Open Shilin Yu-Dissertation.Pdf
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Lecture 18: April 15 Direct Images and Coherence. Last Time, We Defined
89 Lecture 18: April 15 Direct images and coherence. Last time, we defined the direct image functor (for right D-modules) as the composition L op DX Y op Rf op b ⌦ ! b 1 ⇤ b D (DX ) D (f − DY ) D (DY ) f+ where f : X Y is any morphism between nonsingular algebraic varieties. We also ! showed that g+ f+ ⇠= (g f)+. Today, our first◦ task is to◦ prove that direct images preserve quasi-coherence and, in the case when f is proper, coherence. The definition of the derived category b op D (DX )didnot include any quasi-coherence assumptions. We are going to denote b op b op by Dqc(DX ) the full subcategory of D (DX ), consisting of those complexes of right DX -modules whose cohomology sheaves are quasi-coherent as OX -modules. Recall that we included the condition of quasi-coherence into our definition of algebraic b op D-modules in Lecture 10. Similarly, we denote by Dcoh (DX ) the full subcategory b op of D (DX ), consisting of those complexes of right DX -modules whose cohomology sheaves are coherent DX -modules (and therefore quasi-coherent OX -modules). This b op category is of course contained in Dqc(DX ). Theorem 18.1. Let f : X Y be a morphism between nonsingular algebraic ! b op b op varieties. Then the functor f+ takes Dqc(DX ) into Dqc(DY ). When f is proper, b op b op it also takes Dcoh (DX ) into Dcoh (DY ). We are going to deduce this from the analogous result for OX -modules. Recall that if F is a quasi-coherent OX -module, then the higher direct image sheaves j R f F are again quasi-coherent OY -modules. -
A CATEGORICAL INTRODUCTION to SHEAVES Contents 1
A CATEGORICAL INTRODUCTION TO SHEAVES DAPING WENG Abstract. Sheaf is a very useful notion when defining and computing many different cohomology theories over topological spaces. There are several ways to build up sheaf theory with different axioms; however, some of the axioms are a little bit hard to remember. In this paper, we are going to present a \natural" approach from a categorical viewpoint, with some remarks of applications of sheaf theory at the end. Some familiarity with basic category notions is assumed for the readers. Contents 1. Motivation1 2. Definitions and Constructions2 2.1. Presheaf2 2.2. Sheaf 4 3. Sheafification5 3.1. Direct Limit and Stalks5 3.2. Sheafification in Action8 3.3. Sheafification as an Adjoint Functor 12 4. Exact Sequence 15 5. Induced Sheaf 18 5.1. Direct Image 18 5.2. Inverse Image 18 5.3. Adjunction 20 6. A Brief Introduction to Sheaf Cohomology 21 Conclusion and Acknowlegdment 23 References 23 1. Motivation In many occasions, we may be interested in algebraic structures defined over local neigh- borhoods. For example, a theory of cohomology of a topological space often concerns with sets of maps from a local neighborhood to some abelian groups, which possesses a natural Z-module struture. Another example is line bundles (either real or complex): since R or C are themselves rings, the set of sections over a local neighborhood forms an R or C-module. To analyze this local algebraic information, mathematians came up with the notion of sheaves, which accommodate local and global data in a natural way. However, there are many fashion of introducing sheaves; Tennison [2] and Bredon [1] have done it in two very different styles in their seperate books, though both of which bear the name \Sheaf Theory". -
Sheaf Cohomology
Sheaf Cohomology Gabriel Chˆenevert Payman Kassaei September 24, 2003 In this lecture, we define the cohomology groups of a topological space X with coefficients in a sheaf of abelian groups F on X in terms of the derived functors of the global section functor Γ(X; ¢ ). Then we introduce Cechˇ coho- mology with respect to an open covering of X, which permits to make explicit calculations, and discuss under which conditions it can be used to compute sheaf cohomology. 1 Derived functors We first need to review some homological algebra in order to be able to define sheaf cohomology using the derived functors of the global sections functor. Let A be an abelian category, that is, roughly, an additive category in which there exist well-behaved kernels and cokernels for each morphism, so that, for example, the notion of an exact sequence in A makes sense. If X is a fixed object in A and Ab denotes the category of abelian groups, then we have a contravariant functor Hom( ¢ ;X): A ¡! Ab: It is readily seen to be left exact, that is, for any short exact sequence f g 0 ¡! A ¡! B ¡! C ¡! 0 in A , the sequence g¤ f ¤ 0 ¡! Hom(C; X) ¡! Hom(B; X) ¡! Hom(A; X) is exact in Ab. Definition 1.1. An object I of A is said to be injective if the functor Hom( ¢ ;I) is exact. 1 Since Hom( ¢ ;I) is always left exact, we see that an object I of A is injective if and only if for each exact sequence 0 ! A ! B and morphism A ! I, there exists a morphism B ! I making the following diagram commute. -
