Derived categories, perverse sheaves and intermediate extension functor


 Christopher Mathews
 2 years ago
 Views:
Transcription
1 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 Derived categories We wish to give a brief description on how to construct derived categories. First, we need to have an abelian category, C, with some extra properties (e.g. having enough injective objects...). We then write C(C ) to denote the category of complexes in C : C 1 C 0 C 1 D 1 D 0 D 1 where morphisms satisfy the usual commutative squares. We further define C # (C ) C(C ), where # = b, +,, the subcategory of complexes bounded on both sides, bounded on the right, and bounded on the left, respectively. (By bounded on the right, we simply mean that, for some n, all C i = 0 for i n. A complex bounded on the left is defined similarly.) We give the following definitions: 1
2 Definition 1. A map f : X Y is said to be a quasiisomorphism if H n (f) are isomorphisms, n Z. Definition 2. We also define [n] : C(C ) C(C ) to be the functor such that the kth component of the new complex is the same as the (n + k)th component of the original complex, i.e. (C [n]) k = C n+k. We call [n] the nth shift functor. We also note that the differentials of the new complex are given by k [n] = ( 1) n k. Definition 3. We define the homotopy category, K # (C ), to be the category having for objects the same as in C # (C ), and having the same maps as in C # (C ) but modding out all the maps that are homotopic to 0. We note that K # (C ) is not abelian, however it remains additive. It is also triangulated; we have distinguished triangles given by some: X Y Z +1 X [1]. (E.g. X Id X 0 +1 X [1].) We have that H and Hom(, ) are cohomological functors in that they send triangles to long exact sequences. E.g. if we have: X Y Z X [1] in K # (C ), we get a long exact sequence in C : H 0 (X ) H 0 (Y ) H 0 (Z ) H 1 (X ). Definition 4. We say that a family of maps, S, is a multiplicative system, if: Id X S, f, g S f g S, We can always find X and t S such that the following homotopy commutes: g Y X t X f Y where s S and f and g are any morphisms, not necessarily in S. Similarly, s 2
3 Y s Y g f X t X For f, g Hom(X, Y ), it is equivalent to have: and s : Y Y such that s f = s g, t : X Xsuch that f t = g t. With such a multiplicative system it is possible to form fractions in a similar fashion to localisations on rings. We can now form the Derived category, D # (C ), with respect to S := family of quasiisomorphisms: the objects are the same as in K # (C ) and maps are of the following kind: Hom D # (C )(X, Y ) = {(X s W 1 f Y ) s S}/, where the equivalence can be defined in terms of the following commutative diagram: W 1 f 1 s 1 X W 3 s 2 s 3 and composition can be defined looking at the following diagram: W 2 f 2 Y s W 1 u f W 3 h t W 2 X Y Z. (We then write f s 1 for any map in K # (C ).) There is a localisation functor Q : K # (C ) D # (C ), which leaves objects as is and on maps: (i.e. f f/1). f Q(f) = X Id X f Y, 3 g
4 Lemma 1. Let C 0 be a category with S 1 a multiplicative system. Let I 0 C 0 be a full subcategory such that for all X C 0 there is s : X J with s S 1, J I 0. Then I 0 S 1 is a multiplicative system and (I 0 ) I0 S 1 (C 0) S 1, an equivalence of categories. Remark: C D(C ) and D # (C ) D(C ) are fully faithful embeddings. Moreover, D # (C ) is triangulated and Q respects triangles. Definition 5. We say that an abelian subcategory C C is thick if whenever we have an exact X 1 X 2 X 3 X 4 X 5, with X 1, X 2, X 4, X 5 C, then X 3 C. Lemma 2. Let C C be a thick subcategory. Take D C (C ) D(C ) to be the full subcategory whose objects are complexes with cohomology in C. Then D # C (C ) is triangulated. Let F : C D be an additive functor between two abelian categories. Then there is an induced functor on the homotopy categories. We want to know when there exists an extension F : K # (C ) K#F K # (D) Q C D # (C ) F Q D D # (D). Definition 6. A derived functor for a left exact functor F is a pair (RF, τ), where RF is a δfunctor (triangle preserving) and τ Nat(Q D K # F, RF Q C ) is a natural transformation. Definition 7. We say that I C is F injective if: X C we have an exact 0 X I with I I. If 0 X X X 0 is exact with X, X I then X I. F takes exact sequences in I to exact sequences in D. Theorem 1. If F : C D is left exact and there exists an F injective subcategory of C, then RF : D # (C ) D # (D) exists for # = +, b. 4
