Non characteristic finiteness theorems in crystalline cohomology


 Lillian Underwood
 3 years ago
 Views:
Transcription
1 Non characteristic finiteness theorems in crystalline cohomology 1 Non characteristic finiteness theorems in crystalline cohomology Pierre Berthelot Université de Rennes 1 I.H.É.S., September 23, 2015 in honor of Arthur Ogus, for his 70th birthday
2 Non characteristic finiteness theorems in crystalline cohomology 2 1. Introduction 1.1. Torsion coefficients for crystalline cohomology Notation 1. Introduction The purpose of this talk is to give some finiteness results on torsion coefficients for crystalline cohomology. Notation: p = prime number, fixed for the whole talk; N 1 integer. k = perfect field of characteristic p; W N = W N (k), ring of Witt vectors of length N with coefficients in k.
3 Non characteristic finiteness theorems in crystalline cohomology 3 1. Introduction 1.1. Torsion coefficients for crystalline cohomology Finiteness of crystalline cohomology Recall that, for a smooth and proper kscheme X 0, crystalline cohomology with constant coefficients relative to W N is a good cohomology. Here this will mean that we have: 1 Isomorphism with the de Rham cohomology of a smooth lifting over W N, whenever there exists such a lifting; 2 RΓ crys (X 0 /W N ) is a perfect complex of W N modules; 3 RΓ crys (X 0 /W N ) satisfies Poincaré duality. These properties still hold for cohomology with coefficients in a locally free finitely generated crystal. But such crystals do not suffice to describe direct images and to get a relative version of these finiteness theorems, even for a morphism of smooth and proper kschemes.
4 Non characteristic finiteness theorems in crystalline cohomology 4 1. Introduction 1.1. Torsion coefficients for crystalline cohomology Expected generalization We want here to introduce bigger categories of coefficients relative to W N for which similar results hold, and which have a sufficiently reasonable behaviour under inverse and direct image so as to have better relative versions of the finiteness properties. Note however that, since crystalline cohomology itself is not finitely generated over W N in the non proper or non smooth case, and also because of the properties of the Frobenius pullback in arithmetic Dmodules theory, we do not expect a full formalism of Grothendieck s six operations in this context.
5 Non characteristic finiteness theorems in crystalline cohomology 5 1. Introduction 1.2. Main results Categories of coefficients Let X 0 be a smooth kscheme, S = Spec(W N ), and O X0 /S the structural sheaf of the crystalline site Crys(X 0 /W N ). We will construct two triangulated categories of filtered complexes of O X0 /Smodules, called respectively Dperfect and D perfect complexes, such that: 1 Whenever there exists a smooth lifting X of X 0 over W N, these are respectively equivalent and antiequivalent to the category of perfect complexes of filtered D X modules, where D X is the sheaf of PDdifferential operators on X (filtrations will be defined later). 2 Dperfect and D perfect complexes are related by a local biduality theorem.
6 Non characteristic finiteness theorems in crystalline cohomology 6 1. Introduction 1.2. Main results Non characteristic morphisms 3 These complexes have a singular support in the cotangent space T X 0, hence one can define the notion of a noncharacteristic morphism f 0 : X 0 Y 0 with respect to such a complex on X 0 or Y 0. 4 With appropriate noncharacteristic assumptions, one can prove comparison theorems between cohomological operations on perfect filtered complexes of D X modules and Dperfect (resp. D perfect) complexes, as well as finiteness and duality theorems.
7 Non characteristic finiteness theorems in crystalline cohomology 7 1. Introduction 1.2. Main results Remarks Additional comments: The results given here hold for complexes on the PDnilpotent crystalline site, which in some cases forces to assume p 2. Similar results should hold for complexes on the usual crystalline site, but one can expect increased technicalities due to the nilpotency condition needed to associate a crystal to a D X module. If one endows S with the HyodoKato log structure, the theory applies to smooth fine saturated log schemes, satisfying an additional condition on the dualizing complex (see Tsuji s article on log cystalline Poincaré duality).
8 Non characteristic finiteness theorems in crystalline cohomology 8 2. Dperfection Notation 2. Dperfection We now consider a more general situation X where: X 0 S 0 S, S = Z/p N Zscheme, S 0 = V (I) S, I O S = quasicoherent ideal, endowed with a PDnilpotent divided power structure γ; X 0 = smooth S 0 scheme, of relative dimension d (or d X ); X = smooth Sscheme lifting X 0, if assumed to exist; then: D X = sheaf of PDdifferential operators on X /S (= D (0) X ).
9 Non characteristic finiteness theorems in crystalline cohomology 9 2. Dperfection 2.1. Dmodules and crystals The crystal associated to a D X module Assume a smooth lifting X of X 0 is given. Classically, one defines a functor as follows: C X0 : {Left D X modules} {O X0 /Smodules} If (U, T, δ) is a thickening in Crys(X 0 /S), there exists locally on T an Smorphism h : T X extending U X 0 X. If E is a left D X module and h 1, h 2 are two such extensions, the Taylor formula provides a canonical isomorphism of O T modules h2 E h1 E, satisfying a transitivity relation for three extensions. Glueing via these isomorphisms the sheaves h E when (U, T, δ) and h vary, one gets an O X0 /Smodule C X0 (E), functorially in E.
10 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.1. Dmodules and crystals Equivalence between Dmodules and crystals The following basic result is then well known: Proposition 1 The functor C X0 factors through an equivalence {Left D X modules} {Crystals on X 0 /S} C X0 {O X0 /Smodules}. Note that, since C X0 is defined by the inverse image functors h, it is only a right exact functor on the category of left D X modules.
11 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.2. The functor CR X0 Definition of CR X0 We now want a derived category version of Proposition??. We derive C X0 and we shift so as to insure later compatibility with Borel s conventions: we define the functor by setting, for E D (D X ), CR X0 : D (D X ) D (O X0 /S) CR X0 (E ) := LC X0 (E )[d X ], where LC X0 (E ) is obtained by applying C X0 to a resolution of E by a complex of flat D X modules, or more generally by a complex of D X modules which are flat over O X.
