DERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisse-étale and the flat-fppf sites

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1 DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisse-étale and the flat-fppf sites 1 4. Derived categories of quasi-coherent modules 5 5. Derived pushforward of quasi-coherent modules 8 6. Derived pullback of quasi-coherent modules 8 7. Other chapters 9 References Introduction In this chapter we write about derived categories associated to algebraic stacks. This mean in particular derived categories of quasi-coherent sheaves, i.e., we prove analogues of the results in the chapters entitled Derived Categories of Schemes and Derived Categories of Spaces. The results in this chapter are different from those in [LMB00] mainly because we consistently use the big sites. Before reading this chapter please take a quick look at the chapters Sheaves on Algebraic Stacks and Cohomology of Algebraic Stacks where the terminology we use here is introduced. 2. Conventions, notation, and abuse of language We continue to use the conventions and the abuse of language introduced in Properties of Stacks, Section 2. We use notation as explained in Cohomology of Stacks, Section The lisse-étale and the flat-fppf sites The section is the analogue of Cohomology of Stacks, Section 11 for derived categories. Lemma 3.1. Let X be an algebraic stack. Notation as in Cohomology of Stacks, Lemmas 11.2 and (1) The functor g! : Ab(X lisse,étale ) Ab(Xétale ) has a left derived functor Lg! : D(X lisse,étale ) D(Xétale ) which is left adjoint to g 1 and such that g 1 Lg! = id. This is a chapter of the Stacks Project, version 99bc65e, compiled on Sep 21,

2 2 DERIVED CATEGORIES OF STACKS (2) The functor g! : Mod(X lisse,étale, O Xlisse,étale ) Mod(Xétale, O X ) has a left derived functor Lg! : D(O Xlisse,étale ) D(Xétale, O X ) which is left adjoint to g and such that g Lg! = id. (3) The functor g! : Ab(X flat,fppf ) Ab(X fppf ) has a left derived functor Lg! : D(X flat,fppf ) D(X fppf ) which is left adjoint to g 1 and such that g 1 Lg! = id. (4) The functor g! : Mod(X flat,fppf, O Xflat,fppf ) Mod(X fppf, O X ) has a left derived functor Lg! : D(O Xflat,fppf ) D(O X ) which is left adjoint to g and such that g Lg! = id. Warning: It is not clear (a priori) that Lg! on modules agrees with Lg! on abelian sheaves, see Cohomology on Sites, Remark Proof. The existence of the functor Lg! and adjointness to g is Cohomology on Sites, Lemma (For the case of abelian sheaves use the constant sheaf Z as the structure sheaves.) Moreover, it is computed on a complex H by taking a suitable left resolution K H and applying the functor g! to K. Since g 1 g! K = K by Cohomology of Stacks, Lemmas 11.3 and 11.2 we see that the final assertion holds in each case. Lemma 3.2. With assumptions and notation as in Cohomology of Stacks, Lemma We have g 1 Rf = Rf (g ) 1 and L(g )! (f ) 1 = f 1 Lg! on unbounded derived categories (both for the case of modules and for the case of abelian sheaves). Proof. Let F be an abelian sheaf on Xétale (resp. X fppf ). We first show that the canonical (base change) map g 1 Rf F Rf (g ) 1 F is an isomorphism. To do this let y be an object of Y lisse,étale (resp. Y flat,fppf ). Say y lies over the scheme V such that y : V Y is smooth (resp. flat). Since g 1 is the restriction we find that ( g 1 R p f F ) (y) = H p τ (V y,y X, pr 1 F) where τ = étale (resp. τ = fppf), see Sheaves on Stacks, Lemma By Cohomology of Stacks, Equation (11.6.1) for any sheaf H on X lisse,étale (resp. X flat,fppf ) f H(y) = Γ((V y,y X ), (pr ) 1 H) An object of (V y,y X ) can be seen as a pair (x, ϕ) where x is an object of X lisse,étale (resp. X flat,fppf ) and ϕ : f(x) y is a morphism in Y. We can also think of ϕ as a section of (f ) 1 h y over x. Thus (V Y X ) is the localization of the site X lisse,étale (resp. X flat,fppf ) at the sheaf of sets (f ) 1 h y, see Sites, Lemma The morphism pr : (V y,y X ) X lisse,étale (resp. pr : (V y,y X ) X flat,fppf )

