Constructible Derived Category

Size: px
Start display at page:

Download "Constructible Derived Category"

Transcription

1 Constructible Derived Category Dongkwan Kim September 29, Category of Sheaves In this talk we mainly deal with sheaves of C-vector spaces. For a topological space X, we denote by Sh(X) the abelian category of sheaves of C-vector spaces on X. We assume that readers are familiar with basic sheaf theory. (i.e. sheafification, Sh(X) is abelian, etc.) 1.1 Operations on Sheaves Here we recapitulate some operations (i.e. functors) on sheaves. First we define internal Hom and tensor product as follows. (1) Internal Hom. For F, G Sh(X), we define Hom(F, G) = Hom Sh(X) (F, G) Sh(X) as the sheaf defined by U Hom(F U, G U ) = Hom Sh(X) (F U, G U ). (2) Tensor product. For F, G Sh(X), we define F G = F C G Sh(X) as the sheaf associated to the following presheaf U F(U) G(U). Note that the natural map (Hom(F, G)) x Hom(F x, G x ) is in general neither injective nor surjective. However, we have (F G) x F x G x. Now for a continuous map f : X Y, we define operations of sheaves which corresponds to f as follows. (3) Push-forward or direct image. For F Sh(X), we define f F Sh(Y ) such that for open V Y we have f F : V F(f 1 (V )). In particular, if Y is a point, then we denote Γ(X, G) := f G as we identify Sh(Y ) with the category of C-vector spaces and call it the set of global sections of F on X. 1

2 (4) Pull-back or inverse image. For G Sh(Y ), we define f G Sh(X) to be a sheaf associated to the presheaf U lim G(V ). f(u) V Note that the natural map (f F) f(x) F x is not an isomorphism, however (f G) x G f(x). For a subspace f : Z X, using the functors above we have a way to restrict a sheaf F Sh(X) to Z still considered as an element in Sh(X) by taking f f F. We will define two other analogous restriction functors when the subspace is locally closed, which would coincide with f f in some special situations. (5) For a closed subset f : Z X and for F Sh(X), we define F Z := f f F. For an open subset U, we set F U := ker(f F X\U ). Finally, if A is locally closed, say A = U Z for U open and Z closed, then we define F A := (F U ) Z. 1 We briefly mention one of its properties, which would become obvious as we discuss further. Proposition 1. Suppose f : A X is locally closed and g : B = X \ A X. Then for any F Sh(X), f (F A ) = f F, g (F A ) = 0. (6) The sheaf of sections supported by a subspace. For a closed subset f : Z X and F Sh(Z), we let Γ Z (F) be the sheaf defined by V ker(f(v ) F(V \ (V Z))). For A = U Z locally closed with U open and Z closed, define Γ A (F) by Γ A (F)(V ) := Γ A (F U )(U V ). It is straightforward to check that for an open embedding f : U X, Γ U (F) = f f F. So far we defined Hom,, f, f, A, Γ A. If topological spaces are locally compact 2, there is one more functor which is of our interest. (7) Proper push-forward or direct image with proper support. Assume that X, Y are locally compact topological spaces and given a continuous map f : X Y and a sheaf F Sh(X). We denote by f! F the sheaf defined by U {s F(f 1 U) f supp(s) : supp(s) U is proper 3 }. 1 This is well-defined, that is this definition does not depend on the choice of U or Z. 2 In this talk locally compact spaces are always Hausdorff. 3 Here f being proper means that every inverse image of compact set is again compact. 2

3 In particular, if Y is a point then we denote Γ c (X, F) := f! (F) as we identify Sh(Y ) with the cateogory of C-vector spaces and call it the set of compactly (or properly) supported global sections of F on X. Note that we have a natural embedding f! F f F, which is clearly an isomorphism if f is a priori proper. Also we have the following property which would be apparent by the argument later on. It is not in general true for the usual push-forward. Proposition 2. [[KS, Proposition 2.5.2]] If f : X Y is a continuous map between locally compact spaces, then for y Y and F Sh(X) the canonical morphism (f! F) y Γ c (f 1 (y), F f 1 (y)) is an isomorphism. 4 If f : X Y is proper then f f = f! f, which is true especially when f is a closed embedding. One might already expect that we have the following property. Proposition 3. If X is locally compact and f : Z X is a closed embedding, thus in particular Z is also locally compact, then for F Sh(X) we have F Z = f! f F. 1.2 Exactness, Adjunction Formula, and Base Change We state, but omit proofs of, the exactness properties of the operations defined above. Proposition 4. Let f : X Y be a continuous map between topological spaces. Then, (1) Hom : Sh(X) op Sh(X) Sh(X) is left exact. (2) : Sh(X) Sh(X) Sh(X) is exact. 5 (3) f : Sh(X) Sh(Y ) is left exact. Thus in particular, Γ is left exact. (3 ) f : Sh(X) Sh(Y ) is exact if f is a closed embedding. (4) f : Sh(Y ) Sh(X) is exact. (5) F F X : Sh(Y ) Sh(Y ) is exact if f is a locally closed embedding. (6) Γ X : Sh(Y ) Sh(Y ) is left exact if f is a locally closed embedding. 4 This also has an étale analogue which is difficult to prove. See the comments on [M, p.117] after Theorem 17.7 and for more infomation. 5 In general if we consider tensor product over a sheaf of rings then it is right exact. Here we only consider sheaves of C-vector spaces, thus every sheaf is flat. 3

