On the derived category of a regular toric scheme

On the deriv ed category of a regular toric sc he me Thomas H ¨ uttemann Que en ’s University Belfast, Pu r e M athemat ics R ese ar ch Centr e Belfast BT7 1NN , Northern Ir eland, UK e-mail: t.huette mann@qub.ac .uk Let X b e a quasi-compact scheme, equ ipped with an open cov ering by aﬃne sc hemes U σ = Sp ec A σ . A quasi-coheren t s heaf on X gives rise, by taking sections ov er the U σ , to a diagram of mo dules ov er the co ordinate rings A σ , indexed by the intersection poset Σ of th e co vering. If X is a regula r toric scheme o ver an arbitrary comm utative ring, we prov e that the unbound ed d eriv ed category of quasi-coherent sheav es on X can be obtained from a category of Σ op -diagrams of chain complexes of modu les by inv erting maps whic h in duce homology isomorphisms on hyp er-derived inverse limits. Moreo ver, w e show that there is a ﬁnite set of w eak generators. I f Σ is complete, there is exactly one generator for eac h cone in th e fan Σ. The approach taken uses t h e machinery of Bousfield - Hirschhorn colo calisation. The ﬁrst step is to characterise colocal ob jects; these turn out to b e homotop y shea ves in the sense that c hain complexes o ver diﬀerent op en sets U σ agree on intersections up to quasi- isomorphism. In a second step it is show n that the homotopy category of homotopy sheav es is eq uiv alen t to th e derived category of X . (Nov ember 1, 2018) AMS subje ct classiﬁc ation (2000): primary 18F20, secondary 18E30, 18G55 , 55U35 A ddi tional key wor ds: Colocalisation, Quillen model structures, generators of derived category In tro duction A toric sc heme X = X Σ o v er a comm utativ e ring A comes equipp ed with a preferred co v ering by op en aﬃn e sets. F rom a combinato rial p oin t of v iew X is sp eciﬁed by a ﬁ nite fan Σ in Z n ⊗ R ∼ = R n , and eac h cone σ ∈ Σ corresp ond s to an A -algebra A σ and hence to an op en aﬃn e set U σ = Sp ec( A σ ) ⊆ X . By ev aluating on the op en sets U σ w e see that a c hain complex Y of qu asi- coheren t shea v es on X Σ can th us b e sp eciﬁed by a collecti on of A σ -mo dule c hain complexes Y σ for σ ∈ Σ, sub ject to certain compatibilit y conditions. These in clude, among other things, isomorphisms of chain complexes A τ ⊗ A σ Y σ ∼ = Y τ (0.1) for all pairs of cones τ ⊆ σ in Σ; in the language of sh ea ves, this means that we reco v er Y τ b y restricting the sections Y σ o v er U σ to the smaller op en s et U τ . 1 2 T. H ¨ uttemann The main result of this pap er is that the derived catego ry of X Σ can b e describ ed using collecti ons of chain complexes which do not necessarily satisfy the compatibilit y condition (0.1). In m ore tec h n ical parlance, w e will pro v e that the categ ory of (twiste d) diagrams Σ op ✲ c hain complexes , σ 7→ Y σ admits a “colo cal” mo del structur e w hose homotop y category is equiv alen t to the (unb ounded) derive d category D ( Qco ( X Σ )), cf. Theorem 4.6.1. In the pro- cess w e will also identify explicitly a ﬁn ite set of weak generators of D ( Qco ( X Σ )), cf. Construction 3.3.3. In case Σ is a complete fan , the description is partic- ularly simple: It su ﬃces to tak e one lin e bun dle O ( ~ σ ) for eac h cone σ ∈ Σ, cf. Example 3.3.4 and Corollary 4.6.2 . The coﬁbrant ob jects of the colocal mo del structure are c haracterised b y a w eak form of compatibilit y condition (Theorem 3.4.2): In stead of requiring isomorphisms as in (0.1) w e ask for quasi-isomorph ism s A τ ⊗ A σ Y σ ≃ Y τ for all p airs of cones τ ⊆ σ in Σ. W e call the resulting structure a homotopy she af . Clearly ev ery chain complex of qu asi-co herent sheav es is a h omotop y sheaf. A main ingred ien t of the pro of is that the homotop y category of homo- top y shea v es is nothing but th e (unb ounded) derived category of quasi-coheren t shea v es on X Σ (Theorem 4.5.1); this result is v alid for arbitrary toric schemes deﬁned o v er a commuta tiv e ring A , and holds more generally for qu asi-co mpact A -sc hemes equipp ed with a ﬁn ite semi-separating aﬃne co vering. Note that ev- ery quasi-compact separated sc heme can b e equ ipp ed with such a co v ering. The main tec hnical result is th at homotop y shea v es can b e replaced, up to quasi- isomorphism on the co v ering s ets, by quasi-coheren t sheav es (Lemma 4.4.1). The p ap er illustrates th e ph ilosoph y that h omotop y sh ea ves are a ﬂ exi- ble su bstitute for quasi-coheren t shea ve s w hic h allo w f or easier handling in a homotop y-theoretic setting. W e will use the language of Quillen mo del categories as presented b y D wyer and S p alinski [DS95], Hirsc hhorn [Hir03] and Hovey [Ho v99]. An - other essen tial ingredient is the language of toric v arieties, and the corresp ond - ing com binatorial ob jects (cones and fans); a full treatmen t can b e f ound in Ful ton ’s b o ok [F ul93]. W e will also ha v e o ccasion to use v arian ts of dia- gram categ ories and their asso ciated mo del catego ry structur es as introd uced b y R ¨ ondigs and the author [HR]. On the derived category of a regular toric sc heme 3 1 Chain complexes 1.1 Mo del structure and resolutions Let A denote a r ing w ith u nit. The catego ry Ch A of (p ossibly unb ounded) c hain complexes of left A -mo dules will b e considered with th e pr oje ctive mo del structur e : W eak equiv alences are the q u asi-isomorphisms, and ﬁbrations are those maps whic h are surjectiv e in eac h d egree [Ho v99, Th eorem 2.3.11]. A particularly con v enient feature of this mo del structur e is th at al l chain c om- plexes ar e ﬁbr ant . Also of interest is th e full sub cate gory Ch + A of non-negativ e c h ain complexes. It is a mo del categ ory with w eak equiv alences and coﬁbrations as b efore, but with ﬁbrations the maps whic h are surj ectiv e in p ositiv e d egrees [DS95 , The- orem 7.2]. The catego ry Ch + A is equiv alent to the category sMo d A of simp li- cial A -mo dules; the equiv alence is give n by the r educed chain complex fun ctor N : sMo d A ✲ Ch + A and its in v erse, the Dold - Kan functor W . Giv en a c hain complex C ∈ Ch + A the result of applyin g W is the simplicial A -mo dule N ∋ n 7→ h om Ch A ( N ( A [∆ n ]) , C ) where ∆ n denotes the standard n -simplex. The fun ctors N and W p reserv e and detect w eak equiv alences. Note that we can consider N as a fu nctor with v alues in the categ ory C h A . Similarly , the d eﬁnition of W ab o v e make s sense ev en if C is an u n b ounded c hain complex. In this cont ext, th e follo wing is known to b e tru e: 1.1.1 Lemma. L et N : sMo d A ✲ Ch A and W : Ch A ✲ sMo d A b e deﬁne d as ab ove. (1) The functor N is left Quill en with right adjoint W . (2) The functor N pr eserves and dete cts we ak e quivalenc es. (3) A map f of chain c omplexes induc es an H n -isomorphism f or al l n ≥ 0 if and only if W ( f ) is a we ak e qui v alenc e of simplicial mo dules. ✷ 1.1.2 Lemma. The c ate g ory C h A is a c el lular mo del c ate gory in the sense of [Hir03, § 12]; the set of gener ating c oﬁbr ations is I := { S n − 1 ( A ) ✲ D n ( A ) | n ∈ Z } , and the set of gener ating acyclic c oﬁbr ations is J := { 0 ✲ D n ( A ) | n ∈ Z } . Her e S k ( A ) denotes the c hain c omplex which has A in de gr e e k and is trivial everywher e e lse, and D n ( A ) denotes the chain c omplex which has A in de gr e e s n and n − 1 with b oundary map the identity, and is trivial everywher e else. Pro of. This is the con ten t of [Ho v99, Theorem 2.3.11 ]. ✷ 4 T. H ¨ uttemann 1.1.3 Lemma. L et C ∈ C h A b e a c oﬁbr ant chain c omplex. The c osimplicial chain c omplex N ( A [ ∆ • ]) ⊗ A C , i.e., the c osimplicial obje ct N ∋ n 7→ N ( A [∆ n ]) ⊗ A C , deﬁnes a c osimplicial r e solution [Hir03, § 16.1] of C ; the structur e map to the c onstant c osimplicial obje ct cc ∗ C is induc e d by the u ni q ue map ∆ n ✲ ∆ 0 and the natur al isomorph ism N ( A [∆ 0 ]) ⊗ A C ∼ = C . The n -th latching obje ct is the chain c omplex L n ( N ( A [∆ • ]) ⊗ A C ) = N ( A [ ∂ ∆ n ]) ⊗ A C . Pro of. The category of cosimplicial ob jects in C h A carried a R eedy mo del structures [Hir03, § 15.3]. T o pro v e the Lemma, the n on-trivial thing to verify is that N ( A [∆ • ]) ⊗ A C is coﬁbran t with resp ect to this mo del str ucture. The category of cosimplicial simplicial A -mo dules carries a Reedy mo del structure as w ell. The ob ject A [∆ • ] is kno wn to b e coﬁbrant, so for all n ∈ N the latc hin g map [Hir03, Prop osition 16.3.8 (1)] A [ ∂ ∆ n ] = A [∆ • ] ⊗ ∂ ∆ n = L n A [∆ • ] ✲ A [∆ n ] = A [∆ • ] ⊗ ∆ n is a coﬁbration of simplicial A -mo dules. Hence w e hav e a coﬁbration of chain complexes N ( L n A [∆ • ]) ✲ N ( A [∆ n ]) since the fun ctor N is left Quillen b y Lemma 1.1.1. No w the fu nctor N , b eing a left adjoin t, comm utes with colimits so that the source of this m ap is isomorphic to L n N ( A [∆ • ]). T aking tensor p r o duct with a coﬁbrant c h ain complex pr eserv es coﬁbrations and commutes with colimits, so by applying · ⊗ A C w e see th at the latc hin g map L n ( N ( A [∆ • ]) ⊗ A C ) ∼ = L n N ( A [∆ • ]) ⊗ A C ✲ N ( A [∆ n ]) ⊗ A C of N ( A [∆ • ]) ⊗ A C is a coﬁb r ation as required. ✷ 1.2 Homotop y limits of diagrams of cha in complexes 1.2.1 Deﬁnit ion. Let f : C ✲ D b e a m ap of (p ossibly) unb ounded c hain complexes. The c anonic al p ath sp ac e factorisation of f is the factorisation C i ✲ P ( f ) p ✲ D where the d egree n part of P ( f ) is C n × D n +1 × D n with d iﬀeren tial as sp eciﬁed in the f ollo wing diagram: C n × D n +1 × D n C n − 1 ∂ ❄ × D n − ∂ ❄ ✛ = − f ✲ × D n − 1 ∂ ❄ The map i = (id , 0 , f ) is a c hain h omotop y equiv alence (with h omotopy inv erse giv en b y pr 1 ). The map p = p r 3 is lev elwise surjectiv e, h ence p is a ﬁ bration in C h A (in the pr o jectiv e mo del structure). On the derived category of a regular toric sc heme 5 In what follo ws, w e will b e concerned with diagrams indexed by a ﬁ n ite fan Σ. A c one in a ﬁ nite-dimensional real v ector sp ace N R is the p ositiv e span of a ﬁnite set of ve ctors of N R . A fan is a ﬁnite collection of cones Σ = { σ 1 , σ 2 , . . . , σ k } w hic h is closed und er taking f aces, and satisﬁes the condition that th e intersecti on of tw o cones in Σ is a face of b oth cones. W e also require that all the cones are p ointe d , i.e. , ha v e the trivial cone { 0 } as a face. W e consider a fan Σ as a p oset ord ered by inclusion of cones or, equiv alent ly , as a catego ry with morphisms giv en by inclusion of cones. The trivial cone { 0 } is initial in the category Σ .