🍃Sheaf Theory Unit 3 – Stalks and germs

Key Concepts and Definitions

• Sheaf theory studies the global properties of a topological space by analyzing the local properties of its sheaves
• A presheaf $\mathcal{F}$ on a topological space $X$ assigns to each open set $U \subset X$ a set $\mathcal{F}(U)$ and to each inclusion $V \subset U$ a restriction map $\mathcal{F}(U) \to \mathcal{F}(V)$
• A sheaf is a presheaf that satisfies the gluing axiom and the locality axiom
  • The gluing axiom ensures that local sections can be uniquely glued together to form a global section
  • The locality axiom states that a section is determined by its values on any open cover
• The stalk of a sheaf $\mathcal{F}$ at a point $x \in X$, denoted $\mathcal{F}_x$, is the direct limit of the sets $\mathcal{F}(U)$ over all open neighborhoods $U$ of $x$
• A germ of a sheaf $\mathcal{F}$ at a point $x \in X$ is an equivalence class of sections in the stalk $\mathcal{F}_x$
• The étalé space of a sheaf $\mathcal{F}$ is a topological space that encodes the local behavior of the sheaf
• A morphism of sheaves $\varphi: \mathcal{F} \to \mathcal{G}$ is a collection of maps $\varphi(U): \mathcal{F}(U) \to \mathcal{G}(U)$ for each open set $U \subset X$ that commute with the restriction maps

Historical Context and Development

• Sheaf theory originated in the 1940s through the work of Jean Leray and later developed by Henri Cartan, Jean-Pierre Serre, and others
• Initially motivated by problems in topology and complex analysis, such as the study of coherent analytic sheaves on complex manifolds
• The concept of sheaves provided a unified framework for studying local-to-global properties in various mathematical contexts
• Alexander Grothendieck's work in algebraic geometry in the 1950s and 1960s heavily relied on sheaf theory, leading to significant advancements in the field
  • Grothendieck introduced the notion of a scheme, which is a locally ringed space that generalizes both varieties and manifolds
  • He developed the theory of coherent sheaves on schemes, which played a crucial role in his proof of the Riemann-Roch theorem
• Sheaf theory has since found applications in diverse areas of mathematics, including algebraic topology, differential geometry, mathematical physics, and representation theory
• The development of derived categories and the study of perverse sheaves in the 1980s by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne further expanded the scope and depth of sheaf theory

Stalk Construction and Properties

• The stalk of a sheaf $\mathcal{F}$ at a point $x \in X$ is constructed as the direct limit (or colimit) of the sets $\mathcal{F}(U)$ over all open neighborhoods $U$ of $x$
  • The direct limit is a universal object that captures the local behavior of the sheaf near the point $x$
  • Elements of the stalk are called germs, representing the equivalence classes of sections that agree on some open neighborhood of $x$
• The stalk $\mathcal{F}_x$ has a natural ring structure induced by the restriction maps of the sheaf
  • If $\mathcal{F}$ is a sheaf of rings, then $\mathcal{F}_x$ is a local ring with a unique maximal ideal
• Stalks are useful for studying the local properties of a sheaf, such as its rank, support, and vanishing behavior
• The fiber of a sheaf $\mathcal{F}$ at a point $x \in X$ is the set $\mathcal{F}(x) = \mathcal{F}_x / \mathfrak{m}_x \mathcal{F}_x$, where $\mathfrak{m}_x$ is the maximal ideal of the stalk $\mathcal{F}_x$
• A sheaf $\mathcal{F}$ is locally free of rank $n$ if its stalks $\mathcal{F}_x$ are free modules of rank $n$ over the local rings $\mathcal{O}_{X,x}$ for all $x \in X$
• The support of a sheaf $\mathcal{F}$ is the set of points $x \in X$ where the stalk $\mathcal{F}_x$ is non-zero
• A sheaf $\mathcal{F}$ is flasque (or flabby) if for every open set $U \subset X$, the restriction map $\mathcal{F}(X) \to \mathcal{F}(U)$ is surjective