4 (7) f! : Sh(X) Sh(Y ) is left exact if X and Y are locally compact. (7 ) f! : Sh(X) Sh(Y ) is exact if Y is locally compact and f is a locally closed embedding. Also we have the following adjunction formulae. Proposition 5. Let f : X Y be a continuous map between topological spaces and let F, F, F Sh(X), G, G Sh(Y ). Then (a) Hom(F F, F ) Hom(F, Hom(F, F )) (b) Hom(G, f F) f Hom(f G, F) (c) Hom(G X, G ) Hom(G, Γ X (G )) if f is a locally closed embedding. If we take the set of global sections on each formula, we get the usual adjunctions. Meanwhile, suppose we are given the following diagram of locally compact spaces. A g f B g C f D Then for any F Sh(C) we have a well-defined morphism f! F f! g g F by adjunction. Now for U D, we have f! g F(U) = {s F( g 1 f 1 U) f : supp( g s) U is proper} g f! F(U) = {s F( f 1 g 1 U) f : supp(s) g 1 U is proper} In other words, if s f! g F(U) then s F( g 1 f 1 U) = F( f 1 g 1 U) and there exists Z C such that supp(s) g 1 Z and f Z : Z U is proper. Now suppose further that the diagram above is cartesian. Then f : g 1 Z g 1 U is also proper, thus so is supp(s) g 1 U. It defines a natural morphism f! g g f!, and f! F f! g g F g f! g F. Again by adjunction we have g f! F f! g F. Now for x B, we have (g f! F) x = (f! F) g(x) = Γ c (f 1 (g(x)), F f 1 (g(x))) Thus we proved the following theorem = Γ c ( f 1 (x), ( g F) f 1 (x) ) = ( f! g F) x. Theorem 6 (Proper Base Change). If the diagram above is cartesian, we have a natural isomorphism g f! f! g. Now we see that Proposition 1 and 2 are special cases of this theorem together with Proposition 3. 4

5 1.3 Constructible Sheaves The main object with which we are concerned is (cohomologically) constructible sheaves. We briefly discuss the corresponding definitions from now on. Let X be a complex analytic space and P = (X i ) i I be a partition of X which satisfies the following properties. Each X i is nonempty, locally closed and connected. P is locally finite. Each X i \ X i is a union of elements in P. For each X i P, X i and X i \ X i are closed complex analytic subspaces in X. Then we say F Sh(X) is constructible if there exists a partition P which satisfies conditions above such that for each X i P, F Xi is a local system, i.e. a locally constant sheaf of C-vector spaces of finite dimension. In this case we also say that F is constructible with respect to P. We denote by Sh c (X) the full subcategory of constructible sheaves on Sh(X) and by Sh c (X, P) its full subcategory of constructible sheaves with respect to a fixed P. Then it is easy to show that Sh c (X) is an abelian category. Also, it is easy to check that internal Hom and tensor product preserve constructibility. For an analytic morphism f : X Y between complex analytic varieties, one can also easily show that f preserves constructibility. Push-forward is a little tricky and does not preserve the property in general. However, if f is indeed algebraic, then f preserves it using Chevalley constructibility theorem. Similarly, for F Sh c (X) if f is proper on supp F, then f F Sh c (Y ). We postpone going into details about it until we introduce a constructible derived category. Example 7. Suppose X = C and i : Z = {0} X, j : U = X \ Z X. We let F Sh(U) be the local system with monodromy matrix A GL n (C) which has Jordan blocks A 1,, A r with eigenvalues λ 1,, λ r such that λ 1 = = λ s = 1. Then one can easily see that i j F C s, i j! F = Some Questions So far we discussed definitions and some properties of operation on sheaves. Now these questions arise naturally. Is there any description for Γ Z similar to F Z f! f F? Is there any adjunction formula for f!? For a local system, we can easily take its dual. Can we expect such a thing in general? To answer these questions properly, we shall introduce the notion of derived category. 5