—By abuse of language, w e refer to dim( N R ) as the dimension of Σ. 1.2.2 Deﬁnit ion. Let Σ denote a ﬁn ite fan. Giv en a diagram of c hain com- plexes C : Σ op ✲ Ch A , σ 7→ C σ w e deﬁne its c anonic al ﬁbr ant r e plac ement P C : Σ op ✲ Ch A inductiv ely as follo ws. T o b egin w ith, set ( P C ) { 0 } = C { 0 } . F or ev ery 1- dimensional cone ρ ∈ Σ factor the map f : C ρ ✲ ( P C ) { 0 } = C { 0 } as C ρ ✲ P ( f ) ✲ ( P C ) { 0 } , see Deﬁnition 1.2.1, and set ( P C ) ρ = P ( f ). Now con tin ue by in duction on the dimension: Giv en a p ositiv e-dimensional cone σ ∈ Σ, factor the map f : C σ ✲ lim τ ⊂ σ ( P C ) τ as C σ ✲ P ( f ) ✲ lim τ ⊂ σ ( P C ) τ , and deﬁne ( P C ) σ = P ( f ). There resulting map of diagrams C ✲ P C is an ob ject w ise injectiv e w eak equiv alence. By constru ction the d iagram P C is ﬁbrant in th e sens e that for all cones σ ∈ Σ, the map ( P C ) σ ✲ lim τ ⊂ σ ( P C ) τ is surj ectiv e (the limit tak en o v er all cones strictly cont ained in σ ). The ter- minology relates to a m od el structure on the catego ry of Σ op -diagrams in Ch A with ob ject wise weak equiv alences and coﬁbrations. The passage from C to P C is fu nctorial in C and maps ob ject wise w eak equiv alences to ob ject wise w eak equiv alences. 1.2.3 Deﬁnit ion. Let Σ denote a ﬁnite f an as b efore, and let C d enote a diagram of c hain complexes C : Σ op ✲ Ch A , σ 7→ C σ . 6 T. H ¨ uttemann The homotopy limit holim ( C ) = holim Σ op ( C ) of C is d eﬁned as holim( C ) := lim P C . The homology mo dules of h olim( C ) are called the hyp er-derive d inverse limits of the diagram C . 1.2.4 Remark. (1) If Σ has a unique (inclusion-)maximal cone µ , then holim( C ) = lim P C ∼ = ( P C ) µ , so C µ ≃ h olim( C ) induced by the quasi-isomorphism C µ ≃ ✲ ( P C ) µ . (2) If D is a Σ-indexed diagram of A -mo du les, viewed as a diagram of c hain complexes concen trated in degree 0, then th e homotop y limit computes higher d eriv ed in v erse limit: h − k holim( D ) ∼ = lim k ( D ) . Of course lim k ( D ) will b e trivial in this case unless 0 ≤ k ≤ n . The homotop y limit construction is inv arian t under w eak equiv alences of diagrams. Th at is, if f : C ✲ D is an ob ject wise qu asi-isomorphism then th e induced map holim( C ) ✲ holim( D ) is a quasi-isomorphism. 1.2.5 Lemma. L et C b e a chain c omplex of A -mo dules, and let con( C ) denote the c onstant Σ op -diagr am with value C . Then C ≃ holim(con( C )) . Pro of. Since Σ op has terminal ob ject { 0 } , it is easy to see that for σ 6 = { 0 } the map C = con( C ) σ ✲ lim τ ⊂ σ con( C ) τ = C is the ident it y . T his means that con( C ) is ﬁbr ant in the mo del structure men- tioned ab ov e. Hence the canonical map con( C ) ✲ P con( C ) is a w eak equiv- alence of ﬁbrant diagrams. Consequently , the righ t Quillen fun ctor “in v erse limit” yields a quasi-isomorphism C = lim con( C ) ∼ ✲ lim P con( C ) = holim(con( C )) b y application of Bro wn ’s L emm a [DS95, dual of Lemm a 9.9]. ✷ 2 Preshea v es and line bundles on toric sc hemes 2.1 T oric sc hemes Let N ∼ = Z n denote a lattice of rank n . W rite N R = N ⊗ R ∼ = R n . There is an ob vious inclusion N ⊆ N R giv en by identifying p ∈ N with p ⊗ 1 ∈ N R . W e On the derived category of a regular toric sc heme 7 denote the d ual lattice of N by th e letter M , and write M R = M ⊗ R . Clearly M ⊆ M R , and M R is the du al v ector space of N R . Let Σ b e a ﬁnite fan in N R , cf. § 1.2. In addition to the conditions listed there, w e r equire eac h cone in Σ to b e r ational , i.e. , spanned by ﬁnitely many vect ors in N ⊂ N R W e write Σ(1) for the set of 1-cones in Σ. Similarly , if σ ∈ Σ is any cone we write σ (1) for the s et of 1-cones of Σ con tained in σ . Every 1-cone ρ is spanned by a unique primitive elemen t n ρ ∈ N ; the set { n ρ | ρ ∈ σ (1) } is called the set of primitive generators of σ ∈ Σ. A cone σ ∈ Σ then giv es r ise to a p oin ted monoid S σ = { f ∈ M | ∀ ρ ∈ σ (1) : f ( n ρ ) ≥ 0 } + (2.1) where the su bscript “+” means ad d ing a new elemen t ∗ whic h acts lik e a + ∗ = ∗ + a = ∗ for all a ∈ S σ ; this conv enti on w ill b e useful when describin g restriction functors in § 2.4. Th e cone σ th us determines an A -algebra A σ = ˜ A [ S σ ] where A is an y ring with unit (p ossibly non-comm utativ e), and ˜ A [ S σ ] is the reduced monoid algebra A [ S σ ] / A [ ∗ ] of S σ . In case A is a comm utativ e ring, we set U σ = Sp ec( A σ ), and deﬁne the A -sc heme X Σ as the un ion S σ ∈ Σ U σ . By constru ction, U σ ∩ U τ = U σ ∩ τ for all cones σ, τ ∈ Σ. Th e sc heme X Σ is called the toric scheme asso ciate d to Σ. If A is an algebraically closed ﬁ eld, X Σ is an algebraic v ariet y o ver A . See Ful ton [F ul93] for a full treatmen t of toric v arieties, and more details of the construction. 2.2 Preshea v es on toric sc hemes As b efore let Σ denote a ﬁnite fan of rational p ointe d cones, and let A den ote a (p ossibly n on-comm u tativ e) ring w ith unit. F or comm utativ e A this data deﬁnes an A -sc h eme X Σ as indicated in § 2.1. But ev en if A is non -commutati ve w e will sp eak of preshea v es on X Σ : 2.2.1 Deﬁnit ion. Th e categ ory Pre (Σ) of pr eshe aves on the toric scheme X Σ deﬁne d over A has ob jects the diagrams C : Σ op ✲ Ch A , σ 7→ C σ together with additional data that equip eac h entry C σ with the structure of an ob ject of Ch A σ , and such that for eac h inclusion τ ⊆ σ in Σ the s tructure map C σ ✲ C τ is A σ -linear. A particularly useful example of a presheaf is the f unctor O = O ( ~ 0) : Σ op ✲ Ch A , σ 7→ A σ (see § 2.5) where we consider the algebra A σ as an A σ -mo dule c hain complex concen trated in degree 0. 8 T. H ¨ uttemann 2.3 Mo del structures The category Pre (Σ) deﬁned ab ov e is an example of a t wisted diagram cat- egory in th e sense of [HR, § 2.2], formed with r esp ect to an adjun ction b undle similar to the one describ ed in Example 2.5.4 of lo c.cit. (one n eeds to replace “mo dules” with “c hain complexes of mo du les”). W e thus kno w th at the cate- gory Pre (Σ) has t w o Quillen mo del structures, called the f -structure and the c -structure, resp ectiv ely . In b oth cases the w eak equiv alences are th e ob j ect- wise qu asi-isomorphisms. Fibrations an d coﬁbrations are diﬀeren t, as exp lained b elo w. 2.3.1. The f -st ructure [HR, Theorem 3.3.5]. In this mo del structure, a map f : C ✲ D in Pre (Σ) is a coﬁbration if and only if all its comp onents f σ , σ ∈ Σ, are coﬁbrations in their resp ectiv e categories. Fibrations can b e characte rised usin g matching c omplexes . F or σ ∈ Σ deﬁne M σ ( C ) := lim τ ⊂ σ C τ , the limit tak en in th e category Ch A σ o v er all τ ∈ Σ prop erly con tained in σ . T hen f : C ✲ D is a ﬁ bration if and only if for all σ ∈ Σ the ind uced map ι : C σ ✲ M σ ( C ) × M σ ( D ) D σ is a ﬁ bration in Ch A σ ( i.e. , if ι is lev elwise surjectiv e). 2.3.2 Lemma. L et C b e an obje ct of Pre (Σ) . The c anonic al ﬁbr ant r eplac e- ment P C of C as deﬁne d i n 1.2.2 yields an f -ﬁbr ant obje ct of Pre (Σ) . Pro of. The im p ortan t thing to note is th at for eac h inclusion of cones τ ⊆ σ there is an in clusion of algebras A σ ⊆ A τ , so C τ can b e considered as an A σ - mo dule c hain complex b y restriction of s calars. It is then a matter of tracing the deﬁnitions to see that P C ∈ Pre (Σ). Since ﬁbrations are surjections in all relev an t categorie s of c hain complexes, and since sur jectivit y can b e d etected after restricting scalars to the ground rin g A , the Lemma follo ws. ✷ 2.3.3. The c -structure [HR, Theorem 3.2.13]. In th is mo del structure, a map f : C ✲ D in Pre (Σ) is a ﬁb ration if and only if all its comp onen ts f σ , σ ∈ Σ, are ﬁbrations in their resp ectiv e categories ( i.e. , th e comp onents are surj ectiv e in all c hain lev els). Note that all ob jects of Pre (Σ) are c -ﬁbr an t. Coﬁbrations can b e charac terised using latching c omplexes . F or σ ∈ Σ deﬁn e L σ ( C ) := colim τ ⊃ σ A σ ⊗ A τ C τ , the colimit b eing taken o ver all τ ∈ Σ pr op erly con taining σ . Th en f : C ✲ D is a coﬁb ration if and only if for all σ ∈ Σ the map L σ ( D ) ∪ L σ ( C ) C σ ✲ D σ is a coﬁbration in Ch A σ . In particular, D is coﬁbrant if and only if for all σ ∈ Σ the map L σ ( D ) ✲ D σ is a coﬁbration. F or τ ∈ Σ and P ∈ Ch A w e deﬁne the diagram F τ ( P ) : σ 7→ ( 0 if σ 6⊆ τ A σ ⊗ A P if σ ⊆ τ On the derived category of a regular toric sc heme 9 together with the evid ent structure maps induced by the v arious inclusions of A -algebras A σ ✲ A σ ′ . 2.3.4 Lemma. The c -structur e is a c el lular mo del structur e in the sense of [Hir03, § 12.1]. A set of gener ating c oﬁbr ations is give n by I c := { F τ ( i ) | i ∈ I , τ ∈ Σ } wher e I is as in L e mma 1.1.2. Similarly, a set of gener ating acyclic c oﬁbr ations is J c := { F τ ( j ) | j ∈ J, τ ∈ Σ } with J as in L emma 1.1.2. Pro of. This follo ws by direct insp ectio n from L emma 1.1.2. W e omit th e details. ✷ 2.3.5 Lemma. Supp ose C ∈ Pre (Σ) is a c -c oﬁbr ant obje ct (2.3.3). Then A [∆ • ] ⊗ C : Σ op ✲ Ch A , σ 7→ A [∆ • ] ⊗ A C σ is a c osimplicial r esolution of C . Pro of. This follo ws from the fact that A [∆ • ] is Reedy coﬁbrant cosimplicial simplicial mo dule, and the fact the taking tensor p r o ducts comm utes with col- imits. The d etails are similar to L emm a 1.1.3. ✷ 2.4 Restriction and extension b y zero W e w ill use the n otatio n of § 2.1. Let Σ d enote a ﬁ nite fan in N R . Giv en a cone ρ ∈ Σ we deﬁn e the star of ρ as st( ρ ) = { σ ∈ Σ | ρ ⊆ σ } . 2.4.1. A 1-cone ρ ∈ Σ(1) determines a fan Σ /ρ in an ( n − 1)-dimensional v ector space as follo ws . Let Z ρ denote the sub-lattice of N generated by the sp an of ρ . Then ¯ N = N / Z ρ is a lattic e of ran k n − 1. Giv en an y cone σ ∈ st( ρ ) the image ¯ σ of σ u nder th e pr o jection N R ✲ ¯ N R is a p ointed rational p olyhedral cone, and b y v arying σ ∈ st( ρ ) we obtain a fan Σ /ρ of a toric scheme denoted X Σ /ρ = V ρ . Note that this new fan is isomorphic, as a graded p oset, to st( ρ ).