12 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.2. The functor CR X0 Crystalline complexes Notation. For any thickening (U, T, δ) Crys(X 0 /S) and any complex E D(O X0 /S), we denote by E T D(O T ) the complex of Zariski sheaves on T defined by E. Definition. Let E D (O X0 /S). We say that E is crystalline if, for any morphism v : (U, T, δ ) (U, T, δ) in Crys(X 0 /S), the canonical morphism Lv (E T ) E T (obtained by choosing a flat resolution P of E and taking the transition morphism for P ) is an isomorphism of D (O T ). Example: If f 0 : X 0 Y 0 is a smooth morphism, Rf 0 crys (O X0 /S) is a crystalline complex on Crys(Y 0 /S).
13 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.2. The functor CR X0 Triangulated subcategories of D(O X0 /S) Notation. We define some triangulated subcategories of D(O X0 /S): D crys(o X0 /S) = the full subcategory of D(O X0 /S) whose objects are the crystalline complexes; D b ftd (O X 0 /S) = the full subcategory of D(O X0 /S) whose obejcts are the complexes of O X0 /Smodules which are bounded of finite Tor dimension; D qc (O X0 /S) = the full subcategory of D(O X0 /S) whose obejcts are the complexes of O X0 /Smodules E such that E T has O T quasicoherent cohomology sheaves for any thickening (U, T, δ). A sequence of indexes will denote the intersection of the corresponding subcategories.
14 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.2. The functor CR X0 Properties of CR X0 The following properties follow immediately from the definition of CR X0 : Proposition 2 Let E D (D X ). 1 CR X0 (E ) D crys(o X0 /S). 2 CR X0 (E ) D b ftd (O X 0 /S) if and only if E D b ftd (D X ). 3 CR X0 (E ) D qc(o X0 /S) if and only if E D qc(d X ). Our strategy will now be to show that these properties characterize the essential image of DfTd,qc b (D X ) inside D (O X0 /S), and that CR X0 induces an equivalence of categories with this image.
15 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.3. The right adjoint to CR X0 The crystalline bimodule defined by D X We will use the existence of a right adjoint functor to C X0. Recall that the crystalline topos (X 0 /S) crys projects to the Zariski topos X Zar via a morphism of topos u X0 /S : (X 0 /S) crys X Zar. It is characterized by its inverse image functor, defined by Γ((U, T, δ), u 1 X 0 /S(F)) = Γ(U, F) for all sheaves F on X and all thickenings (U, T, δ). The crystalline transfert bimodule. One can apply the functor C X0 to D X viewed as a left D X module over itself. Then, by functoriality, the right action of D X on itself endows C X0 (D X ) with a structure of (O X0 /S, u 1 X 0 /S (D X ))bimodule.
16 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.3. The right adjoint to CR X0 Definition of DM X We define a functor M X : {O X0 /Smodules} {Left D X modules} by setting, for any O X0 /Smodule F, M X (F ) := u XO /S Hom (C OX0 /S X 0 (D X ), F ). The functor M X is right exact, and we set, for F D + (O X0 /S), DM X (F ) := RM X (F )[ d X ] Ru XO /S RHom (C OX0 /S X 0 (D X ), F )[ d X ]. The following statements are then formal:
17 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.3. The right adjoint to CR X0 The adjunction formula Proposition 3 1 For any left D X module E, there is a functorial isomorphism of O X0 /Smodules C X0 (E) C X0 (D X ) u 1 X 0 /S D X u 1 X 0 /S E. 2 The functor M X is right adjoint to the functor C X0. 3 For any E D (D X ), F D + (O X0 /S), there is a canonical isomorphism Ru X0 /S RHom OX0 /S (CR X 0 (E ), F ) RHom DX (E, DM X (F )).
18 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.3. The right adjoint to CR X0 Properties of DM X Proposition 4 Let F Dcrys(O b X0 /S). 1 DM X (F ) DfTd b (D X ) if and only if F DfTd b (O X 0 /S). 2 DM X (F ) Dqc(D b X ) if and only if F Dqc(O b X0 /S). From Propositions?? and??, we get functors CR X0 : DfTd,qc b (D X ) Dcrys,fTd,qc b (O X 0 /S), DM X : Dcrys,fTd,qc b (O X 0 /S) DfTd,qc b (D X ).
19 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.3. The right adjoint to CR X0 The equivalence theorem Theorem 1 The functors CR X0 and DM X are quasiinverse equivalences between DfTd,qc b (D X ) and Dcrys,fTd,qc b (O X 0 /S). Hint: For E DfTd,qc b (D X ), F Dcrys,fTd,qc b (O X 0 /S), the adjunction formula provides canonical morphisms E DM X (CR X0 (E )), CR X0 (DM X (F )) F ). Using the classical computation of Ru X0 /S by means of the ČechAlexander complex, one proves that these are isomorphisms.
20 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.3. The right adjoint to CR X0 Some consequences Corollary 1.1 Let E, F D b ftd,qc (D X ). There is a canonical isomorphism Ru X0 /S RHom OX0 /S (CR X 0 (E ), CR X0 (F )) RHom DX (E, F ). For E = O X [ d X ], we get back the isomorphism between crystalline and de Rham cohomologies: Corollary 1.2 Let F D b ftd,qc (D X ). There is a canonical isomorphism Ru X0 /S (CR X0 (F )) DR(F ) := RHom DX (O X, F )[d X ].
21 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.4. Dperfect complexes Definition of Dperfection Definition. A complex E D b (O X0 /S) is Dperfect if there exists an open covering (V 0,α ) of X 0, and, for each α, a smooth lifting V α of V 0,α over S, a perfect complex E α D b perf (D V α ) and an isomorphism E Crys(V0,α /S) CR V0,α (E α). Special case. Bounded perfect complexes of O X0 /Smodules are Dperfect. Notation. We denote by D b Dperf (X 0/S) D b (O X0 /S) the full subcategory of Dperfect complexes.
