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An Introduction to Sheaf Cohomology in Algebraic Geometry

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An Introduction to Sheaf Cohomology in Algebraic Geometry An Introduction to Sheaf Cohomology in Algebraic Geometry Ryan Mike University of Colorado Boulder Department of Mathematics Defended April 7, 2020 Thesis Advisor: Dr. Sebastian Casalaina-Martin, Department of Mathematics Defense Committee: Dr. Sebastian Casalaina-Martin, Department of Mathematics Dr. Nathaniel Thiem, Department of Mathematics Dr. Rafael Frongillo, Department of Computer Science Contents 0 Introduction and Preliminaries 1 1 Homological Algebra in R-Mod 2 1.1 (Co)chain Complexes . .2 1.2 Injective and Projective Modules . .5 1.3 Resolutions . .7 1.4 Derived Functors . 10 2 Sheaves 12 2.1 Complex Manifolds . 13 2.2 Basic Definitions and Examples . 14 2.3 Homological Algebra in Sh(X).............................. 17 2.4 Cohomology of Sheaves . 19 3 Applications 20 3.1 Relation to Other Cohomology Theories . 20 3.2 The Exponential Sequence and the Complex Logarithm . 22 3.3 The Riemann-Roch Theorem for Line Bundles . 23 0 Introduction and Preliminaries Given a topological space X and an abelian group A, the singular cohomology groups Hn(X; A) provide important algebraic invariants of X which form obstructions to X being contractible. Sheaf cohomology may be thought of as a generalization of singular cohomology (and other cohomology theories) in the sense that it accommodates more general \coefficients" than simply an abelian group A. The abelian group A is replaced by a sheaf of abelian groups F which in turn gives rise to the sheaf cohomology groups Hn(X; F). Broadly speaking, these groups constitute obstructions to solving geometric problems globally, given that they can be solved locally. In this paper, we intend to give an introduction to sheaf cohomology via the derived functors approach of Grothendieck. This paper is expository in nature, and the interested reader can find more information about all of the topics we cover in the list of references we provide at the end. Section 1 covers all of the homological algebra needed to define sheaf cohomology. Assuming familiarity with R-modules, we begin with the basics of cochain complexes, exactness, and coho- mology. This brings us to a discussion of exact functors and injective resolutions which then lets us define derived functors. Section 2 is devoted to developing the theory of sheaves on a topological space. We begin with an informal review of complex manifolds and line bundles. This leads to some examples of sheaves later on, and gives an impression of the types of geometric objects which can be studied using sheaves and cohomology. We then provide basic definitions and examples of sheaves and address how the tools from homological algebra which we developed in section 1 can be transferred to the category of sheaves on a topological space. Finally, we define the sheaf cohomology functors as the derived functors of the global sections functor from the category of sheaves on a topological space to the category of abelian groups. In section 3, we attempt to give the reader a sense for how sheaf cohomology is used in algebraic geometry and more broadly. First, we examine how sheaf cohomology fits within the larger picture 1 of cohomology theories, and in particular, its relation to singular cohomology and de Rham coho- mology. We then introduce the exponential sequence and explore its cohomological consequences to find some interesting properties of the complex logarithm from complex analysis. Finally, we discuss Chern classes and the Euler-Poincar´echaracteristic of a sheaf which lets state the Riemann-Roch theorem for line bundles on curves. Before we begin, we will review a couple preliminaries to set-up the category of R-modules. For our purposes, R will be a commutative ring so we will not distinguish between left and right R-modules. Definition 0.1. Let R be a commutative ring with 1. An R-module is an abelian group M together with an action of R on M (that is, a map R × M ! M) denoted rm for all r 2 R and m 2 M, such that • r(m + n) = rm + rn for all r 2 R and m; n 2 M. • (r + s)m = rm + sm for all r; s 2 R and m 2 M, • r(sm) = (rs)m for all r; s 2 R and m 2 M, and • 1m = m for all m 2 M. Given two R-modules M and N, we say a function φ : M ! N is an R-module homomorphism if it is a homomorphism of the underlying abelian groups which additionally satisfies φ(rm) = rφ(m) for all r 2 R and m 2 M. Note that if R = Z, the ring of integers, then the definition of R-module collapses to the definition of abelian group. Thus Z-modules are abelian groups. There is a category whose objects are R-modules and whose morphisms are R-module homo- morphisms. This is the category of R-modules, denoted R-Mod. For convenience, we will often call a homomorphism of R-modules an R-module map, or just a map when the context is understood. 