—If A = C then V ρ is the closure of the orb it in X Σ corresp onding to ρ , and its is kn o wn that V ρ has co dimens ion 1 in X Σ . F rom no w on we w ill assume that the fan is r e gular , that is, eac h cone of Σ is spanned by part of a Z -basis (whic h d ep ends on the cone u nder consideration) of the lattice N . Th is condition is equiv alen t to the requirement that the toric v ariet y X Σ deﬁned o v er C is s mo oth. 10 T. H ¨ uttemann Giv en ρ ∈ Σ (1) and σ ∈ st( ρ ) let n 1 , . . . , n k denote the p rimitiv e elemen ts of the 1-cones contai ned in σ . Supp ose that n k ∈ ρ (whic h can b e achiev ed by ren umb ering). Let ¯ σ den ote the image of σ in ¯ N R = ( N / Z ρ ) R as b efore, and denote the images of the n j in ¯ N b y ¯ n j . Since σ is r egular, the ¯ n 1 , . . . , ¯ n k − 1 form part of a b asis of the lattice ¯ N , and are p recisely the primitiv e elemen ts of the 1-cones con tained in ¯ σ . S in ce the lattice dual of ¯ N is M ∩ ρ ⊥ , we see that S ¯ σ ∼ = { f ∈ M | f ( n j ) ≥ 0 for 1 ≤ j ≤ k − 1 , and f | ρ = 0 } + (compare to the description (2.1) of the monoid S σ ). Of course f | ρ = 0 is equiv alen t to f ( n k ) = 0.—W e obtain a sur jectiv e map of p oin ted monoids S σ ✲ S ¯ σ , f 7→ ( f if f | ρ = 0 ∗ else (2.2) and, by linearisation, a corresp onding surjectiv e map of A -algebras A σ ✲ A ¯ σ . (2.3) F or comm utativ e A this map exhibits Sp ec( A ¯ σ ) = V ρ ∩ U σ as a closed sub set of U σ ⊆ X Σ . 2.4.2. Recall th at the fan Σ /ρ of V ρ is isomorphic, as a p oset, to st( ρ ) ⊆ Σ. Th us an ob ject C ∈ Pre (Σ /ρ ) can b e consid ered as a functor deﬁned on the p oset st ( ρ ) op , and we deﬁne a diagram ζ ( C ) on Σ op b y setting ζ ( C ) σ := ( 0 if ρ 6⊆ σ C σ if ρ ⊆ σ with stru cture maps induced b y those of C . F or σ ∈ st( ρ ) w e let A σ act on ζ ( C ) σ via the su rjection A σ ✲ A ¯ σ . In this w a y , ζ ( C ) b ecomes an ob ject of Pre (Σ), called the extension by zer o of C . By direct compu tatio n we ve rify: 2.4.3 Lemma. F or C ∈ Pre (Σ /τ ) ther e is an e quality holim Σ op C = h olim st( ρ ) op ζ ( C ) wher e we c onsider the pr eshe aves on left and right hand side as diagr ams with values in the c ate gory of A -mo dules to form the homotopy limits (1.2.3). ✷ 2.4.4. Th e extension functor ζ : Pre (Σ /ρ ) ✲ Pre (Σ) has a left adjoint ε , called r estriction to V ρ . Its eﬀect on C ∈ Pre (Σ) is the follo w ing: As a diagram of A -mod ule c hain complexes, ε ( C ) is given b y ε ( C ) : st( ρ ) op ✲ Ch A , M 7→ A ¯ σ ⊗ A σ C σ , the tensor pro du ct f orm ed with resp ect to the su r jection A σ ✲ A ¯ σ . W e also denote ε ( C ) b y C | V ρ . On the derived category of a regular toric sc heme 11 2.5 Line bundles and t wisting As b efore, let Σ d en ote a regular fan in N R , and r ecall that ev ery 1-cone ρ ∈ Σ is generated by a unique p rimitiv e element n ρ ∈ N . 2.5.1 C onstruction. Fix a vect or ~ k = ( k ρ ) ρ ∈ Σ(1) ∈ Z Σ(1) . Since Σ is regular w e can ﬁnd for eve ry cone σ ∈ Σ an integral linear form f σ : N R ✲ R , uniqu e up to add ing a linear form v anishing on σ , whic h satisﬁes f σ ( n ρ ) = − k ρ for ev ery 1-cone ρ con tained in σ . If τ ∈ Σ is another cone, then f τ and f σ agree on τ ∩ σ (since they agree on 1-cones of τ ∩ σ ), and b oth ± ( f τ − f σ ) are elements of S τ ∩ σ . Consequ ently we ha v e f τ + S τ ∩ σ = f σ + S τ ∩ σ ; in particular, th e set f σ + S σ dep ends on σ and ~ k only (and not the sp eciﬁc c hoice of fu nction f σ ). W e th us obtain a wel l-deﬁned functor O ( ~ k ) : Σ op ✲ A -mo d , τ 7→ ˜ A [ f τ + S τ ] , considered as a d iagram of c hain complexes concentrate d in degree 0. Structur e maps are giv en by inclusions. W e call O ( ~ k ) the line bund le determine d by ~ k . Note that O ( ~ k ) is, in fact, an ob ject of Pre (Σ) (as u sual, w e think of mo dules as c hain complexes concen tr ated in degree 0): T h e action of S τ on f τ + S τ extends to an A τ -mo dule s tructure of ˜ A [ f τ + S τ ], and for ρ ⊆ τ the stru cture maps O ( ~ k ) τ ✲ O ( ~ k ) ρ are easily seen to b e linear with resp ect to th e ring A τ . In eﬀect the v ector ~ k ∈ Z Σ(1) , or rather the collection of the f σ , determines a piecewise linear function on the und erlying space of Σ, and w e ha v e giv en a com binatorial description of the asso ciated line bund le on X Σ . 2.5.2 E xa mple. Let Σ denote the fan of the pro jectiv e line; it is a fan in R with 1-cones the n on-p ositiv e and n on-negativ e r eal n umber s , r esp ectiv ely . F or a vecto r ~ k = ( k 1 , k 2 ) ∈ Z 2 the diagram O ( ~ k ) then has the f orm T k 1 · A [ T − 1 ] ⊂ ✲ A [ T , T − 1 ] ✛ ⊃ T − k 2 · A [ T ] whic h, as a quasi-coherent sh eaf, is isomorphic to the algebraic geometers’ sheaf O P 1 ( k 1 + k 2 ). In general, r ecall that S τ = { g ∈ M | ∀ ρ ∈ τ (1) : g ( n ρ ) ≥ 0 } + . Th e map g 7→ f τ + g deﬁnes an S τ -equiv arian t b ij ection from S τ to B ( ~ k ) τ := f τ + S τ = { g ∈ M | ∀ ρ ∈ τ (1) : g ( n ρ ) ≥ − k ρ } + . (2.4) In particular, O ( ~ k ) τ is a fr e e A τ -mo dule of r ank 1. F rom the construction it is clear that given another v e ctor ~ ℓ ∈ Z Σ(1) with ~ ℓ ≤ ~ k (c omp onentwise ine quality) we have a c anonic al inje ction (inclusion map) O ( ~ ℓ ) ✲ O ( ~ k ). 12 T. H ¨ uttemann 2.5.3 Lemma. Given a 1 -c one ρ ∈ Σ(1) and a c one σ pr op erly c ontaining ρ , let τ ∈ Σ denote the maximal fac e of σ not c ontaining ρ (this is wel l-deﬁne d sinc e Σ is r e gular). L et f ∈ M b e a line ar f orm which takes the value 1 on the primitive gener ator of ρ , and takes the value 0 on the primitive gener ators of τ . Then f ∈ S σ , and S τ = S σ + Z f . In other wor ds, the monoid S τ is obtaine d fr om S σ by i nv e rting the element f . Pro of. Let n 1 , . . . , n k b e th e primitive generators of τ , and let n k +1 b e the primitiv e generator of ρ . A liner form g ∈ M is in S σ if and only if it ev aluates to n on-negativ e n umb er s on primitive generators of σ , i.e. , if and only if g ( n i ) ≥ 0 for 1 ≤ i ≤ k + 1. So f ∈ S σ as claimed. Similarly , we ha v e g ∈ S τ if and only if g ( n i ) ≥ 0 f or 1 ≤ i ≤ k . Thus we ha v e th e in clus ion S τ ⊇ S σ + Z f . F or the r ev er s e in clus ion, let g ∈ S τ . Then  g − g ( n k +1 ) · f  ( n k +1 ) = 0, so g =  g − g ( n k +1 ) · f  + g ( n k +1 ) · f is an element of S σ + Z f as claimed. ✷ 2.5.4 C onstruction. Let ~ k ∈ Z Σ(1) and ρ ∈ Σ(1) b e given. Supp ose that k ρ = 0. Th e v ector ~ k deﬁnes a line bundle on V ρ = X Σ /ρ corresp onding to a v ector ~ ℓ ∈ Z (Σ /ρ )(1) describ ed as follo ws. Since Σ /ρ is isomorphic to st ( ρ ) we can write ~ ℓ = ( ℓ σ ) wh ere σ ranges o ve r the 2-dimensional cones in st( ρ ). F or suc h a cone σ let τ denote the 1-cone contai ned in it diﬀerent from ρ , an d set ℓ σ = k τ . F or ρ ∈ Σ(1) recall that the fan of V ρ is a fan in ( N/ Z ρ ) R ∼ = N/ R ρ , and that N/ Z ρ and M ∩ ρ ⊥ are dual to eac h other. Let ~ k ∈ Z Σ(1) with k ρ = 0. Giv en a cone ¯ σ in the quotien t fan, corresp onding to σ ∈ st ( ρ ), the mo dule O ( ~ ℓ ) ¯ σ is the reduced free A -mo dule with basis n f ∈ M ∩ ρ ⊥ | f ( n τ ) ≥ − k τ for τ ∈ σ (1) \ { ρ } o + = n f ∈ M | f ( n ρ ) = 0 and f ( n τ ) ≥ − k τ for τ ∈ σ (1) \ { ρ } o + . (2.5) Using this explicit description, it is readily v eriﬁed that O ( ~ ℓ ) ¯ σ is isomorph ic to A ¯ σ ⊗ A σ O ( ~ k ) σ , wh ere the tensor pro duct is formed with resp ect to the su rjection A σ ✲ A ¯ σ from (2.3). In fact, A ¯ σ ⊗ A σ O ( ~ k ) σ is the redu ced free A -mo du le on the p ointe d s et S ¯ σ ∧ S σ B ( ~ k ) σ , formed with r esp ect to the sur jection S σ ✲ S ¯ σ from (2.2), which is isomorphic to the set sp eciﬁed in (2.5) ab o ve . 2.5.5 C orollary . F o r ρ ∈ Σ(1) and ~ k ∈ Z Σ(1) with k ρ = 0 , let ~ ℓ denote the ve ctor describ e d in Construction 2.5.4. Then ther e i s an isomorphism O ( ~ k ) | V ρ ∼ = O ( ~ ℓ ) of obje cts in Pre (Σ /ρ ) . In wor ds, the r estriction of the line bund le O ( ~ k ) ∈ Pre (Σ) to X Σ /ρ = V ρ is the line bund le O ( ~ ℓ ) ∈ Pre (Σ /ρ ) . ✷ On the derived category of a regular toric sc heme 13 Note that (2.5) also sp eciﬁes an A -basis of the mo du le ζ  O ( ~ ℓ )  σ in the ex- tension by zero. Using (2.2) we can giv e an explicit d escrip tion of the S σ -action on this s et: The elemen t a ∈ S σ acts by ad d ition if a ( n ρ ) = 0, and acts as th e zero op er ator if a ( n ρ ) 6 = 0. 2.5.6 Prop osition. L et ρ b e a 1 -c one in Σ , and let ~ k ∈ Z Σ(1) b e a ve ctor with k ρ = 0 . Then the c oﬁbr e of the inclusion map i : O ( ~ k − ~ ρ ) ✲ O ( ~ k ) is isomorphic to the extension by zer o of the r e striction of O ( ~ k ) to V ρ . He r e ~ ρ ∈ Z Σ(1) is the ρ -th unit ve c tor, i.e., the ve ctor with ρ - c omp onent 1 and al l other entries zer o. Pro of. Let C denote the coﬁbre of i , and let E = ζ  ε ( O ( ~ k ))  denote the extension b y zero of the restriction. Let σ ∈ Σ \ st( ρ ) so that ρ 6⊆ σ . W e ha ve E σ = 0 by deﬁnition of extension, and w e also hav e C σ = 0 s ince O ( ~ k ) σ = O ( ~ k + e ρ ) σ . So th e σ -comp onents of C and E coincide in this case. No w let σ ∈ st( ρ ). W e kn ow that C σ is a free A -mo d ule with p oint ed basis giv en b y the coﬁb r e of the in clus ion of p oint ed sets B ( ~ k − ~ ρ ) σ ✲ B ( ~ k ) σ , cf. (2.4) for notation. Coﬁbres of p oin ted sets can b e computed b y taking complemen ts and adding a base p oin t. It follo ws by insp ection that C σ has a p oin ted A -basis given by the set describ ed in (2.5 ) which is also a p oin ted A -basis of E σ b y the discus sion b efore. Hence the σ -comp onents of C and E agree in this case as well. The reader can chec k that th e structure maps of C and E corresp ond under these iden tiﬁcations. ✷ 2.5.7 Deﬁnit ion. Give n ~ k ∈ Z Σ(1) and C ∈ Pre (Σ), w e deﬁn e the ~ k -th twist of C , d enoted C ( ~ k ), by C ( ~ k ) σ = O ( ~ k ) σ ⊗ A σ C σ with s tr ucture maps indu ced by those of C and O ( ~ k ). This deﬁnition corresp onds to tensoring a quasi-coheren t sh eaf with the line bund le O ( ~ k ), exp ressed in the language of diagrams. It is easy to c hec k that C ( ~ k )( ~ ℓ ) ∼ = C ( ~ k + ~ ℓ ). F or σ -comp onents this comes from the isomorphism O ( ~ k ) σ ⊗ A σ O ( ℓ ) σ ∼ = O ( ~ k + ~ ℓ ) σ . Since C ( ~ 0) ∼ = C , this pro v es: 2.5.8 Lemma. L et ~ k ∈ Z Σ(1) . The twisting fu nc tor C 7→ C ( ~ k ) is a self- e quivalenc e