22 Non characteristic finiteness theorems in crystalline cohomology Dperfection 2.4. Dperfect complexes Characterizations of Dperfection Proposition 5 Let X be a smooth lifting of X 0 over S, and E D(O X0 /S). Then E is Dperfect if and only if there exists a complex E D b perf (D X ) and an isomorphism E CR X0 (E ). Corollary 5.1 The subcategory D b Dperf (X 0/S) D b (O X0 /S) is triangulated. Proposition 6 Assume that S is locally noetherian. Let E D b crys,ftd,qc (O X 0 /S). Then E is Dperfect relative to S if and only if E Crys(X0 /S 0 ) is Dperfect relative to S 0.
23 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection Direct images under a closed immersion 3. Filtrations, local duality and D perfection Note: if f 0 : X 0 Y 0 is a closed immersion between smooth S 0 schemes, Rf 0 crys dose not preserve Dperfection. Indeed, let (V, T, δ) Crys(Y 0 /S), U = X 0 V. Let K O T be the ideal of U in T, P U (T ) the PDenvelope of K compatible with γ and δ, and K P U (T ) the PDideal generated by K. Then: Rf 0 crys (O X0 /S) T which is not a quasicoherent O T module. lim n P U (T )/K [n]. We will see that Rf 0 crys (O X0 /S) is actually the O X0 /Slinear dual of a Dperfect complex.
24 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.1. Derived categories of filtered modules Filtered rings and modules To deal with such complexes, we will construct a triangulated category of O X0 /Sduals of Dperfect complexes. As D X is locally free of infinite rank over O X, we need to take into account some extra structure on the dual so as to be able to recover the initial complex from its dual via a biduality theorem. To this end, we will now work systematically with filtered complexes. Filtered rings and modules are defined as in [Bourbaki, Alg. Comm., Ch. III, 2, no. 1]. The functors Hom(, ) and are endowed with their natural filtration. This provides filtered versions of the usual functors, e.g. filtered inverse images of filtered modules under a morphism of filtered ringed spaces.
25 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.1. Derived categories of filtered modules Basic examples We will use the following filtrations: 1 O X is filtered by the IPDadic filtration, given by Fil i O X = I [i] O X, with I [i] = O S for i 0. 2 D X is filtered by the tensor product of the filtration by the order of PDdifferential operators with the IPDadic filtration: Fil i D X = I [j] D X, k, j+k=i where, for all n Z, D X,n is the sheaf of PDdifferential operators of order n. 3 O X0 /S is filtered by the J X0 /SPDadic filtration, where (J X0 /S) T = J T.
26 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.1. Derived categories of filtered modules Filtered crystals Definition: A filtered crystal on X 0 /S (called a Tcrystal in [Ogus, Astérisque 221]) is a filtered O X0 /Smodule E such that, for any morphism v : (U, T, δ ) (U, T, δ) of Crys(X 0 /S), the transition morphism v (E T ) E T (where v is the filtered inverse image functor), is a filtered isomorphism. Let X be a smooth Sscheme lifting X 0. With these defiinitions, the functor C X0 extends as an equivalence of categories C X0 : {Filtered left D X modules} {Filtered crystals on X 0 /S}.
27 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.1. Derived categories of filtered modules Derived categories and functors for filtered modules The category of filtered modules over a filtered ring (A, A i ) is not abelian, but it has a natural notion of short exact sequence which turns it into an exact category in the sense of [Quillen, SLNM 341]. We can then apply Laumon s results in [SLNM 1016] to build the derived category DF (A) of complexes of filtered Amodules. One can also define as in [Laumon] the right and left derived functors of an additive functor between categories of filtered modules. Because we work with general filtered modules (without exhaustivity or separatedness assumption), there are enough displayed objects to ensure the derivability of the usual functors. Finally, one can extend the finiteness conditions of [SGA 6, Exposé I] to complexes of filtered modules. This provides the notions of pseudocoherence, finite tor dimension and perfection for complexes of filtered Amodules.
28 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.1. Derived categories of filtered modules The filtered CR X0 functor We use these constructions to define the left derived filtered functor LC X0. All the results of the previous section remain valid in the filtered context. In particular, when X 0 has a smooth lifting X over S, the filtered functor CR X0 := LC X0 [d X ] induces an equivalence of categories CR X0 : D b F ftd,qc (D X ) D b F crys,ftd,qc (O X0 /S). Without liftability assumption, we can define as above the category D b F Dperf (X 0 /S) D b F crys,ftd,qc (O X0 /S): a filtered complex E belongs to D b F Dperf (X 0 /S) if and only if there exists a covering (X α ) of X such that, for each α, the restriction of E to X α belongs to the essential image of D b F perf (D Xα ).
29 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.2. The biduality theorem Exhaustive complexes Definition. If E is a filtered module over a filtered ring A, we denote A f := Fil i A, E f := Fil i E, i Z i Z and we endow A f and E f with the induced filtrations. This turns E f into a filtered A f module. We say that E is exhaustive if E f = E. We say that a complex E DF (A) is exhaustive if the canonical morphism E f E is an isomorphism in DF (A f ). Example: Any Dperfect complex on X 0 /S is exhaustive.
30 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.2. The biduality theorem The finite order biduality morphism Let A be a commutative ring, endowed with an exhaustive filtration, and let E, I be two complexes of filtered Amodules. The classical biduality morphism for E relative to I sits in a commutative diagram E Hom A (Hom A (E, I ), I ) Hom f A (Hom A (E, I ), I ) E f Hom f A (Hom A (E, I ), I ) Hom f f A (Hom A (E, I ), I ). We define the finite order biduality morphism for E relative to I as being the composition of the bottom row of the diagram. This definition extends naturally to DF (A).
31 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.2. The biduality theorem The crystalline dual functor We define the crystalline dualizing complex on X 0 /S by K X0 /S := O X0 /S(d X )[2d X ]. For E DF (O X0 /S), we define its crystalline dual by E := D crys X 0 /S (E ) := RHom f O X0 /S (E, K X0 /S) DF (O X0 /S). If X is a smooth lifting of X 0 over S, we define the functor CR X 0 : D F (D X ) D + F (O X0 /S) by setting, for E D F (D X ), CR X 0 (E ) := (CR X0 (E )).