5 To compute RF (X ) for X D # (C ), we take a quasiisomorphism X I where I is a complex of F injectives. Then RF (X ) = K # F (I ). Remarks: If we have C C C with F, G left exact and I C is F injective, I C is Ginjective, and F (I ) I, then I is G F injective and R(G F ) = RG RF. We also recover the usual derived functor. That is, if we have 0 A B C 0 in C and define R i F so that we get an exact 0 F (A) F (B) F (C) R 1 F (A), then R i F (A) = H i (RF ([A])), where [A] is a complex with A in position 0. 2 The category of sheaves Assume all topological spaces are locally compact and Hausdorff. Let Sh(X) denote the category of sheaves on X and Mod(R) denote the category of Rmodules on X (For a sheaf R of rings on X). Let f : X Y be continuous. Then, for F Sh(X), G Sh(Y ), we have the following functors: Direct image: f : Sh(X) Sh(Y ), (f F )(V ) = F (f 1 V ). f is left exact. We get Γ when Y = {pt}. Proper direct image: f! : Sh(X) Sh(Y ), (f! )(V ) = {s F (f 1 (V )) such that f Supp(s) : Supp(s) V is proper }. f! is left exact. We get Γ c for Y = {pt}. Inverse image: f 1 : Sh(Y ) Sh(X). Consider the presheaf U lim G(V ). V f(u) Then f 1 G is the sheafification. Note that (f 1 G) x = G f(x). This implies that f 1 is exact. Here Γ and Γ c correspond to the global section of the sheaf, or the global section with compact support. For F, G Sh(X), let Hom(F, G) Sh(X), given by U Hom(F U, G U ). This makes Hom(, ) into a left exact bifunctor. If R is sheaf of rings on Y and M Mod(R), N Mod(f 1 R) then Hom R (M, f N)) = Hom f 1 R(f 1 M, N)). 5
6 Thus (f 1, f ) are an adjoint pair. Important: Mod(R) has enough injectives. Let D # (R) := D # (Mod(R)). We will often look at Rf, Rf!, RΓ, RΓ c, f 1, RHom(, ), RHom(, ). One has that Hom D + (R)(L, Rf N ) = Hom D + (f 1 R)(f 1 L, N ), so that (f 1, Rf ) is an adjoint pair. Theorem: PoincaréVerdier duality. Let A be a nice commutative ring. There exists a functor of triangulated categories, f! : D + (A Y ) D + (A X ), called the twisted or shrieck inverse image, such that Rf RHom AX (M, f! N ) = RHom AY (Rf! M, N ) Hom D + (A X )(M, f! N ) = Hom D + (A Y )(Rf! M, N ). Here A Y, A X denote the constant sheaf of A on X and Y respectively. We note that these relations are not defined on the level of sheaves but only on the derived categories. For c : X {pt}, we write ω X := c! (A ) {pt} D# (A X ) and call it the dualising complex. We also define D X (M ) := RHom AX (M, ω X ) and call it the Verdier dual. Definition 8. We say that a sheaf F on X is (algebraically) constructible if F Xα (F X an ) is a locally constant sheaf on X α α (Xα an ). Definition 9. We define D b c(x) to be the full subcategory of D b (Q X an) such that the cohomologies of the complexes are (algebraically) constructible. We have the following properties: D X D X = Id, when restricted to D b c(x). for f : X Y, we have f! ω Y = ω X, f! D Y = D X f 1, Rf D X = D Y Rf!. Let X be a complex analytic space or an algebraic variety. A stratification of X is X = α X α which is locally finite, with X α s smooth and X α is the union of some other X β s. The functors we saw work mostly fine on D b c(x). 6