3 DERIVED CATEGORIES OF STACKS 3 is the localization morphism. In particular, the pullback (pr ) 1 preserves injective abelian sheaves, see Cohomology on Sites, Lemma At this point exactly the same argument as in Sheaves on Stacks, Lemma 20.2 shows that (3.2.1) R p f H(y) = H p τ ((V y,y X ), (pr ) 1 H) where τ = étale (resp. τ = fppf). Since (g ) 1 is given by restriction we conclude that (R p f (g ) F) (y) = H p τ ((V y,y X ), pr 1 F (V y,y X ) ) Finally, we can apply Sheaves on Stacks, Lemma 22.3 to see that H p τ ((V y,y X ), pr 1 F (V y,y X ) ) = Hp τ (V y,y X, pr 1 F) are equal as desired; although we omit the verification of the assumptions of the lemma we note that the fact that V Y is smooth (resp. flat) is used to verify the second condition. The rest of the proof is formal. Since cohomology of abelian groups and sheaves of modules agree we also conclude that g 1 Rf F = Rf (g ) 1 F when F is a sheaf of modules on Xétale (resp. X fppf ). Next we show that for G (either sheaf of modules or abelian groups) on Y lisse,étale (resp. Y flat,fppf ) the canonical map L(g )! (f ) 1 G f 1 Lg! G is an isomorphism. To see this it is enough to prove for any injective sheaf I on Xétale (resp. X fppf ) that the induced map Hom(L(g )! (f ) 1 G, I[n]) Hom(f 1 Lg! G, I[n]) is an isomorphism for all n Z. (Hom s taken in suitable derived categories.) By the adjointness of f 1 and Rf, the adjointness of Lg! and g 1, and their primed versions this follows from the isomorphism g 1 Rf I Rf (g ) 1 I proved above. In the case of a bounded complex G (of modules or abelian groups) on Y lisse,étale (resp. Y fppf ) the canonical map (3.2.2) L(g )! (f ) 1 G f 1 Lg! G is an isomorphism as follows from the case of a sheaf by the usual arguments involving truncations and the fact that the functors L(g )! (f ) 1 and f 1 Lg! are exact functors of triangulated categories. Suppose that G is a bounded above complex (of modules or abelian groups) on Y lisse,étale (resp. Y fppf ). The canonical map (3.2.2) is an isomorphism because we can use the stupid truncations σ n (see Homology, Section 13) to write G as a colimit G = colim G n of bounded complexes. This gives a distinguished triangle n 1 G n n 1 G n G... and each of the functors L(g )!, (f ) 1, f 1, Lg! commutes with direct sums (of complexes). If G is an arbitrary complex (of modules or abelian groups) on Y lisse,étale (resp. Y fppf ) then we use the canonical truncations τ n (see Homology, Section 13) to write G as a colimit of bounded above complexes and we repeat the argument of the paragraph above.

4 4 DERIVED CATEGORIES OF STACKS Finally, by the adjointness of f 1 and Rf, the adjointness of Lg! and g 1, and their primed versions we conclude that the first identity of the lemma follows from the second in full generality. Lemma 3.3. Let X be an algebraic stack. Notation as in Cohomology of Stacks, Lemma (1) Let H be a quasi-coherent O Xlisse,étale -module on the lisse-étale site of X. For all p Z the sheaf H p (Lg! H) is a locally quasi-coherent module with the flat base change property on X. (2) Let H be a quasi-coherent O Xflat,fppf -module on the flat-fppf site of X. For all p Z the sheaf H p (Lg! H) is a locally quasi-coherent module with the flat base change property on X. Proof. Pick a scheme U and a surjective smooth morphism x : U X. By Modules on Sites, Definition 23.1 there exists an étale (resp. fppf) covering {U i U} i I such that each pullback f 1 i H has a global presentation (see Modules on Sites, Definition 17.1). Here f i : U i X is the composition U i U X which is a morphism of algebraic stacks. (Recall that the pullback is the restriction to X /f i, see Sheaves on Stacks, Definition 9.2 and the discussion following.) After refining the covering we may assume each U i is an affine scheme. Since each f i is smooth (resp. flat) by Lemma 3.2 we see that f 1 i Lg! H = Lg i,! (f i ) 1 H. Using Cohomology of Stacks, Lemma 7.5 we reduce the statement of the lemma to the case where H has a global presentation and where X = (Sch/X) fppf for some affine scheme X = Spec(A). Say our presentation looks like O O H 0 j J i I where O = O Xlisse,étale (resp. O = O Xflat,fppf ). Note that the site X lisse,étale (resp. X flat,fppf ) has a final object, namely X/X which is quasi-compact (see Cohomology on Sites, Section 16). Hence we have Γ( i I O) = i I A by Sites, Lemma Hence the map in the presentation corresponds to a similar presentation A A M 0 j J i I of an A-module M. Moreover, H is equal to the restriction to the lisse-étale (resp. flat-fppf) site of the quasi-coherent sheaf M a associated to M. Choose a resolution by free A-modules. The complex... F 2 F 1 F 0 M 0... O A F 2 O A F 1 O A F 0 H 0 is a resolution of H by free O-modules because for each object U/X of X lisse,étale (resp. X flat,fppf ) the structure morphism U X is flat. Hence by construction the value of Lg! H is... O X A F 2 O X A F 1 O X A F Since this is a complex of quasi-coherent modules on Xétale (resp. X fppf ) it follows from Cohomology of Stacks, Proposition 7.4 that H p (Lg! H) is quasi-coherent.