6 2 Derived Category of Sheaves 2.1 Derived Category of Abelian Categories We assume that we are already familiar with the language of a derived category, but we briefly mention some properties. For an abelian category A, we denote by K(A) the homotopy category of chain complexes in A. The derived category of A, usually denoted by D(A), is the category obtained from K(A) by inverting quasi-isomorphisms, which is a morphism of chain complexes which induces isomorphisms on their cohomology. This process is called localization. One of its major properties is that it is a triangulated category. Even though D(A) is no longer abelian, it has an analogue of an exact sequence in an abelian category, called a distinguished triangle. Then a functor between triangulated category is called exact if it is additive, commutes with translations, and sends distinguished triangles to distinguished ones. Note that K(A) is also triangulated and the localization functor K(A) D(A) is exact. Also we may consider at first the category of chain complexes of bounded cohomology, and we let D b (A) be its derived category. Likewise, D + (A), D (A), respectively, is the derived category of bounded below complexes, bounded above complexes, respectively. Sometimes D b (A) is the same as the category of bounded chain complexes if we have some finiteness condition on A. 2.2 Derived Category of Sheaves For a topological space X we may define D(Sh(X)), the derived category of sheaves on X. We assume we already know that Sh(X) has enough injectives. Therefore, it is possible to derive functors which are left exact on Sh(X). Let f : X Y be a continuous map between topological spaces. Then we have RHom : D (Sh(X)) op D + (Sh(X)) D + (Sh(X)) Rf : D + (Sh(X)) D + (Sh(Y )) RΓ X : D + (Sh(Y )) D + (Sh(Y )) if f is a locally closed embedding. Rf! : D + (Sh(X)) D + (Sh(Y )) if X, Y are locally compact. Also, since f,, Z (where Z is a locally closed subset) are already exact, they automatically define functors on the derived category of sheaves. Meanwhile, in general Sh(X) does not have enough projectives as the next example shows. Example 8. Suppose X = C with the usual topology. Then we claim that the skyscraper sheaf C 0 supported by the origin does not admit an epimorphism from a projective sheaf. Thus suppose otherwise and let P C 0 be such a morphism. We will show that P(U) C 0 (U) is a zero map for all neighborhood U of the origin. Choose ε > 0 such that D 2ε U, where D 2ε = {x C x < 2ε} is a disk of radius 2ε. Since we have a surjection C Dε C 0, we should be 6

7 able to lift P C 0 to obtain P C Dε. (Note that C Dε is a constant sheaf on D ε extended by zero on C \ D ε.) However, then C Dε (U) = 0 since U is strictly bigger than D ε, which means P(U) C 0 (U) is a zero map. In other words, P 0 (C 0 ) 0 is zero, which contradicts the assumption. 2.3 Adjunction and Base Change From adjunction of operations on sheaves, we easily deduce the following formulae. For the sake of simplicity, we restrict our situation to the bounded derived category of sheaves. Proposition 9. Let f : X Y be a continuous map between topological spaces and let F, F, F D b (Sh(X)), G, G D b (Sh(Y )). Then (a) RHom(F F, F ) RHom(F, RHom(F, F )) (b) RHom(G, Rf F) Rf RHom(f G, F) (c) RHom(G X, G ) RHom(G, RΓ X (G )) if f is a locally closed embedding. One advantage to consider derived categories is that we may also find adjunction for f!. Theorem 10. Suppose f : X Y is a continuous map between locally compact spaces and f! has finite cohomological dimension. Then we have a well-defined functor f! : D b (Sh(Y )) D b (Sh(X)) such that for F D b (Sh(X)) and G D b (Sh(Y )) the following adjunction formula holds. RHom(Rf! F, G) Rf RHom(F, f! G) Note that f! is unique up to isomorphism by Yoneda lemma. Also in general f! is not a derived functor of a functor on sheaves and only well-defined on the derived category. 6 Now it is a direct consequence from adjunction that if f : X Y is a locally closed embedding then RΓ X Rf f!. For more information one may refer to [KS, Section 3.1]. Meanwhile, we have a derived version of proper base change. Suppose the following diagram is cartesian as before. A f B g g C f D Then we proved that f! g g f! under suitable conditions. Since pull-back is exact, if we derive both sides we get R f! g g Rf!, say as a morphism form D b (Sh(C)) to D b (Sh(B)). (Here we need to verify the fact that g sends injective objects to f! -acyclic objects.) Also by adjunction, it also implies that R f g! g! Rf. 6 If f! were a left adjoint on the category of sheaves, then it should be right exact. 7

8 2.4 Constructible Derived Category In this section we assume X is a complex analytic space. Then we may also consider D b (Sh c (X)), the bounded derived category of constructible sheaves. However, since we mainly deal with the cohomology of a chain complex on the derived category rather than each term on the complex, it is reasonable to consider D b c(sh(x)), the bounded derived category of sheaves of constructible cohomology, or the constructible derived category on X. A striking fact is that when X is a complex algebraic variety, then they are the same 7, which is proved in [N]. 8 From now on we only deal with D b c(sh(x)). The operations we introduced so far preserve cohomological constructibility. In other words, we have the following Proposition 11. Let f : X Y be a morphism between complex analytic spaces and F, F D b c(sh(x)), G D b c(sh(y )). Then, (a) f G, f! G D b c(sh(x)). (b) Rf F, Rf! F D b c(sh(y )) if f is algebraic or f supp F is proper. (c) RHom(F, F ), F F D b c(sh(x)). We omit the proof. 3 Poincare-Verdier Duality 3.1 Duality Functor We will define a duality functor on D b (Sh(X)), which would be a generalization of taking dual of a local system. Thus it would be a contravariant endofunctor on D b (Sh(X)) and it would be nice if it is also an involution. To that end, among the operations we have it is natural to choose RHom(, ) for some D b (Sh(X)). Now we give the definition of Verdier duality as follows. Definition 12. Let f : X Y be a continuous map of finite cohomological dimension, so that f! is well-defined. Then we call ω X/Y := f! C Y D b c(sh(x)) D b (Sh(X)) the relative dualizing complex on X over Y. If Y is a single point, then we set ω X := ω X/Y and call it the dualizing complex on X. Definition 13. Keep assuming the situation above. We define a contravariant endofunctor D = D X : D b (Sh(X)) D b (Sh(X)) : F RHom(F, ω X ) and call it the Verdier duality on D b (Sh(X)). 7 It is not in general true if we fix a partition P on X and consider Sh c(x, P) instead, or X is just a complex analytic space 8 The perverse analogue is proved by Beilinson. 8

