Push-forwards for Witt groups of schemes

PUSH-F OR W ARDS F OR WITT GR OUPS OF SCHEMES BAPTISTE CALM ` ES AND JENS HORNBOSTEL Abstract. Using suitable closed symmetric monoidal structures on derived categories of sc hemes, as we ll as adjunctions of the type (L f ∗ , R f ∗ ) and (R f ∗ , f ! ) ( i.e. Grothendiec k duality theory), w e define push-for wa rds for coheren t Witt groups along prop er morphisms betw een separated no etherian sc hemes. W e also establish fundament al theorems for these push-forwards ( e.g. base c hange and pro jection formula) and provide some computations. Contents Int ro duction 1 1. Closed symmetric mono idal categor ies 5 2. Witt gr oups 7 3. The functors L f ∗ , R f ∗ and f ! 9 4. Pull-back a nd push-forward fo r Witt gr oups 11 5. Prop erties 12 6. Reformulations in specia l cases 15 6.1. Relative dualizing sheaf 15 6.2. Smo oth schemes o ver a base 16 7. Examples 16 7.1. Finite field extensions 17 7.2. Regular embeddings 17 7.3. Pro jective spaces 20 Appendix A. q-flat and q-injective r esolutions 21 References 22 Introduction Push-forwards, also known as transfers o r norm maps, exist for many cohomol- ogy theor ies ov er schemes, e.g. for K -theory , (hig her) Chow g roups and alg e braic cob ordism. They are undoubtedly a use ful to ol for under standing and computing those cohomo logy theories. The present article is ab out the construction o f such push-forward maps for the coherent Witt g roups of sc hemes defined in the seminal work of Balmer 1 . A reader fa milia r with co homology theories mig ht think that constructing a push-for ward is probably straightf orward. He (or she) should b e warned: Witt groups are not an o riented cohomology theo ry . In particular , push- forwards are, in some se ns e, only conditionally defined. F or e x ample, when X and Y are connected noether ia n sc hemes of finite Krull dimension, smo oth ov er a field, 1 The m odern definition of Witt groups using triangulated categories with dualities [ 3 ] can be applied either to the derive d category of complexes of l o cally fr ee shea ves to obtain “lo cally free” Witt gr oups or to the derive d category of complexes with coherent cohomology to obtain “coheren t” Witt groups. As with K -theory , i t is the latter that is naturally co v ariant along prop er morphisms, as we prov e in this article. All schemes considered are ov er Z [1 / 2] so that the derived categories inv olv ed are Z [1 / 2]-linear and the m ac hinery of triangular Witt groups applies. 1 2 BAPTISTE CALM ` ES AND JENS HORNBOSTEL the Witt gro ups dep end on a line bundle L used to define the duality and the push-forward tak es the form (see Theorem 6.7 ) W i +dim X ( X, ω X ⊗ f ∗ L ) → W i +dim Y ( Y , ω Y ⊗ L ) where ω is the canonical bundle (the highest nontrivial ex terior power of the cotan- gent bundle) and L is an arbitr ary line bundle on Y . In particular, if w e pic k a line bundle K ov er X , there is no push-forward starting from W i ( X, K ) if K is not isomo rphic to ω X ⊗ f ∗ L for some L (up to a square M ⊗ 2 , as W i ( Y , K ) ∼ = W i ( Y , K ⊗ M ⊗ 2 ) so only the clas s of K in P ic( X ) / 2 really matters). This fundamen- tal difference with or iented cohomolo gy theories, where the push-forward is alwa ys po ssible, significa nt ly c hanges classica l computations, as one sees e.g. in [ 18 ], [ 39 ], [ 5 ]. The gr oups W dim X − i ( X, ω X ) can be co ns idered as a homolo gy theory a nalo- gous to a no n orient ed complex homology theory in top o lo gy , but the c o nstruction of the push-forward here relies o n triangulated monoida l metho ds. Besides this article and its precurs ors on regula r schemes [ 10 ] and [ 11 ], there are already sev eral articles av a ila ble on the co nstruction of push-fo rwards in special cases. In [ 17 ], Gille defined push-for wards along finite morphisms in the affine case. His approach is quite elementary in the sense tha t he uses direct computations inv olving explicit injectiv e reso lutions etc. It is us eful to get a hand on concrete forms. In [ 31 ] and [ 32 ], Nenashev adapts the oriented co homology techniques o f Panin and Smirnov to the non-oriented case of lo cally free Witt gro ups. He thus obtains push-fo rwards a long pr o jective mo rphisms b etw een smo o th qua si-pro jective v a rieties ov er fields. Still another approach using stable A 1 -representability o f Witt groups can b e fo und in [ 24 ]. W e understand tha t there is also some unpublished work of C. W alter on this sub ject. Our approa ch is different, and it applies to a m uch larg er class of situatio ns: it uses derived functors and Grothendieck duality , so the dualities that appear are canonical and do not dep end on choices as the other constr uctions mentioned ab ov e. If necessa r y , choices can b e made in o r der to compare our co nstructions with others in the sp e cial cases where the la tter are defined. F undamental pro pe r ties s uch a s base change ar e pr oved in a simple a nd conceptual wa y , and w e furthermore obtain the full generality of sing ular sc hemes. An e x ample o f how those prop erties can be used for very concrete computations can b e found in the co mputation of Balmer and the first author of the Witt group of Grassmann v arieties [ 5 ]. Let us now explain wh y w e use triangula ted monoidal metho ds, even thoug h there is no mention of a tensor pro duct in the definition of Witt gro ups of trian- gulated categ ories. In fact, the pro of of ma n y r esults amounts to verifying tha t a certain num ber of diagrams of morphisms of functor s s uch as ( 2 ) b elow ar e com- m utative. It might b e p oss ible to c heck this by hand in every concrete situation; how ever, it would b e extremely painful: try it for exa mple in the simple cas e o f a regular c losed immers ion. Hence, some k ind o f sys tematic method is needed. Our solution to this problem is the use of a conv enient setting inv olving a tensor pro d- uct, an adjoint internal Hom, functors of the t yp e f ∗ , f ∗ and f ! and the adjunction relationships b etw een them, that is so me v ariant o f the s o-called Gro thendieck six functors forma lism in an arbitra ry tr ia ngulated categ ory . In this setting, we hav e shown in [ 12 ] that all the necessar y diagrams co mm ute, whereas this article ex- ploits the existence of this structure on v arious concr ete triangulated c ategories . Here is a brief sket ch of what is involv ed: Witt gro ups ar e defined for triangulated categorie s C equipped with a duality , i.e. with a co nt rav a r iant endofunctor D on C together with a bidual isomorphism of functors Id → D 2 = D ◦ D satisfying D (  A ) ◦  DA = Id DA for all ob jects A o f C . A morphism b et ween Witt groups is naturally induced by an exact functor F : C 1 → C 2 (bo th triang ulated categor ies PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 3 with dualities res p. ( D 1 ,  1 ) a nd ( D 2 ,  2 )) equipped with an isomorphis m o f exa ct functors φ : F D 1 → D 2 F which explains how F “c o mmu tes” w ith the dualities. The fact that the is o morphism φ is not the identit y requir es the analysis of its in- teractions with the other morphis ms of functors inv olved. It is the cen tral pro blem to solve when proving the main theorems. T o start with, this mo rphism φ should make the diagram (1) F F  1 / /  2 F   F D 1 D 1 φD 1   D 2 D 2 F D 2 φ / / D 2 F D 1 commutativ e. In [ 12 ], we discuss such morphisms of functors and diagra ms in the setting of closed symmetric mo no idal categ ories. More precisely , le t C 1 and C 2 be closed symmetric monoidal categ ories, with tensor pr o duct denoted by ⊗ and int ernal Hom denoted by [ − , − ]. Given an o b ject K , the functor D K := [ − , K ] together with the canonica l natura l transfor mation  K : Id → D 2 K defines a weak duality functor. Starting with an exact functor f ∗ : C 1 → C 2 which ha s a left adjoint f ∗ (whic h is monoidal) and a right a djo int f ! , there is a natural tra nsformation ζ : f ∗ D f ! K → D K f ∗ such that the diagra m ( 1 ), which becomes (2) f ∗ f ∗  f ! K / /  K f ∗   f ∗ D f ! K D f ! K ζ D f ! K   D K D K f ∗ D f ! K ζ / / D f ! K f ∗ D K commutes, as shown in [ 12 ]. Therefore, provided  K ,  f ! K and ζ a re isomorphis ms, f ∗ induces a mor phism of Witt groups W( C 1 , D f ! K ,  f ! K ) → W( C 2 , D K ,  K ) The present article is a des c ription of how to apply this abs tract close d monoidal setting to well-chosen derived categories of schemes, with the derived functors L f ∗ , R f ∗ and its rig ht adjoint f ! constructed by Grothendieck duality theory . The main re s ult of this article is the definition of a push-forward alo ng a prop er morphism f : X → Y o f s eparated no e ther ian s chemes. In its most genera l form (Theorem 4.4 ), this push- fo rward is a morphism W i ( X, f ! K ) f ∗ / / W i ( Y , K ) where K is a dualiz ing complex on Y . This push-forward is induced b y the derived functor R f ∗ and a suitable mor phism of functors ζ K : R f ∗ D f ! K → D K R f ∗ . This means that a form on a co mplex A for the duality D f ! K is se nt to a form on the complex R f ∗ A for the duality D K . W e further pr ov e that this push-forward r esp ects comp osition (Theore m 5.3 ). Similarly , for mo rphisms of finite to r-dimension f we define a pull-back (Theorem 4.1 ), that is a morphism W i ( Y , K ) → W i ( X, L f ∗ K ) resp ecting comp ositio n (Theorem 5.2 ). W e a ls o prove a flat base change theorem ( 5.5 ) re la ting the push-for ward a nd the pull-back in cartes ia n diag rams and a pr o jection form ula in the case of regular 4 BAPTISTE CALM ` ES AND JENS HORNBOSTEL schemes (Theorem 5.7 ). Some ex plicit computations o f transfers are provided in the last sectio n. W e assume that schemes are separ ated and no etherian for the following technical reasons : quasi-compac t and separa ted are necessa ry to have an equiv a le nc e betw een the derived catego ry of quasi-co herent sheav es D (Qcoh( X )) and the sub categor y D qc ( X ) of co mplexes with quas i-coherent homology in the derived categor y of all sheav es. No etheria n is used to ensure that the injectives in Qcoh( X ) remain injec- tive in the categ o ry of all O X -mo dules. W or king witho ut those assumptions would probably r equire sig nificant improv ements in the theory of Grothendieck duality; this is be yond the scop e of this ar ticle which only intends to apply this theory to Witt gr oups. Two main ca s es are discussed. The easier case is when a ll schemes co nsidered are regular . Then their derived category D b,c of complexes with coherent and b ounded homology is preserved under the derived tensor pro duct ⊗ L and under RHom, the derived internal Hom. This endo ws D b,c with a na tural structure of symmetric monoidal category . The dualizing complexes (see Definition 2.1 ) ar e line bundles o r shifted line bundles. The coherent Witt g roups are th us defined using the duality RHom( − , L ) for some line bundle L . F urthermor e , the derived pull-ba ck L f ∗ for any morphism, the derived push-forward R f ∗ and its r ight a djoint f ! for pro pe r morphisms also preserve D b,c . Hence the abstract formalism of [ 12 ] applies on the nose, and we therefor e o btain push-forwards and their class ic a l pro p erties of comp osition, base change and pro jection. The general case, when schemes ar e not a ssumed to b e regula r, is more com- plicated. Indeed, in this case ⊗ L or RHom do no t necessarily pr eserve b ounded homology a nd so there is no nice clos ed symmetric monoidal category structure on the categor y D b,c as the following affine exa mple illustr ates. Cho ose a field k a nd set X = Spec ( k [ ǫ ] / ǫ 2 ). Then consider the co mplex with k concentrated in deg r ee zero. A pro jective res olution of k is given by · · · .ǫ / / k [ ǫ ] /ǫ 2 .ǫ / / k [ ǫ ] /ǫ 2 .ǫ / / k [ ǫ ] /ǫ 2 / / k / / 0 Thu s k ⊗ L k is the complex · · · 0 / / k 0 / / k 0 / / k / / 0 which has un b ounded homology . On the other hand, the unbounded derived ca te- gory D qc of complexes with quasi- coherent cohomolog y admits a clos ed symmetric monoidal structure ; this is not completely obvious, see Theorem 1.2 . But this cat- egory is not suitable to define Witt groups , beca use ther e is no obvious (strong ) duality on it and, a nyw ay , as Eilen b erg s windle t ype of argument s s how for K - theory , unbounded categories ar e no t the go o d framew ork to define cohomology theories. Still, the clos e d symmetric mono idal structure on D qc is useful to pr ov e systematically the comm utativity of diagra ms such a s ( 1 ). That is, we can use the framework o f [ 12 ] to pr ov e this commutativit y in the la rge closed symmetric monoidal categor y D qc and then notice that all functors used in the definition of the duality (RHom( − , K ) for some suitable K ) and the push-for ward (R f ∗ ) actu- ally res tr ict to D b,c under mild a dditional ass umptions. Thus, the commutativit y of the diagra ms inv olved is prov ed in large categ ories by general clos ed symmetric monoidal metho ds, but the dia grams actually often live in a smaller ca tegory whose Witt gr oups are in teresting. A technical p oint arising is the construction o f the functor s inv olved in the sy m- metric mo no idal structure as well as L f ∗ , R f ∗ and f ! on the unbounded derived category D qc . This r elies on the work o f Spaltenstein [ 35 ], the articles of Neeman PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 5 [ 29 , 30 ] and on the v ery useful notes of Lipman [ 27 ], which are a reference on Grothendieck dualit y and contain v ery deta iled explanations of a ll construc tio ns. The article is or ganized as follows. In Section 1 , we r ecall the closed sy mmet- ric monoidal structures of the different ca tegories w e use. In Section 2 , we us e these str uctures to define triang ula ted categ ories with dualities and related Witt groups. Section 3 contains results on the deriv ed functors L f ∗ and R f ∗ and on Grothendieck dualit y , i.e. the co nstruction of the right adjoint f ! of R f ∗ . Section 4 co ntains the main result of the pap er, namely Theorem 4.4 . It explains how to use [ 1 2 ] to obtain the definition of push-for wards for the coherent Witt g roups of schemes. It als o contains a de finitio n of the finite to r-dimension ( e.g. flat) pull-back (Theorem 4.1 ). Section 5 explains the b ehavior of the push-for ward a nd the pull- back under comp osition, and prov es a base change fo rmula r elating them. Section 6 explains p ossible re fo rmulations o f the push-forward in differen t contexts and Section 7 s tudies in de ta il the pus h- forward in the ca s e of finite field extensions, regular embeddings and pro jectiv e bundles, which is us eful for computations and also allows a c omparison with the transfer maps of other authors when they are defined. Everything except some sp ecific computatio ns in the last section works bo th for Gr o thendieck-Witt groups GW a nd Witt gr oups W . F or simplicity , w e stated all r esults for W only . The pr esent article is a generalization of the main r esults of the unpublished preprints [ 10 ] and [ 11 ] o n regular schemes. T o keep this article short, s ome appli- cations established in [ 11 ] (d´ evissage/ lo calization, Witt motiv es and partial results ab out t heir decomposition for cellular v arities) are not included here. Most im- po rtant, a ll the abstract theorems ab out triangulated symmetric monoidal functor s and adjunctions b etw een them which are crucial for proving the theorems of this article are proven in the long article [ 12 ]. W e would like to thank Amnon Neema n for his pr ecise explanations ab out his approach to dualizing complexes; it enabled us to g eneralize earlier versions of the results. W e would also like to thank Paul Ba lmer and Bruno Ka hn for their cons tant suppo rt, and the referee for his careful reading a nd detailed comments. 1. Closed symmetric monoidal ca tegories Let S ch denote the ca teg ory o f separated no etheria n schemes and R eg its full sub c ategory of reg ular schemes. F or any scheme X , let K ( X ) (resp. D ( X )) denote the homotop y (r esp. derived) categ ory of ho mological co mplexes of O X - mo dules (without a ny restriction). W e then add subscripts + for b ounded ab ov e, i.e. bounded where the differentials g o, − for bo unded b elow, b for b ounded, i.e. b e- low and ab ov e, qc for quasi-coherent and c for cohere nt to characterise the derived categorie s of complexe s whose homolog y is as the subscript. F or example D b,c ( X ) is the derived catego ry of complexes of O X -mo dules with coher ent and b ounded homology , and D qc ( X ) is the derived categ ory of complexes with qua si-coherent homology . Note that w e work with ho mological nota tion to b e compatible with the literatur e on Witt gr o ups, but it is easy to switch to cohomolo gical notation by moving b ounding subscripts to sup ersc ripts and exchanging + and − i.e . D + = D − . F or any scheme X , the usua l tens or pro duct ⊗ and internal Hom of complexe s together with the ob vious structure morphisms co ming from the co rresp onding ones for sheaves turn K ( X ) into a susp ended c lo sed symmetric monoida l categor y in the sense of [ 12 , Section 3 ]. This is completely cla ssical and is detailed in [ 12 , Appendix], wher e a dis c ussion o n sign choices can be found. In particular , we ha ve a functor T : K ( X ) → K ( X ) given b y ( T A ) n = T ( A n − 1 ). Theorem 1 .1. L et X b e a scheme. 