32 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.2. The biduality theorem Biduality for Dperfect complexes Assume now that E is exhaustive. Taking for I an appropriate resolution of K X0 /S, the previous biduality diagram provides in DF (O X0 /S) a canonical biduality morphism E E. Theorem 2 Let E D b F Dperf (X 0 /S). 1 The complex RHom OX0 /S (E, K X0 /S) is exhaustive, hence isomorphic to E, and bounded. 2 The biduality morphisms E E and E (E ) are isomorphisms.
33 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.3. D perfect complexes Definition of D perfection Definition. A complex F D b F (O X0 /S) is D perfect if there exists an open covering (V 0,α ) of X 0, and, for each α, a smooth lifting V α of V 0,α over S, a perfect complex E α D b F perf (D Vα ) and an isomorphism F Crys(V0,α /S) CR V 0,α (E α). The category of D perfect complexes is a full subcategory of D b F (O X0 /S), denoted by D b F D perf(x 0 /S). The following characterization implies that the condition is independent of the covering (V 0,α ) and of the liftings (V α ).
34 Non characteristic finiteness theorems in crystalline cohomology Filtrations, local duality and D perfection 3.3. D perfect complexes Characterization of D perfection Proposition 7 A filtered complex F D b F (O X0 /S) is D perfect if and only if the following conditions hold: 1 F is exhaustive. 2 F is Dperfect. 3 The biduality morphism F F is an isomorphism in DF (O X0 /S). Corollary 7.1 The category of D perfect complexes is a triangulated subcategory of DF (O X0 /S), which is antiequivalent to D b F Dperf (X 0 /S).
35 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.1. Singular support and non characteristic morphisms Singular support of D and D perfect complexes 4. Comparison and finiteness theorems Dperfect and D perfect complexes have a singular support. It is a closed subset of the cotangent space T X 0, defined as follows. 1 Assume first that X 0 has a smooth lifting X over S. Then gr D X gr O X OX0 gr D X0. As I is a nilpotent ideal, it follows that the topological spaces Spec(gr D X ) and T X 0 can be identified. 2 The associated graded module functor extends as an exact functor gr : DF (D X ) D(gr D X ). If E D b F perf (D X ), then the sheaves H n (gr E ) are quasicoherent gr D X modules, and we denote by H n (gr E ) the corresponding quasicoherent sheaf on the affine X 0 scheme Spec(gr D X ).
36 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.1. Singular support and non characteristic morphisms Singular support of D and D perfect complexes 3 If E D b F Dperf (X 0 /S), let E D b F perf (D X ) be such that E CR X0 (E ). One can define the singular support SS(E ) as the closed subset SS(E ) := n Supp H n (gr E ) T X 0. One checks that this does not depend on the choice of the lifting X. 4 In the general case, one can choose local liftings of X 0 and glue the local constructions. 5 If F D b F D perf(x 0 /S), one defines SS(F ) as being SS(F ).
37 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.1. Singular support and non characteristic morphisms Non characteristic morphisms Let f 0 : X 0 Y 0 be a morphism of smooth S 0 schemes, and let T X 0 ϕ 0 X 0 Y0 T Y 0 g 0 T Y 0 π X0 π X 0 X 0 f 0 Y 0. π Y0 be the associated functoriality diagram for the cotangent bundle. Definitions. If F D b F Dperf (Y 0 /S), we say that f 0 is non characteristic for F if the restriction of ϕ 0 to g0 1 (SS(F )) is proper. NB: This condition is always satisfied when f 0 is smooth. If E D b F Dperf (X 0 /S), we say that f 0 is non characteristic for E if the restriction of f 0 to Supp(E ) is proper ( g 0 ϕ 1 0 (SS(E )) is proper).
38 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.2. Non characteristic finiteness theorems for D X modules Finiteness theorems for Dmodules Assume now that S is locally noetherian. Let f : X Y be a morphism of smooth Sschemes, F D b F perf (D Y ), E D b F perf (D X ). As usual, we define Theorem 3 f! (F ) := D X Y L f 1 (D Y ) f 1 (F )(d X /Y )[d X /Y ], f + (E ) := Rf (D Y X L DX E ). 1 If f is non characteristic for F, then f! (F ) D b F perf (D X ). 2 If f is non characteristic for E, then f + (E ) D b F perf (D Y ). Proof as in [Laumon, Astérique 130], but using more general finiteness properties in derived categories of filtered modules.
39 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.3. Finiteness results for inverse images Inverse images of Dperfect complexes Lemma 8.1 Let f : X Y be a morphism of smooth Sschemes, with reduction f 0 : X 0 Y 0 over S 0. For any F D F (D Y ), there is a canonical isomorphism Lf0 crys(cr Y0 (F )) CR X0 (f! (F ))( d X /Y )[ 2d X /Y ]. Proposition 8 Let f 0 : X 0 Y 0 be a morphism of smooth S 0 schemes. If F D b F Dperf (Y 0 /S) and f 0 is non characteristic for F, then Lf 0 crys (F ) D b F Dperf (X 0 /S).
40 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.3. Finiteness results for inverse images Inverse images of D perfect complexes For inverse images of D perfect complexes, we work with the big crystalline topos. Lemma 9.1 Let f : X Y be a morphism of smooth Sschemes, with reduction f 0 : X 0 Y 0 over S 0. For any F D b F perf (D Y ), there is a canonical isomorphism Lf0 CRYS CR Y 0 (F ) CR X 0 (f! (F )). Proposition 9 Let f 0 : X 0 Y 0 be a morphism of smooth S 0 schemes. If F D b F D perf(y 0 /S) and f 0 is non characteristic for F, then Lf 0 CRYS (F ) D b F D perf(x 0 /S).
41 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.4. Finiteness results for direct images Direct images of Dperfect complexes (smooth case) Lemma 10.1 Let f : X Y be a smooth morphism of smooth Sschemes, with reduction f 0 : X 0 Y 0 over S 0. For any E D b F perf (D X ), there is a canonical isomorphism CR Y0 (f + (E )) Rf 0 crys CR X0 (E ). The proof proceeds by reducing to comparison with de Rham cohomology. Proposition 10 Let f 0 : X 0 Y 0 be a smooth morphism of smooth S 0 schemes. If E D b F Dperf (X 0 /S) and f 0 is non characteristic for E, then Rf 0 crys (E ) D b F Dperf (Y 0 /S).
