ F → j ∗ j ∗ B along the U -isomor phism B → j ∗ j ∗ B . Then P → j ∗ F is a U -is omorphism, and it follows that P → B is a U -quasi- isomorphism. By lemma 9.7 with A the sub- category o f those strictly p erfect complexe s whic h are cohomolo gically supp orted in Z , ther e is a U -quasi-is omorphism B ′ → P with B ′ strictly per fect and coho- mologically supp orted in Z . Since X has a n a mple family of line-bundles a nd U is quasi-c ompact, we can ch o ose line-bundles L i and globa l sections s i ∈ Γ( X , L i ), i = 1 , ..., l such that the set of non-v anishing lo ci X s i , i = 1 , ..., l is an aﬃne op en cov er of U . By Le mma 9 .6, we ca n ﬁnd a n l -tuple o f nega tiv e integers n suc h that the comp osition of U -quasi- isomorphisms A = B ′ ⊗ K ( s ) → B ′ → j ∗ j ∗ M lifts to M .  9.17. Prop osition. L et X b e a scheme which has an ample family of line bund les, let Z ⊂ X b e a close d subscheme with quasi-c omp act op en c omplement, and let j : U ⊂ X b e a quasi-c omp act op en subscheme. L et A X b e a quasi-c oher ent O X - algebr a with involution. Then for any line-bund le L on X and any int e ger n ∈ Z , r estriction of A X ve ctor bund les t o U deﬁnes non-singular exact form functors ( sPerf ( A X on Z ) , U - quis , ♯ n L ) − → ( sPerf ( A U on U ∩ Z ) , U - quis , ♯ n j ∗ L ) which induc e isomorphisms on higher Gr othendie ck-Witt gr oups GW i for i ≥ 1 and a m onomorphism for GW 0 . If, mor e over, we have Z ⊂ U , then the form functors induc e isomorphisms for al l higher Gr othendie ck-Witt gr oups GW i , i ≥ 0 . Pr o of. Let sPerf K 0 ( A U on U ∩ Z ) ⊂ sPerf ( A U on U ∩ Z ) b e the full subc ategory of thos e strictly p erfect complexes of A U -mo dules with cohomologic al s upport in U ∩ Z which hav e class in the image o f K 0 ( A X on Z ) → K 0 ( A U on Z ∩ U ). By coﬁnality 4 .10, the dualit y preserving inclusion (28) sPerf K 0 ( A U on U ∩ Z ) → sP erf ( A U on U ∩ Z ) of ex act categories with weak equiv alenc es the U -quasi- isomorphisms and duality ♯ n j ∗ L induces maps on higher Grothendieck-Witt gr oups GW i which are isomor- phisms for i ≥ 1 and a monomo rphism for i = 0. Restr iction of vector-bundles deﬁnes a no n-singular ex act form functor (29) ( sPerf ( A X on Z ) , U - quis ) → ( sPerf K 0 ( A U on U ∩ Z ) , U - quis ) which induces a homotopy equiv a lence of Grothendieck-Witt spaces b y theore m 5.1, where 5.1 (c) follows from co rollary 9.16 and lemma 5.5; 5.1 (e) and (f ) ar e proved in lemma 9.6; the remaining h yp othesis of theorem 5.1 b eing trivially satisﬁed. If Z ⊂ U , then K 0 ( A X on Z ) = K 0 ( A U on Z ∩ U ), by prop osition 9.15 (c), so that (28) is the identit y inclusion, and the map (29 ), which induces a homotopy equiv alence of Grothendieck-Witt spaces, is the map in the prop osition.  Pr o of of the or em 9.2. By the “change-of-weak-equiv alence theo rem” 4 .2, the se- quence ( sPerf ( A X on Z ) , q uis , ♯ n L ) → ( sPerf ( A X ) , quis , ♯ n L ) → ( sPerf ( A X ) , U - quis , ♯ n L ) 58 MARC O SCHLICHTING induces a ho motop y ﬁbration of Gr othendiec k-Witt spaces. By pro position 9 .17, the form functor ( sPerf ( A X ) , U - quis , ♯ n L ) → ( sPerf ( A U ) , quis , ♯ n j ∗ L ) induces isomorphisms on GW i for i ≥ 1 and a mono morphism for i = 0.  Pr o of of the or em 9.3. The theorem is a sp ecial case of prop osition 9.17.  