6 BAPTISTE CALM ` ES AND JENS HORNBOSTEL (1) The tensor pr o duct on K ( X ) admits a left derive d functor ⊗ L : D ( X ) × D ( X ) → D ( X ) to gether with un it, asso ciativity and symmetry morphisms. (2) It r estricts t o D qc ( X ) × D qc ( X ) → D qc ( X ) . (3) When X ∈ R eg , it furthermor e r estricts to D b,c ( X ) × D b,c ( X ) → D b,c ( X ) . (4) The internal Hom on K ( X ) has a right derive d functor RHom D ( X ) o × D ( X ) → D ( X ) which is a right a djoint t o the derive d tensor pr o duct in the u sual sp e cial sense (natur al in the t hr e e variables). (5) When X ∈ S ch , RHo m r estricts t o D b,c ( X ) o × D b,c ( X ) → D c ( X ) as the usu al RHom (c ompute d by r eplacing the s e c ond variable by a quasi- isomorphi c c omplex of inje ctives in Qcoh( X ) ). (6) When X ∈ R eg , this last r estricte d RHom arrives in D b,c ( X ) . Pr o of. See [ 35 , Theorem A] or [ 27 , 2.5.7] for the existence o f the der ived tenso r pro duct. It is based on the ex istence o f a q-flat (also called K-flat) r esolution for any complex C , i.e. the existence o f a quasi-is o morphism Q C → C wher e Q C is a complex such tha t ( − ⊗ Q C ) pr eserves quasi-isomor phisms. These resolutions can even b e constructed functor ially (see [ 27 , 2.5.5]). The derived tensor pro duct can then b e co nstructed by tak ing q-fla t reso lutions o f b oth v ariables. The former case is used to define the unit mo rphism and the latter cas e to define the a sso ciativity and symmetry morphisms directly from the ones o f K ( X ) (se e [ 35 , Theorem A] or [ 27 , 2.5 .9]). See [ 27 , 2.5 .8] for the fact that ⊗ L restricts to D qc . In the reg ular case, by Point (3 ) of Prop osition A.4 , we can replac e an y complex in D b,c ( X ) b y a bo unded co mplex of lo cally free s heav es, in which case the derived tensor pro duct obviously maps to D b,c ( X ). Similarly , the derived internal Hom is constructed using q-injective (also called K-injective) r esolutions: se e [ 35 , Section 1 ] fo r the de finitio n of a q-injective complex and [ 35 , Theor em A] or [ 27 , 2.4.5 ] for the existence o f RHom. Adjointness is also stated in [ 35 , Theorem A] (see a lso [ 27 , 2 .6 .1] for more details). W e now co nsider RHom( A, B ) with A, B ∈ D b,c ( X ) for X ∈ S ch . By Corollary A.6 , the right der ived functor RH om here is co mputed as the one in [ 23 , Prop. II.3 .3]. This proves p o int (5). In the reg ular case, w e can compute RHom by a lo cally fr ee res o lution of the first v ariable a nd then, up to isomorphism, also replace the second v ariable b y a complex of lo cally free sheaves. As explained ab ov e, b oth these complexes can b e chosen to b e bounded, and since H o m( A, B ) is co herent when A a nd B are [ 19 , 5.3.5 ], this p ov es Poin t (6).  Now the subtle po int is that RHom( M , N ) is not necessa rily an ob ject in D qc ( X ) when M and N are . T o fix this, we us e the quasic oher ator Q : Mo d( X ) → Qcoh( X ) as int ro duced in [ 1 , Lemma 3.2 p.187 ], whic h is r ig ht a djoin t to the inclusio n Qcoh( X ) ⊂ Mo d( X ). On an affine spa ce X = Sp ec( A ), it takes a shea f of O X - mo dules to the quasi- c o herent s heaf asso ciated to the A -mo dule of its global sections by the tilde c o nstruction. Its right derived functor is denoted by R Q , as co nsidered in [ 2 , Remark 0 .4], [ 2 7 , E xercises 4.2 .3 ] or [ 36 , B.1 6]. It is a right adjoint to the inclusion D qc ( X ) ⊆ D ( X ), and A ∼ = R Q ( A ) when A ∈ D − ,qc ( X ) (in particular for PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 7 A ∈ D b,c ( X )) by [ 1 , Exp. II, Pr op. 3.5.2]. An alter native construction o f R Q can be obtained from [ 29 , Theore m 4.1]. Theorem 1 .2. F or any scheme X , (1) the derive d tensor pr o duct ⊗ L to gether with the obvious morphi sms turns D ( X ) into a symmetric monoidal c ate gory, close d by t he RHom , and sus- p en de d in the sense of [ 12 , Section 3] . (2) If X ∈ S ch , the funct or R Q ◦ RHom : D qc ( X ) o × D qc ( X ) → D qc ( X ) is a right adjoint (in the u sual sp e cial way, se e [ 26 , (v) p. 9 7] ) to the r estricte d tensor pr o duct ⊗ L on D qc . This turns D qc ( X ) into a susp ende d close d symmetric monoidal c ate gory. (3) If X ∈ R eg , t he u sual RHom is a right adjoint (in the usual sp e cial way) to the r estricte d tensor pr o duct ⊗ L on D b,c . This turns D b,c ( X ) into a susp ende d close d symmetric monoidal c ate gory. Pr o of. The closed symmetric monoida l structure on D ( X ) ea sily follows from The- orem 1 .1 . The fact that it is susp ended follows, as explained in [ 1 2 , Section 3 ], from the susp ended bifunctor structure of RHom. The symmetric monoidal s tr ucture on D qc ( X ) simply follows from the fact that ⊗ L restricts to it. The fact that it is closed is a formal consequence of the fact that D ( X ) is closed and tha t R Q is a right a djoint to the (monoidal) inclusion ι : D qc ( X ) ⊆ D ( X ): Hom qc ( A ⊗ L B , C ) = Hom( ι ( A ⊗ L B ) , ιC ) ≃ Hom( ιA ⊗ L ιB , ιC ) ≃ Hom( ιA, RHom( ιB , ιC )) ≃ Hom qc ( A, R Q RHom( ιA, ιB )) . (The closedness - th at is the existence of the r ight adjoint to the derived tensor pro duct - ca n also be deduced fro m Brown repr esentabilit y , in the spirit of the ex- amples following [ 29 , Theorem 4 .1].) Poin t (3) follows from the same consider ations, using Theorem 1 .1 (3) and (6).  Notation 1. 3. T o shorten the n otation, let [ − , − ] denote the functor RHom , right adjoint to the tensor pr o duct on t he derive d c ate gory D and let [ − , − ] ′ denote the functor R Q ◦ RHom , right adj oint to the tensor pr o duct on the derive d c ate gory D qc . Since the derived q ua si-cohera tor is the identit y on D − ,qc (see above), if [ A, B ] ∈ D − ,qc then [ A, B ] ∼ = [ A, B ] ′ . W e finish this section by p ointing out a comment o f Neeman: exploiting the fact that for X ∈ S ch , there are enoug h flat ob jects in D qc and his repr esentabilit y result more extensively gives an alternative appr oach for co nstructing a clo sed symmetric monoidal structure on D qc ( X ) directly without passing throug h D ( X ). 2. Witt groups F rom now o n, w e ass ume that all schemes are defined over Z [1 / 2 ]. T o define a Witt group, we need a strong duality on a triang ulated catego ry . Using the pr e vious fr amework of tr ia ngulated closed symmetric mo noidal categor ies, we rec a ll how [ − , K ] and [ − , K ] ′ define dualities. The purp o se of this section is to compare the restrictions o f these dualities to the sub categor y D b,c and to discuss when these dualities ar e stro ng o n it. F or any ob ject K , let ♯ K (resp. ♯ ′ K ) denote the contrav ariant exact functor [ − , K ] (r esp. [ − , K ] ′ ). F ollowing [ 12 , Sec tio n 3.2], applied to the closed symmetric monoidal structure on D ( X ) with X a n arbitrar y sc heme, we may define the bidual morphism  K : Id → ♯ K ♯ K 8 BAPTISTE CALM ` ES AND JENS HORNBOSTEL as a mor phism of tr ia ngulated endofunctors of D ( X ). F rom [ 12 , Cor . 3.2 ], we obtain that ( D ( X ) , ♯ K ,  K ) is a triangulated categor y with weak dualit y (in the sense o f [ 12 , Definition 2 .1.1], so  K is not necessarily an isomorphism). Similarly , when X ∈ S ch , we obtain a triang ulated categor y with weak duality ( D qc ( X ) , ♯ ′ K ,  ′ K ). Definition 2. 1. Let K b e an ob ject of D qc ( X ). It is a dualizing c omplex (or it is dualizing ) if - the functor [ − , K ] preserves D b,c ( X ) and - the bidual mor phism  K is an iso morphism on D b,c ( X ). If furthermor e it ha s finite injectiv e dimensio n, i.e. it is quas i-isomorphic to a finite complex of injectives, we say it is an inje ct ively b ounde d dualizing c omplex . In the terminology of [ 12 , Definition 2.1.1], the second condition says that ♯ K is a strong duality on the sub categ ory D b,c ( X ). Note that fo r any X ∈ S ch , a dualizing complex K is a utomatically in D b,c ( X ) since the natural morphism K → [ O X , K ] coming from the monoidal structure is an isomorphism and O X is coher ent . In particular, our definition is exactly the “mo dern” [ 30 , Definition 3 .1], b y Le mma 3.5 of loc. cit. Also note that our injectiv ely b ounded dualizing complexes are the “old” dualiz ing complexes of [ 23 , V. § 2] Prop ositio n 2 . 2. L et X ∈ S ch and K ∈ D qc ( X ) b e a dualizing c omplex. Then the functors ♯ K and ♯ ′ K c oinci de and the bidual m orphisms  K and  ′ K ar e e qual on the sub c ate gory D b,c ( X ) . Pr o of. Since [ A, K ] ∈ D b,c ( X ) for any A ∈ D b,c ( X ), we hav e [ A, K ] ′ ∼ = [ A, K ] by the remark after Notation 1.3 which pr ov es that ♯ ′ K ∼ = ♯ K . The bidual mor phisms are then equal by the large co mm utative dia g ram cons idered in the pro of of [ 12 , Theorem 4.1.2], in whic h the f ∗ should be replaced b y the inclusio n D qc ( X ) ⊂ D ( X ), which is monoidal by definition of the tenso r pro duct o n D qc ( X ).  