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A Primer on Sheaf Theory and Sheaf Cohomology

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A Primer on Sheaf Theory and Sheaf Cohomology A PRIMER ON SHEAF THEORY AND SHEAF COHOMOLOGY SAAD SLAOUI Abstract. These notes are intended to provide an introduction to basic sheaf theory and sheaf cohomology. The first few sections introduce some key defi- nitions and technical tools leading to the definition of sheaf cohomology in the language of derived functors. The exposition is motivated whenever possible with ideas drawing from elementary manifold theory and algebraic topology. The last two sections initiate a discussion of ways in which the computation of sheaf cohomology groups can be simplified. Some familiarity with basic no- tions of category theory including (co)limits is assumed, as well as some degree of acquaintance with Abelian categories and chain complexes. Contents 1. Introduction 1 2. Sheaves on a Topological Space 3 3. Interlude: Smooth Manifolds as Ringed Spaces 6 4. Stalks and Sheafification 7 5. A Few Notable Adjunctions 13 6. Some Homological Algebra 16 7. Defining Sheaf Cohomology 21 8. Flasque Sheaves 22 9. Cechˇ Cohomology 25 Acknowledgments 28 References 28 1. Introduction \C'´etaitcomme si le bon vieux `m`etre cohomologique' standard dont on disposait jusqu’`apr´esentpour `arpenter' un espace, s'´etaitsoudain vu multiplier en une mul- titude inimaginablement grande de nouveaux `m`etres' de toutes les tailles, formes et substances imaginables, chacun intimement adapt´e`al'espace en question, et dont chacun nous livre `ason sujet des informations d'une pr´ecisionparfaite, et qu'il est seul `apouvoir nous donner." - Grothendieck, R´ecoltes et Semailles Let X be a topological space, and denote by Top(X) the category with objects the open sets of X and morphisms given by inclusions (i.e. HomTop(X)(U; V ) = {∗} if U ⊆ V , empty otherwise). As a rule of thumb, one can hope to learn more about a given geometric object by studying functions from or into that object. Sheaves are one way in which this practice can be formalized for topological spaces: a sheaf F on a space X can often be thought of as assigning to each open U in X a space of 1 2 SAAD SLAOUI functions on U such that the assignment is compatible with taking restrictions and going from local to global sections. In fact, sheaves prove to be quite successful at realizing the rule of thumb suggested above: one can show further down the line that Top(X) can be identified as a subcategory of the category Sh(X) of sheaves of sets on X, and that this information is sufficient to recover X up to homeomor- phism, provided precisely that X is a sober space (i.e. every irreducible subset of X is of the form fxg for a unique x 2 X). This is the case, in particular, if X is just assumed to be Hausdorff. (For more details, see for instance [11].) Sheaves also offer an alternative framework in which to view familiar objects such as (smooth) manifolds; this new perspective will naturally lead us to introducing the notion of schemes. We shall elaborate on this viewpoint in Section 2, as soon as we have established the rudiments of the subject. The purpose of this article is to provide an introduction to basic concepts in sheaf theory and sheaf cohomology, with some emphasis put in developing the homolog- ical algebra framework in which the latter can be naturally expressed. Sections 2 through 4 present elementary constructions that can be performed on sheaves. Sections 5 and 6 contain a streamlined treatment of relevant notions from cate- gory theory and homological algebra, with the aim to arrive at a definition of sheaf cohomology in Section 7. Section 8 presents an alternative way to compute sheaf cohomology groups, and hints at further constructions of a similar kind. Finally, Section 9 provides an introductory account to Cechˇ cohomology, which often sim- plifies computations of sheaf cohomology groups. As hinted at by the opening quote, sheaf cohomology theory appears to be a remarkable generalization of more conventional cohomology theories. For a given space X, any sheaf of abelian groups, i.e. any compatible assignment of abelian groups to elements of Top(X) in a sense to be made precise later, gives rise to a family of cohomology groups of X with coefficients in that sheaf. In particular, for a fixed abelian group A, one may consider the sheaf that comes as close as possible to asssigning A to every open in X, while still satisfying certain compatibility con- ditions. We call this sheaf the constant sheaf on X associated to A, and denote it ∗ by AX . We may then restrict