9 Thus in particular, ω X = DC X. We investigate some interesting properties of Verdier duality. First, we need the following lemmas. Lemma 14 (Projection Formula). For continuous f : X Y where X, Y are locally compact and F D b (Sh(X)), G D b (Sh(Y )), we have Rf! F G Rf! (F f G). Lemma 15. Keep the setting above and assume further that f has finite cohomological dimension. Then for G, G D b (Sh(Y )) we have f! RHom(G, G ) RHom(f G, f! G ). Proof. For any F D b (Sh(X)) we have Hom(F, f! RHom(G, G )) = Hom(Rf! F, RHom(G, G )) from which the result follows. = Hom(Rf! F G, G ) = Hom(Rf! (F f G), G ) = Hom(F f G, f! G ) = Hom(F, RHom(f G, f! G )) Proposition 16. Keep the setting above, then we have f! D = Df. Proof. For any G D b (Sh(Y )), we have f! DG = f! RHom(G, ω Y ) = RHom(f G, f! ω Y ) = RHom(f G, ω X ) = Df G. Thus the result follows. Proposition 17. Keep the setting above, then we have DRf! = Rf D. Proof. It follows from Hom(G, DRf! F) = Hom(Rf! F G, ω Y ) = Hom(Rf! (F f G), ω Y ) and Yoneda lemma. = Hom(F f G, f! ω Y ) = Hom(F f G, ω X ) = Hom(f G, DF) = Hom(G, Rf DF) From now on we investigate properties of the dualizing complex. Suppose f : X Y is a morphism of finite cohomological dimension between locally compact spaces, and G, G D b (Sh(Y )). Then by adjunction we have a natural morphism Rf! f! G G, from which we also have Rf! f! G G G G. By projection formula it is equivalent to Rf! (f! G f G ) G G, thus by adjunction again we have f! G f G f! (G G ). If we set G = C Y, then f! G = ω X/Y, thus we have ω X/Y f G f! G. Now we are ready to state the following theorem. 9

10 Theorem 18. Suppose f : X Y is a topological submersion with fiber dimension d. Then, (a) H k (ω X/Y ) = 0 if k d and H d (ω X/Y ) is a local system of rank 1. When Y is a point it is called the orientation sheaf of X. (b) The natural morphism ω X/Y f G f! G above is an isomorphism. (c) If X and Y are orientable manifolds, then ω X/Y C X [d]. Thus in particular f [d] f!. Proof. To show (a) it suffices to consider f : R n {pt}. For (b) we need to keep track of the construction of this morphism. (c) is more complicated, see [KS, Lemma 3.3.7]. Example 19 (Poincaré duality). Suppose X is a topological manifold. Then for f : X {pt} we apply DRf! Rf D to C X to get DRf! C X Rf DC X, or DRΓ c (X, C X ) RΓ(X, ω X ). Thus by taking ( i)-th cohomology, we have H i c(x, C X ) H i (X, ω X ). If furthermore X is orientable, then ω X C X [d] where d = dim X, thus we have which is the Poincaré duality of X. H i c(x, C X ) H d i (X, C X ) Example 20 (Alexander duality). Suppose X is an orientable topological manifold and Z X is a closed subset. Then if we denote the closed embedding of Z to X by f : Z X, we have RΓ Z D Rf f! D Rf Df DRf! f Thus if we apply this to C X ω X [ d] then we have RΓ Z C X [d] DC Z. By taking ( i)-th cohomology on both sides we have which is the Alexander duality. H d i Z (X, C X) Hc(Z, i C Z ) 3.2 Verdier Duality on Constructible Derived Category Unfortunately, D is not a duality on the whole D b (Sh(X)). However, we have Theorem 21. Suppose X is a complex analytic space. Then the restriction of D is a contravariant endofunctor on D b c(sh(x)). Furthermore, D is an involution on D b c(sh(x)). We omit the proof. 10