5 DERIVED CATEGORIES OF STACKS 5 4. Derived categories of quasi-coherent modules Let X be an algebraic stack. As the inclusion functor QCoh(O X ) Mod(O X ) isn t exact, we cannot define D QCoh (O X ) as the full subcategory of D(O X ) consisting of complexes with quasi-coherent cohomology sheaves. In stead we define the category as follows. Definition 4.1. Let X be an algebraic stack. Let M X Mod(O X ) denote the category of locally quasi-coherent O X -modules with the flat base change property. Let P X M X be the full subcategory consisting of parasitic objects. We define the derived category of O X -modules with quasi-coherent cohomology sheaves as the Verdier quotient 1 D QCoh(O X ) = D (O MX X )/D (O PX X ) This definition makes sense: By Cohomology of Stacks, Proposition 7.4 we see that M X is a weak Serre subcategory of Mod(O X ) hence D MX (O X ) is a strictly full, saturated triangulated subcategory of D(O X ), see Derived Categories, Lemma Since parasitic modules form a Serre subcategory of Mod(O X ) (by Cohomology of Stacks, Lemma 8.2) we see that P X = Parasitic M X is a weak Serre subcategory of Mod(O X ) and hence D PX (O X ) is a strictly full, saturated triangulated subcategory of D(O X ). Since clearly D PX (O X ) D MX (O X ) we conclude that the first is a strictly full, saturated triangulated subcategory of the second. Hence the Verdier quotient exists. A morphism a : E E of D MX (O X ) becomes an isomorphism in D QCoh (O X ) if and only if the cone C(a) has parasitic cohomology sheaves, see Derived Categories, Section 6 and especially Lemma Consider the functors D MX (O X ) Hi Q M X QCoh(OX ) Note that Q annihilates the subcategory P X, see Cohomology of Stacks, Lemma 9.2. By Derived Categories, Lemma 6.8 we obtain a cohomological functor (4.1.1) H i : D QCoh (O X ) QCoh(O X ) Moreover, note that E D QCoh (O X ) is zero if and only if H i (E) = 0 for all i Z. Note that the categories P X and M X are also weak Serre subcategories of the abelian category Mod(Xétale, O X ) of modules in the étale topology, see Cohomology of Stacks, Proposition 7.4 and Lemma 8.2. Hence the statement of the following lemma makes sense. Lemma 4.2. Let X be an algebraic stack. The comparison morphism ɛ : X fppf Xétale induces a commutative diagram D PX (O X ) D MX (O X ) D(O X ) ɛ D PX (Xétale, O X ) ɛ D MX (Xétale, O X ) ɛ D(Xétale, O X ) 1 This definition is different from the one in the literature, see [Ols07, 6.3], but it agrees with that definition by Lemma 4.3.