of Pre (Σ) with inverse C 7→ C ( − ~ k ) . ✷ 14 T. H ¨ uttemann F or σ ∈ Σ there is an S σ -equiv arian t bijection B ( ~ k ) σ ✲ S σ , cf. (2.4). Note that this bijection is not canonical: It ma y b e mo diﬁed by adding or subtr acting a ﬁxed in ve rtible elemen t of S σ . By p assing to fr ee A -mo dules, w e obtain a non-canonical isomorph ism O ( ~ k ) σ ∼ = A σ and consequently a non-canonical isomorphism C ( ~ k ) σ ∼ = C σ . This implies that t w isting pr eserv es and detects w eak equiv alences of pr eshea ves, pr eserv es c -ﬁbrations (ob j ectwise su rjections), and preserves f -coﬁbrations (ob ject wise coﬁbrations). F rom Lemma 2.5.8 we th us conclude: 2.5.9 C orollary . L et ~ k ∈ Z Σ(1) . (1) The twisting f u nctor C 7→ C ( ~ k ) is a left and right Quillen functor with r esp e ct to the c -structur e; in p articular, if C ∈ Pre (Σ) is c -c oﬁbr ant so is C ( ~ k ) . (2) The twisting f u nctor C 7→ C ( ~ k ) is a left and right Quillen functor with r esp e ct to the f - structur e; in p articular, if C ∈ Pre (Σ) i s f -ﬁbr ant so is C ( ~ k ) . ✷ 2.5.10 Lemma. F or ~ k ∈ Z Σ(1) and C ∈ Pre (Σ) ther e ar e isomorphisms hom Pre (Σ)  O ( ~ k ) , C  ∼ = hom Pre (Σ)  O , C ( − ~ k )  ∼ = lim C ( − ~ k ) . These isomorphisms ar e natur al i n C . Pro of. Th is follo w s from insp ection, using the trivial f act that O σ = A σ is the free A σ -mo dule of rank 1. ✷ 3 Shea ve s, homotop y shea v es, and colo calisation 3.1 Shea ves and homotop y shea ves 3.1.1 Deﬁnit ion. An ob ject C ∈ Pre (Σ) is called a (strict) she af if for all inclusions σ ⊆ τ in Σ the map A σ ⊗ A τ C τ ✲ C σ , (3.1) adjoin t to the str u cture map C τ ✲ C σ , is an isomorphism. W e call C a homoto py she af if th e map (3.1) is a quasi-isomorphism for all σ ⊆ τ in Σ. Ev ery strict sheaf is a homotop y sheaf. Im p ortan t examples of strict sh ea ves are the fun ctors O ( ~ k ) deﬁned in § 2.5. 3.1.2 Lemma. The notion of a homotopy she af is homotopy invariant: Given a we ak e qu ivalenc e C ✲ D in Pre (Σ) , the pr eshe af C is a homotopy she af if and only if D is a homotopy she af. On the derived category of a regular toric sc heme 15 Pro of. F or all σ ⊆ τ in Σ the monoid S σ is obtained from S τ b y inv erting an elemen t of S τ , cf. [F ul93, § 2.1, Prop osition 2], s o that A σ is a lo calisat ion of A τ . Since lo calisation is exact b oth vertica l maps in th e follo wing square diagram are quasi-isomorphisms: A σ ⊗ A τ C τ ✲ C σ A σ ⊗ A τ D τ ❄ ✲ D σ ❄ This p ro v es that the upp er h orizon tal map is a qu asi-isomorphism if and only if the lo wer h orizon tal m ap is a qu asi-isomorphism. ✷ 3.1.3 Lemma. Supp ose we have a short exact se quenc e 0 ✲ B ✲ C ✲ D ✲ 0 of obje cts in Pre (Σ) . Then i f two of the thr e e pr eshe aves B , C and D ar e homoto py she aves, so is the thir d. Pro of. Let σ ⊆ τ b e an inclusion of cones in Σ. Consider the follo wing comm utativ e ladder diagram: 0 ✲ A σ ⊗ A τ B τ ✲ A σ ⊗ A τ C τ ✲ A σ ⊗ A τ D τ ✲ 0 0 ✲ B σ ❄ ✲ C σ ❄ ✲ D σ ❄ ✲ 0 The b otto m r o w is exact by hyp othesis. Since A σ is a lo calisation of A τ the top ro w is exact as well. Moreo ver, by hyp othesis tw o of the vertica l maps are quasi-isomorphisms. The ﬁve lemma, applied to the asso ciated inﬁnite ladder diagram of homology m od ules, guarante es that the third vertica l map is a qu asi- isomorphism as well . ✷ Since a r etract of a quasi-isomorphism is a quasi-isomorphism, w e also ha v e: 3.1.4 Lemma. Supp ose that C is a r etr act, in the c ate g ory Pre (Σ) , of the homoto py she af D . Then C is a homotopy she af. ✷ 3.1.5 Prop osition. L et ρ b e a 1 -c one in Σ . (1) The r estriction functor ε : Pre (Σ) ✲ Pre (Σ /ρ ) , deﬁne d in § 2.4.4, is a left Quill en functor with r esp e ct to the c -structur e (2.3.3). (2) The fu nc tor ε pr eserves strict she aves and f - c oﬁbr ant (2.3.1) homotopy she aves. 16 T. H ¨ uttemann Pro of. P art (1) is true s ince the r ight adjoint ζ of ε clearly pr eserv es ﬁ brations and acyclic ﬁb rations in the c -structure. F or (2) s u pp ose that C ∈ Pre (Σ) is a strict s heaf. An inclusion of cones ¯ σ ⊆ ¯ τ in Σ /ρ corresp onds to an in clusion of cones σ ⊆ τ in st( ρ ). T he commuta tiv e diagram A σ ✲ A ¯ σ A τ ✻ ✲ A ¯ τ ✻ then in duces the top horizon tal isomorphism in th e follo wing diagram: A ¯ σ ⊗ A ¯ τ ε ( C ) ¯ τ = A ¯ σ ⊗ A ¯ τ A ¯ τ ⊗ A τ C τ ∼ = ✲ A ¯ σ ⊗ A σ A σ ⊗ A τ C τ ε ( C ) ¯ σ ❄ = ✲ A ¯ σ ⊗ A σ C σ ∼ = ❄ (3.2) The r igh t v ertical map is an isomorp h ism as C is a strict s heaf. Hence the left v ertical map is an isomorph ism as we ll, w hic h p ro v es that ε ( C ) is a strict sh eaf as claimed. No w supp ose that C is an f -coﬁbran t homotopy sheaf. W e wan t to pro v e that ε ( C ) is an f -coﬁbr an t homotop y sheaf as w ell. Fix σ ∈ s t ( ρ ). Since C is f -coﬁbran t we know that C σ is coﬁbrant in the category of A σ -mo dule c hain complexes. Hence ε ( C ) ¯ σ = A ¯ σ ⊗ A σ C σ is coﬁbr an t in the catego ry of A ¯ σ -mo dule chain complexes. As this is true for all σ ∈ st( ρ ) w e kn o w that ε ( C ) is f -coﬁbr an t. W e are left to c hec k that for all σ ⊆ τ in st( ρ ) the left v ertical map in the diagram (3.2) is a wea k equiv alence. B y hypothesis, th e map A σ ⊗ A τ C τ ✲ C σ is a weak equiv alence of coﬁbrant ob jects. Hence the right vertic al map of diagram (3.2), obtained by base change , is a w eak equiv alence as well, p ro ving the assertion. ✷ 3.2 Colo cal ob jects and colo cal equiv alences 3.2.1 Nota tion. F or ~ k ∈ Z r and ℓ ∈ Z w e let O ( ~ k )[ ℓ ], cf. § 2.5. 1 , den ote th e sheaf O ( ~ k ) considered as a chain complex concentrate d in c hain degree ℓ . W e denote by ˆ O ( ~ k ) the c -coﬁbrant replacemen t ˆ O ( ~ k ) ∼ ✲ ✲ O ( ~ k ) with source consist- ing of b oun ded c hain complexes of ﬁn itely generated free mo dules; more sp eciﬁ- cally , we us e a mapp ing cylinder factorisation construction dual to the canonical path sp ace factorisation discus s ed earlier. Note that ˆ O ( ~ k )[ ℓ ] ✲ O ( ~ k )[ ℓ ] then is a c -coﬁbrant replacemen t as well with source a strict sheaf in the sen se of Deﬁnition 3.1.1. On the derived category of a regular toric sc heme 17 F or a giv en c hain complex M of A -bimo du les, we deﬁne the pr esheaf M ⊗ O ( ~ k )[ ℓ ] : σ 7→ M ⊗ A O ( ~ k )[ ℓ ] σ , and similarly for ˆ O ( ~ k )[ ℓ ]. Th e resulting presh ea ves are in fact strict sh ea ves as is easily chec ke d b y insp ection. 3.2.2 Deﬁnit ion. A map f : C ✲ D in Pre (Σ) is called an ˆ O ( ~ k )[ ℓ ] -c olo c al e quivalenc e , cf. [Hir03, Deﬁnition 3.1.8 (1)], if the ind u ced map hom Pre (Σ) ( N A [∆ • ] ⊗ ˆ O ( ~ k )[ ℓ ] , C ) ✲ hom Pre (Σ) ( N A [∆ • ] ⊗ ˆ O ( ~ k )[ ℓ ] , D ) is a we ak homotopy equiv alence of simplicial sets. Here N A [∆ • ] is the cosimpli- cial A -bimo du le c hain complex n 7→ N A [∆ n ] with N the reduced c h ain complex functor. 3.2.3 Prop osition. Fix ℓ ∈ Z and ~ k ∈ Z r . A map f : C ✲ D of obje c ts i n Pre (Σ) is an ˆ O ( ~ k )[ ℓ ] -c olo c al e q u ivalenc e if and only if the c orr esp onding map of A -mo dule chain c omplexes holim C ( − ~ k ) ✲ holim D ( − ~ k ) induc es isomorphisms on homolo gy in de gr e es ≥ ℓ . Pro of. Le t C ∼ ✲ P C denote the canonical f -ﬁbr an t replacement for C , cf. 1.2.2, and r ecall that h olim C = lim P C . Similarly , we ha v e a wea k equiv a- lence D ∼ ✲ P D . T he map f indu ces a corresp ondin g map ˜ f : P C ✲ P D . Consider the h uge diagram of Fig. 1 . W e claim that the ve rtical maps are wea k equiv alences or isomorph isms of simplicial sets as marked. W e list the reasons for eac h of the squares: Squar e 1: W e kn o w that N A [∆ • ] ⊗ ˆ O ( ~ k ) is a cosimplicial resolution of ˆ O ( k ) with resp ect to the c -structur e of Pre (Σ), and that C , P C , D and P D are c -ﬁbrant . It follo ws f r om [Hir03, C orollary 16.5.5 (2)] that the vertical maps are w eak equ iv alences. Squar e 2: This follo w s immediately from [Hir03, Corollary 16.5.5 (1)] since P C and P D are f -ﬁbr an t b y construction, and since the map N A [∆ • ] ⊗ O ( ~ k )[ ℓ ] ✲ N A [∆ • ] ⊗ ˆ O ( ~ k )[ ℓ ] is a Reedy w eak equiv alence of cosimplicial resolutions for the f -structure of Pre (Σ). Squar e 3: Use adjointness of tensor p ro duct and hom complex for eac h en try of the diagrams inv olved. Note that O ( ~ k )[ ℓ ] is a c hain-complex with non-trivial en tries in d egree ℓ only . Squar e 4: This uses the isomorphism of fu nctors from Lemma 2.5.10. 18 T. H ¨ uttemann hom Pre (Σ)  N A [∆ • ] ⊗ ˆ O ( ~ k )[ ℓ ] , C  f ∗ ✲ hom Pre (Σ)  N A [∆ • ] ⊗ ˆ O ( ~ k )[ ℓ ] , D  1 hom Pre (Σ)  N A [∆ • ] ⊗ ˆ O ( ~ k )[ ℓ ] , P C  ∼ ❄ ˜ f ∗ ✲ hom Pre (Σ)  N A [∆ • ] ⊗ ˆ O ( ~ k )[ ℓ ] , P D  ∼ ❄ 2 hom Pre (Σ)  N A [∆ • ] ⊗ O ( ~ k )[ ℓ ] , P C  ∼ ✻ ˜ f ∗ ✲ hom Pre (Σ)  N A [∆ • ] ⊗ O ( ~ k )[ ℓ ] , P D  ∼ ✻ 3 hom C h A  N A [∆ • ] , hom ( O ( ~ k )[ ℓ ] , P C )  ∼ = ❄ ˜ f ∗ ✲ hom C h A  N A [∆ • ] , hom ( O ( ~ k )[ ℓ ] , P D )  ∼ = ❄ 4 hom C h A  N A [∆ • ] , lim( P C )( − ~ k )[ − ℓ ]  ∼ = ❄ ˜ f ∗ ✲ hom C h A  N A [∆ • ] , lim( P D )( − ~ k )[ − ℓ ]  ∼ = ❄ 5 hom C h A  N A [∆ • ] , holim C ( − ~ k )[ − ℓ ]  ∼ ❄ ˜ f ∗ ✲ hom C h A  N A [∆ • ] , holim D ( − ~ k )[ − ℓ ]  ∼ ❄ 6 W (holim C ( − ~ k )[ − ℓ ]) = ❄ W (holim f ( − ~ k )) ✲ W (holim D ( − ~ k )[ − ℓ ]) = ❄ Figure 1: Diagram Squar e 5: R ecall that C ✲ P C is an f -ﬁbrant replacemen t, hence so is its ( − ~ k )th t wist C ( − ~ k ) ✲ ( P C )( − ~ k ) b y Corollary 2.5.9. But C ( − ~ k ) ✲ P ( C ( − ~ k )) is another f -ﬁbrant r eplacemen t, so w e k n o w that ( P C )( − ~ k ) and P ( C ( − ~ k )) are w eakly equiv alen t. S in ce b oth ob jects are f -ﬁb ran t they are ﬁbrant as diagrams of A -mo dule chain complexes. In p articular, applicatio n of th e inv erse limit functor yields we akly equiv alen t c hain complexes. The left v ertical map then is kno wn to b e a w eak equiv alence b y [Hir03, Corollary 16.5.5 (1)], applied to th e category Ch A with th e pro jectiv e mo del structur e; for th e target, n ote that lim P ( C ( − ~ k )) = holim C ( − ~ k ) b y deﬁnition of homotop y limits.