7 3 tstructures Definition 10. Let D be a triangulated category, D 0, D 0 be full subcategories of D. We say that (D 0, D 0 ) is a tstructure if: D 1 D 0 and D 1 D 0, If X D 0 and Y D 1 then Hom(X, Y ) = 0, For all X D there is a triangle: with X 0 D 0, X 1 D 1, X 0 X X 1 +1, where D n = D 0 [ n] and D n = D 0 [ n] for all n Z. Example: Standard tstructure on D(C ): D 0 (C ) = {X D(C ) H j (X) = 0 for j < 0}, D 0 (C ) = {X D(C ) H j (X) = 0 for j > 0}. We note that here C = D 0 D 0, and in general, we call D 0 D 0 the heart of the tstructure. Hence using a different tstructure gives us a different heart. Proposition 1. Let i : D n D be the inclusion.then there exists a (truncation) functor, τ n, such that for all Y D n and X D, we have So (i, τ n ) is an adjoint pair. Hom D n(y, τ n X) = Hom D (i(y ), X). Proposition 2. Let i : D n D be the inclusion.then there exists a (truncation) functor, τ n, such that for all Y D n and X D, we have So (τ n, i) is an adjoint pair. Hom D n(τ n X, Y ) = Hom D (X, i(y )). Proposition 3. The heart is an abelian category. 7
8 Example: 1 d 1 Let s look at X X 0 d0 X 1. Then, Similarly, we define: τ 0 (X ) = 0 Imd 1 X 0 d 0 X 1 d 1. τ 0 (X ) = X 1 d 1 X 0 d 0 Coker(d 1 ) 0. From the third axiom, there is a triangle: τ n X X τ n+1 X +1. Here, we see that H n (X ) = τ 0 τ 0 X[n]. In general, we define cohomological functors to be H n (X ) := τ 0 τ 0 X[n], for different tstructures. So in general, if D = D(C ), then it is not necessarily true that H n (X ) = H n (X ), where the right hand side is the cohomology of X. Suppose we have a functor F : D 1 D 2 and tstructures (C 1, D 0 1, D 0 1 ) and (C 2, D 0 2, D 0 2 ), where the C i denote their respective hearts. Then we have p F : C 1 i D 1 F D 2 H0 C 2. (1) Definition 11. F is left texact if F (D 0 1 ) D 0 2 ; it is right texact if F (D 0 1 ) D 0 2. Proposition 4. If F is left (respectively right) texact then p F is a left (respectively right) exact functor of abelian categories. Proposition 5. If D i D i, F : D 1 D 2, G : D 2 D 1. With (F, G) an adjoint pair, then: If F (D 1 ) D 2 and F (D 0 1 ) D d 2, then G(D 0 2 ) D d 1 (whenever G takes an object of D 0 2 to D 1 ). If G(D 0 2 ) D d 1, then F (D 0 1 ) D d 2. 4 Perverse sheaves Inside D b c(x), we define subcategories: p Dc 0 (X) : F p D 0 c (X) if dim Supp H j (F ) j, j Z. 8
9 p Dc 0 (X) : F p D 0 c (X) if dim Supp H j (D X F ) j, j Z. Perv(Q X ) := p D 0 c (X) p D 0 (X). c Proposition 6. Take F Dc(X) b and X = α X α a stratification such that the X α s are connected and i 1 X α F and i! X α F have locally constant cohomology sheaves for all α, where i Xα : X α X is the inclusion map. Then, 1. F p D 0 c (X) iff H j (i 1 X α F ) = 0, α, j > dim X α. 2. F p D 0 c (X) iff H j (i! X α F ) = 0, α, j < dim X α. Proof. We prove 1. We have dim Supp H j (F ) j sup α {dim(x α supph j (F ))} j X α Supp H j (F ) =, whenever dim X α > j H j (F Xα ) = 0, whenever dim X α > j. It can be shown that ( p D 0 c (X), p D 0 c (X)) is a tstructure, and hence the associated perverse cohomology groups: p H n (A ) = p τ 0 p τ 0 (A [n]). We can define, as in (1), the perversifications p Rf, p Rf!, p f 1, and ( p f 1, p Rf ), ( p Rf!, p f! ) are adjoint pairs. Proposition 7. Let f : Y X be a continuous function with dim(f 1 (x)) d for x X. Then: 1. If F p D 0 c (X) then f 1 F p D d c (Y ). 2. If F p D 0 c (X) then f! F p D d c (Y ). Proof. 1. Let F p D 0 c (X). We have dim SuppH j (f 1 F [d])) = dim(f 1 (SuppH j+d (F )) dim SuppH j+d (F ) + d j d + d = j. So f 1 F [d] p D 0 c (Y ), i.e. f 1 F p D d c (Y ). 2. f! = D Y f 1 D X. If i : Z X is locally closed then Ri ( p D 0 (Z)) p D 0 (X); and Ri! ( p D 0 (Z)) p D 0 (X). 9
10 If j : U X is open then j 1 = j! is texact. If i : Z X is closed then i! = i. For instance, if we write Perv Z (Q X ) for the perverse sheaves on X supported on Z, then we have: Perv Z (Q X ) p i 1 = p i! Perv(Q Z ); Perv(Q Z ) p i = p i! Perv Z (Q X ). Intermediate extension: We write i, j as above where Z = X \ U. We have j! F j F. Then we have passing through p H 0 : p j! F p j F. p j! F Then p j! F is called the intermediate extension. We have the following: D X p j! F = p j! D U F, Given mild conditions on U U X, the extension is transitive. Characterisations: The following are equivalent: for F Perv (Q U ), 1. G = p j! F, 2. G Perv(Q X ) satisfies G U = F, i 1 G p D 1 c (Z), i! G p D 1 c (Z). 3. G Perv(Q X ) and G U = F and G has neither subobjects nor quotients supported on Z. If F Perv(Q U ) is simple then p j! F is simple in Perv(Q X ). We also have that p j! preserves monomorphisms and epimorphisms, but is not exact. If U X is open and smooth, let L Loc(U), local system on U; in this case L[d U ] Perv(Q U ), where d U := dim U. If L is irreducible, then IC(X, L) := p j! (L[d U ]) is simple in Perv(Q X ). If Z X is closed, Z 0 Z open, then for L Loc(Z 0 ), i Z ( p j Z0! L[d Z ]) = (i Z ) IC(Z, L). The structure theorem says that simple objects in the category Perv(C X ) are all of this form. 10