11 3.3 Formalism of Six Functors Suppose f : X Y is an analytic morphism between complex analytic spaces. So far we defined the following functors. (1) RHom : D b c(sh(x)) op D b c(sh(x)) D b c(sh(x)) (2) : D b c(sh(x)) D b c(sh(x)) D b c(sh(x)) (3) Rf : D b c(sh(x)) D b c(sh(y )) (4) Rf! : D b c(sh(x)) D b c(sh(y )) (5) f : D b c(sh(y )) D b c(sh(x)) (6) f! : D b c(sh(y )) D b c(sh(x)). Also we have the duality functor D : D b c(sh(x)) D b c(sh(x)) op which is an involution, i.e. DD = id. So far we showed that they satisfy the following properties. f Rf, Rf! f!, and Tensor-Hom adjunction The adjunctions above are functorial, thus each f Rf, Rf!, f, f! functor. is a There exists a natural morphism Rf! Rf which is an isomorphism if f is proper. If f is smooth of relative complex dimension d, then there exists a natural isomorphism f f! [ 2d] If the following diagram is cartesian, A f B g g C f D then there exist natural isomorphisms R f! g g Rf! and R f g! g! Rf. For F D b c(sh(x)) and G D b c(sh(y )), we have the projection formula F Rf! G Rf! (f F G). DRf! Rf D, DRf Rf! D, Df! f D, Df f! D. This is a typical example of so-called Grothendieck s six operations. For another example, which corresponds to D-module theory, we refer readers to [E]. 11

12 References [CD] Denis-Charles Cisinski and Frédéric Déglise, Triangulated categories of mixed motives, arxiv: [math.ag] (2009), available at [CG] Neil Chriss and Victor Ginzburg, Representation Theory and Complex Geometry, Modern Birkhäuser Classics, Birkhäuser, [D] Alexandru Dimca, Sheaves in Topology, Springer, [E] Pavel Etingof, Formalism of six functors, available at ~etingof/dmodfactsheet.pdf. [GW1] Iordan Ganev and Robin Walters, The derived category of constructible sheaves (2014), available at [GW2], The derived category of constructible sheaves (2014), available at [I] Birger Iversen, Cohomology of Sheaves, Springer-Verlag, [KS] Masaki Kashiwara and Pierre Schapira, Sheaves on Manifolds, Grundlehren der mathematischen Wissenschaften, vol. 292, Springer-Verlag Berlin Heidelberg, [M] J.S. Milne, Lectures on Etale Cohomology (2013), available at org/math/coursenotes/lec.pdf. [N] Madhad V. Nori, Constructible Sheaves, Algebra, arithmetic and geometry, Part I, II(Mumbai, 2000), 2002, pp

Derived categories, perverse sheaves and intermediate extension functor

Derived 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 t-structures 7 4 Perverse sheaves 8 1

More information

1. THE CONSTRUCTIBLE DERIVED CATEGORY

1. 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 information

VERDIER DUALITY AKHIL MATHEW

VERDIER 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 information

AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES

AFFINE 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 C-linear. 1.

More information

PERVERSE SHEAVES: PART I

PERVERSE 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 information

GK-SEMINAR SS2015: SHEAF COHOMOLOGY

GK-SEMINAR SS2015: SHEAF COHOMOLOGY GK-SEMINAR 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 information

SOME OPERATIONS ON SHEAVES

SOME OPERATIONS ON SHEAVES SOME OPERATIONS ON SHEAVES R. VIRK Contents 1. Pushforward 1 2. Pullback 3 3. The adjunction (f 1, f ) 4 4. Support of a sheaf 5 5. Extension by zero 5 6. The adjunction (j!, j ) 6 7. Sections with support

More information

8 Perverse Sheaves. 8.1 Theory of perverse sheaves

8 Perverse Sheaves. 8.1 Theory of perverse sheaves 8 Perverse Sheaves In this chapter we will give a self-contained account of the theory of perverse sheaves and intersection cohomology groups assuming the basic notions concerning constructible sheaves

More information

There are several equivalent definitions of H BM (X), for now the most convenient is in terms of singular simplicies. Let Ci

There are several equivalent definitions of H BM (X), for now the most convenient is in terms of singular simplicies. Let Ci 1. Introduction The goal of this course is to give an introduction to perverse sheaves, to prove the decomposition theorem and then to highlight several applications of perverse sheaves in current mathematics.

More information

Elementary (ha-ha) Aspects of Topos Theory

Elementary (ha-ha) Aspects of Topos Theory Elementary (ha-ha) Aspects of Topos Theory Matt Booth June 3, 2016 Contents 1 Sheaves on topological spaces 1 1.1 Presheaves on spaces......................... 1 1.2 Digression on pointless topology..................