6 6 DERIVED CATEGORIES OF STACKS Moreover, the left two vertical arrows are equivalences of triangulated categories, hence we also obtain an equivalence D MX (Xétale, O X )/D PX (Xétale, O X ) D QCoh (O X ) Proof. Since ɛ is exact it is clear that we obtain a diagram as in the statement of the lemma. We will show the middle vertical arrow is an equivalence by applying Cohomology on Sites, Lemma 22.6 to the following situation: C = X, τ = fppf, τ = étale, O = O X, A = M X, and B is the set of objects of X lying over affine schemes. To see the lemma applies we have to check conditions (1), (2), (3), (4). Conditions (1) and (2) are clear from the discussion above (explicitly this follows from Cohomology of Stacks, Proposition 7.4). Condition (3) holds because every scheme has a Zariski open covering by affines. Condition (4) follows from Descent, Lemma 8.4. We omit the verification that the equivalence of categories ɛ : D MX (Xétale, O X ) D MX (O X ) induces an equivalence of the subcategories of complexes with parasitic cohomology sheaves. It turns out that D QCoh (O X ) is the same as the derived category of complexes of modules with quasi-coherent cohomology sheaves on the lisse-étale or flat-fppf site. Lemma 4.3. Let X be an algebraic stack. Let F be an object of D MX (O X ). (1) With g as in Cohomology of Stacks, Lemma 11.2 for the lisse-étale site we have (a) g 1 F is in D QCoh (O Xlisse,étale ), (b) g 1 F = 0 if and only if F is in D PX (O X ), (c) Lg! H is in D MX (Xétale, O X ) for H in D QCoh (O Xlisse,étale ), and (d) the functors g 1 and Lg! define mutually inverse functors D QCoh (O X ) g 1 Lg! D QCoh (O Xlisse,étale ) (2) With g as in Cohomology of Stacks, Lemma 11.2 for the flat-fppf site we have (a) g 1 F is in D QCoh (O Xlisse,étale ), (b) g 1 F = 0 if and only if F is in D PX (O X ), (c) Lg! H is in D MX (O X ) for H in D QCoh (O Xflat,fppf ), and (d) the functors g 1 and Lg! define mutually inverse functors D QCoh (O X ) g 1 Lg! D QCoh (O Xflat,fppf ) Proof. The functor g 1 is exact, hence (a) and (b) follow from Cohomology of Stacks, Lemmas 12.3 and The construction of Lg! in Lemma 3.1 (via Cohomology on Sites, Lemma 27.1 which in turn uses Derived Categories, Proposition 28.2) shows that Lg! on any object H of D(O Xlisse,étale ) is computed as Lg! H = colim g! K n = g! colim K n (termwise colimits) where the quasi-isomorphism colim K n H induces quasiisomorphisms K n τ n H. Since M X Mod(Xétale, O X ) (resp. M X Mod(O X ))

7 DERIVED CATEGORIES OF STACKS 7 is preserved under colimits we see that it suffices to prove (c) on bounded above complexes H in D QCoh (O Xlisse,étale ) (resp. D QCoh (O Xflat,fppf )). In this case to show that H n (Lg! H ) is in M X we can argue by induction on the integer m such that H i = 0 for i > m. If m < n, then H n (Lg! H ) = 0 and the result holds. In general consider the distinguished triangle τ m 1 H H H m (H )[ m]... (Derived Categories, Remark 12.4) and apply the functor Lg!. Since M X is a weak Serre subcategory of the module category it suffices to prove (c) for two out of three. We have the result for Lg! τ m 1 H by induction and we have the result for Lg! H m (H )[ m] by Lemma 3.3. Whence (c) holds. Let us prove (2)(d). By (a) and (b) the functor g 1 = g induces a functor c : D QCoh (O X ) D QCoh (O Xflat,fppf ) see Derived Categories, Lemma 6.8. Thus we have the following diagram of triangulated categories D MX (O X ) Lg! g 1 q D QCoh (O Xflat,fppf ) c D QCoh (O X ) where q is the quotient functor, the inner triangle is commutative, and g 1 Lg! = id. For any object of E of D MX (O X ) the map a : Lg! g 1 E E maps to a quasiisomorphism in D(O Xflat,fppf ). Hence the cone on a maps to zero under g 1 and by (b) we see that q(a) is an isomorphism. Thus q Lg! is a quasi-inverse to c. In the case of the lisse-étale site exactly the same argument as above proves that D MX (Xétale, O X )/D PX (Xétale, O X ) is equivalent to D QCoh (O Xlisse,étale ). Applying the last equivalence of Lemma 4.2 finishes the proof. The following lemma tells us that the quotient functor D MX (O X ) D QCoh (O X ) is a Bousfield colocalization (insert future reference here). Lemma 4.4. Let X be an algebraic stack. Let E be an object of D MX (O X ). There exists a canonical distinguished triangle E E P E [1] in D MX (O X ) such that P is in D PX (O X ) and for all P in D PX (O X ). Hom D(OX )(E, P ) = 0 Proof. Consider the morphism of ringed topoi g : Sh(X flat,fppf ) Sh(X fppf ). Set E = Lg! g 1 E and let P be the cone on the adjunction map E E. Since g 1 E g 1 E is an isomorphism we see that P is an object of D PX (O X ) by Lemma 4.3 (2)(b). Finally, Hom(E, P ) = Hom(Lg! g 1 E, P ) = Hom(g 1 E, g 1 P ) = 0 as g 1 P = 0. Uniqueness. Suppose that E E P is a second distinguished triangle as in the statement of the lemma. Since Hom(E, P ) = 0 the morphism E E factors