1 Homological Algebra in R-Mod The purpose of this section is to establish the algebraic preliminaries needed to develop sheaf cohomology. Homological algebra unfolds in the abstract setting of abelian categories. Loosely speaking, these are categories equipped with sensible notions of product, coproduct, zero object, kernel, cokernel, and image, in which the sets of morphisms are enriched with an abelian group structure. General abelian categories can look very foreign at first glance, making results more difficult to state and prove. For this reason, we will restrict attention to the category R-Mod, where we assume our reader will have more familiarity. This is the prototypical example of an abelian category, and a good place to develop intuition and understanding for homological algebra. We emphasize, however, that all of the key results we prove extend to the more general setting of abelian categories. Indeed, this will be an important point when we move to the category of sheaves on a topological space in the next section. For the remainder of this section, we will fix a commutative ring R with 1. 1.1 (Co)chain Complexes In many areas of mathematics, objects often appear not by themselves, but in sequences with other objects of the same type. Given a collection of R-modules M1;M2;:::;Mn and R-module maps 2 fi : Mi ! Mi+1 for 1 ≤ i < n, we can form the sequence f1 f2 f3 fn−1 M1 M2 M3 ··· Mn: One curiosity about such a sequence is how the image of the incoming map fi−1 interacts with the kernel of the outgoing map fi in the module Mi. We say the sequence is exact at Mi if these submodules are equal, that is, if ker fi = im fi−1. The sequence is called exact if it is exact at Mi for 1 < i < n. Exactness also tells us something about the composite maps fi ◦ fi−1. In particular, exactness at Mi implies fi ◦ fi−1 = 0. Our next definition singles out precisely this case. Definition 1.1. A chain complex C• of R-modules is a family fCngn2Z of R-modules together with R-module maps fdn : Cn ! Cn−1gn2Z called differentials satisfying the property that dn ◦dn+1 = 0 for all n 2 Z. Because dn ◦ dn+1 = 0 we have 0 ⊆ im dn+1 ⊆ ker dn ⊆ Cn th and we define the n homology module of Cn to be the quotient Hn(C•) := ker dn= im dn+1. It is common to depict a chain complex C• by the diagram d d d d ··· Cn+1 Cn Cn−1 ··· and omit indices on the differentials for convenience. A chain map f : C• ! D• is family of R-module maps ffn : Cn ! Dngn2Z which commute with the differentials, that is, so that following diagram commutes d d d d ··· Cn+1 Cn Cn−1 ··· fn+1 fn fn−1 d d d d ··· Dn+1 Dn Dn−1 ··· : Note that if C• is exact at Cn, then Hn(C•) = 0. In this sense, we can view the homology modules Hn(C•) as obstructions to C• being exact. Examples 1.1. • The sequence consisting of all zero modules and zero differentials ··· 0 0 0 ··· is a chain complex which we will call the zero complex and often simply denote by 0. • More generally, any arbitrary family of R-modules fCng with zero differentials is a chain complex. • The sequence ·2 ·2 ·2 ·2 ··· Z=4 Z=4 Z=4 ··· with all differentials as `times-two' maps x 7! 2x is a chain complex which is exact. Anyone familiar with some category theory will be well-acquainted with the `co-' prefix, indi- n n n n+1 cating arrow reversals. This gives a variant in notation: C = C−n and d = d−n : C ! C . 3 • n Definition 1.2. A cochain complex C of R-modules is a family fC gn2Z of R-modules together n n n+1 n n−1 th with R-module maps fd : C ! C gn2Z such that d ◦ d = 0 for all n 2 Z. The n cohomology module of Cn is the quotient Hn(C•) := ker dn= im dn−1. Maps of cochain complexes are defined in the same way as chain maps and are called cochain maps. Elements of Hn(C•) are called cohomology classes and for a representative c 2 ker dn, we use square brackets to denote its cohomology class [c] 2 Hn(C•). Note on conventions. When we develop sheaf cohomology later in this paper, we will be chiefly concerned with cochain complexes and, as the reader may have guessed, cohomology.
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