More information

PERVERSE SHEAVES. Contents

PERVERSE SHEAVES. Contents PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a

More information

Non characteristic finiteness theorems in crystalline cohomology

Non characteristic finiteness theorems in crystalline cohomology 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

More information

Some remarks on Frobenius and Lefschetz in étale cohomology

Some 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 information

1 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

1 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 pro-étale topology for schemes In this lecture we consider replete topoi This is a nice class of topoi that include the pro-étale topos, and whose derived categories

More information

Introduction to Chiral Algebras

Introduction to Chiral Algebras Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument

More information

ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY

ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up

More information

Basic Facts on Sheaves

Basic Facts on Sheaves Applications of Homological Algebra Spring 2007 Basic Facts on Sheaves Introduction to Perverse Sheaves P. Achar Definition 1. A sheaf of abelian groups F on a topological space X is the following collection

More information

DERIVED CATEGORIES OF COHERENT SHEAVES

DERIVED 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 information

LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES

LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain

More information

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction 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 information

IND-COHERENT SHEAVES AND SERRE DUALITY II. 1. Introduction

IND-COHERENT SHEAVES AND SERRE DUALITY II. 1. Introduction IND-COHERENT 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 information

Algebraic Geometry Spring 2009

Algebraic 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 information

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

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

More information

Solutions to some of the exercises from Tennison s Sheaf Theory

Solutions to some of the exercises from Tennison s Sheaf Theory Solutions to some of the exercises from Tennison s Sheaf Theory Pieter Belmans June 19, 2011 Contents 1 Exercises at the end of Chapter 1 1 2 Exercises in Chapter 2 6 3 Exercises at the end of Chapter

More information

AN INTRODUCTION TO PERVERSE SHEAVES AND CHARACTER SHEAVES

AN INTRODUCTION TO PERVERSE SHEAVES AND CHARACTER SHEAVES AN INTRODUCTION TO PERVERSE SHEAVES AND CHARACTER SHEAVES ANNE-MARIE AUBERT Abstract. After a brief review of derived categories, triangulated categories and t-structures, we shall consider the bounded

More information

INTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES

INTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES INTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES 1. Why correspondences? This part introduces one of the two main innovations in this book the (, 2)-category of correspondences as a way to encode

More information

Algebraic Geometry Spring 2009

Algebraic 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 information

IndCoh Seminar: Ind-coherent sheaves I

IndCoh Seminar: Ind-coherent sheaves I IndCoh Seminar: Ind-coherent 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 information

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

More information

1. Algebraic vector bundles. Affine Varieties

1. 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 quasi-projective varieties over a field k Affine Varieties 1.

More information

IC of subvarieties. Logarithmic perversity. Hyperplane complements.

IC of subvarieties. Logarithmic perversity. Hyperplane complements. 12. Lecture 12: Examples of perverse sheaves 12.1. IC of subvarieties. As above we consider the middle perversity m and a Whitney stratified space of dimension n with even dimensional strata. Let Y denote

More information

Lecture 3: Flat Morphisms

Lecture 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, 1-5]. 1.1 Open and Closed Subschemes If (X, O X ) is a

More information

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc

More information

Show that the second projection Ñ Fl n identifies Ñ as a vector bundle over Fl n. In particular, Ñ is smooth. (Challenge:

Show that the second projection Ñ Fl n identifies Ñ as a vector bundle over Fl n. In particular, Ñ is smooth. (Challenge: 1. Examples of algebraic varieties and maps Exercise 1.1 Let C be a smooth curve and f : C P 1 a degree two map ramified at n points. (C is called a hyperelliptic curve.) Use the n ramification points

More information

Journal of Pure and Applied Algebra

Journal of Pure and Applied Algebra Journal of Pure and Applied Algebra 214 (2010) 1384 1398 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa Homotopy theory of

More information

Direct Limits. Mathematics 683, Fall 2013

Direct Limits. Mathematics 683, Fall 2013 Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry

More information

NOTES ON PERVERSE SHEAVES AND VANISHING CYCLES. David B. Massey

NOTES ON PERVERSE SHEAVES AND VANISHING CYCLES. David B. Massey NOTES ON PERVERSE SHEAVES AND VANISHING CYCLES David B. Massey 0. Introduction to Version 2-16 These notes are my continuing effort to provide a sort of working mathematician s guide to the derived category

More information

Lecture 2 Sheaves and Functors

Lecture 2 Sheaves and Functors Lecture 2 Sheaves and Functors In this lecture we will introduce the basic concept of sheaf and we also will recall some of category theory. 1 Sheaves and locally ringed spaces The definition of sheaf

More information

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three 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 information

ALGEBRAIC GROUPS JEROEN SIJSLING

ALGEBRAIC GROUPS JEROEN SIJSLING ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined

More information

Basic results on Grothendieck Duality

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 information

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of D-modules Mainz July 9, 2018 1 The

More information

Micro-support of sheaves

Micro-support of sheaves Micro-support of sheaves Vincent Humilière 17/01/14 The microlocal theory of sheaves and in particular the denition of the micro-support is due to Kashiwara and Schapira (the main reference is their book

More information

Algebraic Geometry Spring 2009

Algebraic 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 information

Nonabelian Poincare Duality (Lecture 8)

Nonabelian Poincare Duality (Lecture 8) Nonabelian Poincare Duality (Lecture 8) February 19, 2014 Let M be a compact oriented manifold of dimension n. Then Poincare duality asserts the existence of an isomorphism H (M; A) H n (M; A) for any

More information

Modules over a Ringed Space

Modules over a Ringed Space Modules over a Ringed Space Daniel Murfet October 5, 2006 In these notes we collect some useful facts about sheaves of modules on a ringed space that are either left as exercises in [Har77] or omitted