Example 2 .3 . (1) A dualizing complex tensored by a shifted line bundle is still a dualizing complex. In fact, this is the only freedom: by [ 30 , Lemma 3.9] (see also [ 2 3 , Theor em V.3.1 ] for the injectiv ely bo unded case), a dualizing complex is unique up to tensor ing by shifted line bundles (the shift ca n b e different on different connected comp onent o f X ). (2) On a Gorenstein scheme X ( e.g. regular ), O X itself is dualizing, so by the previous p oint, the only dualizing co mplexes are the shifted line bundles. Note that on a regula r scheme, the categor y D b,c ( X ) itself is close d sy mmetric monoidal. It follows that dualizing complexes a re dualizing ob jects in the sense of [ 12 , Definition 3 .2.2] in the ca teg ory D b,c ( X ), for X ∈ S c h . Theorem 2.4. L et X ∈ S c h and K b e dualizing. Then ( D b,c ( X ) , ♯ K ,  K ) is a triangulate d c ate gory with str ong duality in the sense of [ 12 , Def. 2.1.1] . L et it b e denote d by C K and its Witt gr oups [ 12 , Definition 2.1 .5] by W i ( X, K ) , i ∈ Z . Pr o of. The functor ♯ K = ♯ ′ K is a contra v a riant endofunctor of D b,c ( X ) a nd  K =  ′ K is an isomorphism on this categor y by definit ion of dualizing complexes b y Prop ositio n 2.2 . The necessary commutativ e diag r ams that ♯ ′ K and  ′ K m ust sat- isfy simply follow fro m the fact that they ar e already satisfied in D qc ( X ) since ( D qc ( X ) , ♯ ′ K ,  ′ K ) is a triangulated categor y with weak dua lit y .  W e ma y th us think of the triangula ted c ategory with dualit y ( D b,c ( X ) , ♯ K ,  K ) as being restricted from ( D ( X ) , ♯ K ,  K ) or from ( D qc ( X ) , ♯ ′ K ,  ′ K ), b oth structures coinciding on D b,c ( X ). PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 9 R emark 2.5 . In [ 3 ], a ll dualities conside r ed are strict, i.e. they strictly commu te with the susp e nsion, but this as sumption is only there for simplicity . Instead, in [ 12 , Def. 2.1.1], we only assume commutativit y up to a natural isomor phism, and all theorems in [ 3 ] are still true in this more genera l situation. R emark 2.6 . Recall (see e.g. [ 40 , Def. 10.5.1 ]) that for a left exact functor f betw een exact catego ries, the rig ht derived functor rea lly is a couple (R f , s ) with s : q f → (R f ) q and q the morphism from the ho mo topy category to the der ived category . It is o nly the c ouple (R f , s ) which is unique up to unique iso morphism and therefor e deser ves b eing calle d t he rig ht der ived functor, des pite the standard abbreviated notatio n R f . Co nsequently , the v arious derived functors, for exam- ple RHom( − , K ) (use d to define t he dua lit y) and R f ∗ (used below to define the push-forward) together with all the morphisms of functor s defining the sy mmetric monoidal s tructure ca n be considered as abstract exact functor s and morphisms of exact funct or s . With them, it is p os sible to define co herent Witt g roups and push-forwards by the metho ds discussed in this a r ticle, since these metho ds only inv olve the abstra ct triang ulated categ o ries and functors, i.e. the framework of [ 12 ]. But as such, there is no uniqueness of all these constr uctions. It is only if w e keep as extra data all the structural morphisms of the derived functors (the s part of the couples), and thus the rela tio nship b etw een the closed symmetric mono ida l structure on K ( X ) and the one on D ( X ), that the who le derived construction b e- comes unique up to unique isomo rphism, thus in par ticula r the induced dualities, pull-backs and push-forwards. 3. The functors L f ∗ , R f ∗ and f ! W e now introduce the functors L f ∗ , R f ∗ and f ! asso ciated to a mor phism of schemes f and explain how they b ehav e with res pec t to the mono idal str uctures. The firs t tw o functors ar e derived functors , wher e a s the third one is rig ht adjoint to R f ∗ at the level of der ived catego ries, but is not the der ived functor o f some underlying functor on the categor y of O X -mo dules. The constructio n of f ! is the heart of Gro thendieck duality theor y , for which we refer the rea der to [ 23 ], [ 3 7 ], [ 29 ], [ 13 ] o r [ 27 ]. Prop ositio n 3.1. L et f : X → Y b e a morphism of schemes. (1) The functor f ∗ admits a left derive d functor L f ∗ : D ( Y ) → D ( X ) which r estricts to D qc ( Y ) → D qc ( X ) . (2) If f is of fi n ite tor-dimension (se e e.g. [ 27 , Examples (2.7 .6)] ) or if X , Y ∈ R eg , then L f ∗ r esticts to D b,c ( Y ) → D b,c ( X ) . Pr o of. F or existence, s ee [ 35 , Theorem A (iii) or Pr op. 6.7 ] or [ 27 , Exa mple 2 .7.3]. F or the fact that it restricts to D qc , see [ 27 , 3.9.1 ]. It restricts to D b,c in the finite tor-dimension case b ecause L f ∗ is then b ounded and it res pec ts the co herence of the cohomolo g y by [ 23 , Prop o s ition I I.4.4], b ear ing in mind Pr op osition A.7 . The case X , Y ∈ R eg follows from Point (3) of Pro po sition A.4 and Pr o p o sition A.7 .  Prop ositio n 3.2. The usual i somorphism f ∗ ( A ⊗ B ) → f ∗ A ⊗ f ∗ B induc es an isomorphi sm of triangulate d bifunctors ( in the sense of [ 12 , Def. 1.4.14] ) α : L f ∗ ( − ⊗ − ) → L f ∗ ( − ) ⊗ L f ∗ ( − ) which turns L f ∗ into a susp ende d symm et ric monoidal fu n ctor in the sense of [ 12 , Section 4] . Pr o of. See [ 35 , Prop. 6.8]. The mo rphism α is defined as the corr esp onding one on K ( X ) after having replaced b oth v a riables by q-fla t resolutions . It is already an isomorphism on K ( X ). The comm utative dia grams required (compatibilit y 10 BAPTISTE CALM ` ES AND JENS HORNBOSTEL with the asso cia tivit y , unit and s ymmetry of the monoidal structures) then easily follow from the cor resp onding ones on K ( X ), using Prop osition A.3 , Poin ts (1) and (3).  By [ 12 , P rop osition 4 .1.1] applied to the symmetric monoidal s tructure a nd L f ∗ on D ( X ), there is a natural mo rphism β : L f ∗ [ − , − ] → [L f ∗ ( − ) , L f ∗ ( − )] . W e also obtain a morphism β ′ : L f ∗ [ − , − ] ′ → [L f ∗ ( − ) , L f ∗ ( − )] ′ . using D qc ( X ) instead of D ( X ). Prop ositio n 3.3 . L et X , Y ∈ S ch and A, B ∈ D qc ( Y ) . Assuming [ A, B ] ∈ D qc ( Y ) and [L f ∗ A, L f ∗ B ] ∈ D qc ( X ) , the morphisms β A,B and β ′ A,B c oinci de. In p articular, when K and L f ∗ K ar e dualizing, β K and β ′ K c oinci de. Pr o of. This follows from the c ommut ative diagram L f ∗ [ A, B ] ′ β ′ / /   [L f ∗ A, L f ∗ B ] ′   L f ∗ [ A, B ] β / / [L f ∗ A, L f ∗ B ] in which the v ertical maps be c ome identities under the a ssumptions. This diagram is formally obtained fro m the definitions of β and β ′ out of the clo s ed monoidal structures.  Prop ositio n 3.4. When X , Y ∈ S ch and f : X → Y is of finite tor-dimension or when X , Y ∈ R eg and for any f : X → Y , the natur al morphism β is an isomorphi sm on obje cts in D b,c . Pr o of. This follows from [ 27 , Pro po sition 4.6.6 ] for f o f finite tor-dimension, the first v ar iable co herent and the second in D − , so in pa rticular for both in D b,c . Note that the ρ o f lo c. cit. coincides with our β b y definition (compare [ 27 , (3.5 .4.5)] and [ 12 , Pr o p osition 4.1.1]). In the regula r case, by Point (3) of Pro po sition A.4 , we can assume o ur ob jects ar e bounded complexes of lo cally free sheav es, in which case the r esult follows from [ 27 , Pro p o sition 4.6.7].  Prop ositio n 3.5. L et f : X → Y b e a morphism of schemes. (1) The functor f ∗ admits a right derive d functor R f ∗ : D ( X ) → D ( Y ) . (2) The functor R f ∗ r estricts to D qc ( X ) → D qc ( Y ) when f is quasi-c omp act and sep ar ate d, in p articular if X and Y ar e in S ch , se e [ 22 , Cor. 6.1.10] . (3) The fun ct or R f ∗ r estricts to D b,c ( X ) → D b,c ( Y ) when f is pr op er and Y is quasi-c omp act. Pr o of. F or existence, see [ 35 , Theorem A (iii)] or [ 27 , Examples 2.7.3]. F or the fact that it res tr icts to D qc ( − ) see [ 27 , 3 .9.2]. In the prop er case with Y quasi- compact, it restricts to D b,qc ( − ) by [ 2 7 , 3.9 .2] and it then pres erves coher ence o f the cohomolog y [ 21 , Theorem 3 .2 .1]. Note that we use [ 19 , Definition 5 .3.1] to define coherent modules on no n necessa rily no etherian schemes.  Prop ositio n 3.6. F or any morph ism f of schemes, the functor R f ∗ is a right adjoint to L f ∗ on D ( − ) and c onse quently on al l ful l sub c ate gories to whi ch b oth functors r estrict. Pr o of. See [ 35 , Theorem A (iii)] or [ 27 , P rop osition 3.2.1].  PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 11 By [ 12 , Prop osition 4.2.5 ] applied to the monoidal structure and the functors on the catego ries D ( − ), w e obtain the pro jection morphism π : R f ∗ ( − ) ⊗ − → R f ∗ ( − ⊗ L f ∗ ( − )) Theorem 3. 7 . L et f : X → Y b e qu asi-c omp act and quasi-sep ar ate d e.g. pr op er. Then the pr oje ction morphism π is an isomorphism on D qc . Pr o of. This is [ 27 , Pro p o sition 3.9.4].  