8 8 DERIVED CATEGORIES OF STACKS as E E E, see Derived Categories, Lemma 4.2. Similarly, the morphism E E factors as E E E. Consider the composition ϕ : E E of the maps E E and E E. Note that ϕ 1 : E E fits into the commutative diagram E E ϕ 1 E E hence factors through P [ 1] E. Since Hom(E, P [ 1]) = 0 we see that ϕ = 1. Whence the maps E E and E E are inverse to each other. 5. Derived pushforward of quasi-coherent modules As a first application of the material above we construct the derived pushforward. In Examples, Section 50 the reader can find an example of a quasi-compact and quasi-separated morphism f : X Y of algebraic stacks such that the direct image functor Rf does not induce a functor D QCoh (O X ) D QCoh (O Y ). Thus restricting to bounded below complexes is necessary. Proposition 5.1. Let f : X Y be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor Rf induces a commutative diagram D + P X (O X ) 0 D + M X (O X ) D(O X ) Rf and hence induces a functor Rf Rf D + P Y (O Y ) D + M Y (O Y ) D(O Y ) Rf QCoh, : D + QCoh (O X ) D + QCoh (O Y) on quotient categories. Moreover, the functor R i f QCoh of Cohomology of Stacks, Proposition 10.1 are equal to H i Rf QCoh, with H i as in (4.1.1). Proof. We have to show that Rf E is an object of D + M Y (O Y ) for E in D + M X (O X ). This follows from Cohomology of Stacks, Proposition 7.4 and the spectral sequence R i f H j (E) R i+j f E. The case of parasitic modules works the same way using Cohomology of Stacks, Lemma 8.3. The final statement is clear from the definition of H i in (4.1.1). 6. Derived pullback of quasi-coherent modules Derived pullback of complexes with quasi-coherent cohomology sheaves exists in general. Proposition 6.1. Let f : X Y be a morphism of algebraic stacks. The exact functor f induces a commutative diagram D MX (O X ) D(O X ) f D MY (O Y ) f D(O Y )