42 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.4. Finiteness results for direct images Comparison theorem for D perfect complexes To study the direct images of D perfect complexes, we need to make stronger assumptions on S, in order to use Grothendieck s duality theory for coherent sheaves in the crystalline context: from now on, we assume that S is a quotient of a discrete valuation ring. The following comparison theorem can be viewed as a relative duality theorem for the CR X0 functor: Theorem 4 Let f : X Y be a proper morphism between smooth Sschemes, and let E D b F perf (D X ). There exists in D b F (O Y0 /S) a canonical isomorphism CR Y 0 (f + (E )) Rf 0 crys (CR X 0 (E )).
43 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.4. Finiteness results for direct images Direct images of D perfect complexes Theorem 5 Let X 0, Y 0 be proper and smooth S 0 schemes, and f 0 : X 0 Y 0 an S 0 morphism. If E D b F D perf(x 0 /S), then Rf 0 crys (E ) D b F D perf(y 0 /S). The proof uses the graph factorization to deal separately with the cases of a closed immersion, which follows from Theorem??, and of a proper and smooth morphism, which uses the next duality theorem.
44 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.4. Finiteness results for direct images Relative Poincaré duality Theorem 6 Let f 0 : X 0 Y 0 be a proper and smooth morphism of smooth S 0 schemes. 1 There exists in D b F (O YO /S) a trace morphism Tr f0 : Rf 0 crys (K X0 /S) K Y0 /S, whose value on the thickening (Y 0, Y 0 ) can be identified with the de Rham trace morphism. 2 Let E D b F Dperf (X 0 /S). The pairing Rf 0 crys (E ) L OY0 /S Rf 0 crys (E ) K Y0 /S induced by Tr f0 is a perfect pairing.
45 Non characteristic finiteness theorems in crystalline cohomology Comparison and finiteness theorems 4.4. Finiteness results for direct images Crystalline cohomology of Dperfect complexes Theorem 7 Let k be a perfect field of characteristic p 2, S 0 = Spec(k), S = Spec(W N (k)). Let X 0 be a proper and smooth kscheme, and let E D b F Dperf (X 0 /S). 1 The crystalline cohomology complexes RΓ(X 0 /S, E ) and RΓ(X 0 /S, E ) are perfect complexes of W N (k)modules. 2 The crystalline trace morphism induces a perfect pairing RΓ(X 0 /S, E ) L WN (k) RΓ(X 0 /S, E ) W N (k).
Basic results on Grothendieck Duality
Basic results on Grothendieck Duality Joseph Lipman 1 Purdue University Department of Mathematics lipman@math.purdue.edu http://www.math.purdue.edu/ lipman November 2007 1 Supported in part by NSA Grant
More informationarxiv: v1 [math.ag] 25 Feb 2018
ON HIGHER DIRECT IMAGES OF CONVERGENT ISOCRYSTALS arxiv:1802.09060v1 [math.ag] 25 Feb 2018 DAXIN XU Abstract. Let k be a perfect field of characteristic p > 0 and W the ring of Witt vectors of k. In this
More informationRational points over finite fields for regular models of algebraic varieties of Hodge type 1
Annals of Mathematics 176 (2012), 413 508 http://dx.doi.org/10.4007/annals.2012.176.1.8 Rational points over finite fields for regular models of algebraic varieties of Hodge type 1 By Pierre Berthelot,
More informationIntroduction and preliminaries Wouter Zomervrucht, Februari 26, 2014
Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationHochschild homology and Grothendieck Duality
Hochschild homology and Grothendieck Duality Leovigildo Alonso Tarrío Universidade de Santiago de Compostela Purdue University July, 1, 2009 Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality
More informationDuality, Residues, Fundamental class
Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class
More informationLifting the Cartier transform of OgusVologodsky. modulo p n. Daxin Xu. California Institute of Technology
Lifting the Cartier transform of OgusVologodsky modulo p n Daxin Xu California Institute of Technology RiemannHilbert correspondences 2018, Padova A theorem of DeligneIllusie k a perfect field of characteristic
More informationLecture 21: Crystalline cohomology and the de RhamWitt complex
Lecture 21: Crystalline cohomology and the de RhamWitt complex Paul VanKoughnett November 12, 2014 As we ve been saying, to understand K3 surfaces in characteristic p, and in particular to rediscover
More informationA padic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1
A padic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection
More informationREFLEXIVITY AND RIGIDITY FOR COMPLEXES, II: SCHEMES
REFLEXIVITY AND RIGIDITY FOR COMPLEXES, II: SCHEMES LUCHEZAR L. AVRAMOV, SRIKANTH B. IYENGAR, AND JOSEPH LIPMAN Abstract. We prove basic facts about reflexivity in derived categories over noetherian schemes;
More informationAn introduction to the RiemannHilbert Correspondence for Unit F Crystals.