Theorem 3.8. F or any sep ar ate d morphism f : X → Y with X and Y sep ar ate d and quasi-c omp act, the fun ctor R f ∗ : D qc ( X ) → D qc ( Y ) has a right adjoint f ! . Pr o of. See [ 29 , Exa mple 4.2] and use that D qc ( − ) and D (Qco h( − )) are equiv alent for separa ted quasi-compact schemes by Prop ositio n A.4 (1).  Prop ositio n 3.9. L et f : X → Y b e a pr op er morphism of sep ar ate d no etherian schemes and let K b e a du alizing c omplex on Y . Then f ! K is a dualizing c omplex on X . If K is an inje ctively b ounde d dualizing c omplex i.e. dualizing in the s ense of [ 23 , V. § 2] , then f ! K is inje ctively b ounde d t o o. Pr o of. F or the case o f injectiv ely b ounded complexes, see [ 23 , V, § 8] or [ 37 , Cor ol- lary 3]. F or the general ca se, we repro duce here a pro of of Neeman. Since the question of whether f ! K is dualizing is loca l on X , we ma y assume Y is affine and restrict to an a ffine op en set U of X . As f is o f finite type , we hav e a fac- torization U → A n × Y → Y for some n wher e the left arrow is a clos ed embed- ding. T a king the c losure of U in P n × Y , we see that U ca n b e embedded as an op en subset of a clo sed subset of so me Y × P n . Hence we only have to show that closed immersio ns, op en immersions a nd pro jections Y × P n → Y resp ect dualizing complexes. The case of clos e d immersio ns is done in [ 30 , Theorem 3.1 4, Remar k 3.17 and Lemma 3.18]; c losed immersions are finite. The case of open immer - sions is [ 30 , Theor em 3.12]. F or pro jective morphisms f : P n Y → Y , one use s that RHom( A, f ! K ) ∼ = RHom( A, f ! O ⊗ f ∗ K ) ∼ = RHom( A, f ∗ K ) ⊗ f ! O , us ing Lemma 5.6 below and that f ! O is a s hifted line bundle b y [ 23 , Section VI I.4]. Then one chec ks the conditions of Definition 2.1 on ob jects of the form f ∗ B and O ( i ) which by a theorem of B e ilinson [ 8 ] genera te D b,c ( P n Y ) as a thick triang ulated categor y .  4. Pull-back and push-f or w ard for Witt groups W e can now state the main result of this ar ticle: the definition of the push- forward fo r coheren t Witt groups a lo ng pro pe r morphisms (Theor em 4.4 ). This section also cont ains a definition of the pull- backs for morphisms of finite tor- dimension (Theorem 4.1 ). Let f : X → Y b e a morphism o f schemes. By [ 12 , Theor em 4.1 .2] applied to the mo no idal ca teg ories D ( − ), β K : L f ∗ ♯ K → ♯ L f ∗ K L f ∗ defines a duality pre s erv- ing functor { L f ∗ , β K } b et ween triangulated categ ories with weak dua lities, from ( D ( Y ) , ♯ K ,  K ) to ( D ( X ) , ♯ L f ∗ K ,  L f ∗ K ), for any ob ject K of D qc ( Y ). Theorem 4 .1. L et f : X → Y b e a morphism of schemes such that - the obje cts K and L f ∗ K ar e dualizing. - L f ∗ pr eserves D b,c , - β K is an isomorphism on D b,c ( Y ) , Then { L f ∗ , β K } induc es a morphism on Witt gr oups f ∗ : W i ( Y , K ) → W i ( X, L f ∗ K ) that we c al l pul l-b ack. This pul l-b ack ther efor e exists in p articular if K and L f ∗ K ar e dualizing and 12 BAPTISTE CALM ` ES AND JENS HORNBOSTEL - f is of fin ite t or-dimension and X , Y ∈ S ch or - for any f and X , Y ∈ R eg in which c ase K dualizing imples L f ∗ K dualizing by Example 2.3 . Pr o of. This follows fro m [ 12 , Theorem 4.1.2 and Lemma 2.2 .6 (1)]. The Theore m of lo c. cit. ensures the existence o f the appropr iate comm utative diagrams in D ( X ). The req uir ements in the Lemma of lo c. c it. that the dualities given by K and f ∗ K restrict as stro ng dualities to D b,c are satisfied by assumption, and the r equirement that β K is an iso morphism when restr icted to D b,c follows from Pr op osition 3.4 .  R emark 4.2 . Note that w e o btain the very same pull-back when sta rting with the monoidal structure on D qc instead of D . This follows from Prop ositio n 3 .3 . R emark 4.3 . In [ 30 , Theo rem 3.12], it is proved that if f is an op en immersion, then L f ∗ K is automatically dualizing if K is dualizing. Let X , Y ∈ S ch , let K ∈ D qc ( Y ) and let f : X → Y b e a s e pa rated mo rphism. F rom [ 12 , Theorem 4.2 .9] applied to the closed mono ida l catego ry D qc ( X ), we obtain a morphism of functors ζ K : R f ∗ ♯ ′ f ! K → ♯ ′ K R f ∗ . By loc. cit., the pair { R f ∗ , ζ K } is duality preserving , i.e . Diagram ( 1 ) commutes. Theorem 4.4. L et X, Y ∈ S ch and f : X → Y b e a sep ar ate d morph ism such that R f ∗ pr eserves D b,c . L et K and f ! K b e dualizing. Then { R f ∗ , ζ K } induc es a morphisms of Witt gr oups f ∗ : W i ( X, f ! K ) → W i ( Y , K ) that we c al l pus h-forwar d. This push-forwar d is ther efor e define d in p articular if f is pr op er and K is dualizing ( se e Pr op ositio n 3.9 ). Pr o of. This follows from [ 12 , Theorem 4.2.9 and Lemma 2.2.6 . (1)]. F o r the The- orem of lo c. c it., consider the triang ula ted clo sed monoidal category D qc . The fact that ζ K is a n isomorphism follows from [ 12 , Prop. 4.3 .3] using The o rem 3.7 . Then, apply the Lemma of lo c. cit. to the sub categ ories D b,c , to which the dualities restrict by definition of a dualizing ob ject. Note that when X and Y are reg ular, the complete pr o of works using dir ectly D b,c as the tria ng ulated clo sed monoidal category in [ 12 , Theore m 4.2.9].  5. Proper ties W e now show that b oth push-forwards and pull-backs r esp ect comp o sition and that they c o mm ute in an appr opriate wa y (“base change”) pr ovided ce rtain stan- dard conditions ho ld. W e also prove a pro jection formula for regular schemes. Theorem 5 .1. F or any f : X → Y and g : Y → Z , (1) ther e is an isomorphism L f ∗ ◦ L g ∗ → L( g ◦ f ) ∗ b etwe en fun ctors on D ( − ) which is asso ciative in the us u al sense. (2) Ther e is an isomorphim R( g ◦ f ) ∗ → R g ∗ ◦ R f ∗ b etwe en fun ctors on D ( − ) which is asso ciative in the usual sense, and re sp e cts the adjoint c ouple (L( − ) ∗ , R( − ) ∗ ) in the sense of [ 12 , Def. 5.1.5] . (3) When the schemes ar e sep ar ate d and quasi-c omp act, and b oth f and g ar e sep ar ate d, ther e is an isomorphism f ! ◦ g ! → ( g ◦ f ) ! b etwe en functors on D qc ( − ) which is asso cia tive in the usual sense, and which r esp e cts the ad- joint c ouple (R( − ) ∗ , ( − ) ! ) in the sense of [ 12 , Def. 5.1.5] . Pr o of. F or the functors L f ∗ on D , the isomor phism is in [ 35 , Theorem A (iii)] or [ 27 , 3.6 .4]. F or a pro of that it is asso ciative, see [ 27 , Scholium 3.6.1 0 ]. The other po int s follow from the first one by [ 12 , Lemma 5.1.6].  PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 13 Theorem 5 .2. The pul l-b ack r esp e cts c omp osition: the diagr am W i ( Z, K ) g ∗ / / ( gf ) ∗ + + X X X X X X X X X X X X X X X X X X X X X X X X W i ( Y , L g ∗ K ) f ∗ / / W i ( X, L f ∗ L g ∗ K ) ≀ W i ( X, L( g f ) ∗ K ) c ommutes, under the c onditions for the existenc e of the pul l-b acks f ∗ and g ∗ of The or em 4.1 . Pr o of. This follows fro m [ 12 , Theor em 5.1.3 and Cor . 5 .1.4] applied to the s truc- tures on D ( − ).  Theorem 5 .3. The push-forwar d r esp e cts c omp osition: the diagr am W i ( X, f ! g ! K ) ≀ f ∗ / / W i ( Y , g ! K ) f ∗ / / W i ( Z, K ) W i ( X, ( g f ) ! K ) ( gf ) ∗ 3 3 g g g g g g g g g g g g g g g g g g g g g g c ommutes, u nder the c onditions for the existenc e of t he push-forwar d of The or em 4.4 . Pr o of. This follows fro m [ 12 , Theore m 5.1.9 and Cor . 5.1.10] applied to the struc- tures on D qc ( − ).  W e now prove a ba s e change formula. L et us consider a pull-back diagram V ¯ f   ¯ g / / Y f   X g / / Z By [ 12 , Section 5.2], w e o btain a morphism of functors ε : L f ∗ R g ∗ → R ¯ g ∗ L ¯ f ∗ betw een functors on D ( X ). Prop ositio n 5.4. If al l schemes ar e in S ch and the diagr am is t or-indep endent, e.g. f flat, the morphism ε is an isomorphism on D qc ( X ) . Pr o of. The cas e where f is flat is [ 27 , Pr o p osition 3 .9.5] (all maps be t ween s chemes in S ch are “concentrated” in the se ns e o f lo c. c it.). The more gener al ca se is [ 27 , Theorem 3.10 .3].  Then, when the schemes a re in S ch , still by [ 12 , Section 5.2], ε induces a mo r- phism γ : L ¯ f ∗ g ! → ¯ g ! L f ∗ betw een functors on D qc ( X ). It is an isomorphism on the sub catego ry D − ,qc by [ 27 , Cor o llary 4.4 .3]. In par ticular, γ K is an isomor phism when K is dualizing (and th us in D b,c ( Z )). Theorem 5.5 (Base change) . Under the assu mptions of Pr op osition 5.4 and the ones for the pul l-b acks alo ng f and ¯ f and the push-forwar ds along g and ¯ g to 14 BAPTISTE CALM ` ES AND JENS HORNBOSTEL exist (The or ems 4.1 and 4.4 ), the pul l- b ack and push-forwar d satisfy a b ase change formula: the diagr am W i ( V , ¯ g ! L f ∗ K ) ¯ g ∗ / / W i ( Y , L f ∗ K ) W i ( V , L ¯ f ∗ g ! K ) ∼ γ 5 5 k k k k k k k k k k W i ( X, g ! K ) ¯ f ∗ O O g ∗ / / W i ( Z, K ) f ∗ O O c ommutes. Pr o of. This follows from [ 12 , Theo rem 5.2.1 and Coro lla ry 5.2.2 ] a pplied to the structures on D qc ( − ), keeping in mind Remar k 4.2 .  