Lecture 10 - Direct and Inverse Images, Stalks, and Sheafification
Lecture 10 - Direct and Inverse Images, Stalks, and Sheafification October 29, 2014 1 Generalities on Abelian Categories and Limits In this lecture we define and explore the categories of sheaves and presheaves and functors between them. As such we begin by reviewing some basics that will come in handy later. This section exists primarily for completeness and ease of reference, and may(should) be skipped on a first reading. 1.1 Abelian Categories Definition. An Abelian category is a Category C with the following prop- erties: • It has an initial object Z 2 C. That is, every other object has a unique morphism from Z, and a unique morphism to Z. • All hom-sets are equipped with the structure of an abelian group such that the composition of morphisms is bilinear. • All finite objects have a biproduct. That is, for every set A1;:::;Ak of objects there exists an object B with maps pm to Am and im from Am such that pm ◦ ik is the zero morphism for m 6= k, and the identity for m = k, such that the morphisms pm make B the coproduct of the Ak and the morphisms ik make B the product of the Ak. One should think of B as the direct sum AND the direct product of the Ak. • C has all kernels and cokernels. Recall that for a morphism φ : A ! B a kernel of φ is a morphism k : K ! A such that φ ◦ k = 0 and any other morphism k0 : K0 ! A such that φ ◦ k0 = 0 factors through k.A cokernel is the dual notion. -
Relation Between Two Twisted Inverse Image Pseudofunctors in Duality Theory
Relation between two twisted inverse image pseudofunctors in duality theory Srikanth B. Iyengar, Joseph Lipman and Amnon Neeman Abstract Grothendieck duality theory assigns to essentially-finite-type maps f of noetherian × ! schemes a pseudofunctor f right-adjoint to Rf∗, and a pseudofunctor f agreeing with f × when f is proper, but equal to the usual inverse image f ∗ when f is ´etale.We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by \compactly supported" versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes. Introduction The relation in the title is given by a canonical pseudofunctorial map :(−)× ! (−)! between \twisted inverse image" pseudofunctors with which Grothendieck duality theory is concerned. These pseudofunctors on the category E of essentially-finite-type separated maps of noethe- rian schemes take values in bounded-below derived categories of complexes with quasi-coherent homology, see 1.1 and 1.2. The map , derived from the pseudofunctorial \fake unit map" ! id ! (−) ◦ R(−)∗ of Proposition 2.1, is specified in Corollary 2.1.4. A number of concrete exam- ples appear in x3. For instance, if f is a map in E, then (f) is an isomorphism if f is proper; but if f is, say, an open immersion, so that f ! is the usual inverse image functor f ∗ whereas f × is right-adjoint to Rf∗ , then (f) is usually quite far from being an isomorphism (see e.g., 3.1.2, 3.1.3 and 3.3). -
Talk 3: Review of Constructible Sheaves and Operations on Sheaves
Talk 3: Review of constructible sheaves and operations on sheaves Good resources for the material presented today include [KS13], [Dim04], [Ive86], and [NZ09]. 3.1 Constructible sheaves Throughout this section, X will be a closed, real analytic manifold of dimension n, unless otherwise noted. As indicated by the name, constructible sheaves are those that we can build from very simple pieces using relatively simple methods. We start with the simplest possible sheaf (of C-modules) on X, which is the constant sheaf CX . This is the sheafification of Id CX (U) = C; (V,! U) 7! (C −! C) for open sets V ⊂ U ⊂ X. Notice that for any other sheaf F on X we have an obvious map CX × F ! F; defined by the fact that F(U) is a C-module for each U ⊂ X. This motivates the notation mod(CX ) for sheaves on X. Also note that we have a constant sheaf MX 2 mod(CX ) for each M 2 mod(C). Next, we allow ourselves to patch constant sheaves together, perhaps with monodromy. A locally constant sheaf is a sheaf F 2 mod(CX ) such that each x 2 X admits an open neighborhood U ⊂ X for which FjU is a constant sheaf. Example 4. Let π : Y ! X be a finite-sheeted covering space with fiber F . We define a sheaf F by assigning F(U) = ChΓ(π : Y ! U)i; with the obvious restriction maps. Given x 2 X, we can choose an open neighborhood U ⊂ X such that π−1(U) ' U, so Γ(π : Y ! U) ' F . -