11 Definition 12. If L is an irreducible local system on Z 0 Z, closed, then IC(Z, L) is simple, so i Z IC(Z, L) is a simple perverse sheaf on X. We call these sheaves DeligneGoreskiMacPherson complexes (DGMcomplexes). Theorem: Structure theorem for perverse sheaves (BBD). Let X be a variety. We have: Perv(Q X ) is a full subcategory of D b c(x) which is abelian, stable by extensions and Verdier duality. The simple objects are DGMcomplexes. All objects are finite successive extensions of simple objects (Perv(Q X ) is artinian and noetherian.) References [1] Hotta, Takeuchi and Tanisaki: DModules, Perverse Sheaves, and Representation Theory [2] De Cataldo and Migliorni: The decomposition theorem, perverse sheaves and the topology of algebraic maps 11
8 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 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 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 informationThere 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 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 informationAN INTRODUCTION TO PERVERSE SHEAVES AND CHARACTER SHEAVES
AN INTRODUCTION TO PERVERSE SHEAVES AND CHARACTER SHEAVES ANNEMARIE AUBERT Abstract. After a brief review of derived categories, triangulated categories and tstructures, we shall consider the bounded
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 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 informationPerverse sheaves learning seminar: Perverse sheaves and intersection homology
Perverse sheaves learning seminar: Perverse sheaves and intersection homology Kathlyn Dykes November 22, 2018 In these notes, we will introduce the notion of tstructures on a triangulated cateogry. The
More informationPart II: Recollement, or gluing
The Topological Setting The setting: closed in X, := X. i X j D i DX j D A tstructure on D X determines ones on D and D : D 0 = j (D 0 X ) D 0 = (i ) 1 (D 0 X ) Theorem Conversely, any tstructures on
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 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 informationIC 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 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 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 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 informationdi 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 informationNOTES ON PERVERSE SHEAVES AND VANISHING CYCLES. David B. Massey
NOTES ON PERVERSE SHEAVES AND VANISHING CYCLES David B. Massey 0. Introduction to Version 216 These notes are my continuing effort to provide a sort of working mathematician s guide to the derived category
More informationPERVERSE 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 informationModules 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 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 informationMotivic integration on Artin nstacks
Motivic integration on Artin nstacks 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 informationDerived 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 informationDirect 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 informationABSTRACT 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 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 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 informationANNALESDEL INSTITUTFOURIER
ANNALESDEL INSTITUTFOURIER ANNALES DE L INSTITUT FOURIER Daniel JUTEAU Decomposition numbers for perverse sheaves Tome 59, n o 3 (2009), p. 11771229.
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 informationGood tilting modules and recollements of derived module categories, II.