10. Extension to nega tive G r othendieck-Witt gr oups F or an op en s ubsc heme U ⊂ X , the res triction map GW 0 ( X ) → GW 0 ( U ) is no t surjective, in general, not even if X is regular. The pur pose of this section is to extend the long exact s equence asso ciated with the homotopy ﬁbration of theo rem 9.2 to negative degrees. Theor ems 9.2 and 9.3 will b e extended to a ﬁbr ation and a weak eq uiv a lence of no n-connective sp ectra. 10.1. Cone and s usp ension of A X . The c one ring is the ring C o f inﬁnite ma- trices ( a i,j ) i,j ∈ N with co eﬃcients a i,j in Z for which each row a nd e ac h c olumn has only ﬁnitely many no n-zero e n tries. T ransp osition of matrices t ( a i,j ) = ( a j,i ) makes C into a ring with inv olution. As a Z -mo dule C is tor sion free , hence ﬂat. The su sp ension ring S is the factor ring of C by the tw o sided ideal M ∞ ⊂ C of those matrice s which have only ﬁnitely man y non-zero entries. T ransp osition also makes S in to a ring with inv o lution such that the quotient map C ։ S is a map of rings with in v olution. F o r a nother des cription o f the suspensio n ring S , consider the ma trices e n ∈ C , n ∈ N , with entries ( e n ) i,j = 1 for i = j ≥ n and zero otherwise. They are symmetric idemp oten ts, i.e., t e n = e n = e 2 n , a nd they form a m ultiplicativ e subset of C which s atisﬁes the Øre condition, that is, the m ultiplicative subset satisﬁes the axioms for a calculus o f fractions . One chec ks that the quotient map C ։ S identiﬁes the susp ension ring S with the lo calization of the cone ring C with resp ect to the elements e n ∈ C , n ∈ N . In particular , the susp ension r ing S is also a ﬂat Z -mo dule. Let X b e a quasi- compact and quasi-separated scheme. F or a quasi-coher en t sheaf A X of O X -algebra s, write C A X and S A X for the quasi-c oherent sheaves of O X -algebra s ass ociated with the presheaves C ⊗ Z A X and S ⊗ Z A X . On quasi- compact o pen subsets U ⊂ X , we hav e ( C A X )( U ) = C ⊗ Z A X ( U ) and S A X = S ⊗ Z A X ( U ), b y ﬂatness of C and S . If A X is a sheaf of algebras with inv olutions, then the inv olutions on C and on S ma k e C A X and S A X int o sheav es o f O X - algebras with in volution. Let ǫ = 1 − e 1 ∈ C b e the sy mmetric idemp otent with entries 1 at (0 , 0) and zero other wise. The image C ǫ o f the r igh t multiplication map × ǫ : C → C is a ﬁnitely gene rated pro jective left C - module. It is equipp ed with a symmetric form ϕ : C ǫ ⊗ Z ( C ǫ ) op → C : x ⊗ y op 7→ x · t y . The ide mpotent ǫ makes ( C ǫ, ϕ ) into a dire ct factor of the unit symmetric form ( C, µ ), see 7.8 7.8. Therefore, tenso r pro duct ( C ǫ, ϕ ) ⊗ Z ? deﬁnes a non-sing ular exact form functor ι : (sPerf ( A X ) , ♯ n L ) → (sPerf ( C A X ) , ♯ n L ) : V 7→ C ǫ ⊗ Z V . Since S is a ﬂat C -algebra , the quotient map C → S induces an exact functor ρ : C A X -Mo d → S A X -Mo d : M 7→ S ⊗ C M on categor ies of mo dules which sends THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 59 vector bundles to vector bundles. The tw o functor s ι and ρ yield a s equence of non-singular exact form functors (30) (sPerf ( A X ) , ♯ n L ) ι − → (sPerf ( C A X ) , ♯ n L ) ρ − → (sPerf ( S A X ) , ♯ n L ) . The functors satisfy ρ ◦ ι = 0 b ecause S ⊗ C C ǫ = im ( × ǫ : S → S ) = 0 as 0 = ǫ ∈ S . The following theorem will allow us to extend the results of § 9 to nega tiv e Grothendieck-Witt gro ups. 10.2. Theorem. L et X b e a scheme with an ample family of line-bund les, let Z ⊂ X b e a close d su bset with quasi-c omp act op en c omplement X − Z , and let A X b e a quasi-c oher en t O X -algebr a with involution. Then for any line bu n d le L on X , and any inte ger n ∈ Z , the s e quenc e (30) induc es a homotopy ﬁbr ation of Gr othendie ck- Witt s p ac es with c ont r actible total sp ac e GW n ( A X on Z , L ) → GW n ( C A X on Z , L ) → GW n ( S A X on Z , L ) . The pro of of theor em 10.2 will o ccup y sec tions 10 .3 to 10.9. 