More information

Smooth morphisms. Peter Bruin 21 February 2007

Smooth morphisms. Peter Bruin 21 February 2007 Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,

More information

Section Higher Direct Images of Sheaves

Section 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 information

Algebraic Geometry Spring 2009

Algebraic 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 information

Derivations and differentials

Derivations and differentials Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra,

More information

Weil-étale Cohomology

Weil-étale Cohomology Weil-étale Cohomology Igor Minevich March 13, 2012 Abstract We will be talking about a subject, almost no part of which is yet completely defined. I will introduce the Weil group, Grothendieck topologies

More information

1 Notations and Statement of the Main Results

1 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 information

which is a group homomorphism, such that if W V U, then

which is a group homomorphism, such that if W V U, then 4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV

More information

Lectures on Grothendieck Duality. II: Derived Hom -Tensor adjointness. Local duality.

Lectures 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 Left-derived functors. Tensor and Tor. 1 2 Hom-Tensor adjunction. 3 3 Abstract

More information

PART II.2. THE!-PULLBACK AND BASE CHANGE

PART II.2. THE!-PULLBACK AND BASE CHANGE PART II.2. THE!-PULLBACK AND BASE CHANGE Contents Introduction 1 1. Factorizations of morphisms of DG schemes 2 1.1. Colimits of closed embeddings 2 1.2. The closure 4 1.3. Transitivity of closure 5 2.

More information

Homology and Cohomology of Stacks (Lecture 7)

Homology and Cohomology of Stacks (Lecture 7) Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the l-adic homology and cohomology of algebro-geometric objects of a more general nature than algebraic

More information

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when

More information

PART II.1. IND-COHERENT SHEAVES ON SCHEMES

PART II.1. IND-COHERENT SHEAVES ON SCHEMES PART II.1. IND-COHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Ind-coherent sheaves on a scheme 2 1.1. Definition of the category 2 1.2. t-structure 3 2. The direct image functor 4 2.1. Direct image

More information

REFLEXIVITY AND RIGIDITY FOR COMPLEXES, II: SCHEMES

REFLEXIVITY 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 information

Lectures on Grothendieck Duality II: Derived Hom -Tensor adjointness. Local duality.

Lectures 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 information

Poincaré duality for étale cohomology

Poincaré duality for étale cohomology Poincaré duality for étale cohomology Tony Feng February 1, 2017 Contents 1 Statement of Poincaré duality 1 2 The Trace map 3 3 Derived categories 10 4 The Duality Theorem 16 A Some technical results concerning

More information

AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES

AN 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 information

Lecture 9: Sheaves. February 11, 2018

Lecture 9: Sheaves. February 11, 2018 Lecture 9: Sheaves February 11, 2018 Recall that a category X is a topos if there exists an equivalence X Shv(C), where C is a small category (which can be assumed to admit finite limits) equipped with

More information

Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and

More information

PERVERSE SHEAVES ON A TRIANGULATED SPACE

PERVERSE SHEAVES ON A TRIANGULATED SPACE PERVERSE SHEAVES ON A TRIANGULATED SPACE A. POLISHCHUK The goal of this note is to prove that the category of perverse sheaves constructible with respect to a triangulation is Koszul (i.e. equivalent to

More information

What are stacks and why should you care?

What are stacks and why should you care? What are stacks and why should you care? Milan Lopuhaä October 12, 2017 Todays goal is twofold: I want to tell you why you would want to study stacks in the first place, and I want to define what a stack

More information

De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)

De 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 information

arxiv: v1 [math.ag] 13 Sep 2015

arxiv: v1 [math.ag] 13 Sep 2015 ENHANCED PERVERSITIES ANDREA D AGNOLO AND ASAKI KASHIWARA arxiv:1509.03791v1 [math.ag] 13 Sep 2015 Abstract. On a complex manifold, the Riemann-Hilbert correspondence embeds the triangulated category of

More information

Derived Categories Of Sheaves

Derived Categories Of Sheaves Derived Categories Of Sheaves Daniel Murfet October 5, 2006 We give a standard exposition of the elementary properties of derived categories of sheaves on a ringed space. This includes the derived direct

More information

Lecture 9 - Faithfully Flat Descent

Lecture 9 - Faithfully Flat Descent Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,

More information

An introduction to derived and triangulated categories. Jon Woolf

An 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 information

Hochschild homology and Grothendieck Duality

Hochschild 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 information

Matrix factorizations over projective schemes

Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University of California, Los Angeles January 11, 2013 Matrix factorizations Let Q be a commutative ring and f an element of Q. Matrix

More information

1.5.4 Every abelian variety is a quotient of a Jacobian

1.5.4 Every abelian variety is a quotient of a Jacobian 16 1. Abelian Varieties: 10/10/03 notes by W. Stein 1.5.4 Every abelian variety is a quotient of a Jacobian Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety.