—A similar argumen t applies to the right v ertical map. Squar e 6: This is just the deﬁnition of the Dold - Kan functor W . On the derived category of a regular toric sc heme 19 In particular, f is an ˆ O ( ~ k )[ ℓ ]-colocal equiv alence if and only if the top horizon tal map f ∗ is a wea k equiv alence if and only if W (holim f ( − ~ k )) is a w eak equiv alence if and only if holim f ( − ~ k )[ − ℓ ] is a quasi-isomorphism in non- negativ e degrees. ✷ 3.2.4 Deﬁnit ion. Let R ⊆ Z Σ(1) b e a non-empt y sub set. (1) A map f ∈ Pre (Σ) is called an R -c olo c al e quivalenc e if it is an ˆ O ( ~ k )[ ℓ ]- colocal equiv alence in the sense of Deﬁnition 3.2.2 for all ~ k ∈ R and ℓ ∈ Z . In other w ords, f is an R -colo cal equiv alence if and only if it is a colo cal equiv alence in the sense of [Hir03, Deﬁnition 3.1.8 (1)] with resp ect to the set ˆ O ( R ) := { ˆ O ( ~ k )[ ℓ ] | ~ k ∈ R, ℓ ∈ Z } . (2) An ob ject B ∈ Pre (Σ) is called R -c olo c al if it is ˆ O ( R )-colocal in th e sense of [Hir03, Deﬁnition 3.1.8 (2)] with resp ect to the c -structur e of Pre (Σ); equiv alen tly , if B is c -coﬁbrant and ˆ O ( R )-cellular [Hir03, T heorem 5.1.5]. If the set R is unders to o d we will drop it f r om the notation and s im p ly sp eak of colocal equiv alences and colocal ob jects. More explicitly , a map f : C ✲ D in Pre (Σ) is an R -colo cal equ iv alence if for all ~ k ∈ R and all ℓ ∈ Z the map hom Pre (Σ)  N A [∆ • ] ⊗ ˆ O ( ~ k )[ ℓ ] , C  f ∗ ✲ hom Pre (Σ)  N A [∆ • ] ⊗ ˆ O ( ~ k )[ ℓ ] , D  is a w eak equiv alence of simp licial sets. The ob ject B ∈ Pre (Σ) is R -colo cal if it is c -coﬁbrant , and if for all R -colocal maps f : C ✲ D in Pre (Σ) th e map hom Pre (Σ)  B , C  ✲ hom Pre (Σ)  B , D  is a wea k equ iv alence of simp licial sets, where B den otes a cosimplicial resolution [Hir03, Deﬁnition 16.1.20 (1)] of B with resp ect to the c -structure of Pre (Σ). 3.2.5 C orollary . A map f in Pre (Σ) is an ˆ O ( { ~ k } ) -c olo c al e quiv alenc e if and only if holim( f ( − ~ k )) is a quasi-isomorphism. Pro of. Th is f ollo ws from Prop osition 3.2.3, together with the fact that a map g of c hain complexes is a we ak equiv alence if and only if W ( g [ ℓ ]) is a we ak equiv alence of simplicial sets for all ℓ ∈ Z . ✷ 3.3 Colo cally acyclic ob jects 3.3.1 Deﬁnit ion. Let R ⊆ Z Σ(1) b e a non-empty subset. An ob ject B ∈ Pre (Σ) is called R -c olo c al ly acyclic if the unique map B ✲ 0 is an R -colocal equiv alence. If the set R is u ndersto o d w e will drop it from the notation and simply sp eak of colocally acyclic ob jects. 20 T. H ¨ uttemann 3.3.2 Nota tion. (1 ) F or a cone σ ∈ Σ let ~ σ ∈ Z Σ(1) denote the vecto r whose ρ -comp onen t is 1 if ρ ⊆ σ , and is 0 otherwise. Note that the zero-cone corresp onds to the zero-v ector. (2) Similarly , w e write − ~ σ for the v ector whose ρ -comp onent is − 1 if ρ ⊆ σ , and is 0 otherwise. 3.3.3 C onstruction. T o the regular fan Σ w e asso ciate a ﬁnite set R Σ ⊂ Z Σ(1) as follo ws: (1) If Σ has a uniqu e in clusion-maximal cone (so X Σ is aﬃne), we set R Σ := { ~ 0 } . This co v ers the unique fan in R 0 as a sp ecial case. (2) Su pp ose that Σ do es n ot hav e a unique inclusion-maximal cone. Let ρ ∈ Σ(1) b e a 1-cone. W e consider Z (Σ /ρ )(1) as a su bset of Z Σ(1) in the follo w ing wa y: A 1-cone ¯ σ ∈ Σ /ρ corresp onds to a 2-cone σ ∈ Σ whic h con tains exactly tw o 1-cones: T he cone ρ and a cone τ 6 = ρ . W e iden tify the ¯ σ -comp onen t of Z (Σ /ρ )(1) with the τ -comp onent of Z Σ(1) . All other comp onen ts will b e set to 0.—Using this identiﬁca tion, we set R Σ := [ ρ ∈ Σ(1) R Σ /ρ ∪ [ ρ ∈ Σ(1)  ~ ρ + R Σ /ρ  where ~ ρ + R Σ /ρ = { ~ ρ + ~ k | ~ k ∈ R Σ /ρ } . Note that R Σ /ρ is deﬁned b y induction on the dimension of Σ . 3.3.4 E xa mple. If Σ is complete then R Σ = { ~ σ | σ ∈ Σ } . 3.3.5 Prop osition. If C ∈ Pre (Σ) is an R Σ -c olo c al ly acyclic c -c oﬁbr ant ho- motopy she af on X Σ , then C ≃ 0 in the c -structur e (i.e., al l c omplexes C σ ar e acyclic). Se e 3.3.3 for a deﬁnition of R Σ . Pro of. The statemen t is tru e if the fan Σ con tains a u nique inclusion-maximal cone µ (so X Σ = U µ is aﬃne). Indeed, b y Remark 1.2.4 we h a v e a qu asi- isomorphism C µ ✲ holim( C ). If C is R Σ -colocally acyclic, then holim( C ) ≃ 0 (since ~ 0 ∈ R Σ ), hence C µ ≃ 0. Since C is a h omotop y sh eaf, th is implies th at all its comp onents C τ ≃ A τ ⊗ A µ C µ are acyclic as w ell.—In particular, the Prop osition is tru e for the u nique fan in R 0 . If Σ do es n ot con tain a unique inclusion-maximal cone, we pro ceed by in- duction on the dimension. Induction h yp othesis: The theorem h olds for ob jects of Pre (∆) for all regular fans ∆ with dim ∆ < dim Σ = n . Step 1: The map C ( − ~ ρ ) ✲ C ( ~ 0) ∼ = C is a weak equiv alence for eac h ρ ∈ Σ(1) . Fix a 1-cone ρ ∈ Σ, and ﬁ x ~ k ∈ R Σ /ρ ⊂ R Σ , the inclusion of sets as explained in Construction 3.3.3 (2). Th en ~ ρ + ~ k ∈ R Σ b y construction. On the derived category of a regular toric sc heme 21 The in clusion O  − ( ~ ρ + ~ k )  ✲ O ( − ~ k ) induces a short exact sequence of ob jects in Pre (Σ) 0 ✲ C  − ( ~ ρ + ~ k )  i ✲ C ( − ~ k ) ✲ Q ( − ~ k ) ✲ 0 . (3.3) If σ is a cone not con taining ρ then all comp onents of the v ectors ~ k and ~ ρ + ~ k cor- resp onding to 1-cones in σ (1) v anish. Hence the σ -comp onen t of the in clusion i is the identit y , so th at Q ( − ~ k ) σ = 0 in this case. F rom the ab ov e sequence we obtain a short exact sequence of A -mo d u le c hain complexes 0 ✲ holim C  − ( ~ ρ + ~ k )  ✲ holim C ( − ~ k ) ✲ holim Q ( − ~ k ) ✲ 0 . No w sin ce C ✲ 0 is an R Σ -colocal equiv alence by hyp othesis, Corollary 3.2.5 (applied to the v ectors ~ k and ~ ρ + ~ k in R Σ ) yields that holim C  − ( ~ ρ + ~ k )  ≃ 0 ≃ holim C ( − ~ k ) . W e conclud e that holim Q ( − ~ k ) ≃ 0 as well. F rom Pr op ositio n 2.5.6 it is easy to conclude that Q ( − ~ k ) = ζ  ε ( C ( − ~ k ))  is nothing b ut the extension b y zero of the restriction C ( − ~ k ) | V ρ of C ( − ~ k ) to V ρ = X Σ /ρ . Since t wisting comm utes w ith restriction, Q ( − ~ k ) could equally b e describ ed as the extension b y zero of the ( − ~ k )th t wist of th e restriction C | V ρ . In other w ords, we ha v e shown that f or all ~ k ∈ R Σ /ρ the chain complex holim (Σ /ρ ) op C | V ρ ( − ~ k ) is acyclic w h ere we ha v e used L emma 2.4.3 to restrict to the smaller ind exing category s t( ρ ) op in the h omotop y limit. F rom Corol- lary 3.2.5 w e infer that the map C | V ρ ✲ ∗ in Pre (Σ /ρ ) is an R Σ /ρ -colocal equiv alence. But by the indu ction hyp othesis we then kn o w that C | V ρ ≃ 0. Since Q ( ~ 0) = ζ ( C | V ρ ) this implies that Q ( ~ 0) ≃ 0. F rom the short exact sequence (3.3), applied to ~ k = ~ 0 ∈ R Σ /ρ w e then see that the map C ( − ~ ρ ) ✲ C ( ~ 0) ∼ = C is a w eak equiv alence as claimed. Step 2: All the structure maps C σ ✲ C τ of C are qua si-isomor- phisms. L et τ ⊂ σ b e a co dimension-1 inclusion of cones in Σ. Let ρ denote the uniqu e 1-cone con tained in σ \ τ . W e w ant to iden tify the σ -comp onen t of the ﬁrst map in the sequence (3.3) for ~ k = ~ 0: By deﬁn ition, it is the natural inclusion m ap O ( − ~ ρ ) σ ⊗ A σ C σ ✲ O ( ~ 0) σ ⊗ A σ C σ ∼ = C σ . (3.4) Since Σ is r egular we can c ho ose f ∈ M su c h that f v anishes on the primitiv e generators of τ , and suc h that f tak es the v alue 1 on the primitive generator of ρ . Then f ∈ S σ , and there is an isomorph ism of A σ -mo dules O ( ~ 0) σ ✲ O ( − ~ ρ ) σ describ ed by b 7→ b + f on elements of the canonical A -basis. W e can thus rewrite the map (3.4) up to isomorph ism as C σ f ✲ C σ . 22 T. H ¨ uttemann The mo dule c hain complex A τ ⊗ A σ C σ is obtained from C σ b y in ve rting the action of the element f (Lemma 2.5.3), i.e. , by forming the colimit of the sequence C σ f ✲ C σ f ✲ C σ f ✲ . . . . No w f acts b y quasi-isomorphism on C σ b y the results of S tep 1; indeed, as ju st seen ab o v e f is the σ -comp onent of the w eak equ iv alence C ( − ~ ρ ) ✲ C ( ~ 0) ∼ = C . Hence the canonical map C σ ✲ A τ ⊗ A σ C σ is a quasi-isomorphism. Since C is a homotop y sh eaf, the map A τ ⊗ A σ C σ ✲ C τ is a qu asi-isomorphism. The com bination of th ese t w o statemen ts sho ws that the structure map C σ ✲ C τ is a quasi-isomorphism. As an y inclusion of cones in Σ can b e written as a sequence of co dimension-1 inclusions, it follo ws th at all structure map s of C are qu asi-isomorphisms as claimed. Step 3: All en tries of the diagram C are a cyclic. W rite con( B ) for the constant Σ op -diagram with v alue B . Fix a cone σ ∈ Σ. The s tructure maps of C assem ble to maps of diagram C ✲ con( C { 0 } ) ✛ con( C σ ) ; b oth these maps are weak equiv alences of diagrams of A -modu le chain complexes b y S tep 2. Application of th e homotop y limit functor giv es a c hain of quasi- isomorphisms (we use Lemma 1.2.5 in the last step) holim C ∼ ✲ holim con( C { 0 } ) ✛ ∼ holim con( C σ ) ≃ C σ . But sin ce ~ 0 ∈ R Σ w e k n o w by Corollary 3.2.5 that holim C ≃ 0, so C σ ≃ 0 as required. ✷ 3.4 Homotop y shea v es as coﬁbran t ob jects 3.4.1 Prop osition. (Colo cal mo del structure of Pre (Σ) ) L e t R ⊆ Z Σ(1) . The c ate gory Pre (Σ) has a mo del structur e, c al le d the R -colo cal mo del struc- ture , wher e a map f is a we ak e quivalenc e if and only if it is an R -c olo c al e quivalenc e (Deﬁnition 3.2.4), and a ﬁbr ation if and only if it is a ﬁbr ation in the c -structur e of Pre (Σ) . The mo del structur e is right pr op er, and every obje ct is ﬁbr ant. Pro of. This is [Hir03, Theorem 5.1.1], app lied to the c -structure of Pre (Σ). ✷ 3.4.2 The orem. L et R Σ ⊂ Z Σ(1) denote the ﬁnite set sp e ciﬁe d in Construc- tion 3.3.3. (1) If C is an R Σ -c olo c al obje c t of Pre (Σ) , then C is a c -c oﬁbr ant homotop y she af. On the derived category of a regular toric sc heme 23 (2) If C is a c - c oﬁbr ant homotopy she af on X Σ , then C is R Σ -c olo c al. Pro of. W e consider the category Pre (Σ) equip p ed with the R Σ -colocal mo del structure of Prop osition 3.4.1. P art (1) follo ws f rom the description of colo cal ob jects in the general theory of righ t Bous field lo calisation. W e h a ve to intro d uce some auxiliary notation and results ﬁr st. Recall th at the c -structure of Pre (Σ) h as a set J c = { F τ (0 ✲ D n ( A )) | n ∈ Z , τ ∈ Σ } of generating coﬁbrations as sp eciﬁed in Lemma 2.3.4. S ince the chain com- plexes D n ( A ) are acyclic so are all the entries in the diagrams F τ ( D n ( A )). Consequent ly , all maps in J c are injectiv e maps of h omotop y s hea ves, and