10.3. Lemma. The functor ι in ( 30 ) is ful ly faithful. Pr o of. The image ǫC o f the left m ultiplica tion map ǫ × : C → C is a r igh t C -mo dule. W e have a Z -bimo dule isomorphis m η : Z → ǫC ⊗ C C ǫ : 1 7→ ǫ ⊗ C ǫ = 1 ⊗ ǫ = ǫ ⊗ 1 and a C -bimo dule ma p µ : C ǫ ⊗ Z ǫC → C : Aǫ ⊗ ǫB 7→ AǫB such that the comp ositions C ǫ ∼ = C ǫ ⊗ Z Z id ⊗ η − → C ǫ ⊗ Z ǫC ⊗ C C ǫ µ ⊗ id − → C ⊗ C C ǫ ∼ = C ǫ and ǫC ∼ = Z ⊗ Z ǫC η ⊗ id − → ǫC ⊗ C C ǫ ⊗ Z ǫC id ⊗ µ − → ǫC ⊗ C C ∼ = ǫC are the iden tit y maps. It follo ws that η and µ deﬁne unit and c ounit of an adjunction betw een the functors A X -Mo d → C A X -Mo d : M 7→ C ǫ ⊗ Z M and C A X -Mo d → A X -Mo d : N 7→ ǫC ⊗ C N . Since the unit η is an isomorphism, the ﬁrst functor is fully faithful. In particular , ι is fully faithful.  10.4. Prop osition. The se quenc e of t riangulate d c ate gories D b V ect( A X ) ι − → D b V ect( C A X ) ρ − → D b V ect( S A X ) is exact up to dir e ct factors. Pr o of. The multiplication map µ : C ǫ ⊗ Z ǫC → C factors thr ough M ∞ ⊂ C and induces an iso morphism µ : C ǫ ⊗ Z ǫC → M ∞ (it is a ﬁltered colimit of iso morphisms of ﬁnitely generated free Z -mo dules). The ex act functors Qcoh( A X ) ι − → Qcoh( C A X ) ρ − → Qcoh( S A X ) hav e exact r igh t adjoints κ : Qcoh( C A X ) → Qco h( A X ) : M 7→ ǫC ⊗ C M and σ : Qco h ( S A X ) → Qcoh( C A X ) : N 7→ N such that for a le ft C A X -mo dule M the adjunt ion maps ικ → id a nd id → σ ρ are part of a functorial exact sequence (31) 0 → ικM → M → σ ρM → 0 which is the tensor pro duct (over C ) of M with the exact sequence of ﬂat C -mo dules 0 → M ∞ → C → S → 0. It follows that the sequenc e of tria ngulated categ ories D Qcoh( A X ) ι − → D Qcoh( C A X ) ρ − → D Qcoh( S A X ) 60 MARC O SCHLICHTING is exact as κ a nd ρ induce r igh t adjoint functor s o n derived ca tegories, and (31) induces a functorial dis tinguished triang le for every ob ject o f D Qcoh( C A X ). By prop osition 9.1 2 , these triangula ted categories a re compactly generated. Since ι and ρ pr eserve co mpact ob jects, the asso ciated s equence of c ompact ob jects – which is the sequence in prop osition 10.4 – is exact up to factor s 9.11.  Let sPerf S ( C A X ) ⊂ sPerf ( C A X ) be the full sub category of those complexes V for which S ⊗ C V is a cyclic. This sub category is clo sed under the inv olution ♯ n L , so that sP erf S ( C A X ) inherits the structure of a n exact category with weak equiv alences a nd duality from sPerf ( C A X ). Since ρι = 0 , ι induces a non-singula r exact form functor ι : sPerf ( A X ) → sPerf S ( C A X ). 