[Page 1] An introduction to the RiemannHilbert Correspondence for Unit F Crystals. Matthew Emerton and Mark Kisin In memory of Bernie Dwork Introduction Let X be a smooth scheme over C, and (M, ) an
More informationSeminar on Crystalline Cohomology
Seminar on Crystalline Cohomology Yun Hao November 28, 2016 Contents 1 Divided Power Algebra October 31, 2016 1 1.1 Divided Power Structure.............................. 1 1.2 Extension of Divided Power
More informationDERIVED CATEGORIES OF COHERENT SHEAVES
DERIVED CATEGORIES OF COHERENT SHEAVES OLIVER E. ANDERSON Abstract. We give an overview of derived categories of coherent sheaves. [Huy06]. Our main reference is 1. For the participants without bacground
More informationCrystalline Cohomology and Frobenius
Crystalline Cohomology and Frobenius Drew Moore References: Berthelot s Notes on Crystalline Cohomology, discussions with Matt Motivation Let X 0 be a proper, smooth variety over F p. Grothendieck s etale
More informationFormal completion and duality
Formal completion and duality Leovigildo Alonso Tarrío Atelier International sur la Théorie (Algébrique et Analytique) des Résidus et ses Applications, Institut Henri Poincaré, Paris May, 1995 Abstract
More informationTowards an overconvergent DeligneKashiwara correspondence
Towards an overconvergent DeligneKashiwara correspondence Bernard Le Stum 1 (work in progress with Atsushi Shiho) Version of March 22, 2010 1 bernard.lestum@univrennes1.fr Connections and local systems
More information6. DE RHAMWITT COMPLEX AND LOG CRYS TALLINE COHOMOLOGY
6. DE RHAMWITT COMPLEX AND LOG CRYS TALLINE COHOMOLOGY α : L k : a fine log str. on s = Spec(k) W n (L) : Teichmüller lifting of L to W n (s) : W n (L) = L Ker(W n (k) k ) L W n (k) : a [α(a)] Example
More informationSERRE FINITENESS AND SERRE VANISHING FOR NONCOMMUTATIVE P 1 BUNDLES ADAM NYMAN
SERRE FINITENESS AND SERRE VANISHING FOR NONCOMMUTATIVE P 1 BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X bimodule of rank
More informationNotes on pdivisible Groups
Notes on pdivisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More information8 Perverse Sheaves. 8.1 Theory of perverse sheaves
8 Perverse Sheaves In this chapter we will give a selfcontained account of the theory of perverse sheaves and intersection cohomology groups assuming the basic notions concerning constructible sheaves
More informationIwasawa algebras and duality
Iwasawa algebras and duality Romyar Sharifi University of Arizona March 6, 2013 Idea of the main result Goal of Talk (joint with Meng Fai Lim) Provide an analogue of PoitouTate duality which 1 takes place
More informationPERVERSE SHEAVES. Contents
PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on Dmodules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a
More informationDERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisseétale and the flatfppf sites
DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisseétale and the flatfppf sites 1 4. Derived categories of quasicoherent modules 5
More informationConstructible isocrystals (London 2015)
Constructible isocrystals (London 2015) Bernard Le Stum Université de Rennes 1 March 30, 2015 Contents The geometry behind Overconvergent connections Construtibility A correspondance Valuations (additive
More informationNOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE
NOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE GUFANG ZHAO Contents 1. Introduction 1 2. What is a Procesi bundle 2 3. Derived equivalences from exceptional objects 4 4. Splitting of the
More informationOn Faltings method of almost étale extensions
Proceedings of Symposia in Pure Mathematics On Faltings method of almost étale extensions Martin C. Olsson Contents 1. Introduction 1 2. Almost mathematics and the purity theorem 10 3. Galois cohomology
More informationPART II.1. INDCOHERENT SHEAVES ON SCHEMES
PART II.1. INDCOHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Indcoherent sheaves on a scheme 2 1.1. Definition of the category 2 1.2. tstructure 3 2. The direct image functor 4 2.1. Direct image
More information1. THE CONSTRUCTIBLE DERIVED CATEGORY
1. THE ONSTRUTIBLE DERIVED ATEGORY DONU ARAPURA Given a family of varieties, we want to be able to describe the cohomology in a suitably flexible way. We describe with the basic homological framework.
More information0.1 Spec of a monoid
These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.
More informationAn overview of Dmodules: holonomic Dmodules, bfunctions, and V filtrations
An overview of Dmodules: holonomic Dmodules, bfunctions, and V filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of Dmodules Mainz July 9, 2018 1 The
More informationOdds and ends on equivariant cohomology and traces
Odds and ends on equivariant cohomology and traces Weizhe Zheng Columbia University International Congress of Chinese Mathematicians Tsinghua University, Beijing December 18, 2010 Joint work with Luc Illusie.
More informationAFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES
AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are Clinear. 1.
More informationConstructible Derived Category
Constructible Derived Category Dongkwan Kim September 29, 2015 1 Category of Sheaves In this talk we mainly deal with sheaves of Cvector spaces. For a topological space X, we denote by Sh(X) the abelian
More informationON WITT VECTOR COHOMOLOGY FOR SINGULAR VARIETIES
ON WITT VECTOR COHOMOLOGY FOR SINGULAR VARIETIES PIERRE BERTHELOT, SPENCER BLOCH, AND HÉLÈNE ESNAULT Abstract. Over a perfect field k of characteristic p > 0, we construct a Witt vector cohomology with
More informationDerived categories, perverse sheaves and intermediate extension functor
Derived categories, perverse sheaves and intermediate extension functor Riccardo Grandi July 26, 2013 Contents 1 Derived categories 1 2 The category of sheaves 5 3 tstructures 7 4 Perverse sheaves 8 1
More informationA minicourse on crystalline cohomology
A minicourse on crystalline cohomology June 15, 2018 Haoyang Guo Abstract This is the lecture notes for the minicourse during June 1115, 2018 at University of Michigan, about the crystalline cohomology.