Good tilting modules and recollements of derived module categories, II. Hongxing Chen and Changchang Xi Abstract Homological tilting modules of finite projective dimension are investigated. They generalize
More informationNon 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 informationShow 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 informationTRIANGULATED CATEGORIES, SUMMER SEMESTER 2012
TRIANGULATED CATEGORIES, SUMMER SEMESTER 2012 P. SOSNA Contents 1. Triangulated categories and functors 2 2. A first example: The homotopy category 8 3. Localization and the derived category 12 4. Derived
More informationElementary (haha) Aspects of Topos Theory
Elementary (haha) 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 informationGeneralizations of intersection homology and perverse sheaves with duality over the integers
Generalizations of intersection homology and perverse sheaves with duality over the integers Greg Friedman April 10, 2018 Contents 1 Introduction 2 2 Some algebraic preliminaries 9 3 Torsiontipped truncation
More information(iv) Whitney s condition B. Suppose S β S α. If two sequences (a k ) S α and (b k ) S β both converge to the same x S β then lim.
0.1. Stratified spaces. References are [7], [6], [3]. Singular spaces are naturally associated to many important mathematical objects (for example in representation theory). We are essentially interested
More informationWeilétale Cohomology
Weilétale 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 informationMIXED HODGE MODULES PAVEL SAFRONOV
MIED HODGE MODULES PAVEL SAFRONOV 1. Mixed Hodge theory 1.1. Pure Hodge structures. Let be a smooth projective complex variety and Ω the complex of sheaves of holomorphic differential forms with the de
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationCATEGORY THEORY. Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths.
CATEGORY THEORY PROFESSOR PETER JOHNSTONE Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths. Definition 1.1. A category C consists
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 informationwhich 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 informationTHE DECOMPOSITION THEOREM AND THE TOPOLOGY OF ALGEBRAIC MAPS
THE DECOMPOSITION THEOREM AND THE TOPOLOGY OF ALGEBRAIC MAPS Abstract. Notes from fives lectures given by Luca Migliorini in Freiburg in February 2010. Notes by Geordie Williamson. 1. Lecture 1: Hodge
More informationarxiv: 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 RiemannHilbert correspondence embeds the triangulated category of
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 informationNOTES ON PERVERSE SHEAVES AND VANISHING CYCLES. David B. Massey
NOTES ON PERVERSE SHEAVES AND VANISHING CYCLES David B. Massey 0. Introduction to Version 707 These notes are my continuing effort to provide a sort of working mathematician s guide to the derived category
More informationGENERALIZED tstructures: tstructures FOR SHEAVES OF DGMODULES OVER A SHEAF OF DGALGEBRAS AND DIAGONAL tstructures.
GENERLIZED tstructures: tstructures FOR SHEVES OF DGMODULES OVER SHEF OF DGLGEBRS ND DIGONL tstructures ROY JOSHU bstract. tstructures, in the abstract, apply to any triangulated category. However,
More informationALEXANDER INVARIANTS OF HYPERSURFACE COMPLEMENTS. Laurentiu G. Maxim. A Dissertation. Mathematics
ALEXANDER INVARIANTS OF HYPERSURFACE COMPLEMENTS Laurentiu G. Maxim A Dissertation in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements
More informationNotes on Partial Resolutions of Nilpotent Varieties by Borho and Macpherson
Notes on Partial Resolutions of Nilpotent Varieties by Borho and Macpherson Chris Elliott January 14th, 2014 1 Setup Let G be a complex reductive Lie group with Lie algebra g. The paper [BM83] relates
More informationAUSLANDER REITEN TRIANGLES AND A THEOREM OF ZIMMERMANN
Bull. London Math. Soc. 37 (2005) 361 372 C 2005 London Mathematical Society doi:10.1112/s0024609304004011 AUSLANDER REITEN TRIANGLES AND A THEOREM OF ZIMMERMANN HENNING KRAUSE Abstract A classical theorem
More informationWIDE SUBCATEGORIES OF dcluster TILTING SUBCATEGORIES
WIDE SUBCATEGORIES OF dcluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,
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 informationBasic 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 informationTHE DERIVED CATEGORY OF A GRADED GORENSTEIN RING