10.5. Prop osition. F or any line bund le L on X , and any n ∈ Z , the functor ι induc es a homotopy e quivalenc e GW n (sPerf ( A X ) , quis , L ) ∼ − → GW n (sPerf S ( C A X ) , quis , L ) . Pr o of. The pro of is a cons equence of theorem 5.1 (or of lemma 5.6). Since ι is fully faithful, conditions (e) a nd (f ) ar e satisﬁed. Since ι induces a fully faithful functor on derived ca tegories, by prop osition 10.4, condition (b) is also satisﬁed. The only non-trivial condition to check is (c). By lemma 5.5, we only need to show that for every M ∈ s P erf S ( C A X ) there is a n A ∈ s P erf ( A X ) and a quasi-isomo rphism C ǫ ⊗ Z A → M . Let M b e a stric tly p erfect complex of C A X mo dules with S ⊗ C M a cyclic. By prop osition 1 0.4, there is a zigzag of quasi-isomo rphisms C ǫ ⊗ Z B ← N → M in sPerf S ( C A X ) with B ∈ sPerf ( A X ). Since N ∈ sPerf S ( C A X ), prop osition 10.4 implies that the counit of adjunction C ǫ ⊗ Z ǫC ⊗ C N → N is a quasi- isomorphism. W e apply lemma 9.7 with C A X in pla ce of A X and U = X , A = sPerf ( A X ) to the quasi-isomo rphism ǫC ⊗ C N → ǫC ⊗ C C ǫ ⊗ Z B ∼ = B , and obtain a strictly perfect complex A of A X -mo dules and a qua si-isomorphism A → ǫC ⊗ C N . Finally , the comp osition C ǫ ⊗ Z A → C ǫ ⊗ Z ǫC ⊗ C N = M ∞ ⊗ C N → N → M is a quasi- isomorphism.  F or a quasi- coherent O X -algebra A X , call an A X -mo dule M qu asi-fr e e if it is isomorphic to a ﬁ nite direct sum L i A X ⊗ L i of A X -mo dules of the form A X ⊗ L i for some line bundles L i on X . Note that a q uasi-free A X -mo dule is a vector bundle. 10.6. Lemma. L et X b e a quasi-c omp act and quasi-sep ar ate d s cheme, and let A X b e a quasi-c oher ent she af of O X -algebr as. L et A, M b e quasi-c oher ent C A X -mo dules with A quasi-fr e e. Then the fol lowing hold. (a) F or every map f : ρA → ρM of S A X -mo dules, ther e ar e maps s : B → A and g : B → M of C A X -mo dules with B quasi-fr e e su ch that f ◦ ρ ( s ) = ρ ( g ) and ρ ( s ) an isomorphism. (b) F or any two maps f , g : A → M of C A X -mo dules such that ρ ( f ) = ρ ( g ) ther e is a map s : B → A of of quasi-fr e e C A X -mo dules such that f ◦ s = g ◦ s and ρ ( s ) is an isomorphism. THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 61 Pr o of. The pro of reduces to A = C A X ⊗ L with L a line-bundle o n X . F o r such an A , the map H om C A X ( A, M ) → H om S A X ( ρS, ρM ) ca n b e iden tiﬁed with the map on global sections Γ( X , M ⊗ L − 1 ) → Γ( X , S ⊗ C M ⊗ L − 1 ) = S ⊗ C Γ( X , M ⊗ L − 1 ). This map is surjective s ince C → S is, proving (a). The map is a lso a lo calization by a calculus of fra ctions with resp ect to the set of elements e n ∈ C , n ∈ N , of 10 .1 . This s ho ws that (b) also holds.  10.7. Lemma. L et X b e a quasi-c omp act and quasi-sep ar ate d scheme, L a line bund le on X , and let A X b e a quasi-c oher ent she af of O X -algebr as with involution. Then for every n ∈ Z , t he Gr othendie ck-Witt s p ac e GW n ( C A X , L ) is c ontr actible. Pr o of (co mpare [Kar7 0 ]). W e will deﬁne a C -bimo dule M , which is ﬁnitely ge ner- ated pro jective as left C -mo dule, together with a symmetr ic form ϕ : M ⊗ C M op → C in C -Bimo d who se adjoint M → [ M op , C ] C is an is omorphism. F urthermor e, we will cons truct a n is ometry ( C, µ ) ⊥ ( M , ϕ ) ∼ = ( M , ϕ ) of sy mmetric fo rms in C -Bimo d. Therefore, tensor pro duct ( M , ϕ ) ⊗ C ? deﬁnes a non-singular ex- act form functor ( F, ϕ ) : (sPerf ( C A X ) , quis , ♯ n L ) → (sPerf ( C A X ) , quis , ♯ n L ) which satisﬁes id ⊥ ( F, ϕ ) ∼ = ( F, ϕ ), so that on higher Grothendieck-Witt gro ups w e hav e GW n i ( id ) + GW n i ( F, ϕ ) = GW n i ( F, ϕ ) which implies GW n i ( id ) = 0 , that is, GW n i ( C A X , L ) = 0, hence GW n ( C A X , L ) is