More information1 Replete topoi. X = Shv proét (X) X is locally weakly contractible (next lecture) X is replete. D(X ) is left complete. K D(X ) we have R lim
Reference: [BS] Bhatt, Scholze, The proétale topology for schemes In this lecture we consider replete topoi This is a nice class of topoi that include the proétale topos, and whose derived categories
More informationRigid Geometry and Applications II. Kazuhiro Fujiwara & Fumiharu Kato
Rigid Geometry and Applications II Kazuhiro Fujiwara & Fumiharu Kato Birational Geometry from Zariski s viewpoint S U D S : coherent (= quasicompact and quasiseparated) (analog. compact Hausdorff) U
More informationCohomology theories III
Cohomology theories III Bruno Chiarellotto Università di Padova Sep 11th, 2011 Road Map Bruno Chiarellotto (Università di Padova) Cohomology theories III Sep 11th, 2011 1 / 27 Contents 1 pcohomologies
More informationBRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,
CONNECTIONS, CURVATURE, AND pcurvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and pcurvature, in terms of maps of sheaves
More informationTHE SIX OPERATIONS FOR SHEAVES ON ARTIN STACKS II: ADIC COEFFICIENTS
THE SIX OPERATIONS FOR SHEAVES ON ARTIN STACKS II: ADIC COEFFICIENTS YVES ASZO AND MARTIN OSSON Abstract. In this paper we develop a theory of Grothendieck s six operations for adic constructible sheaves
More informationladic Representations
ladic Representations S. M.C. 26 October 2016 Our goal today is to understand ladic Galois representations a bit better, mostly by relating them to representations appearing in geometry. First we ll
More informationIdentification of the graded pieces Kęstutis Česnavičius
Identification of the graded pieces Kęstutis Česnavičius 1. TP for quasiregular semiperfect algebras We fix a prime number p, recall that an F p algebra R is perfect if its absolute Frobenius endomorphism
More informationREVISITING THE DE RHAMWITT COMPLEX
REVISITING THE DE RHAMWITT COMPLEX BHARGAV BHATT, JACOB LURIE, AND AKHIL MATHEW Abstract. The goal of this paper is to offer a new construction of the de RhamWitt complex of smooth varieties over perfect
More informationDeformation theory of representable morphisms of algebraic stacks
Deformation theory of representable morphisms of algebraic stacks Martin C. Olsson School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, molsson@math.ias.edu Received:
More informationWhat is an indcoherent sheaf?
What is an indcoherent sheaf? Harrison Chen March 8, 2018 0.1 Introduction All algebras in this note will be considered over a field k of characteristic zero (an assumption made in [Ga:IC]), so that we
More informationSynopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].
Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasicoherent O Y module.
More informationCohomology and Base Change
Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is halfexact if whenever 0 M M M 0 is an exact sequence of Amodules, the sequence T (M ) T (M)
More informationAPPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP
APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP In this appendix we review some basic facts about étale cohomology, give the definition of the (cohomological) Brauer group, and discuss
More informationGrothendieck duality for affine M 0 schemes.
Grothendieck duality for affine M 0 schemes. A. Salch March 2011 Outline Classical Grothendieck duality. M 0 schemes. Derived categories without an abelian category of modules. Computing Lf and Rf and
More informationSynopsis of material from EGA Chapter II, 5
Synopsis of material from EGA Chapter II, 5 5. Quasiaffine, quasiprojective, proper and projective morphisms 5.1. Quasiaffine morphisms. Definition (5.1.1). A scheme is quasiaffine if it is isomorphic
More informationON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DGMANIFOLDS
ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DGMANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical
More informationON A LOCALIZATION FORMULA OF EPSILON FACTORS VIA MICROLOCAL GEOMETRY
ON A LOCALIZATION FORMULA OF EPSILON FACTORS VIA MICROLOCAL GEOMETRY TOMOYUKI ABE AND DEEPAM PATEL Abstract. Given a lisse ladic sheaf G on a smooth proper variety X and a lisse sheaf F on an open dense
More informationRigid cohomology and its coefficients
Rigid cohomology and its coefficients Kiran S. Kedlaya Department of Mathematics, Massachusetts Institute of Technology padic Geometry and Homotopy Theory Loen, August 4, 2009 These slides can be found
More informationAN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES
AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at
More informationIndCoh Seminar: Indcoherent sheaves I
IndCoh Seminar: Indcoherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means category ). This section contains a discussion of
More informationVERDIER DUALITY AKHIL MATHEW
VERDIER DUALITY AKHIL MATHEW 1. Introduction Let M be a smooth, compact oriented manifold of dimension n, and let k be a field. Recall that there is a natural pairing in singular cohomology H r (M; k)
More informationDualizing complexes and perverse sheaves on noncommutative ringed schemes
Sel. math., New ser. Online First c 2006 Birkhäuser Verlag Basel/Switzerland DOI 10.1007/s0002900600224 Selecta Mathematica New Series Dualizing complexes and perverse sheaves on noncommutative ringed
More informationPERVERSE SHEAVES: PART I
PERVERSE SHEAVES: PART I Let X be an algebraic variety (not necessarily smooth). Let D b (X) be the bounded derived category of Mod(C X ), the category of left C X Modules, which is in turn a full subcategory
More informationThe overconvergent site
The overconvergent site Bernard Le Stum 1 Université de Rennes 1 Version of April 15, 2011 1 bernard.lestum@univrennes1.fr 2 Abstract We prove that rigid cohomology can be computed as the cohomology
More informationGKSEMINAR SS2015: SHEAF COHOMOLOGY
GKSEMINAR SS2015: SHEAF COHOMOLOGY FLORIAN BECK, JENS EBERHARDT, NATALIE PETERNELL Contents 1. Introduction 1 2. Talks 1 2.1. Introduction: Jordan curve theorem 1 2.2. Derived categories 2 2.3. Derived
More informationHomotopy types of algebraic varieties
Homotopy types of algebraic varieties Bertrand Toën These are the notes of my talk given at the conference Theory of motives, homotopy theory of varieties, and dessins d enfants, Palo Alto, April 2326,
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationSome remarks on Frobenius and Lefschetz in étale cohomology
Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)
More information1 Moduli spaces of polarized Hodge structures.
1 Moduli spaces of polarized Hodge structures. First of all, we briefly summarize the classical theory of the moduli spaces of polarized Hodge structures. 1.1 The moduli space M h = Γ\D h. Let n be an
More informationLectures on Grothendieck Duality II: Derived Hom Tensor adjointness. Local duality.
Lectures on Grothendieck Duality II: Derived Hom Tensor adjointness. Local duality. Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu February 16, 2009 Joseph Lipman (Purdue
More informationTORSION IN THE CRYSTALLINE COHOMOLOGY OF SINGULAR VARIETIES
TORSION IN THE CRYSTALLINE COHOMOLOGY OF SINGULAR VARIETIES BHARGAV BHATT ABSTRACT. This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties
More informationSection Higher Direct Images of Sheaves
Section 3.8  Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will
More informationDe Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)
II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the
More informationTunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society
Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society Grothendieck Messing deformation theory for varieties of K3 type Andreas Langer and Thomas Zink
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves
More informationALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES
ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasicoherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open
More informationMODULI TOPOLOGY. 1. Grothendieck Topology
MODULI TOPOLOG Abstract. Notes from a seminar based on the section 3 of the paper: Picard groups of moduli problems (by Mumford). 1. Grothendieck Topology We can define a topology on any set S provided
More informationLectures on Grothendieck Duality. II: Derived Hom Tensor adjointness. Local duality.