their coﬁbres are homotopy shea ve s. The s et Λ( R Σ ) := { L n ( N A [∆ • ]) ⊗ ˆ O ( ~ k ) ✲ N A [∆ n ] ⊗ ˆ O ( ~ k ) | ~ k ∈ R Σ } is a f ull set of horn s on ˆ O ( R Σ ) in the sense of [Hir03 , Deﬁnition 5.2.1]; here N denotes the redu ced c hain complex f u nctor as usual. This follo ws from th e fact th at N A [∆ • ] ⊗ ˆ O ( ~ k ) is a cosimplicial resolution of ˆ O ( ~ k ) b y L emm a 2.3.5. Note that Λ( R Σ ) is a set of in jectiv e maps of h omotop y shea v es; the coﬁbr es are the ob jects N A [∆ n /∂ ∆ n ] ⊗ ˆ O ( ~ k ) n ≥ 0 , ~ k ∈ R Σ whic h are homotop y s hea ves as wel l. No w su pp ose that C an R Σ -colocal ob ject of Pre (Σ). F rom [Hir03, C orol- lary 5.3.7] we kno w that C is a retract of a c -coﬁbrant ob ject X ∈ Pre (Σ) whic h admits a weak equiv alence X ∼ ✲ Y to an ob ject Y ∈ Pre (Σ) whic h is a cell complex with resp ect to the maps in J c ∪ Λ( R Σ ). Sin ce the coﬁbres of all the maps in this set are h omotop y shea v es as observ ed ab o ve , it follo w s from (transﬁnite) ind uction on the num b er of cells in Y that the presheaf Y is a ho- motop y sh eaf. The induction step works as follo ws: S upp ose that f : A ✲ B is an injectiv e map of preshea v es su c h that its coﬁbre B / A is a homotop y sh eaf, and supp ose that Z is a homotop y sheaf. Then there is a short exact sequence in Pre (Σ) 0 ✲ Z ✲ Z ∪ A B ✲ B / A ✲ 0 where Z and B / A are homotop y s hea ves. It follo ws from L emma 3.1.3 that Z ∪ A B is a homotop y sheaf as wel l. Since Y is a h omotop y sh eaf so is the presheaf X b y Lemma 3.1.2; conse- quen tly , its retract C is a homotop y sheaf as we ll (Lemma 3.1.4). P art (2): Let ˜ Y ∼ co ✲ ✲ C b e a coﬁbran t r ep laceme nt with resp ect to the colocal m o d el structure, constructed b y factorising the map 0 ✲ C as a 24 T. H ¨ uttemann colocally acyclic coﬁbration follo we d b y a c -ﬁb ration. Then Y is R -colocal. W e will show that the map Y ∼ co ✲ ✲ C is a w eak equ iv alence (in the c -structure); then C is colocal as w ell by [Hir03, Prop osition 3.2.2 (2)]. The m ap Y ∼ co ✲ ✲ C is a c -ﬁb ration, hen ce sur jectiv e. W e thus h a ve a sh ort exact sequence of ob jects in Pre (Σ) 0 ✲ ˜ K ✲ Y ∼ co ✲ ✲ C ✲ 0 . (3.5) The map ˜ K ✲ 0 is the pullbac k of Y ∼ co ✲ ✲ C , so ˜ K ✲ 0 is a colo cally acyclic ﬁbration, hence ˜ K is colo cally acyclic. By consid er in g the long exact homology sequence asso ciate d to (3.5) we are redu ced to showing ˜ K ≃ 0. Let K ∼ ✲ ✲ ˜ K denote a c -coﬁbrant r eplacemen t. It is enough to pro v e that K ≃ 0. Note that K is R Σ -colocally acyclic since ˜ K is s o, and sin ce every w eak equ iv alence is a colocal equiv alence [Hir03, Prop osition 3.1.5]. By hypothesis and part (1), b oth Y and C are homotop y sheav es. Hence ˜ K , b eing the k ernel of a surjection Y ✲ C , is a homotop y sheaf as w ell by Lemma 3.1.3. Consequen tly , K is a c -coﬁbrant homotop y sh eaf whic h satisﬁes the h yp otheses of Prop osition 3.3.5 whic h pro v es K ≃ 0 as required . ✷ 3.4.3 C orollary . L et R Σ ⊂ Z Σ(1) denote the ﬁnite set sp e ci ﬁe d in Construc- tion 3.3.3. L et f : X ✲ Y b e a map of homoto py she aves. Then f is a we ak e quivalenc e if and only if the induc e d map of chain c omplexes holim f ( − ~ k ) : holim X ( − ~ k ) ✲ holim Y ( − ~ k ) is a quasi-isomorphism for al l ~ k ∈ R Σ . Pro of. By c -coﬁbran t approxi mation and lifting, w e can construct a comm u- tativ e square X c f c ✲ Y c X ∼ ❄ f ✲ Y ∼ ❄ (3.6) where b oth vertic al maps are wea k equiv alences, and with c -coﬁbrant preshea v es X c and Y c . Th en f is a weak equiv alence if and only if f c is. No w f c is a map of R Σ -colocal ob jects by Lemma 3.1.2 and Theorem 3.4.2. Hence f c is a wea k equiv alence if and only if f c is an R Σ -colocal map [Hir03, Theorem 3.2.13 (2)]. By Corollary 3.2.5 this is equiv alen t to saying th at the map holim f c ( − ~ k ) is a quasi-isomorphism for all ~ k ∈ R Σ . Ho wev er, since holim and t wisting b oth preserve w eak equiv alences, this is equiv alen t, in view of diagram (3.6) ab ov e, to the condition that holim f ( − ~ k ) is a quasi-isomorphism for all ~ k ∈ R Σ . ✷ On the derived category of a regular toric sc heme 25 4 The deriv ed category Our n ext goal is to prov e that for a large class of sc hemes the (unb ounded) deriv ed category of qu asi-co herent sheav es can b e obtained as the h omotopy catego ry of homotop y shea v es. The material in this section will app ly to any regular toric sc heme X deﬁned o v er a commutat iv e rin g A ; more generally , it will b e enough to assume that X is a scheme equ ipp ed with a ﬁ nite semi-separating co v er [TT90, § B.7] as sp eciﬁed in Deﬁnition 4.1.1 b elo w. Then th e categories of chain complexes of quasi-coheren t sheav es on U σ and X , resp ectiv ely , admit the inje ctive mo del structur e with coﬁbr ations the lev elwise injectiv e maps, and the categorie s of c hain complexes of quasi-coheren t sh ea ves on U σ admit the pr oje ctive mo del structur e with ﬁbrations the lev elwise surjectiv e maps. Finally , all the inclusions U σ ⊆ X are aﬃne maps and hence induce exact p ush-forwa rd functors. 4.1 Co v erings indexed by a fan 4.1.1 Deﬁnit ion. Let A b e a comm utativ e ring, let Σ b e a ﬁn ite fan in N R , and let X b e an A -sc heme. A collection ( U σ ) σ ∈ Σ of op en subsc hemes of X is called a Σ -c overing if S σ ∈ Σ U σ = X , and if for all τ , σ ∈ Σ w e hav e U τ ∩ U σ = U τ ∩ σ . If all the U σ are aﬃne, we call ( U σ ) σ ∈ Σ an aﬃne Σ -c overing . If the A -scheme X admits an aﬃne Σ-co v ering, for some ﬁn ite fan Σ, then X is necessarily quasi-compact and semi-separated [TT90, § B.7], hence in p ar- ticular quasi-separated. Th ese f acts are relev an t as they guaran tee the existence of certain mo del category s tr uctures, cf. § 4.3. 4.1.2 E xa mple. Eve ry quasi-compact separated sc heme X adm its an aﬃne Σ-co v ering for some fan Σ. Indeed, let U 0 , U 1 , . . . , U n b e an op en aﬃn e co ve r of X . L et Σ denote the usu al fan of n -d imensional pro jectiv e space, describ ed as follo ws. Let e 1 , e 2 , . . . , e n denote the unit vecto rs of R n , set e 0 = − e 1 − e 2 − . . . − e n , and d eﬁne M := { 0 , 1 , . . . , n } . Then Σ is the collection of cones σ E = cone  { e i | i ∈ E }  ⊂ R n for pr op er subs ets E ⊂ M . Giv en suc h a set E deﬁn e U σ E := T i ∈ M \ E U i ; these int ersections are aﬃne since X is separated. Then ( U σ ) σ ∈ Σ is an aﬃne Σ-co v ering of X by construction. More generally , if X is quasi-compact, and the sets U 0 , U 1 , . . . , U n form a semi-separating co vering of X , the ab o v e construction pro vides an aﬃne Σ- co v ering for X . 4.2 Shea ves and homotop y shea ves F rom no w on we will assume that A is a comm utativ e ring, that Σ is a ﬁnite fan in N R , and th at X is an A -sc heme equipp ed with an aﬃne Σ-co v ering 26 T. H ¨ uttemann ( U σ ) σ ∈ Σ (Deﬁnition 4.1.1); a fortiﬁori , X is quasi-compact and semi-separated. F or an y op en sub sc heme Y ⊆ X w e write Qco ( Y ) for the category of quasi- coheren t sh ea ves of O Y -mo dules, and Ch Qco ( Y ) for the category of (p ossibly unboun ded) c hain complexes in Qco ( Y ).—In wh at follo ws w e w ill consider a presheaf to ha v e v alues in the categories Ch Qco ( U σ ) rather than in chain com- plexes of mo du les: 4.2.1 Deﬁnit ion. Th e category Pre (Σ) of preshea v es on X is the category of Σ op -diagrams C which assign to eac h σ ∈ Σ an ob ject C σ ∈ C h Qco ( U σ ), and to eac h inclusion τ ⊆ σ in Σ a map C σ | U τ ✲ C τ , whic h is the iden tit y for τ = σ , sub ject to the condition that for ν ⊆ τ ⊆ σ in Σ the comp osition C σ | U ν =  C σ | U τ  | U ν ✲ ( C τ ) | U ν ✲ C ν coincides w ith the structure map corresp onding to the inclusion ν ⊆ σ . The category Pre (Σ) is another examp le of a t wisted diagram category in the sense of [HR, § 2.2] , formed with r esp ect to the adjunction b undle Σ op ✲ Cat , σ 7→ Ch Qco ( U σ ) and s tructural adjun ctions giv en b y r estriction (the left adjoin ts) and pu sh- forw ard along inclusions. W e can th us app eal to the general mac hinery of t wisted diagrams again to equip Pre (Σ) with v arious mo del structur es. W e d eﬁne the notions of strict and homotopy shea v es for Pre (Σ) in analogy to Deﬁn ition 2.2.1: 4.2.2 Deﬁnit ion. Give n an ob ject C ∈ Pre (Σ) w e call C a strict she af if f or all inclusions τ ⊆ σ in Σ the structure m ap C σ | U τ ✲ C τ is an isomorphism; w e call C a homotopy she af if for all inclusions τ ⊆ σ in Σ th e stru cture map C σ | U τ ✲ C τ is a quasi-isomorphism. 4.2.3 Remark. Since restriction to the op en su bset U σ is an exact functor, Lemmas 3.1.2, 3.1.3 and 3.1.4 app ly mutatis mutandis . That is, if f : C ✲ D is a map in Pre (Σ) w hic h is a quasi-isomorphism on eac h U σ , we kno w that C is a h omotop y sheaf if and only if D is a homotop y sheaf. Moreo v er, the class of homotop y shea v es is closed under kernels, cok ernels, extensions, and retracts. 4.2.4 Remark. In the case of a toric sc heme the catego ries Pre (Σ) (Deﬁni- tion 2.2.1) and Pre (Σ) co dify the same in formation. Recall that for an aﬃne sc heme U = Sp ec( B ) the category of quasi-coherent shea v es on U is equiv alen t, via the exact global sections functor, to the category of B -mo dules. Conse- quen tly , if X = X Σ is a r e gular toric scheme with fan Σ , the functor Pre (Σ) ✲ Pre (Σ) , C 7→  σ 7→ Γ( C ; U σ )  On the derived category of a regular toric sc heme 27 is an e quiv alenc e of c ate gories. It maps strict she aves to strict she aves, and homoto py she aves to homotopy she aves. The diﬀerence b et w een Pre (Σ) and Pre (Σ) is of a pur ely tec hnical n atur e; the c hoice of which category to use is mostly dictated by conv enience rather than necessit y . Our p revious resu lts on homotop y sh ea ves and colocalisation th us apply mutatis mutandis for a regular toric sc heme X Σ . 