contractible. T o construct ( M , ϕ ) and the bimo dule iso metry ( C, µ ) ⊥ ( M , ϕ ) ∼ = ( M , ϕ ) we choose a bijection σ : N ∼ = → N × N : n 7→ ( σ 1 ( n ) , σ 2 ( n )) and deﬁne a homomorphism of r ings with involutions σ : C → C : a 7→ σ ( a ) with σ ( a ) ij =        a σ 1 ( i ) ,σ 1 ( j ) if σ 2 ( i ) = σ 2 ( j ) 0 otherwise . The C -bimo dule M is C as a left mo dule, and ha s right m ultiplication deﬁned by M × C → M : ( x, a ) 7→ x · I ( a ). The symmetric form ϕ is the C -bimo dule ma p M ⊗ C M op → C : x ⊗ y op 7→ x · t y . Since, as a left C -mo dule, ( M , ϕ ) is just the unit symmetric for m ( C, µ ) (see 7 .8 7 .8), the adjoint M → [ M op , C ] C of ϕ is an isomorphism. In o rder to deﬁne the bimo dule isometry ( C, µ ) ⊥ ( M , ϕ ) ∼ = ( M , ϕ ), co nsider the elements γ , δ ∈ C deﬁned by γ ij =        1 if σ ( j ) = ( i, 0) 0 other wise , and δ ij =        1 if σ ( j ) = σ ( i ) + (0 , 1) 0 other wise . The ho momorphism I a nd the element s γ , δ ∈ C are related by the following iden- tities δ · t γ = 0 , γ · t γ = δ · t δ = 1 , t γ · γ + t δ · δ = 1 , a · γ = γ · I ( a ) , I ( a ) · δ = δ · I ( a ) 62 MARC O SCHLICHTING for all a ∈ C . Therefor e, the map C ⊕ M → M : ( a, x ) 7→ a · γ + x · δ is a C -bimo dule isomorphism with inv erse the ma p M → C ⊕ M : x 7→ ( x · t γ , x · t δ ). It pr eserves forms b ecause ( aγ + xδ ) · t ( bγ + y δ ) = a · t b + x · t y .  W rite sPerf 0 ( A X ) ⊂ sPerf ( A X ) for the full sub category o f those strictly p erfect complexes of A X -mo dules whic h are degree-wise quasi-free. Note that this category is closed under the duality ♯ n L . W e equip the categor y sPerf 0 ( A X ) with the degree- wise split exact s tructure. T ogether with the set o f quasi-iso morphisms of complexes of A X vector bundles, it b ecomes a categor y of complexes in the sense of Deﬁnition 5.4. 10.8. Lemma. L et X b e a scheme with an ample family of line bund les. Th e inclusion of quasi-fr e e m o dules into the c ate gory of ve ctor bu n d les induc es a (ful ly faithful) c oﬁnal triangle fun ctor D ( sPerf 0 ( A X ) , quis) ⊂ D ( sPerf ( A X ) , quis) . Mor e over, for every strictly p erfe ct c omplex M of A X -mo dules with class [ M ] in the image of the map K 0 ( sPerf 0 ( A X ) , quis) → K 0 ( sPerf ( A X ) , quis) t her e is a quasi- isomorphi sm A → M of c omplexes of A X -mo dules with A a b ounde d c omplex of quasi-fr e e mo dules. Pr o of. The tr iangle functor in the lemma is fully faithful, by lemma 9.7 with U = X and A = sPerf 0 ( A X ). It is co ﬁnal, b y Neeman’s theor em 9.11 (a) and prop osition 9.12. Let sPerf K 0 ( A X ) ⊂ sPerf ( A X ) b e the full sub category o f tho se strictly p erfect complexes of A X -mo dules M whose class [ M ] is in the image of the map K 0 ( sPerf 0 ( A X ) , quis) → K 0 ( sPerf ( A X ) , quis). Then the inclusion ( sPerf 0 ( A X ) , quis) ⊂ (sPerf K 0 ( A X ) , quis) of exact ca tegories with weak equiv a - lences induces a n equiv alence o f derived categ ories, so that another application of lemma 9.7 with U = X and A = sPerf 0 ( A X ) ﬁnishes the pro of o f the cla im.  