W e conclude this s e ction with a pro jection formula for Witt gr o ups, in the c ase of regular schemes. F or this, we fir st need to intro duce another natura l morphis m that will a nyw ay be of some use e ven in the c a se of non r egular schemes. When f : X → Y is a separa ted morphism in S ch , using the functors L f ∗ (monoidal), R f ∗ and f ! betw een the catego ries D qc ( X ) and D qc ( Y ), and the fact that the pro jection mor phism π is an is omorphism by Theo rem 3.7 , we obtain a morphism of functor s θ : f ! ( − ) ⊗ L L f ∗ ( − ) → f ! ( −⊗ L − ) by [ 12 , Prop osition 4.3.1 ]. Lemma 5.6. The morphism θ : f ! A ⊗ L L f ∗ B → f ! ( A ⊗ L B ) is an isomorphism when B is a p erfe ct c omplex. Pr o of. The morphism θ is compatible with op en immersions b y Diagram 41 of [ 12 , Prop ositio n 5.2.5], a nd so w e can restrict to the ca se of bounded co mplexes of vector bundles, then to vector bundles, then a gain using op e n immersions to the trivial bundle O Y . In that cas e, o ne ca n show that θ A, O Y coincides with the unit mor phism of the monoidal str uctur e f ! ( A ) ⊗ L O X → f ! ( A ), and is therefor e an isomor phism. By this coincide nc e w e mea n that the left diagram f ! A ⊗ L L f ∗ O Y θ / / f ! ( A ⊗ L O Y ) f ! A ⊗ L O X ∼ ≀ f ! A ≀ R f ∗ B ⊗ L O Y π / / R f ∗ ( B ⊗ L L f ∗ O Y ) R f ∗ B ∼ ≀ R f ∗ ( B ⊗ L O X ) ≀ is commutativ e, in which the left vertical morphism is the fac t that L f ∗ is monoidal and in particular resp ects units, and the bo ttom and right maps ar e unit morphisms of the monoida l s tructures. By following the definition of θ given in [ 12 , Pro po sition 4.3.1] the commutativit y of the left diagr a m follows from the o ne o f the r ight hand side, which is in turn implied, using the definition o f π in [ 12 , Pr op osition 4 .2 .5], by the c ompatibility of the unit and monoidal structure mo r phisms for L f ∗ .  F or any scheme X in R eg , the der ived tensor pro duct preserves D b,c ( X ) (Theo - rem 1 .1 (3)). This gives tw o different pro ducts on Witt groups b y the formalism of [ 18 ], us ing [ 12 , Pr op osition 4.4.6 and Cor o llary 4.4 .7] applied to the clo sed monoidal structure of D b,c ( X ). W e fix one of these pro ducts (say , the left o ne) for the fol- lowing results. When K and L a re shifted line bundles, thus dualizing complexes , the pro duct is a pairing W i ( X, K ) × W j ( X, L ) → W i + j ( X, K ⊗ L ) . PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 15 Theorem 5.7 (Pro jection formula) . F or any pr op er morphism f : X → Y with X , Y ∈ R e g , the pul l-b ack and push-forwar d satisfy a pr oje ction formula: If K , L ar e shifte d line bun d les on Y , x ∈ W i ( X, f ! K ) and y ∈ W j ( Y , L ) , then f ∗ ( I ( x.f ∗ y )) = f ∗ ( x ) .y in W i + j ( Y , K ⊗ L ) with I the isomorphism fr om W i ( X, f ! K ⊗ f ∗ L ) to W i ( X, f ! ( K ⊗ L )) induc e d by θ K,L . Pr o of. First note that L being a shifted line bundle explains the absence of der iv a - tions in the pull-back and tensor pro ducts ab ov e. Then, the morphism θ K,L is an isomorphism by Lemma 5.6 , th us the r esult follows from [ 12 , Theore m 5.5.1 a nd Corollar y 5 .5 .2] applied to the closed monoidal str ucture on D b,c .  6. Ref ormula tions in special cases In this s ection, we give o ther canonical w ays of writing the push-forward, under additional assumptions . Notation 6. 1. F or an e qu idimensional morphism f : X → Y of r elative dimension n , let ω f denote the obje ct f ! ( O Y )[ − n ] := T − n f ! ( O Y ) . This notation is motiv ated b y the fact that in s e veral cases, this ob ject ca n be identified with a geometric ob ject called a relative dualizing sheaf and usually denoted ω f : see sec tio ns 7 .2 a nd 7.3 for the ex amples o f regular embeddings and pro jective spa ces. 6.1. Relative dualizing sheaf. Theorem 6.2 . Le t f : X → Y b e a pr op er morphism in S ch , K a dualizing c omplex on Y such that f ! K is a dualizing c omplex and assume t hat θ O Y ,K : f ! O Y ⊗ L L f ∗ K → f ! K is an isomorphism. Then we c an r ewrite the pus h-forwar d of The or em 4.4 as f ∗ : W i + d ( X, ω f ⊗ L L f ∗ K ) → W i ( Y , K ) . In p articular, the hyp otheses and ther efor e the c onclusion hold if K is dualizing and either of the two fol lowing c onditions hold. - f is quasi-p erfe ct (se e b elow, e.g. of finite tor-dimension) and f ! K is dual- izing. - Y is a Gor ens tein scheme, e.g. r e gular. Pr o of. The refor m ulation o f the push- fo rward is [ 12 , Definitions 6 .1 .3 and 6.1 .4 ]. When f is quasi-p erfect, [ 28 , Prop os ition 2.1] shows that θ is an is omorphism o n all D qc ( Y ). Exa mple 2.2 in lo c. cit. shows that if f is of finite to r -dimension, it is quasi-p erfect. When Y is Gorenstein, the only dualizing c o mplexes a re shifted line bundles, for which θ is an is o morphism by Lemma 5.6 .  Let g : Y → Z b e another pro pe r morphisms in S ch and M a dualizing complex on Z and let ι f ,g : ω f ⊗ L L f ∗ ( ω g ⊗ L L g ∗ M ) → ω gf ⊗ L L( g f ) ∗ M be the mor phism defined in [ 12 , Theore m 6 .1.5]. Theorem 6.3. The push-forwar d of The or em 6.2 resp e cts c omp osition: the mor- phism ι f ,g is an isomorph ism and if I denotes the isomorphism of Witt gr oups induc e d by ι f ,g , then the push-forwar d on Witt gr oups define d ab ove satisfies that g ∗ f ∗ = ( g f ) ∗ I . Pr o of. This follows from [ 1 2 , Theorem 6.1.5 , L emma 2.2.6 (2)]. Note tha t ι f ,g is an isomorphism b eca use it is a co mpo sition o f is omorphisms under the a ssumptions for the re fo rmulated push-forward to exist.  16 BAPTISTE CALM ` ES AND JENS HORNBOSTEL Theorem 6.4. In the situation of The or em 5.5 and un der the assumptions of the r eformulatio n of the push-forwar d ab ove for t he morphisms g and ¯ g , the b ase change the or em 5.5 b e c omes f ∗ g ∗ = ¯ g ∗ I ¯ f ∗ wher e I is the isomorphism of Witt gr oups induc e d by the isomorphism ι : L ¯ f ∗ ( ω g ⊗ L L g ∗ K ) → ω ¯ g ⊗ L L ¯ g ∗ L f ∗ K. Pr o of. This fo llows from [ 12 , Theorem 6.1.7]. Note t hat γ O Z is an isomorphism (see b efore Theo rem 5.5 ).  In the regula r case, the pro jection form ula 5.7 b ecomes the following. Theorem 6. 5 . L et f : X → Y b e a pr op er e quidimensio nal morphism of r elative dimension d with X , Y ∈ R eg . Then the push-forwar d of The or em 6.2 and the pul l-b ack of The or em 4.1 satisfy f ∗ I ( x.f ∗ ( y )) = f ∗ ( x ) .y in W i + j ( Y , L ⊗ K ) for any x ∈ W i + d ( X, ω f ⊗ f ∗ L ) and y ∈ W j ( Y , K ) . Pr o of. See [ 12 , Theorem 6.1.9 and Corolla ry 6 .1.10].  6.2. Smo oth sc hemes ov er a base. W e now fix a base scheme S ∈ S ch w ith a dualizing co mplex K S and cons ide r the categor y S mP r /S of schemes in S ch that are smo oth, equidimensiona l and proper over S . F or such a sc heme X , le t the structural mor phism b e denoted b y p X : X → S and its r elative dimension ov er S by d X . Note that any separated mor phism b e tween s chemes in S mP r /S is pro per , being the co mpo sition of a closed embedding , its g raph, and a pr o p er pro jectio n. Notation 6.6. L et X ∈ S mP r / S . We set ω X = p ! X ( K S )[ − d X ] . Observe that ω X = ω p X if K S = O S . Theorem 6 .7. L et f : X → Y b e a sep ar ate d morphism, X , Y ∈ S mP r / S and let L b e a line bun d le on Y . The push-forwar d c an b e written f ∗ : W i + d X ( X, ω X ⊗ f ∗ L ) → W i + d Y ( Y , ω Y ⊗ L ) Pr o of. First, let us note that when pulling back or tensoring b y a line bundle, there is nothing to derive. This is why no L app ear in front of f ∗ and ⊗ . W e then use Definitions [ 12 , Definitions 6 .3.3 and 6.3 .4]. W e need to chec k that the mo rphism ω X ⊗ L ≃ f ! ω Y ⊗ f ∗ L → f ! ( ω Y ⊗ L ) is an iso morphism. This is the ca s e by Lemma 5.6 .  Theorem 6 .8. The push-forwar d of The or em 6.7 r esp e cts c omp osition. Pr o of. See [ 12 , Theorem 6.3.5 and Corolla ry 6 .3.6].  Theorem 6 .9. The push-forwar d of The or em 6.7 satisfies flat b ase change. Pr o of. See [ 12 , Theorem 6.3.7 and Corolla ry 6 .3.8].  