Germ Theory in Sheaf Theory

• Germs are the basic building blocks of sheaves, representing the local behavior of sections near a point
• A germ of a sheaf $\mathcal{F}$ at a point $x \in X$ is an equivalence class of sections in the stalk $\mathcal{F}_x$
  • Two sections are equivalent if they agree on some open neighborhood of $x$
  • The equivalence class of a section $s \in \mathcal{F}(U)$ is denoted by $[s]_x$ or $\operatorname{germ}_x(s)$
• The germs of a sheaf form a sheaf themselves, called the étalé space or the sheaf of germs
  • The étalé space of a sheaf $\mathcal{F}$ is denoted by $\operatorname{Et}(\mathcal{F})$ or $\mathcal{F}^+$
  • The projection map $\pi: \operatorname{Et}(\mathcal{F}) \to X$ sends a germ $[s]_x$ to its base point $x$
• Germ theory allows for the local study of sheaves and the construction of global objects from local data
• Operations on sheaves, such as tensor products, direct sums, and sheaf cohomology, can be defined using germs
• The sheaf of germs of a presheaf satisfies the gluing axiom, providing a way to construct a sheaf from a presheaf (sheafification)

Relationship Between Stalks and Germs

• Stalks and germs are closely related concepts in sheaf theory, both capturing the local behavior of a sheaf near a point
• The stalk $\mathcal{F}_x$ of a sheaf $\mathcal{F}$ at a point $x \in X$ is the set of germs of $\mathcal{F}$ at $x$
  • Elements of the stalk are equivalence classes of sections, i.e., germs
• The germs of a sheaf $\mathcal{F}$ form a sheaf themselves, called the étalé space or the sheaf of germs $\operatorname{Et}(\mathcal{F})$
  • The stalk of the étalé space at a point $x$ is isomorphic to the stalk of the original sheaf: $\operatorname{Et}(\mathcal{F})_x \cong \mathcal{F}_x$
• A morphism of sheaves $\varphi: \mathcal{F} \to \mathcal{G}$ induces a morphism of stalks $\varphi_x: \mathcal{F}_x \to \mathcal{G}_x$ for each $x \in X$
  • The induced morphism on stalks is obtained by applying the direct limit functor to the morphism of sheaves
• The support of a sheaf $\mathcal{F}$ can be characterized using stalks or germs
  • $x \in \operatorname{Supp}(\mathcal{F})$ if and only if the stalk $\mathcal{F}_x$ is non-zero
  • Equivalently, $x \in \operatorname{Supp}(\mathcal{F})$ if and only if there exists a non-zero germ of $\mathcal{F}$ at $x$
• The étalé space provides a useful geometric interpretation of a sheaf, with the germs as the fibers of the projection map $\pi: \operatorname{Et}(\mathcal{F}) \to X$

Applications in Algebraic Geometry

• Sheaf theory is a fundamental tool in algebraic geometry, providing a language to study geometric objects and their local-to-global properties
• Schemes, the central objects of study in modern algebraic geometry, are defined as locally ringed spaces that locally resemble the spectrum of a ring
  • The structure sheaf $\mathcal{O}_X$ of a scheme $X$ encodes the algebraic properties of the scheme
  • Morphisms of schemes are defined as morphisms of locally ringed spaces
• Coherent sheaves on a scheme $X$ generalize the notion of vector bundles and provide a framework for studying modules over the structure sheaf $\mathcal{O}_X$
  • The category of coherent sheaves on $X$ is an abelian category, allowing for the use of homological algebra techniques
• Sheaf cohomology, particularly coherent cohomology, plays a crucial role in the study of schemes and their properties
  • The cohomology groups $H^i(X, \mathcal{F})$ measure the global sections and obstructions to extending local sections of a sheaf $\mathcal{F}$
  • Vanishing theorems for coherent cohomology, such as the Kodaira vanishing theorem, provide powerful tools for understanding the geometry of schemes
• The theory of perverse sheaves, developed in the 1980s, has found important applications in the study of intersection cohomology and the topology of singular algebraic varieties
• Sheaf theory has also been used to study moduli spaces, such as the moduli space of vector bundles on a curve or the moduli space of stable sheaves on a projective variety