Pr o of of the or em 10.2. By theo rem 9.2, we only need to trea t the case Z = X . In this ca se, the tota l spa ces are contractible, by lemma 1 0.7. Let sPerf K 0 ( S A X ) ⊂ sPerf ( S A X ) be the full sub category of those s trictly p er- fect complexes of S A X -mo dules E whose class [ E ] is zero in the Gro thendiec k group K 0 (sPerf ( S A X ) , quis) of S A X -vector bundles. F urthermor e, call a map f of strictly p erfect complexes of C A X -mo dules an S -qua si-isomorphism if ρ ( f ) is a quasi-isomo rphism o f complex es of S A X -mo dules. The set o f S -quasi- isomorphisms is denoted by S - q uis. Consider the comm utative diag ram of ex act categor ies with weak equiv a lences and dualit y ♯ n L induced by inclusions and the map of ring s with inv o lution C → S (sPerf 0 ( C A X ) , quis) / /   (sPerf 0 ( C A X ) , S - quis) ρ / /   (sPerf 0 ( S A X ) , quis)   (sPerf ( C A X ) , quis) / / (sPerf ( C A X ) , S - quis) ρ / / (sPerf K 0 ( S A X ) , quis) . Note tha t K 0 of all categor ies with weak equiv ale nces in the dia gram is 0 . F or the t wo left hand categor ies, this follows from lemma 10.7 a nd lemma 10.8, since for coﬁnal tr iangle functors T 0 ⊂ T , the map K 0 ( T 0 ) → K 0 ( T ) is injective. Since the left ho rizontal maps a re surjective on K 0 , the middle tw o c ategories with weak THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 63 equiv alences have trivia l Grothendieck g roup K 0 . F or the upp er right corner , v an- ishing of K 0 follows mor eo ver fr om the fa ct that its K 0 is generated by cla sses of complexes concentrated in degree 0 and the fact that every quasi-free S A X -mo dule is the image o f a q uasi-free C A X -mo dule, so that the upp er hor izon tal map is sur- jective on K 0 . Ther efore, the r igh t vertical and the lower right horizo n tal functors, which – a priori – hav e ima ges in sPerf ( S A X ), hav e indeed image in sPerf K 0 ( S A X ). W e will show that the upper right horizontal and middle and rig h t v ertical functors induce equiv alences of Grothendieck-Witt spaces (for a n y dualit y ♯ n L ), so that the low er right horizontal functor will induce an eq uiv a lence, to o. The upp er rig h t horizontal functor is a lo calization by a calculus of r igh t fr ac- tions, by lemma 10.6 and 5.12 (c). Therefore, theorem 5.11 shows that it induces a homotopy equiv alence of Grothendieck-Witt spaces. F or the r igh t vertical functor , the resolution lemma 5.6 (which w e can apply b ecause of lemma 10.8) s ho ws that it induces an equiv a lence of Grothendieck-Witt spa ces. Similarly , by lemma 10.8, for every strictly perfect complex of C A X -mo dules M , there is a bounded complex A of quasi-free C A X -mo dules and a q uasi-isomorphism A → M . A quasi-is omorphism of co mplexes of C A X -mo dules is, a fortior i, a n S -quasi-iso morphism, so that the resolution lemma applies to show that the middle v ertical functor induces an equiv- alence of Grothendieck-Witt spaces. Summariz ing, we have s ho wn that the lower right ho rizont al functor induces an equiv alence of Gr othendiec k-Witt s paces. By the “ch ange of weak equiv a lence theor em” (theor em 4.2), the seq uence of exact categor ies with weak equiv alences and duality ♯ n L ( sPerf S ( C A X ) , quis ) → ( sPerf ( C A X ) , quis ) → ( sPerf ( C A X ) , S - quis ) induces a homo top y ﬁbr ation of Gro thendiec k-Witt spaces. Using prop osition 10.5 we can replace the left hand ter m wit h ( sPerf ( A X ) , quis ). Using the equiv alence of Grothendieck-Witt spaces of the low er righ t horizontal functor above and coﬁnalit y (theorem 4.10) a pplied to the inclusion o f exa ct ca tegories with weak equiv alences and duality ( sPerf K 0 ( S A X ) , quis ) ⊂ ( sPerf ( S A X ) , quis ), w e ca n replace the righ t hand ter m in the s equence by ( sPerf ( S A X ) , quis ).  