our attention to the sheaf cohomology H (X; AX ) of X with coefficients in AX . If X is locally contractible and paracompact, we are en- ∗ ∼ ∗ sured the existence of a natural isomorphism H (X; AX ) = H (X; A) between sheaf cohomology with coefficients in AX and singular cohomology taking coefficients in A (see for instance [15]). So restricting one's attention to CW-complexes (which are paracompact and locally contractible), which is not an unreasonable thing to do due to the fact that all spaces are weakly equivalent to a CW-complex, one can recover all singular cohomology groups from sheaf cohomology. Immediate from the de Rham theorem and the above discussion is also the fact that we have, for any ∗ ∼ ∗ manifold X, an isomorphism with de Rham cohomology HdeR(X) = H (X; RX ). Hence we have recovered standard cohomological constructions without even looking at sheaf cohomology with coefficients in non-constant sheaves. If one allows for any sheaf of coefficients, the situation rapidly takes a flavor of its own. For instance, sheaf cohomology may no longer be homotopy invariant. This tells us that homotopy theory has \limited leverage" on sheaf cohomology, whereas singular A PRIMER ON SHEAF THEORY AND SHEAF COHOMOLOGY 3 cohomology can be entirely introduced in homotopy-theoretic terms (via the use of Eilenberg-Maclane spectra). 2. Sheaves on a Topological Space We formulate a definition for the notion of a sheaf on a space X with values in a fixed concrete category C, meaning that C can be thought of as a category of sets with some additional structure (so C may for instance denote the category of sets, abelian groups or commutative rings): Definition 2.1. A presheaf F on a space X with values in a category C is a F contravariant functor F : Top(X) ! C. We write resV;U : F (V ) ! F (U) (where res stands for \restriction morphism") for the image of the inclusion morphism U,! V under F , or simply resV;U when F is clear from the context. A sheaf F is a presheaf on X with values in a concrete category C satisfying the following local-to-global condition: Given an open cover fUig of U 2 Top(X) and elements ffi 2 F (Ui)g such that resUi;Uij fi = resUj ;Uij fj for all i; j, there exists a unique element f 2 F (U) satisfying resU;Ui f = fi for all i. We follow the convention that a presheaf is required to map the empty set to the terminal object in C, if there is one. Notice that the definition of a presheaf makes sense with values in an arbitrary category, and that a presheaf on X with values in C may be alternatively defined as a covariant functor F : Top(X)op ! C, where Top(X)op denotes the opposite category (reversed arrows) of Top(X). Hence, presheaves naturally fit in the cate- gory Fun(Top(X)op; C) with objects: functors F : Top(X)op ! C and morphisms: natural transformations of such functors. We extend this terminology to sheaves, and give the corresponding categories names in two important cases: Definition 2.2. A sheaf morphism between two sheaves F; G on X with values in C is a natural transformation η : F ! G of the underlying presheaves; namely, a sheaf morphism consists of a collection of morphisms ηU : F (U) ! G(U) in C such that for each inclusion U,! V we have a commutative square F resV;U F (V ) / F (U) ηV ηU G(V ) G(U) G / resV;U Sheaves on X with values in a concrete category C then fit into a category (properly speaking, the full subcategory of the category of C-valued preshaves satisfying the sheaf axiom). We write Ab(X) (resp. Sh(X)) to denote the resulting category of sheaves of abelian groups on X (resp. sheaves of sets on X). We shall sometimes refer to elements of Ab(X) as abelian sheaves. Noting that we have essentially created a way of associating \algebraic data" to a given topological space, a reader with some background in Algebraic Topology may be wondering whether there is a natural way to associate to any continuous map f : X ! Y a corresponding \functor" between the associated sheaf categories, 4 SAAD SLAOUI in the same way for instance that a map f : X ! Y induces (graded) group ∗ ∗ ∗ homomorphisms f∗ : H∗(X) ! H∗(Y ) on homology and f : H (Y ) ! H (X) on cohomology, with the assignment being functorial in both these instances. Such assignments may indeed be carried out in the settings of, say, sheaves of abelian groups: Definition 2.3. Let f : X ! Y be a continuous map of spaces. Define the pushforward of f to be the map: f∗ : Ab(X) ! Ab(Y), −1 F 7! (f∗F: V 7! F(f (V )); for any V 2 Top(Y)), with behavior with respect to inclusions inherited from F. We refer to f∗F 2 Ab(Y) as the pushforward of F under f.
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