More information

THE SMOOTH BASE CHANGE THEOREM

THE SMOOTH BASE CHANGE THEOREM THE SMOOTH BASE CHANGE THEOREM AARON LANDESMAN CONTENTS 1. Introduction 2 1.1. Statement of the smooth base change theorem 2 1.2. Topological smooth base change 4 1.3. A useful case of smooth base change

More information

Duality, Residues, Fundamental class

Duality, 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 information

Representable presheaves

Representable presheaves Representable presheaves March 15, 2017 A presheaf on a category C is a contravariant functor F on C. In particular, for any object X Ob(C) we have the presheaf (of sets) represented by X, that is Hom

More information

APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP

APPENDIX 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 information

Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

More information

Vector Bundles on Algebraic Varieties

Vector Bundles on Algebraic Varieties Vector Bundles on Algebraic Varieties Aaron Pribadi December 14, 2010 Informally, a vector bundle associates a vector space with each point of another space. Vector bundles may be constructed over general

More information

THE SIX OPERATIONS FOR SHEAVES ON ARTIN STACKS II: ADIC COEFFICIENTS

THE 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 information

Schemes via Noncommutative Localisation

Schemes via Noncommutative Localisation Schemes via Noncommutative Localisation Daniel Murfet September 18, 2005 In this note we give an exposition of the well-known results of Gabriel, which show how to define affine schemes in terms of the

More information

What is an ind-coherent sheaf?

What is an ind-coherent sheaf? What is an ind-coherent 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 information

Modules over a Scheme

Modules over a Scheme Modules over a Scheme Daniel Murfet October 5, 2006 In these notes we collect various facts about quasi-coherent sheaves on a scheme. Nearly all of the material is trivial or can be found in [Gro60]. These

More information

Topological K-theory

Topological K-theory Topological K-theory Robert Hines December 15, 2016 The idea of topological K-theory is that spaces can be distinguished by the vector bundles they support. Below we present the basic ideas and definitions

More information

Seminar on Étale Cohomology

Seminar on Étale Cohomology Seminar on Étale Cohomology Elena Lavanda and Pedro A. Castillejo SS 2015 Introduction The aim of this seminar is to give the audience an introduction to étale cohomology. In particular we will study the

More information

di Scienze matematiche, fisiche e naturali Corso di Laurea in Matematica

di Scienze matematiche, fisiche e naturali Corso di Laurea in Matematica Università degli studi di Padova Facoltà di Scienze matematiche, fisiche e naturali Corso di Laurea in Matematica Tesi Magistrale Perversity of the Nearby Cycles Relatore: Ch.mo Prof. Bruno Chiarellotto

More information

3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X). 3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

More information

Motivic integration on Artin n-stacks

Motivic integration on Artin n-stacks Motivic integration on Artin n-stacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of

More information

Sheaves, Co-Sheaves, and Verdier Duality

Sheaves, Co-Sheaves, and Verdier Duality Sheaves, Co-Sheaves, and Verdier Duality Justin M. Curry March 16, 2012 Abstract This note demonstrates that the derived category of cellular sheaves is covariantly equivalent to the derived category of

More information

Exercises 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 Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

More information

TOPICS IN ALGEBRA COURSE NOTES AUTUMN Contents. Preface Notations and Conventions

TOPICS IN ALGEBRA COURSE NOTES AUTUMN Contents. Preface Notations and Conventions TOPICS IN ALGEBRA COURSE NOTES AUTUMN 2003 ROBERT E. KOTTWITZ WRITTEN UP BY BRIAN D. SMITHLING Preface Notations and Conventions Contents ii ii 1. Grothendieck Topologies and Sheaves 1 1.1. A Motivating

More information

Synopsis of material from EGA Chapter II, 5

Synopsis of material from EGA Chapter II, 5 Synopsis of material from EGA Chapter II, 5 5. Quasi-affine, quasi-projective, proper and projective morphisms 5.1. Quasi-affine morphisms. Definition (5.1.1). A scheme is quasi-affine if it is isomorphic

More information

sset(x, Y ) n = sset(x [n], Y ).

sset(x, Y ) n = sset(x [n], Y ). 1. Symmetric monoidal categories and enriched categories In practice, categories come in nature with more structure than just sets of morphisms. This extra structure is central to all of category theory,

More information

1 Categorical Background

1 Categorical Background 1 Categorical Background 1.1 Categories and Functors Definition 1.1.1 A category C is given by a class of objects, often denoted by ob C, and for any two objects A, B of C a proper set of morphisms C(A,

More information

Iwasawa algebras and duality

Iwasawa 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 Poitou-Tate duality which 1 takes place

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday

More information

UNIVERSAL DERIVED EQUIVALENCES OF POSETS

UNIVERSAL DERIVED EQUIVALENCES OF POSETS UNIVERSAL DERIVED EQUIVALENCES OF POSETS SEFI LADKANI Abstract. By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for

More information

DESCENT THEORY (JOE RABINOFF S EXPOSITION)

DESCENT THEORY (JOE RABINOFF S EXPOSITION) DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,

More information

CHAPTER 1. Étale cohomology

CHAPTER 1. Étale cohomology CHAPTER 1 Étale cohomology This chapter summarizes the theory of the étale topology on schemes, culminating in the results on l-adic cohomology that are needed in the construction of Galois representations

More information