4.3 Mo del structures F or eve ry quasi-separated and quasi-compact scheme Y the category Qco ( Y ) of qu asi-coherent O Y -mo dule sh ea ves is a Grothendieck ab el ian category [TT90, § B.3] w hic h, in particular, satisﬁes axiom AB5 (“ﬁltered colimits are exact”) . It is w ell-kno wn [Ho v 01 ] that therefore the category Ch Qco ( Y ) of (p ossibly unb ounded) c hain complexes of qu asi-co herent sheav es on Y adm its the inje ctive mo del structur e with we ak equiv alences the quasi-isomorphisms, and coﬁbrations the lev elwise injections. Since a semi-separated sc heme is automatically quasi-separated, and quasi- separatedness is stable under passage to op en subs c h emes, this applies to our sc heme X as w ell as to all the co v ering sets U σ . The full su b catego ry of Pre (Σ) spann ed by the strict sheav es is equiv alen t to the category Ch Qco ( X ) of (un b ound ed) c hain complexes of quasi-coheren t shea v es on X Σ . Its d eriv ed category D ( Qco ( X )) can b e obtained as the h omo- top y catego ry of the injectiv e mo del stru cture of Ch Qco ( X ) describ ed ab o v e. 4.3.1 Lemma. L et U ⊆ X b e an op en subset. The functor Ch Qco ( X ) ✲ Ch Qco ( U ) , F 7→ F | U is a left Quillen functor with right adjoint given b y push-forwar d along the i n- clusion U ✲ X . (Her e we e quip Ch Qco ( U ) with the inje ctiv e mo del structur e as wel l.) Pro of. This follo ws from th e fact that restriction to op en su bsets is exact, hence preserves wea k equiv alences (quasi-isomorphisms) and coﬁbrations (in- jections). ✷ 4.3.2 Lemma. The c ate gory Pre (Σ) has a mo del structur e wher e a map is a we ak e quivalenc e if it is an obje ctwise quasi-isomorphism, and a c oﬁbr ation if it is obje ctwise and levelwise inje ctiv e . Pro of. This is the f -structur e of [HR, Theorem 3.3.5], based on the in j ectiv e mo del stru cture of the categories Ch Qco ( U σ ). ✷ 28 T. H ¨ uttemann Fibrations in this mo del structure can b e characte rised us in g matching c om- plexes : Given C ∈ Pre (Σ) and σ ∈ Σ deﬁne M σ C = lim τ ⊂ σ i τ ∗ ( C τ ) where i τ : U τ ⊆ U σ is the inclus ion, and the limit is tak en o ve r all τ ∈ Σ strictly con- tained in σ . Th en f : C ✲ D is a ﬁ bration if and only if for all σ ∈ Σ th e induced map C σ ✲ M σ C × M σ D D σ (4.1) is a ﬁbr ation in the category Ch Qco ( U σ ).—If f is a ﬁ bration then in particular all the comp onen ts f σ : C σ ✲ D σ are ﬁb rations in their resp ectiv e categories. 4.4 Strictifying homotop y shea v es No w consider the “constan t diagram” functor, deﬁn ed b y Φ : Ch Qco ( X ) ✲ Pre (Σ) , F 7→  σ 7→ F | U σ  . With resp ect to the mo del structure of Lemma 4.3.2 the functor Φ is left Quillen (b y exactness of restriction to op en s u bsets) with righ t adjoint give n b y Ξ : Pre (Σ) ✲ Ch Qco ( X ) , C 7→ lim σ ∈ Σ op j σ ∗ ( C σ ) where the j σ : U σ ✲ X are the v arious inclusion maps. By constru ction we ha v e canonical maps Ξ ( C ) ✲ j σ ∗ C σ whic h giv e rise, up on restriction to U σ , to map s r σ : (Ξ( C )) | U σ ✲ ( j σ ∗ C σ ) | U σ = C σ . These maps are natural in σ in the sens e that for eac h inclusion τ ⊆ σ of cones in Σ the map r τ equals the comp osite map Ξ( C ) | U τ = (Ξ ( C ) | U σ ) | U τ r σ | U τ ✲ C σ | U τ ✲ C τ . (4.2) In other wo rd s , the maps r σ assem ble to a map of preshea v es r : Φ ◦ Ξ( C ) ✲ C whic h is the counit of the adj unction of Φ and Ξ. Recall that an ob ject C ∈ Pre (Σ) is a h omotop y sheaf (Deﬁnition 4.2.2) if th e structure maps C σ | U τ ✲ C τ are qu asi-isomorph isms for all inclusions τ ⊆ σ in Σ. Th e follo wing Lemma shows ho w the functor Ξ can b e used to strictify homotopy s hea ves, i.e. , h o w to replace a homotopy s h eaf by w eakly equiv alen t strict sheaf: 4.4.1 Lemma. Every homo topy she af ¯ C ∈ Pre (Σ) is we akly e quivalent to a strict she af. Mor e pr e cisely, let C ✛ ≃ ¯ C denote a ﬁbr ant r eplac ement. Then for e ach σ ∈ Σ the c anonic al map r σ : Ξ( C ) | U σ ✲ C σ On the derived category of a regular toric sc heme 29 is a q uasi-isomorph ism in C h Qco ( U σ ) . In other wor ds, we have a chain of we ak e quivalenc es of homotopy she aves Φ ◦ Ξ( C ) ≃ r ✲ C ✛ ≃ ¯ C wher e Φ ◦ Ξ( C ) is, in fact, a strict she af. Pro of. First n ote that C , b eing weakly equiv alen t to the homotopy sh eaf ¯ C , is a homotop y sheaf by Remark 4.2.3. W e ha ve to pr o v e that the m ap r σ : Ξ( C ) | U σ ✲ C σ is a weak equiv alence in the categ ory Ch Qco ( U σ ). In fact, it is enough to p ro v e the claim for all maximal cones σ : Given any τ ∈ Σ choose a maximal cone σ cont aining τ . By (4.2), th e map r τ then is the comp osition of the r estriction of the we ak equiv alence r σ to U τ with the stru cture m ap C σ | U τ ✲ C τ . Th e latter is a quasi-isomorphism since C is a homotop y sheaf, th e former is a quasi-isomorphism since restriction is exact. Hence r τ is a we ak equiv alence. So let σ ∈ Σ b e a maximal cone. W e w an t to sh o w th at the top h orizon tal map t = r σ in the follo w ing d iagram is a w eak equiv alence (wh ere j τ : U τ ✲ X denotes the inclusion map as b efore): (Ξ( C )) | U σ = lim τ ∈ Σ op ( j τ ∗ C τ ) | U σ t ✲ lim τ ⊆ σ ( j τ ∗ C τ ) | U σ ∼ = C σ lim τ 6 = σ ( j τ ∗ C τ ) | U σ ❄ h ✲ lim τ ⊂ σ ( j τ ∗ C τ ) | U σ p ❄ (4.3) The diagram is cartesian: It arises from ﬁ rst re-writing the limit deﬁnin g Ξ( C ) as a pullbac k of limits ind exed o ve r smaller categ ories, then applying the exact restriction functor ( · ) | U σ . Moreo v er, the map p is a ﬁbration since C is a ﬁbrant ob ject; ind eed, p is n othing b u t the map (4.1) corresp ond ing to σ ∈ Σ for the map C ✲ 0. Hence by right pr op erness of the injectiv e mo del structure of Ch Qco ( U σ ) it is enough to sh o w that the low er horizon tal map h is a wea k equiv alence. F or ν ⊆ σ let i ν : U ν ✲ U σ and j ν : U ν ✲ X d enote the inclusions. Then we hav e an equalit y  j ν ∗ ( F )  | U σ = i ν ∗ ( F ) for F ∈ Qco ( U ν ) , (4.4) and if τ ⊇ ν is another cone,  j τ ∗ ( G )  | U σ = i ν ∗ ( G ) | U ν for G ∈ Qco ( U τ ) . (4.5) W e emb ed the map h of d iagram (4.3) ab o v e into the larger diagram (4.6) b elo w. W e h av e u sed (4.4) for the upp er vertica l map on the righ t, and (4.5) f or 30 T. H ¨ uttemann the upp er ve rtical map on the left (recall also that restriction and p ush forw ard are exact functors, h en ce commute with ﬁ nite limits). The map f is indu ced b y the structure m ap s C τ | U τ ∩ σ ✲ C τ ∩ σ of C . lim τ 6 = σ ( j τ ∗ C τ ) | U σ h ✲ lim τ ⊂ σ ( j τ ∗ C τ ) | U σ lim τ 6 = σ i τ ∩ σ ∗ ( C τ | U τ ∩ σ ) = ❄ ✲ lim τ ⊂ σ i τ ∗ ( C τ ) = ❄ lim τ 6 = σ i τ ∩ σ ∗ ( C τ ∩ σ ) ≃ f ❄ ∼ = g ✲ lim τ ⊂ σ i τ ∗ ( C τ ) = ❄ (4.6) The map g is easily seen to b e an isomorphism: In the diagram τ 7→ i τ ∩ σ ∗ ( C τ ∩ σ ) all structure maps corresp onding to th e inclusions τ ∩ σ ⊆ τ are isomorphisms, hence all terms with τ 6⊆ σ are redun dan t w hen form ing th e limit, and the map g is giv en by forgetting the r edundant terms. W e are th us redu ced to sho wing that the map f is a qu asi-isomorphism whic h will follo w f r om an application of Bro wn ’s L emm a [DS95 , dual of L emm a 9.9]. W e n eed some preliminary remarks. R ecall that sin ce U σ is aﬃne, sa y U σ = Sp ec A σ , the category Ch Qco ( U σ ) is equiv alen t to th e category of A σ - mo dules. Hence Ch Qco ( U σ ) is equ iv alen t to the categ ory Ch A σ , whic h implies that w e can equ ip the category Ch Qco ( U σ ) with the pr oje ctive mo del struc- tur e : Fibr ations are the lev elwise surjectiv e maps, and weak equiv alences are the qu asi-isomorphisms. A coﬁbration in the pro jectiv e mo del structur e turns out to b e lev elwise injectiv e (ev en lev elwise split injectiv e), but this condition do es n ot c haracterise coﬁbrations. W e w ill denote the categ ory of fu nctors (Σ \ { σ } ) op ✲ Ch Qco ( U σ ) b y C := F un  (Σ \ { σ } ) op , Qco ( U σ )  . The category C carries a mo del str ucture where a map is a w eak equiv alence ( r esp. , coﬁbration) if and only if it is an ob ject wise weak equiv alence ( r esp. , coﬁbration in the pro jectiv e mo del structure). A diagram D ∈ C is ﬁbrant if and only if for all ν ∈ Σ \ { σ } the map D ν ✲ lim τ ⊂ ν D τ is a ﬁbr ation in the pro jectiv e mo del structure ( i. e . , is lev elwise sur jectiv e), the limit taken ov er all cones τ ∈ Σ \ σ strictly con tained in ν . On the derived category of a regular toric sc heme 31 With resp ect to the pr o jectiv e mo d el structure of Ch Qco ( U σ ) the inv erse limit f unctor lim : C ✲ Ch Qco ( U σ ) , D 7→ lim Σ \{ σ } ) op ( D ) is right Quille n with left adjoin t giv en b y th e constant diagram functor ∆ : Ch Qco ( U σ ) ✲ C , C 7→  ∆( C ) : τ 7→ C  ; note that ∆ preserves w eak equiv alences and coﬁbrations as these notions are de- ﬁned ob ject wise in C . Thus, us in g Bro wn ’s Lemma [DS95 , dual of Lemma 9.9], w e know that if f i s a we ak e quivalenc e in C with sour c e and tar get ﬁbr ant di- agr ams, then lim ( f ) is a we ak e q uivalenc e in Ch Qco ( U σ ). W e will apply this last observ ation to the map f in th e diagram (4.6): W e kno w that f is a weak equiv alence pro vided w e can v erify the follo wing thr ee assertions: (1) Th e natural transformation of diagrams deﬁning f consists of we ak equiv- alences (quasi-isomorphisms) (2) Th e diagram τ 7→ i τ ∩ σ ∗ ( C τ | U τ ∩ σ ) (the source of f ) is a ﬁ bran t ob ject of C (3) Th e diagram τ 7→ i τ ∩ σ ∗ ( C τ ∩ σ ) (the target of f ) is a ﬁbrant ob ject of C Assertion (1) is easy to verify . The map f is induced by the structure maps C τ | U τ ∩ σ ✲ C τ ∩ σ whic h are w eak equiv alences since C is a h omotop y sheaf b y hypothesis?; note also that the functor i τ ∩ σ ∗ is exact since the inclusion U τ ∩ σ ⊆ U σ is aﬃne. F or assertion (2) we ha v e to v erify that f or eac h ν ∈ Σ \ σ the map i ν ∩ σ ∗ ( C ν | U ν ∩ σ ) ✲ lim τ ⊂ ν i τ ∩ σ ∗ ( C τ | U τ ∩ σ ) (4.7) is lev elwise su rjectiv e. By hyp othesis C is a ﬁbrant ob ject (Lemma 4.3.2) of Pre (Σ), so the map C ν ✲ lim τ ⊂ ν k τ ∗ ( C τ ) (with k τ b eing the inclusion U τ ⊆ U ν ) is a ﬁbration in the injectiv e mo del structure of Ch Qco ( U ν ); in particular, this map is leve lwise surjectiv e. Since restriction to op en subsets is exact, it follo ws that the m ap C ν | U ν ∩ σ ✲ lim τ ⊂ ν ( k τ ∗ ( C τ )) | ν ∩ σ = lim τ ⊂ ν ℓ τ ∩ σ ∗ ( C τ | U τ ∩ σ ) is lev elwise sur j ectiv e, wh ere n o w ℓ τ ∩ σ denotes the inclusion U τ ∩ σ ⊆ U ν ∩ σ . W e can now apply the exact fu nctor i ν ∩ σ ∗ ; since i ν ∩ σ ∗ ◦ ℓ τ ∩ σ ∗ = i τ ∩ σ ∗ w e conclud e that the map (4.7) is lev elwise surjectiv e as claimed. 