Advanced Topics and Extensions

• Derived categories and the derived functor formalism provide a more general framework for studying sheaves and their cohomology
  • The derived category $D(X)$ of a scheme $X$ is obtained by localizing the category of complexes of sheaves on $X$ with respect to quasi-isomorphisms
  • Derived functors, such as the derived tensor product and derived sheaf cohomology, are obtained by applying the localization functor to the corresponding classical functors
• Perverse sheaves are a special class of complexes of sheaves that satisfy certain support and dimensional conditions
  • The category of perverse sheaves on a variety $X$ is an abelian category, providing a rich structure for studying the topology of $X$
  • The intersection cohomology of a singular variety can be defined using perverse sheaves, leading to important applications in representation theory and the study of singularities
• Microlocal sheaf theory, developed by Masaki Kashiwara and Pierre Schapira, extends the ideas of sheaf theory to the cotangent bundle and provides a framework for studying singularities and wave propagation
  • The microlocal support of a sheaf captures the directions in which the sheaf fails to propagate
  • Microlocal sheaf theory has found applications in the study of D-modules, linear partial differential equations, and the Riemann-Hilbert correspondence
• Sheaves on topoi, introduced by Grothendieck, generalize the notion of sheaves on topological spaces to arbitrary categories with a Grothendieck topology
  • A topos is a category that behaves like the category of sheaves on a topological space
  • The theory of sheaves on topoi has connections to logic, set theory, and the foundations of mathematics
• Homotopical algebra and $\infty$-categories provide a framework for studying sheaves and their cohomology in a homotopy-theoretic setting
  • The $\infty$-category of sheaves on a topological space or a site can be defined using the language of simplicial sets or quasi-categories
  • Homotopical methods have led to the development of new invariants and techniques in sheaf theory, such as the study of factorization algebras and the Fukaya category

Problem-Solving Techniques

• When working with sheaves, it is often useful to consider the stalks or germs of the sheaf to study its local behavior
  • To prove a property of a sheaf, it is sometimes sufficient to prove the property for each stalk (e.g., injectivity, surjectivity, isomorphism)
  • The support of a sheaf can be determined by examining where its stalks are non-zero
• The gluing axiom and the locality axiom are crucial for constructing global sections or objects from local data
  • To construct a global section of a sheaf, first construct local sections on an open cover and then use the gluing axiom to patch them together
  • The locality axiom ensures that the global section is unique and well-defined
• Exact sequences and long exact sequences are powerful tools for studying the relationships between sheaves and their cohomology groups
  • The short exact sequence of sheaves $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ induces a long exact sequence in cohomology
  • Splitting lemmas and the snake lemma can be used to extract information from exact sequences
• Spectral sequences, such as the Leray spectral sequence or the Grothendieck spectral sequence, provide a systematic way to compute sheaf cohomology by breaking it down into simpler pieces
  • The $E_2$ page of the Leray spectral sequence for a map $f: X \to Y$ and a sheaf $\mathcal{F}$ on $X$ is given by $E_2^{p,q} = H^p(Y, R^q f_* \mathcal{F})$
  • The Grothendieck spectral sequence relates the cohomology of a composition of functors to the cohomology of the individual functors
• Čech cohomology provides a concrete way to compute sheaf cohomology using open covers and Čech cochains
  • The Čech complex associated to an open cover $\mathcal{U} = \{U_i\}$ of a space $X$ and a sheaf $\mathcal{F}$ is given by $\check{C}^p(\mathcal{U}, \mathcal{F}) = \prod_{i_0 < \cdots < i_p} \mathcal{F}(U_{i_0} \cap \cdots \cap U_{i_p})$
  • The Čech cohomology groups are the cohomology groups of the Čech complex, and they are isomorphic to the sheaf cohomology groups for a sufficiently fine cover
• Sheafification is a process that converts a presheaf into a sheaf by adding the missing local sections required by the gluing and locality axioms
  • The sheafification of a presheaf $\mathcal{F}$ is denoted by $\mathcal{F}^+$ and is characterized by a universal property
  • Sheafification can be used to construct sheaves from simpler objects, such as presheaves or étale spaces