9 DERIVED CATEGORIES OF STACKS 9 The composition D MY (O Y ) f D MX (O X ) q X D QCoh (O X ) is left deriveable with respect to the localization D MY (O Y ) D QCoh (O Y ) and we may define LfQCoh as its left derived functor Lf QCoh : D QCoh (O Y ) D QCoh (O X ) (see Derived Categories, Definitions 15.2 and 15.9). If f is quasi-compact and quasi-separated, then LfQCoh and Rf QCoh, satisfy the following adjointness: Hom DQCoh (O X )(Lf QCohA, B) = Hom DQCoh (O Y )(A, Rf QCoh, B) for A D QCoh (O Y ) and B D + QCoh (O X ). Proof. To prove the first statement, we have to show that f E is an object of D MX (O X ) for E in D MY (O Y ). Since f = f 1 is exact this follows immediately from the fact that f maps M Y into M X. Set D = D MY (O Y ). Let S be the collection of morphisms in D whose cone is an object of D PY (O Y ). Set D = D QCoh (O X ). Set F = q X f : D D. Then D, S, D, F are as in Derived Categories, Situation 15.1 and Definition Let us prove that LF (E) is defined for any object E of D. Namely, consider the triangle E E P E [1] constructed in Lemma 4.4. Note that s : E E is an element of S. We claim that E computes LF. Namely, suppose that s : E E is another element of S, i.e., fits into a triangle E E P E [1] with P in D PY (O Y ). By Lemma 4.4 (and its proof) we see that E E factors through E E. Thus we see that E E is cofinal in the system S/E. Hence it is clear that E computes LF. To see the final statement, write B = q X (H) and A = q Y (E). Choose E E as above. We will use on the one hand that Rf QCoh, (B) = q Y (Rf H) and on the other that Lf QCoh (A) = q X (f E ). Hom DQCoh (O X )(Lf QCohA, B) = Hom DQCoh (O X )(q X (f E ), q X (H)) = colim H H Hom D(OX )(f E, H ) = colim H H Hom D(OY )(E, Rf H ) = Hom D(OY )(E, Rf H) = Hom DQCoh (O Y )(A, Rf QCoh, B) Here the colimit is over morphisms s : H H in D + M X (O X ) whose cone P (s) is an object of D + P X (O X ). The first equality we ve seen above. The second equality holds by construction of the Verdier quotient. The third equality holds by Cohomology on Sites, Lemma Since Rf P (s) is an object of D + P Y (O Y ) by Proposition 5.1 we see that Hom D(OY )(E, Rf P (s)) = 0. Thus the fourth equality holds. The final equality holds by construction of E. 7. Other chapters Preliminaries (1) Introduction (2) Conventions (3) Set Theory (4) Categories

10 10 DERIVED CATEGORIES OF STACKS (5) Topology (6) Sheaves on Spaces (7) Sites and Sheaves (8) Stacks (9) Fields (10) Commutative Algebra (11) Brauer Groups (12) Homological Algebra (13) Derived Categories (14) Simplicial Methods (15) More on Algebra (16) Smoothing Ring Maps (17) Sheaves of Modules (18) Modules on Sites (19) Injectives (20) Cohomology of Sheaves (21) Cohomology on Sites (22) Differential Graded Algebra (23) Divided Power Algebra (24) Hypercoverings Schemes (25) Schemes (26) Constructions of Schemes (27) Properties of Schemes (28) Morphisms of Schemes (29) Cohomology of Schemes (30) Divisors (31) Limits of Schemes (32) Varieties (33) Topologies on Schemes (34) Descent (35) Derived Categories of Schemes (36) More on Morphisms (37) More on Flatness (38) Groupoid Schemes (39) More on Groupoid Schemes (40) Étale Morphisms of Schemes Topics in Scheme Theory (41) Chow Homology (42) Adequate Modules (43) Dualizing Complexes (44) Étale Cohomology (45) Crystalline Cohomology (46) Pro-étale Cohomology Algebraic Spaces (47) Algebraic Spaces (48) Properties of Algebraic Spaces (49) Morphisms of Algebraic Spaces (50) Decent Algebraic Spaces (51) Cohomology of Algebraic Spaces (52) Limits of Algebraic Spaces (53) Divisors on Algebraic Spaces (54) Algebraic Spaces over Fields (55) Topologies on Algebraic Spaces (56) Descent and Algebraic Spaces (57) Derived Categories of Spaces (58) More on Morphisms of Spaces (59) Pushouts of Algebraic Spaces (60) Groupoids in Algebraic Spaces (61) More on Groupoids in Spaces (62) Bootstrap Topics in Geometry (63) Quotients of Groupoids (64) Simplicial Spaces (65) Formal Algebraic Spaces (66) Restricted Power Series (67) Resolution of Surfaces Deformation Theory (68) Formal Deformation Theory (69) Deformation Theory (70) The Cotangent Complex Algebraic Stacks (71) Algebraic Stacks (72) Examples of Stacks (73) Sheaves on Algebraic Stacks (74) Criteria for Representability (75) Artin s Axioms (76) Quot and Hilbert Spaces (77) Properties of Algebraic Stacks (78) Morphisms of Algebraic Stacks (79) Cohomology of Algebraic Stacks (80) Derived Categories of Stacks (81) Introducing Algebraic Stacks Miscellany (82) Examples (83) Exercises (84) Guide to Literature (85) Desirables (86) Coding Style (87) Obsolete (88) GNU Free Documentation License (89) Auto Generated Index

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