7. Examples Note that f ! is unique up to unique isomorphism whenever it is defined, because it is always defined as a right a djoin t to R f ∗ . This allows us to us e c o mputations of f ! from [ 23 ] and o ther sources in the examples b elow. PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 17 7.1. Finite fie l d extens ions. The simplest example of a pro pe r morphism is the case of a finite field extension E /F giving rise to a finite mo rphism f : X = Spe c( E ) → Spec( F ) = Y . The tilde construction giv es equiv a le nces of c ategories Mo d( F ) ≃ Qcoh( Y ) and Mo d( E ) ≃ Qco h( X ), and the sub categor ies of finite dimensional v ector spaces cor- resp ond to coherent sheaves of mo dules. W e th us describ e all o b jects and functors through these equiv a lences of catego ries. The o nly dualizing c o mplex (up to shifts and isomorphisms) on Y is F itself. The functors f ∗ = ( − ⊗ F E ) a nd f ∗ = ( − ) | F are exact, there is no thing to derive. The functor f ! is g iven by [ 23 , I I I § 6] as H om F ( E , − ) (mapping to E -vector spaces) and the unit and counit of the adjunc- tion ( f ∗ , f ! ) are r esp ectively given b y  V → H om F ( E , V | F ) a 7→ ( e 7→ e.a )  H om F ( E , V ′ ) | F → V ′ φ 7→ φ (1) for a n E -vector space V and an F -vector space V ′ . F or fields, the only nonz e ro Witt gro up mo dulo 4 is W 0 which is the clas sical Witt gro up of the field. So we are re duce d to study push-forward fo r forms on vector spa ces, i.e. co mplexes concentrated in degree zero . F ollowing the constructio n, it is easy to chec k that for any E -vector space V , the morphism ζ : f ∗ [ V , f ! F ] → [ f ∗ V , F ] c o incides w ith the Cartan isomo rphism of F -vector spa ces H om E ( V , H om F ( E , F )) | F ≃ H om F ( V | F , F ) which sends a morphism φ : V → H om F ( E , F ) to the mor phis m ( a 7→ φ ( a )(1)). Thu s, the push-forward f ∗ : W 0 ( E , H om F ( E , F )) → W 0 ( F ) is a Scharlau tr a nsfer (see [ 34 , p. 48]) with resp ect to the usual trace T r : E → F . T o see this, note that T r factor s as E ≃ → H om F ( E , F ) → F where the isomorphism is given b y e 7→ ( x 7→ T r ( e.x )) and H om F ( E , F ) → F is the ev aluation at 1 . 7.2. Regular em b eddings . Let F b e a vector bundle of r ank d > 0 over X with a regular section s : O X → F , i.e. such that the co rresp onding e m b edding f : Z ⊂ X of the zer o lo cus is a clo s ed reg ular embedding of co dimension d . In that case the augmented Koszul res olution is exact [ 15 , IV, § 2, P rop osition 3.1] and th us yields a quasi-iso morphism (3) K F qis   = ( 0 / / Λ d F ∨ / / Λ d − 1 F ∨ / / · · · / / F ∨ / / O X   / / 0 ) f ∗ O Z = ( 0 / / f ∗ O Z / / 0 ) from the Koszul co mplex K F to f ∗ O Z concentrated in degree 0. Since f is a closed embedding, thus finite, f ∗ is exact and coincides with R f ∗ . Let ∆ F = Λ d F b e the determinant of F . In this situation, we hav e f ! A = f ∗ ∆ F [ − d ] ⊗ L f ∗ A for all A ∈ D b,c ( X ); this ma y be extracted from [ 23 , I I I § 7], see also [ 37 , Prop os ition 1], after applying Lemma 5 .6 and using that F is dual to the co ta ngent s heaf. By tensoring the augmen ted Koszul resolution with ∆ F and using the canonical isomorphisms Λ i F ∨ ⊗ ∆ F ∼ = Λ d − i F a nd f ∗ O Z ⊗ ∆ F ∼ = f ∗ f ∗ ∆ F , we obtain the trace map f ∗ f ! O X → O X (counit of the a djunction (R f ∗ , f ! )) in the derived categor y as 18 BAPTISTE CALM ` ES AND JENS HORNBOSTEL the comp osition of a usua l map of complexes follow ed by the inv erse of a quasi- isomorphism ( O X is in degr ee 0 and f ∗ f ∗ ∆ F in degree − d ): f ∗ f ! O X   = ( 0 / / 0 / / · · · / / 0 / / f ∗ f ∗ ∆ F id   / / 0 ) ( 0 / / F / / · · · / / ∆ F / / f ∗ f ∗ ∆ F / / 0 ) O X ( 0 / / O X s O O / / 0 / / · · · / / 0 / / 0 ) Now a ssume Z is Gorenstein. Then O Z is dualizing, and the isomor phism O Z → [ O Z , O Z ] adjoin t to O Z ⊗ L O Z ≃ O Z defines a for m on O Z , denoted b y 1 Z . On the other hand, there is a well-kno wn for m θ F : K F → H om( K F , ∆ ∨ F [ d ]) (see [ 7 , § 4 ]) given b y the canonical isomorphism Λ i F ∨ ≃ (Λ d − i F ∨ ) ∨ ⊗ Λ d F ∨ in deg ree i , with a s ign chosen so that when F = ⊕ L i is a dire c t sum of line bundles, this form is the tensor pr o duct of forms θ L i on Ko szul complexes of leng th one (4) K L i θ L i   = ( 0 / / L ∨ i − 1   s ∨ i / / O X 1   / / 0 ) H om( K L i , L ∨ i [1]) = ( 0 / / L ∨ i − s ∨ i / / O X / / 0 ) representing elements in W 1 ( X, L ∨ i ). The following pr o p o sition ca n b e considered as a concr ete description of the push-forward o f 1 Z along f . Prop ositio n 7.1. L et Z and X b e Gor enstein schemes and f : Z → X b e a close d r e gular emb e dding of c o dimension d define d as the zer o lo cu s of a r e gular se ction of a ve ctor bund le F of r ank d whose determinant Λ d F is denote d by ∆ F . The n the image of the form 1 Z : O Z ≃ → [ O Z , O Z ] (adjoint to O Z ⊗ L O Z ≃ O Z ) u nder the c omp osition W 0 ( Z, O Z ) ≃ W d ( Z, f ! ∆ ∨ F ) f ∗ / / W d ( X, ∆ ∨ F ) is a form φ such that the fol lowing diagr am in D ( X ) c ommutes. K F θ   qis ≃ / / f ∗ O Z φ   H om( K F , ∆ ∨ F [ d ]) qis ≃ / / [ f ∗ O Z , ∆ ∨ F [ d ]] Pr o of. Let δ F : K F ⊗ K F → K F in D ( X ) b e the compo sition K F ⊗ K F qis ⊗ q is / / f ∗ O Z ⊗ L f ∗ O Z λ / / f ∗ ( O Z ⊗ L O Z ) ≃ / / f ∗ O Z qis − 1 / / K F where λ is the morphism fro m [ 12 , Prop os ition 4.2.1]. Note that δ F is in fact represented by a mo rphism of complexes (not just a fractio n): one can check that the map fro m K F ⊗ K F to K F in degree i is a s um of the c a nonical morphisms Λ k F ⊗ Λ i − k F → Λ i F with appropria te signs . W e a lso co nsider the ma p σ F : K F → PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 19 ∆ ∨ F [ d ] given b y K F σ F   = ( 0 / / Λ d F ∨ / / Λ d − 1 F ∨ / / · · · / / F ∨ / / O X / / 0 ) ∆ ∨ F [ d ] = ( 0 / / ∆ ∨ F / / 0 ) The following three lemmas toge ther clea r ly imply the prop osition. Lemma 7 . 2. The morphism of c omplexes x F : K F → H om( K F , ∆ ∨ F [ d ]) define d as the c omp osition K F / / H om( K F , K F ⊗ K F ) ( δ F ) ∗ / / H om( K F , K F ) ( σ F ) ∗ / / H om( K F , ∆ ∨ F [ d ]) c oinci des with the form θ wher e the first map is the unit of the adjunction of the tensor pr o duct and the int ernal Hom in the homotopy c ate gory. Lemma 7 . 3. The form φ : f ∗ O Z → [ f ∗ O Z , ∆ ∨ F [ d ]] c oincid es with the c omp osition f ∗ O Z / / [ f ∗ O Z , f ∗ O Z ⊗ L f ∗ O Z ] λ / / [ f ∗ O Z , f ∗ ( O Z ⊗ L O Z )] ≀   [ f ∗ O Z , ∆ ∨ F [ d ]] [ f ∗ O Z , f ∗ f ! ∆ ∨ F [ d ]] o o [ f ∗ O Z , f ∗ O Z ] ≃ o o wher e the first map is the u n it of the adjunction of the tensor pr o duct and the internal H om in D ( X ) , the p enultimate one is the identific ation O Z ≃ f ! ∆ ∨ F [ d ] and the last one is induc e d by the c ounit of t he adjunction ( f ∗ , f ! ) , i.e. the t ra c e map describ e d ab ove. Lemma 7.4. The c omp osition of L emma 7.2 c oincid es with the one of L emm a 7.3 when f ∗ O Z is identifie d with K F using q is . Pr o of of L emma 7.2 . As we are dealing with honest morphisms of complexes we may first reduce to op en subsets on which F is a sum o f line bundles L i (note that t wo morphisms in D ( X ) are not nec e s sarily eq ua l if they a re e q ual when restricted to all o p en sets o f an affine c overing, s e e for exa mple [ 6 ]). W e then reduce to the case of codimensio n d = 1, by multiplicativit y o f Kos zul complexes: let d = d 1 + d 2 and let F = F 1 ⊕ F 2 where F 1 (resp. F 2 ) is the sum of the first d 1 line bundles (resp. last d 2 ). Le t f 1 : O Z 1 → X (re sp. f 2 : O Z 2 → X ) b e the cor resp onding regular subschemes. Then ∆ F ≃ ∆ F 1 ⊗ ∆ F 2 and K F ≃ K F 1 ⊗ K F 2 . W e leav e it to the reader to show that the diagra m K F 1 ⊗ K F 2 x F 1 ⊗ x F 2   ≃ / / K F x F / / H om( K F , ∆ ∨ F [ d ]) ≃   H om( K F 1 , ∆ ∨ 1 [ d 1 ]) ⊗ H o m( K F 2 , ∆ ∨ 2 [ d 2 ]) τ / / H om( K F 1 ⊗ K F 2 , ∆ ∨ 1 [ d 1 ] ⊗ ∆ ∨ 2 [ d 2 ]) commutes, where τ is the mo rphism de fined as in [ 12 , Definition 4.4 .1], using the monoidal structure on the homotopy c a tegory . By definition, θ F , θ F 1 and θ F 2 make the sa me dia gram commutativ e when they replace x F , x F 1 and x F 2 . Hence it suffices to show the lemma for one line bundle L and its a sso ciated Koszul complex of length one, which can b e check ed b y hand.  Pr o of of L emma 7.3 . By definition, the form φ is given by the comp ositio n f ∗ O Z f ∗ 1 Z / / f ∗ [ O Z , O Z ] ≃ / / f ∗ [ O Z , f ! ∆ ∨ F [ d ]] ζ / / [ f ∗ O Z , ∆ ∨ F [ d ]] . 20 BAPTISTE CALM ` ES AND JENS HORNBOSTEL One prov es using the c losed monoidal structure that it co incides with the co mpo - sition f ∗ O Z ≃ / / f ∗ f ! ∆ ∨ F [ d ] / / f ∗ [ O Z , f ! ∆ ∨ F [ d ]] ζ / / [ f ∗ O Z , ∆ ∨ F [ d ]] where the seco nd ma p is adjoint to the unit mo rphism of the mo noidal structure. Then, lo oking ba ck at the definition of ζ and µ in [ 12 , P rop osition 4 .2.2 a nd The- orem 4 .2.9], one sees that φ is the comp os ition ar o und the low er left corner of the commutativ e diagram f ∗ O Z / /   [ f ∗ O Z , f ∗ O Z ⊗ L f ∗ O Z ]   / / [ f ∗ O Z , f ∗ ( O Z ⊗ L O Z )]   / / [ f ∗ O Z , f ∗ O Z ]   f ∗ f ! ∆ ∨ F [ d ]   / / [ f ∗ O Z , f ∗ f ! ∆ ∨ F [ d ] ⊗ L f ∗ O Z ]   / / [ f ∗ O Z , f ∗ ( f ! ∆ ∨ F [ d ] ⊗ L O Z )]   / / [ f ∗ O Z , f ∗ f ! ∆ ∨ F [ d ]]   f ∗ [ O Z , f ! ∆ ∨ F [ d ]] - - [ f ∗ O Z , f ∗ [ O Z , f ! ∆ ∨ F [ d ]] ⊗ L f ∗ O Z ] . . [ f ∗ O Z , f ∗ ([ O Z , f ! ∆ ∨ F [ d ]] ⊗ L O Z )] 6 6 l l l l l l l l l l l l l [ f ∗ O Z , ∆ ∨ F [ d ]] which thus proves the lemma (all squares in this diagr am a re commutativ e by obvious functoria l reasons , a nd the tr ia ngle b y adjunction).  