Base Change and Grothendieck Duality for Cohen–Macaulay
Compositio Math. 140 (2004) 729–777 DOI: 10.1112/S0010437X03000654 Base change and Grothendieck duality for Cohen–Macaulay maps Pramathanath Sastry Abstract Let f : X → Y be a Cohen–Macaulay map of finite type between Noetherian schemes, and g : Y → Y amap,withY Noetherian. Let f : X → Y be the base change of f under g and g : X → X thebasechangeofg under f. We show that there is a canonical f ∗ isomorphism θg : g ωf ωf ,whereωf and ωf are the relative dualizing sheaves. The f map θg is easily described when f is proper, and has a more subtle description when f is f ∗ r r not. If f is smooth we show that θg corresponds to the canonical identification g Ωf =Ωf of differential forms, where r is the relative dimension of f. This work is closely related to B. Conrad’s work on base change. However, our approach to the problems and our viewpoint are very different from Conrad’s: dualizing complexes and their Cousin versions, residual complexes, do not appear in this paper. 1. Introduction Our approach to Grothendieck duality is the approach of Deligne and Verdier [Del66, Ver69]with important elaborations by Alonso et al..[AJL99]. In particular, we do not use residual complexes or dualizing complexes, which are crucial ingredients in the approach laid out in Hartshorne’s voluminous book [Har66]. We will be examining the behavior of relative duality for a Cohen– Macaulay map f : X → Y of Noetherian schemes. Our intent is to show that the main result of Conrad’s recent book [Con00] on Grothendieck duality concerning base change for relative dualizing sheaves and traces for Cohen–Macaulay maps (see [Con00, pp. -
FIBRATIONS and HOMOTOPY COLIMITS of SIMPLICIAL SHEAVES 1. Introduction Dold and Thom [3] Introduced a Class of Maps Called Quasi
FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES CHARLES REZK Abstract. We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the notion of a quasi-fibration of spaces, and seems to be of independent interest. 1. Introduction Dold and Thom [3] introduced a class of maps called quasi-fibrations. A map f : X Y of topological spaces is called a quasi-fibration if for each point y Y ! 1 2 the fiber f − (y) is naturally weakly equivalent to the homotopy fiber of f over y. Thus, quasi-fibrations behave for some purposes of homotopy theory very much like other types of fibrations; for example, there is a long exact sequence relating the 1 homotopy groups of X, Y , and f − (y). A notable feature of quasi-fibrations is that (as shown by Dold and Thom) quasi-fibrations defined over the elements of an open cover of a space Y can sometimes be \patched" together to give a quasi-fibration mapping to all of Y . In this paper we study a class of maps called sharp maps. In our context, a map f : X Y will be called sharp if for each base-change of f along any map into the base Y!the resulting pullback square is homotopy cartesian. We are particularly interested in sharp maps of sheaves of simplicial sets. -
Arxiv:1209.2995V6 [Math.CT] 14 Feb 2017 ..Eatctgr Fcnrhrn Ohae 44 40 40 46 14 of 14 Cosheaves Contraherent 27 Cosheaves Contraherent 2.5
CONTRAHERENT COSHEAVES LEONID POSITSELSKI Abstract. Contraherent cosheaves are globalizations of cotorsion (or similar) modules over commutative rings obtained by gluing together over a scheme. The category of contraherent cosheaves over a scheme is a Quillen exact category with exact functors of infinite product. Over a quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull dimension (in a different version—over any lo- cally Noetherian scheme), it also has enough projectives. We construct the derived co-contra correspondence, meaning an equivalence between appropriate derived cat- egories of quasi-coherent sheaves and contraherent cosheaves, over a quasi-compact semi-separated scheme and, in a different form, over a Noetherian scheme with a dualizing complex. The former point of view allows us to obtain an explicit con- struction of Neeman’s extraordinary inverse image functor f ! for a morphism of quasi-compact semi-separated schemes f : X −→ Y . The latter approach provides an expanded version of the covariant Serre–Grothendieck duality theory and leads to Deligne’s extraordinary inverse image functor f ! (which we denote by f +) for a morphism of finite type f between Noetherian schemes. Semi-separated Noetherian stacks, affine Noetherian formal schemes, and ind-affine ind-schemes (together with the noncommutative analogues) are briefly discussed in the appendices. Contents Introduction 3 1. Contraadjusted and Cotorsion Modules 14 1.1. Contraadjusted and very flat modules 14 1.2. Affine geometry of contraadjusted and very flat modules 17 arXiv:1209.2995v6 [math.CT] 14 Feb 2017 1.3. Cotorsion modules 20 1.4. Exact categories of contraadjusted and cotorsion modules 23 1.5. -