Lectures on Grothendieck Duality II: Derived Hom Tensor adjointness. Local duality. Joseph Lipman February 16, 2009 Contents 1 Leftderived functors. Tensor and Tor. 1 2 HomTensor adjunction. 3 3 Abstract
More informationINTRO TO TENSOR PRODUCTS MATH 250B
INTRO TO TENSOR PRODUCTS MATH 250B ADAM TOPAZ 1. Definition of the Tensor Product Throughout this note, A will denote a commutative ring. Let M, N be two Amodules. For a third Amodule Z, consider the
More informationMath 248B. Applications of base change for coherent cohomology
Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the socalled cohomology and base change theorem: Theorem 1.1 (Grothendieck).
More informationINDCOHERENT SHEAVES AND SERRE DUALITY II. 1. Introduction
INDCOHERENT SHEAVES AND SERRE DUALITY II 1. Introduction Let X be a smooth projective variety over a field k of dimension n. Let V be a vector bundle on X. In this case, we have an isomorphism H i (X,
More informationGeometric motivic integration
Université Lille 1 Modnet Workshop 2008 Introduction Motivation: padic integration Kontsevich invented motivic integration to strengthen the following result by Batyrev. Theorem (Batyrev) If two complex
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasiprojective varieties over a field k Affine Varieties 1.
More informationINTEGRAL TRANSFORMS FOR COHERENT SHEAVES
INTEGRAL TRANSFORMS FOR COHERENT SHEAVES DAVID BENZVI, DAVID NADLER, AND ANATOLY PREYGEL Abstract. The theory of integral, or FourierMukai, transforms between derived categories of sheaves is a well
More informationÉTALE π 1 OF A SMOOTH CURVE
ÉTALE π 1 OF A SMOOTH CURVE AKHIL MATHEW 1. Introduction One of the early achievements of Grothendieck s theory of schemes was the (partial) computation of the étale fundamental group of a smooth projective
More informationDerived intersections and the Hodge theorem
Derived intersections and the Hodge theorem Abstract The algebraic Hodge theorem was proved in a beautiful 1987 paper by Deligne and Illusie, using positive characteristic methods. We argue that the central
More informationConvergent isocrystals on simply connected varieties
Convergent isocrystals on simply connected varieties Hélène Esnault and Atsushi Shiho April 12, 2016 Abstract It is conjectured by de Jong that, if X is a connected smooth projective variety over an algebraically
More informationForschungsseminar padic periods and derived de Rham Cohomology after Beilinson
Forschungsseminar padic periods and derived de Rham Cohomology after Beilinson Organizers: Lars Kindler and Kay Rülling Sommersemester 13 Introduction If X is a smooth scheme over the complex numbers,
More informationAlgebraic varieties and schemes over any scheme. Non singular varieties
Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two
More informationSUMMER COURSE IN MOTIVIC HOMOTOPY THEORY
SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY MARC LEVINE Contents 0. Introduction 1 1. The category of schemes 2 1.1. The spectrum of a commutative ring 2 1.2. Ringed spaces 5 1.3. Schemes 10 1.4. Schemes
More informationCATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS. Contents. 1. The ring K(R) and the group Pic(R)
CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS J. P. MAY Contents 1. The ring K(R) and the group Pic(R) 1 2. Symmetric monoidal categories, K(C), and Pic(C) 2 3. The unit endomorphism ring R(C ) 5 4.
More informationDMODULES: AN INTRODUCTION
DMODULES: AN INTRODUCTION ANNA ROMANOVA 1. overview Dmodules are a useful tool in both representation theory and algebraic geometry. In this talk, I will motivate the study of Dmodules by describing
More informationAn introduction to derived and triangulated categories. Jon Woolf
An introduction to derived and triangulated categories Jon Woolf PSSL, Glasgow, 6 7th May 2006 Abelian categories and complexes Derived categories and functors arise because 1. we want to work with complexes
More information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More informationWild ramification and the characteristic cycle of an ladic sheaf
Wild ramification and the characteristic cycle of an ladic sheaf Takeshi Saito March 14 (Chicago), 23 (Toronto), 2007 Abstract The graded quotients of the logarithmic higher ramification groups of a local
More informationExercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti
Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY Email address: alex.massarenti@sissa.it These notes collect a series of
More informationAPPENDIX 3: AN OVERVIEW OF CHOW GROUPS
APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss
More informationLecture 3: Flat Morphisms
Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 15]. 1.1 Open and Closed Subschemes If (X, O X ) is a
More informationRELATIVE AMPLENESS IN RIGIDANALYTIC GEOMETRY
RELATIVE AMPLENESS IN RIGIDANALYTIC GEOMETRY BRIAN CONRAD 1. Introduction 1.1. Motivation. The aim of this paper is to develop a rigidanalytic theory of relative ampleness for line bundles, and to record
More informationTHE GROTHENDIECK GROUP OF A QUANTUM PROJECTIVE SPACE BUNDLE
THE GROTHENDIECK GROUP OF A QUANTUM PROJECTIVE SPACE BUNDLE IZURU MORI AND S. PAUL SMITH Abstract. We compute the Grothendieck group of noncommutative analogues of projective space bundles. Our results
More informationMATH 233B, FLATNESS AND SMOOTHNESS.
MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)
More informationpadic ÉTALE COHOMOLOGY AND CRYSTALLINE COHOMOLOGY FOR OPEN VARIETIES
padic ÉTALE COHOMOLOGY AND CRYSTALLINE COHOMOLOGY FOR OPEN VARIETIES GO YAMASHITA This text is a report of a talk padic étale cohomology and crystalline cohomology for open varieties in a symposium at
More informationON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF
ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF MATTHEW H. BAKER AND JÁNOS A. CSIRIK Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical
More information