Since the to tal space in the ﬁbr ation of theorem 1 0.2 is contractible, we obtain a homotopy e quiv alence of spaces (32) GW n ( A X on Z, L ) ≃ − → Ω GW n ( S A X on Z, L ) . 10.9. Deﬁnition. Let X b e a scheme with an ample family of line-bundles, A X be a quasi-co heren t shea f of O X -algebra s with in volution, L a line bundle on X , Z ⊂ X a closed subscheme with qua si-compact open co mplemen t X − Z a nd n ∈ Z an in teger. T he Gr othendie ck-Witt sp e ctrum G W n ( A X on Z, L ) of symmetric spaces over A X with co eﬃcients in the n -th shifted line bundle L [ n ] and suppo rt in Z is the sequence GW n ( S k A X on Z, L ) , k ∈ N , of Grothendieck-Witt spaces together with the b onding maps given by the ho mo- topy equiv alence (32). As usual, if Z = X , n = 0 , A X = O X or L = O X , we omit the lab el cor resp onding to Z , n , A , or L , res pectively . 64 MARC O SCHLICHTING By constructio n, we hav e π i G W n ( A X on Z , L ) =        GW n i ( A X on Z , L ) for i ≥ 0 GW n 0 ( S − i A X on Z , L ) for i ≤ 0 . 10.10. Remark. By prop osition 8.4, there are natura l homotopy equiv alences of sp ectra G W n ( A X on Z, L ) ≃ G W n +4 ( A X on Z , L ). Finally , we a re in p osition to prov e the main theo rems o f this article. 10.11. Theorem (Lo calization). L et X b e a scheme with an ample family of line- bund les, let Z ⊂ X b e a close d subscheme with quasi-c omp act op en c omplement j : U ⊂ X , and let L a line bund le on X . L et A X b e a quasi-c oher ent she af of O X -algebr as with involut ion. Then for every n ∈ Z , t he fol lowing se qu enc e is a homotopy ﬁbr ation of Gr othendie ck-Witt sp e ctr a G W n ( A X on Z, L ) − → G W n ( A X , L ) − → G W n ( A U , j ∗ L ) . Pr o of. This is b ecause the sequences GW n ( S i A X on Z , L ) − → GW n ( S i A X , L ) − → GW n ( S i A U , j ∗ L ) are ho motop y ﬁbrations fo r i ∈ N , by theorem 9 .2.  10.12. Theorem (Za riski-excision). L et X b e a scheme with an ampl e family of line-bund les, let Z ⊂ X b e a close d subscheme with quasi-c omp act op en c omplement, let L b e a line bund le on X and let A X b e a qu asi-c oher ent she af of O X -algebr as with involution. Then for every n ∈ Z and every qu asi-c omp act op en subscheme j : V ⊂ X c ontaining Z , r estriction of ve ctor-bund les induc es a homotopy e quivalenc e of Gr othendie ck-Witt sp e ctr a G W n ( A X on Z, L ) ∼ − → G W n ( A V on Z , j ∗ L ) . Pr o of. This is b ecause the maps GW n ( S i A X on Z, L ) − → GW n ( S i A V on Z , j ∗ L ) . are ho motop y equiv alences for i ∈ N , by theorem 9.3.  10.13. Corollary (Mayer-Vietoris for op en covers). L et X = U ∪ V b e a scheme with an ample family of line-bund les which is c over e d by two op en quasi-c omp act subschemes U, V ⊂ X . L et A X b e a quasi-c oher ent O X -mo dule with involution. Le t L b e a line-bund le on X , and n ∈ Z . Then r estriction of ve ctor bund les induc es a homotopy c artesian squar e of Gr othendie ck-Witt sp e ctr a G W n ( A X , L ) / /   G W n ( A U , L )   G W n ( A V , L ) / / G W n ( A U ∩ V , L ) . THE MA YER-VIETORIS PRINCIPLE FOR GR OTHENDIECK-WITT GR OUPS 65 Pr o of. 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The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

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