Pr o of of L emma 7.4 . This follows fr o m the computation of the resolution of f ∗ O Z by K F when computing the derived functor s [ f ∗ O Z , − ] and −⊗ L f ∗ O Z .  This finishes the pro of of Pro po sition 7.1 .  R emark 7.5 . If F = F ′ ⊕ L 1 , with L 1 a line bundle, s = ( s ′ , s 1 ), s ′ and s 1 transverse, the push-forward of 1 Z is zero: decompo se the inclusion Z ⊂ X as Z ⊂ Z ( s ′ ) ⊂ X where Z ( s ′ ) is the zero lo cus of s ′ . Push-forwards r esp ect co mpo sition and the push-forward of 1 Z along Z ⊂ Z ( s ′ ) is a lready zero since it is the form ( 4 ) which is the cone o f a (degenera te) form s : L ∨ 1 → O X . On the other hand, an example when this push-forward is nonzero can b e extracted from [ 5 ]. Let k b e a field and let Gr k (2 , 4) be the Gr a ssmannian of 2 - planes in A 4 k . A nonzero map fro m A 4 k to A k induces a sec tio n of the dual W ∨ of the universal subbundle W ⊂ A 4 k of rank 2 . Its zero lo cus is a copy of P 2 regular ly embedded in Gr (2 , 4). The push-forward of the unit form of P 2 to Gr (2 , 4) is nonzero by [ 5 ] where it is proved tha t it is an element of a ba sis o f the total Witt group of Gr (2 , 4) as a W( k )-mo dule. 7.3. Pro jectiv e spaces. Let Y ∈ S ch b e a Gorenstein scheme, let E be a vector bundle of rank r + 1 on Y and let us examine when the unit form on X = P Y ( E ) can b e pus hed forward to Y alo ng f : X → Y . Since f is smo oth (thus flat), we can use sectio n 6.1 . In the case of a smoo th morphism f , the ob ject ω f of 6.1 is a line bundle, and it is the maxima l exterior pow er of the relative c o tangent bundle (see [ 2 3 , C h. VI I § 4]). Here, since f is a pro jective bundle, it is g iven by ω f = f ∗ (∆ E ) ∨ ⊗ O ( − r − 1 ) (se e e.g. [ 14 , Appendix B.5 .8]). If r + 1 is even, we can push-forward the unit form 1 X : O X ≃ [ O X , O X ] from W 0 ( X, O X ) by using the comp osition W 0 ( X, O X ) ≃ W 0 ( X, O ( − r − 1)) = W 0 ( X, ω f ⊗ f ∗ (∆ E )) → W − r ( Y , ∆ E ) where the first isomor phis m is giv en by tensoring with the canonica l form φ r = [ O ( − ( r + 1) / 2 ) ≃ → O (( r + 1) / 2) ⊗ O ( − ( r + 1)) ∼ = H om O ( O ( − ( r + 1) / 2) , O ( − ( r + 1 ))] and the last map is the push-forward in the form of Theorem 6.2 . Co mputing the image of 1 X through this comp ositio n mea ns therefore co mputing the image of φ r by the push- forward. The complex on whic h f ∗ ( φ r ) lives is R f ∗ ( O ( − ( r + 1) / 2)). But this complex is zero b y [ 20 , 2 .1.15], so f ∗ ( φ r ) = 0. If r + 1 is o dd, there is no PUSH-FOR W ARDS FOR WITT GR OUPS OF SCHEMES 21 push-forward induced by f with source W 0 ( X, O X ) be cause then there is no line bundle K on Y suc h that O X is equa l to f ∗ (∆ ∨ E ) ⊗ O ( − r − 1) ⊗ f ∗ ( K ) up to a square in P ic ( Y ). In other w ords , pushing for ward the unit form of P Y ( E ) to Y is not very interesting: whenever it is po ssible, w e get z e r o. Of co ur se, there a re other forms on P r ( E ) not mapping to 0 under the push-for ward, a s we will see in the following r emark. R emark 7.6 . Let us explain a p otential source of confusion. Let i : Sp ec k → P r k be a r ational p oint and L a line bundle o n P r k . Since P ic(Spec k ) = 0, using first an isomorphism O k ≃ ω i ⊗ i ∗ ( L ), we can push-forward from W 0 (Spec k , O k ) to W r ( P r k , L ) for a ny L . But for different L , we get very differ ent push-forwards. Indeed, for example W r ( P r k , O ( − r )) = 0 for o dd r (b y [ 39 ] o r [ 4 ]) so a ny push- forward to there is ob viously zero, wherea s since ω k ≃ O k the push-forward (written as in Theo rem 6.7 ) W 0 (Spec k , O k ) ≃ W 0 (Spec k , ω k ) → W r ( P r k , ω P r k ) is certainly nonzero , b ecause we can further comp ose it by a push-forward back to W 0 (Spec k , ω k ) a nd sinc e the push-forward resp ects comp osition, the comp osite is the identit y . No te that this last case also g ives a n example of a form on P r k whose push-forward to Spec k is not zero. Mor e genera lly , this phenomeno n of differe nt push-forwards star ting from the same gr oup can happ en whenever f ∗ : Pic( Y ) → Pic( X ) is not injective. Appendix A. q-fla t and q-injective resolutions F or the conv enience of the reader , w e include here well-known facts on q-flat o r q-injective ob jects, that are rep eatedly used in the pro ofs of this a rticle. Most of them are due to Spaltenstein. Definition A.1. Let X b e a sc heme and A b e an o b ject in the homotopy catego ry K ( X ). W e s ay that A is q- flat (or K-flat) if the tr iangulated functor ( − ⊗ A ) : K ( X ) → K ( X ) pr eserves quasi-isomo rphisms. W e say that A is q-injective (or K-injective) if the triang ulated functor H om • ( − , A ) : K ( X ) o → K ( X ) prese rves quasi-isomo rphisms. Example A.2 . A b ounded ab ove complex of flat O X -mo dules is q-flat. A bounded below complex of injectiv es is q-injective. A discus sion of q-flat and q-injective complexes can be found in [ 35 , Sections 1 and 5]. See in particular Prop ositions 1.5 and 5.3 . Prop ositio n A.3. L et A and B b e obje ct s in K ( X ) or K ( Y ) and let f : X → Y b e a morphism of schemes. (1) If A and B ar e q-fl at, then so is A ⊗ B . (2) If A is q-flat and B is q-inje ctive, then H om • ( A, B ) is q-inje ctive. (3) If A is q-flat, then f ∗ A is q-flat. Pr o of. See [ 35 , Prop. 5.3 and 5.4].  The fo llowing t wo pro po sitions s umma r ize the equiv alences of categories and the prop erties of injectives that we need. Let Qco h( X ) denote the ab elian ca tegory of quasi-coher ent sheaves on X and C b (V ect( X )) (re s p. C b (Coh( X ))) the ca teg ory of bo unded complexes o f lo cally free (resp. coheren t) sheav es on X . Prop ositio n A.4. L et X ∈ S ch . (1) The natur al functor D (Qco h( X )) → D qc ( X ) is an e quivalenc e of c ate gories and thu s the same is true for their homo lo gic al ly b ounde d, b ounde d b elow or b ounde d ab ove sub c ate gories and the sub c ate gories with c oher ent homolo gy. 22 BAPTISTE CALM ` ES AND JENS HORNBOSTEL (2) The natur al functor D b (Coh( X )) → D b,c (Qcoh( X )) is an e quivalenc e. (3) If X ∈ R eg , then the natur al functor D (C b (V ect( X ))) → D b,c ( X ) is an e quivale nc e of c ate gories. Pr o of. Point (1) is [ 9 , Cor. 5.5]. In point (2), fully faithful follows fr om [ 25 , Theorem 12.1], se cond part: F or affine s chemes, use [ 25 , Exa mple 1 2.3]. In g eneral, take a finite affine cov er of the no etherian scheme X and then take the direct sum ov er the coherent sheav es on X o btained b y extending the cohere nt shea ves on the affine subsc hemes using [ 22 , I.6.9.7]. Essentially surjective can b e found in [ 16 , Example 2.5 .2 ] which fo llows fro m [ 38 , Lemme 2.1 .2 .c)] and an induction a rgument. Poin t (3) can then b e prov ed as follows. Let C b (Coh( X )) b e the category of b ounded complexes of coherent sheav es. Deco mpo se the functor D (C b (V ect( X ))) → D b,c ( X ) as D (C b (V ect( X ))) / / D (C b (Coh( X ))) / / D b (Coh( X )) / / D b,c (Qcoh( X )) / / D b,c ( X ) . All these functors ar e equiv alences of categ ories: the first one by the fact since X is regular , every co herent sheaf has a finite resolutio n by lo cally free s he aves by [ 33 , § 7 Poin t 1], the sec ond one by [ 25 , Lemma 11 .7], the third one by Point (2) and the fourth b y Poin t (1).  Prop ositio n A.5. L et X ∈ S ch . (1) The c ate gory Q c oh( X ) has enough inje ctives by [ 36 , B.3] . (2) The natu ra l inclusion Qcoh( X ) → Mo d( X ) pr eserves inje ctives by [ 3 6 , B.4] . (3) Every b oun de d b elow c omplex of quasi-c oher ent O X -mo dules admits a quasi- isomorphi sm into a c omplex of Qc oh( X ) -inje ctives by (1) and [ 23 , I.4.6] . (4) Every b ounde d b elow c omplex of Qcoh( X ) -inje ct ives is q-inje ctive in b oth K (Q c oh( X )) and K ( X ) by (2). Corollary A.6. L et X ∈ S c h . On obje cts in D − ,qc ( X ) or D − ( X ) , the un b ounde d right derive d fun ctors c ompute d by q- inje ctive r esolutions (as in [ 35 ] ) c oincide with the mor e classic al b oun de d b elow right derive d functors c ompute d by u sing r esolu- tions by b ounde d b elow c omplexes of inje ctives (as in [ 23 ] ). Pr o of. By the prop osition, any ob ject A ∈ D − ,qc ( X ) is qua si-isomor phic to a bo unded b elow complex of Q c oh( X )-injectives which are als o Mo d( X )-injective (resp. a bounded below complex of Mod( X )- injectives) and this complex is q- injectiv e.  Similarly , for q-flat resolutions , w e hav e a weak er statement, sufficient for our purp oses. Prop ositio n A.7. L et X ∈ S ch . On obje cts in D + ,qc ( X ) or D + ( X ) , the un- b ounde d left derive d functors c ompute d u sing q-flat r esolutions c an b e c ompute d by using b ou n de d ab ove r esolutions c onsisting of flat O X -mo dules. Pr o of. 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