THE DERIVED CATEGORY OF A GRADED GORENSTEIN RING JESSE BURKE AND GREG STEVENSON Abstract. We give an exposition and generalization of Orlov s theorem on graded Gorenstein rings. We show the theorem holds
More informationModules over a Scheme
Modules over a Scheme Daniel Murfet October 5, 2006 In these notes we collect various facts about quasicoherent sheaves on a scheme. Nearly all of the material is trivial or can be found in [Gro60]. These
More informationLectures on Homological Algebra. Weizhe Zheng
Lectures on Homological Algebra Weizhe Zheng Morningside Center of Mathematics Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing 100190, China University of the Chinese Academy
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 informationLECTURE 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 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 informationSolutions 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 informationSOME 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 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 informationABSOLUTELY PURE REPRESENTATIONS OF QUIVERS
J. Korean Math. Soc. 51 (2014), No. 6, pp. 1177 1187 http://dx.doi.org/10.4134/jkms.2014.51.6.1177 ABSOLUTELY PURE REPRESENTATIONS OF QUIVERS Mansour Aghasi and Hamidreza Nemati Abstract. In the current
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 informationSuperperverse intersection cohomology: stratification (in)dependence
Superperverse intersection cohomology: stratification (in)dependence Greg Friedman Yale University July 8, 2004 typeset=august 25, 2004 Abstract Within its traditional range of perversity parameters, intersection
More informationMatrix 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 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 informationThe perverse tstructure
The perverse tstruture Milan Lopuhaä Marh 15, 2017 1 The perverse tstruture The goal of today is to define the perverse tstruture and perverse sheaves, and to show some properties of both. In his talk
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 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 informationProperties of Triangular Matrix and Gorenstein Differential Graded Algebras
Properties of Triangular Matrix and Gorenstein Differential Graded Algebras Daniel Maycock Thesis submitted for the degree of Doctor of Philosophy chool of Mathematics & tatistics Newcastle University
More informationPerverse poisson sheaves on the nilpotent cone
Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2010 Perverse poisson sheaves on the nilpotent cone Jared Lee Culbertson Louisiana State University and Agricultural
More informationHodge Theory of Maps
Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1  Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar
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 informationCategories and functors
Lecture 1 Categories and functors Definition 1.1 A category A consists of a collection ob(a) (whose elements are called the objects of A) for each A, B ob(a), a collection A(A, B) (whose elements are called
More informationLecture 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 informationArithmetic of certain integrable systems. University of Chicago & Vietnam Institute for Advanced Study in Mathematics
Arithmetic of certain integrable systems Ngô Bao Châu University of Chicago & Vietnam Institute for Advanced Study in Mathematics System of congruence equations Let us consider a system of congruence equations
More informationDerived Categories. Mistuo Hoshino
Derived Categories Mistuo Hoshino Contents 01. Cochain complexes 02. Mapping cones 03. Homotopy categories 04. Quasiisomorphisms 05. Mapping cylinders 06. Triangulated categories 07. Épaisse subcategories
More informationOVERVIEW OF SPECTRA. Contents
OVERVIEW OF SPECTRA Contents 1. Motivation 1 2. Some recollections about Top 3 3. Spanier Whitehead category 4 4. Properties of the Stable Homotopy Category HoSpectra 5 5. Topics 7 1. Motivation There
More informationGraduate algebraic Ktheory seminar
Seminar notes Graduate algebraic Ktheory seminar notes taken by JL University of Illinois at Chicago February 1, 2017 Contents 1 Model categories 2 1.1 Definitions...............................................