Ultracategories Seminar Talk 1: Sheafs on Topological Spaces
Ultracategories seminar Talk 1: Sheafs on topological spaces Julie Zangenberg Rasmusen February 2019 Contents 1 Introduction to sheaves 1 2 Sheaf of cross-sections 3 3 Bundles of germs 4 4 Étale spaces 6 5 Stalks 10 6 Direct and inverse image sheaves 12 These notes are based on chapter 2 of "Sheaves in Geometry and Logic" by Saunders Mac Lane and leke Moerdijk. 1 Introduction to sheaves Denition 1.1. Let X be a topological space. Then we dene the category O(X) to have all open subsets U of X as objects and dening the morphisms V ! U to be the inclusion V ⊂ U. Denition 1.2. A presheaf is a functor P : O(X)op ! Sets. This means that every inclusion of open sets V ⊂ U in X determines a functor P (V ⊂ U): P (U) ! P (V ) which we will often denote as t 7! tjV for each t 2 P (U) (as if they were restriction of ordinary maps) 1 1 INTRODUCTION TO SHEAVES Denition 1.3. A sheaf of sets on a topological space X is a presheaf F : O(X)op ! Sets, such that each open covering U = [iUi; i 2 I of an open set U of X yields an equalizer diagram p e Q / Q FU / FUi / F (Ui \ Uj) i q i;j where for t 2 FU e(t) = ftjUi ji 2 Ig and for a family ti 2 FUi, pftig = ftijUi\Uj g; qftig = ftjjUi\Uj g Denition 1.4. Let X be a topological space. We dene a new category Sh(X): Objects: All sheaves F of sets on X Morphisms F ! G: Natural transformations F ) G. -
Sheaf Theory W. D. Gillam
Sheaf Theory W. D. Gillam Department of Mathematics, Brown University E-mail address: [email protected] Contents Introduction 7 Chapter I. Sheaves 9 1. Open sets 9 2. Presheaves 9 3. Representable presheaves 10 4. Generators 11 5. Sheaves 11 6. Stalks 13 7. Sheafification 15 8. Espaces ´etal´e 16 9. Sieves 18 10. Properties of Sh(X) 20 11. Examples 22 12. Exercises 22 Chapter II. Direct and Inverse Images 23 1. Functoriality of Ouv(X) 23 2. Direct images 23 3. Skyscraper sheaves 23 4. Inverse images 24 5. Sheaf Hom 26 6. Constant sheaves 27 7. Representable case 28 8. Inverse images and espaces ´etal´e 28 9. Faithfulness of X 7! Sh(X) 29 Chapter III. Descent 31 1. Canonical topology 31 2. Open maps and variants 31 3. Descent for spaces 31 4. Descent for sheaves 31 Chapter IV. Cohomology 33 1. Abelian categories 33 2. Enough injectives 33 3. Homological algebra 33 4. Abelian Sheaves 33 3 4 CONTENTS 5. Ringed spaces 36 6. Sheaf Ext 37 7. Supports 38 8. Sheaf cohomology 39 9. Acyclic sheaves 39 10. Flasque sheaves 39 11. Godement resolution 42 12. Soft sheaves 42 13. Filtered direct limits 42 14. Vanishing theorems 46 15. Higher direct images 46 16. Leray spectral sequence 47 17. Group quotients 48 18. Local systems 48 19. Examples 48 20. Exercises 48 Chapter V. Base Change 51 1. Proper maps 51 2. Comparison map 55 3. Homotopy invariance 57 Chapter VI. Duality 59 1. Adjoints for immersions 59 2. Extension by zero 61 3. -
1. General Sheaf Theory Exercises 1.1
SHEAF THEORY EXERCISES PETER J. HAINE 1. General Sheaf Theory Exercises 1.1. Exercise. Let 푋 be a space. Prove that the opposite of the poset Op(푋) of open subsets of 푋 is filtered. 1.2. Exercise. Let 푋 be a space and 푥 ∈ 푋. Write Op푥(푋) for the full subposet of Op(푋) op spanned by those open subsets containing the point 푥. Prove that Op푥(푋) is filtered. 1.3. Exercise. Prove that the subcategory of CRing of domains is stable under filtered col- imits, i.e., that if 퐹∶ 퐼 → CRing be a filtered diagram of commutative rings, where 퐹(푖) is a domain for all 푖 ∈ 퐼, then colim퐼 퐹 is a domain (where the colimit is taken in CRing). You’ll need to use the explicit description of filtered colimits in Set, which also describes the underlying set of the filtered colimit in CRing. You should think about what the ring structure should be. More generally, this description gives the underlying set of a filtered colimit in Mod푅 or Alg푅, where 푅 is a commutative ring (taking 푅 = 퐙, we recover the special cases of Ab = Mod퐙 and CRing = Alg퐙). 1.4. Exercise. Let 퐶 be a category and 푓∶ 푋 → 푌 be a morphism in 퐶. Prove 푓 is a monomorphism if and only if the square 푋 푋 푓 푋 푌 푓 is a pullback square. 1.5. Exercise. Suppose that 퐶 and 퐷 are categories and 퐹, 퐺∶ 퐶 → 퐷 functors. Prove that if 훼∶ 퐹 ⇒ 퐺 is a natural transformation whose components are monomorphisms in 퐷, then 훼 is a monomorphism in 퐷퐶.