32 T. H ¨ uttemann W e n o w discuss assertion (3). W e h a ve to show that for eac h ν ∈ Σ \ σ the map i ν ∩ σ ∗ ( C ν ∩ σ ) ✲ lim τ ⊂ ν i τ ∩ σ ∗ ( C τ ∩ σ ) (4.8) is leve lwise surjectiv e (where i µ : U µ ✲ U σ as b efore). Consider the diagram D : { τ ⊂ ν } op ✲ Ch Qco ( U σ ) , τ 7→ i τ ∩ σ ∗ ( C τ ∩ σ ) , its limit b eing the target of the map (4.8 ). If ν ⊂ σ then the map (4.8 ) arises b y application of the exact functor i ν ∗ = i ν ∩ σ ∗ to the map C ν = C ν ∩ σ ✲ lim τ ⊂ ν ℓ τ ∗ C τ (4.9) where ℓ τ : U τ ✲ U ν = U ν ∩ σ is the inclus ion map. Now C is a ﬁbrant ob ject of Pre (Σ) by h yp othesis, so (4.9 ) is a ﬁ bration in the in jectiv e mo del structure, hence lev elwise surjectiv e. It follo w s that (4.8 ) is lev elwise surjectiv e as w ell. It remains to deal with the case ν 6⊆ σ . Let τ b e a prop er face of ν . The structure maps of D corresp onding to th e inclusions τ ∩ σ ⊆ τ are identi t y maps : i τ ∩ σ ∗ ( C τ ∩ σ ) = i ( τ ∩ σ ) ∩ σ ∗ ( C ( τ ∩ σ ) ∩ σ ) ✲ i τ ∩ σ ∗ ( C τ ∩ σ ) It follo ws that the limit of D is isomorph ic to the limit of the restriction of D to faces of the form τ ∩ σ for τ ⊂ ν . So deﬁne Q := { τ ∩ σ | τ ⊂ ν } . In fact, Q is the p oset of pr op er faces of ν wh ich are also faces of σ . No w s in ce ν 6⊆ σ we kno w that Q has maximal element ν ∩ σ ⊂ ν . With this notation, the map (4.8 ) can b e em b edded into a c h ain i ν ∩ σ ∗ ( C ν ∩ σ ) ( 4.8 ) ✲ lim τ ⊂ ν i τ ∩ σ ∗ ( C τ ∩ σ ) ∼ = ✲ lim τ ∈ Q op i τ ∩ σ ∗ ( C τ ∩ σ ) ∼ = i ν ∩ σ ∗ ( C ν ∩ σ ) with comp osition the id en tit y map. It follo ws that the map (4.8) is lev elwise surjectiv e as claimed. ✷ 4.5 The derived category via homotop y shea v es W e h a v e constru cted a pair of adjoint fun ctors Φ : Ch Qco ( X ) ✲ Pre (Σ) and Ξ : Pre (Σ) ✲ Ch Qco ( X ) , the fun ctor Φ b eing the left adjoin t. Moreo ver, the pair (Φ , Ξ) is a Quill en pair w ith resp ect to the injectiv e mo del structure on Ch Qco ( X ), and the mo del structure describ ed in Lemma 4.3.2 on Pre (Σ). F rom general m o d el category theory , we obtain an adjoin t pair of total deriv ed fun ctors L Φ : Ho Ch Qco ( X ) ✲ Ho Pre (Σ) and R Ξ : Ho Pre (Σ) ✲ Ho Ch Qco ( X ) whic h w e can use to giv e a d escription of the derived category D  Qco ( X )  = Ho Ch Qco ( X ) via homotop y shea ve s: On the derived category of a regular toric sc heme 33 4.5.1 The orem. L et H denote the ful l sub c ate g ory of Ho Pre (Σ) sp anne d by the homoto py she aves. The Quillen p air (Φ , Ξ) induc es an e quivalenc e of c ate gories L Φ : Ho C h Qco ( X ) ✲ H with inverse given by R Ξ . Pro of. W e ﬁrst ha v e to ve rify that L Φ tak es v alues in H . Eve ry ob ject F of Ch Qco ( X ) is coﬁbrant in the inj ectiv e mo del stru ctur e, hence L Φ( F ) ∼ = Φ( F ) in Ho Pre (Σ), and the relev an t stru cture maps Φ( F ) σ | U τ = ( F | U σ ) | U τ = F | U τ = Φ ( F ) τ are iden tities, hence wea k equiv alences. T his sho ws that L Φ( F ) is a homotop y sheaf, so L Φ( F ) ∈ H . Giv en C ∈ H th e counit map of the adjunction of L Φ and R Ξ is m od elled b y the p oin t-set level counit map of (Φ , Ψ ) at C f , ǫ C f : Φ(Ξ( C f )) ✲ C f where C ∼ ✲ C f denotes a ﬁ bran t replacement in Pre (Σ). Fix a cone σ ∈ Σ. The σ -comp onen t of ǫ C f is nothing but the map r σ of Lemma 4.4.1 applied to C f . Since C f is a homotop y sh eaf Lemma 4.4.1 applies, and we conclude that ǫ C f is a w eak equiv alence. Hence L Φ ◦ R Ξ( C ) ✲ C is an isomorphism in H . Giv en F ∈ C h Qco ( X ) th e unit map of the adj u nction of L Φ and R Ξ is mo delled by the comp osition F η F ✲ Ξ(Φ( F )) Ξ( a ) ✲ Ξ(Φ( F ) f ) (4.10) where a : Φ( F ) ∼ ✲ Φ( F ) f denotes a ﬁbrant replacemen t of Φ ( F ) in Pre (Σ), and w here η F is the p oint- set lev el adjunction u nit of (Φ , Ξ ). Since the f u nctor Φ detects we ak equ iv alences it is enough to sho w that the comp osition of the tw o top horizon tal maps in the follo w ing diagram is a w eak equiv alence: Φ( F ) Φ( η F ) ✲ Φ(Ξ(Φ( F ))) Φ ◦ Ξ( a ) ✲ Φ(Ξ(Φ( F ) f )) Φ( F ) ǫ Φ( F ) ❄ ∼ a ✲ = ✲ Φ( F ) f ǫ Φ( F ) f ∼ ❄ The vertical maps are p oin t-set lev el counit maps for Φ( F ) and Φ( F ) f , resp ec- tiv ely; hence the square comm utes b y n aturalit y . The right-hand v ertical map is a w eak equiv alence b y Lemma 4.4.1, applied to the ﬁbrant h omotop y sheaf Φ( F ) f . Th e map a is the ﬁbr an t-replacemen t map, hence a w eak equiv alence, 34 T. H ¨ uttemann and the d iagonal m ap is the iden tit y b y the theory of adjun ctions (triangle iden- tities [Mac71, § IV, p. 83]). This prov es that the comp osition (4.10) is a weak equiv alence as claimed. W e ha ve sho wn that b oth u nit and counit maps of the adju nction ( L Φ , R Ξ) are isomorphisms in the h omotop y categ ories in question. Hence they giv e an equiv alence of categ ories of D  Qco ( X )  = Ho Ch Qco ( X ) and H as claimed. ✷ 4.6 The derived category of a r egular toric sc heme 4.6.1 The orem. L et A b e a c ommutative ring with unit. Supp ose that Σ is a r e gular fan, and denote the asso ci ate d A -scheme by X Σ . L et R Σ ⊂ Z Σ(1) denote the ﬁnite set of inte g r al v e ctors as sp e ciﬁe d i n Construction 3.3.3. The derive d c ate gory D ( Qco ( X Σ )) c an b e obtaine d fr om the twiste d diagr am c ate gory Pre (Σ) deﬁne d in 2.2.1 by inverting al l those maps X ✲ Y which induc e quasi-isomorphisms holim X ( − ~ k ) ∼ ✲ holim Y ( − ~ k ) for al l ~ k ∈ R Σ . (4.11) Mor e pr e cisely, the homotopy c ate gory of the c olo c al mo del structur e as describ e d in Pr op osition 3.4.1 is e quivalent to D ( Qco ( X Σ )) . W i th r esp e ct to this mo del structur e, the c oﬁbr ant obje cts ar e pr e cisely the c -c oﬁbr ant homotopy she aves, and a map of c oﬁbr ant obje cts is an obje ctwise we ak e quivalenc e if and only if it satisﬁes the c ondition (4.11). Pro of. The c haracterisations of coﬁbrant ob jects and their colo cal equ iv a- lences are giv en in Prop osition 3.4.1 and Corollary 3.4.3. The homotop y cate- gory of the colo cal m o d el structure is equiv alen t to its sub category A spann ed by homotop y sh ea ves (since every homotop y s h ea ves is isomorphic, via c -coﬁbrant replacemen t, to a colocal ob ject). Th e categ ory A is equiv alen t to the su b cate- gory H of Ho Pre (Σ) span n ed by th e homotop y sheav es, cf. Remark 4.2.4. Th e catego ry H , in tu rn, is equiv alen t to D ( Qco ( X Σ )) according to Theorem 4.5.1. This ﬁnished the pro of. ✷ 4.6.2 C orollary . In the situation of The or em 4.6.1, the diagr ams O ( ~ k ) , ~ k ∈ R Σ form a set of we ak gener ators of D ( Qco ( X Σ )) : A morphism f : C ✲ D in the c ate gory D ( Qco ( X Σ )) i s an i somorphism if and only i f for al l ~ k ∈ R Σ and al l ℓ ∈ Z , the map hom( O ( ~ k )[ ℓ ] , f ) : hom( O ( ~ k )[ ℓ ] , C ) f ∗ ✲ hom( O ( ~ k )[ ℓ ] , D ) is an isomorphism of abe l ian g r oups. H er e O ( ~ k )[ ℓ ] denotes the diagr am O ( ~ k ) c onsider e d as a chain c omplex c onc entr ate d in de gr e e ℓ . On the derived category of a regular toric sc heme 35 Pro of. By Th eorem 4.6.1 it is enough to pr o ve the corresp ond ing statement for th e h omotopy category of the colo cal mo d el str u cture on Pre (Σ), cf. Pr op o- sition 3.4.1. Moreo v er, replacing C by a coﬁbr an t ob ject we may assume that f is r epresen ted by an actual map g : C ✲ D in Pre (Σ). Th e morph ism f is an isomorph ism if and only if g is an R Σ -colocal equiv alence. Morphism sets in the homotop y category can b e describ ed as the set of path comp onen ts of m ap p ing sp aces; we are thus reduced to sho wing that g is an R Σ -colocal equiv alence if and only if the m ap hom Pre (Σ)  ˆ O ( ~ k )[ ℓ ] ⊗ N A [∆ • ] , C  g ∗ ✲ hom Pre (Σ)  ˆ O ( ~ k )[ ℓ ] ⊗ N A [∆ • ] , D  induces a bijection after app licatio n of the functor π 0 for all ℓ ∈ Z and all ~ k ∈ R Σ . Ho we ve r, it follo ws from th e pr oof of Prop osition 3.2.3 that g ∗ is a π 0 -isomorphism if and only if the m ap holim C ( − ~ k ) ✲ holim D ( − ~ k ) is an H ℓ -isomorphism. This ﬁnishes the p r o of in view of Corollary 3.2.5 ✷ In the sp ecial case of pro jectiv e n -space the fan Σ has n + 1 d iﬀeren t 1-cones. The set R Σ ⊂ Z n +1 as deﬁned in Construction 3.3.3 then consists of all the p ossible (0 , 1)-v ectors with at most n non-zero en tries, cf. Example 3.3.4, and for any ~ k ∈ Z n +1 the line b undle O ( ~ k ) is isomorp hic to the line bund le u sually denoted O P n ( ℓ ) where ℓ = | ~ k | is the sum of the entries of ~ k . In other w ords, w e reco v er the classical results that the shea v es O P n ( ℓ ), 0 ≤ ℓ ≤ n , generate the d eriv ed category . Note that Construction 3.3.3 giv es an exp licit algorithm to construct generators for the deriv ed category of any regular toric sc heme, deﬁned ov er an arb itrary comm utativ e ring A . References [DS95] W. G. Dwyer and J . Spali ´ ns k i. Ho motop y theo ries and mo del categ ories. In Handb o ok of algebr aic top olo gy , pages 73–1 26. North-Holland, Amsterda m, 1995. [F ul93] William F ulton. Intro duction to toric varieties , volume 131 of Annals of Mathematics Stu dies . Princeton University Pr ess, Princ e ton, NJ, 199 3. The William H. Ro ever Lectures in Geometry . [Hir03] Philip S. Hirschhorn. Mo del c ate gories and their lo c alizatio ns , volume 99 of Mathematic al Surveys and Mono gr aphs . Amer ic an Mathematica l So ciet y , Providence, RI, 20 03. [Hov99] Ma rk Hov ey . Mo del c ate gories , volume 63 of Mathematic al Surveys and Mono gr aphs . Amer ic an Mathematical So ciet y , P rovidence, RI, 1999. [Hov01] Ma rk Hov ey . Mo del catego r y structures o n chain c omplexes of s heav es. T r ans. Amer. Math. So c. , 353(6 ):2441–2 457 (elec tronic), 2 0 01. 36 T. H ¨ uttemann [HR] Thomas H ¨ uttemann and Oliver R¨ ondigs. Twis ted diag rams and homo to p y sheav es. Preprint, Queen’s University Belfast (200 8), http:/ /www.q ub.ac.uk/puremaths/Preprints/Preprints_2008.html , arXiv:080 5.4076. [Mac71] Sa unders MacLane . Cate gories for the working mathematician . Springer-V erlag , New Y ork, 197 1. Gra duate T exts in Mathematics, V ol. 5. [TT90] R.W. Thomaso n and Thoma s T robaugh. Higher alg ebraic K -theor y of schemes and of derived categories . In The Gr othendie ck F estschrift, Col le ct. Artic. in Honor of the 60th Birthday of A. Gr othendie ck. V ol. III , pag es 247–4 35. Birk h¨ auser, Bosto n, 1990.

On the derived category of a regular toric scheme

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