More informationDerived Categories Part II
Derived Categories Part II Daniel Murfet October 5, 2006 In this second part of our notes on derived categories we define derived functors and establish their basic properties. The first major example
More informationCALABIYAU ALGEBRAS AND PERVERSE MORITA EQUIVALENCES
CALABIYAU ALGEBRAS AND PERERSE MORITA EQUIALENCES JOSEPH CHUANG AND RAPHAËL ROUQUIER Preliminary Draft Contents 1. Notations 2 2. Tilting 2 2.1. tstructures and filtered categories 2 2.1.1. tstructures
More informationCohomology jump loci of local systems
Cohomology jump loci of local systems Botong Wang Joint work with Nero Budur University of Notre Dame June 28 2013 Introduction Given a topological space X, we can associate some homotopy invariants to
More informationWEIGHT STRUCTURES AND SIMPLE DG MODULES FOR POSITIVE DG ALGEBRAS
WEIGHT STRUCTURES AND SIMPLE DG MODULES FOR POSITIVE DG ALGEBRAS BERNHARD KELLER AND PEDRO NICOLÁS Abstract. Using techniques due to Dwyer Greenlees Iyengar we construct weight structures in triangulated
More informationAuslanderYoneda algebras and derived equivalences. Changchang Xi ( ~) ccxi/
International Conference on Operads and Universal Algebra, Tianjin, China, July 59, 2010. Auslander and derived Changchang Xi ( ~) xicc@bnu.edu.cn http://math.bnu.edu.cn/ ccxi/ Abstract In this talk,
More information2 Passing to cohomology Grothendieck ring Cohomological FourierMukai transform Kodaira dimension under derived equivalence 15
Contents 1 Fourier Mukai Transforms 3 1.1 GrothendieckVerdier Duality.................. 3 1.1.1 Corollaries......................... 3 1.2 Fourier Mukai Transforms.................... 4 1.2.1 Composition
More informationCharacter sheaves and modular generalized Springer correspondence Part 2: The generalized Springer correspondence
Character sheaves and modular generalized Springer correspondence Part 2: The generalized Springer correspondence Anthony Henderson (joint with Pramod Achar, Daniel Juteau, Simon Riche) University of Sydney
More informationMicrosupport of sheaves
Microsupport of sheaves Vincent Humilière 17/01/14 The microlocal theory of sheaves and in particular the denition of the microsupport is due to Kashiwara and Schapira (the main reference is their book
More informationINTERSECTION SPACES, PERVERSE SHEAVES AND TYPE IIB STRING THEORY
INTERSECTION SPACES, PERVERSE SHEAVES AND TYPE IIB STRING THEORY MARKUS BANAGL, NERO BUDUR, AND LAURENŢIU MAXIM Abstract. The method of intersection spaces associates rational Poincaré complexes to singular
More informationAN INTRODUCTION TO PERVERSE SHEAVES. Antoine ChambertLoir
AN INTRODUCTION TO PERVERSE SHEAVES Antoine ChambertLoir Antoine ChambertLoir Université ParisDiderot. Email : Antoine.ChambertLoir@math.univparisdiderot.fr Version of February 15, 2018, 10h48 The
More informationDERIVED EQUIVALENCES AND GORENSTEIN PROJECTIVE DIMENSION
DERIVED EQUIVALENCES AND GORENSTEIN PROJECTIVE DIMENSION HIROTAKA KOGA Abstract. In this note, we introduce the notion of complexes of finite Gorenstein projective dimension and show that a derived equivalence
More informationHungry, Hungry Homology
September 27, 2017 Motiving Problem: Algebra Problem (Preliminary Version) Given two groups A, C, does there exist a group E so that A E and E /A = C? If such an group exists, we call E an extension of
More informationarxiv: v1 [math.kt] 27 Jan 2015
INTRODUCTION TO DERIVED CATEGORIES AMNON YEKUTIELI arxiv:1501.06731v1 [math.kt] 27 Jan 2015 Abstract. Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the old
More informationThe RiemannRoch Theorem
The RiemannRoch Theorem TIFR Mumbai, India Paul Baum Penn State 7 August, 2015 Five lectures: 1. Dirac operator 2. AtiyahSinger revisited 3. What is Khomology? 4. Beyond ellipticity 5. The RiemannRoch
More informationPART I. Abstract algebraic categories
PART I Abstract algebraic categories It should be observed first that the whole concept of category is essentially an auxiliary one; our basic concepts are those of a functor and a natural transformation.
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 informationIntersection homology duality and pairings: singular, PL, and sheaftheoretic
Intersection homology duality and pairings: singular, PL, and sheaftheoretic Greg Friedman and James McClure December 17, 2018 Contents 1 Introduction 2 2 Conventions and some background 9 2.1 Pseudomanifolds
More informationAn Introduction to Spectral Sequences
An Introduction to Spectral Sequences Matt Booth December 4, 2016 This is the second half of a joint talk with Tim Weelinck. Tim introduced the concept of spectral sequences, and did some informal computations,
More informationarxiv:math/ v3 [math.ag] 5 Mar 1998
arxiv:math/9802004v3 [math.ag] 5 Mar 1998 Geometric methods in the representation theory of Hecke algebras and quantum groups Victor GINZBURG Department of Mathematics University of Chicago Chicago, IL
More informationThe RiemannRoch Theorem
The RiemannRoch Theorem Paul Baum Penn State Texas A&M University College Station, Texas, USA April 4, 2014 Minicourse of five lectures: 1. Dirac operator 2. AtiyahSinger revisited 3. What is Khomology?
More information