2.4 Sections of a sheaf

Sections of a sheaf are the backbone of sheaf theory, allowing us to study local and global properties of mathematical structures. They represent consistent choices of elements from the stalks over open sets, capturing the essence of how local data fits together.

Understanding sections is crucial for grasping sheaves' behavior and applications. From defining local and global sections to exploring their properties and operations, sections provide a powerful tool for analyzing geometric and algebraic objects across various mathematical fields.

Definition of sections

• Sections are a fundamental concept in sheaf theory that describe how the local data of a sheaf can be consistently chosen over open sets
• Sections capture the idea of a "continuous selection" of elements from the stalks of a sheaf, allowing us to work with the sheaf globally

Sections as morphisms

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• Sections can be viewed as morphisms from an open set $U$ to the sheaf $\mathcal{F}$
• More precisely, a section over $U$ is a morphism $s: U \to \mathcal{F}$ such that $\pi \circ s = id_U$, where $\pi: \mathcal{F} \to X$ is the projection map
• This morphism perspective allows us to study sections using tools from category theory

Local sections

• Local sections are sections defined over a specific open set $U \subset X$
• For each open set $U$, the set of local sections over $U$ is denoted by $\mathcal{F}(U)$
• Local sections capture the local behavior of the sheaf and are the building blocks for global sections

Global sections

• Global sections are sections defined over the entire space $X$
• The set of global sections is denoted by $\mathcal{F}(X)$ or $\Gamma(X, \mathcal{F})$
• Global sections provide a way to study the sheaf as a whole and can be used to extract invariants and properties of the sheaf

Properties of sections

• Sections have several important properties that reflect the local-to-global nature of sheaves
• These properties allow us to manipulate and study sections effectively

Restrictions of sections

• Given a section $s \in \mathcal{F}(U)$ and an open subset $V \subset U$, we can restrict the section $s$ to $V$ to obtain a section $s|_V \in \mathcal{F}(V)$
• Restriction is a natural operation that allows us to study the behavior of sections on smaller open sets
• The restriction maps $\rho_{U, V}: \mathcal{F}(U) \to \mathcal{F}(V)$ are a key part of the sheaf axioms

Extensions of sections

• Given a section $s \in \mathcal{F}(U)$ and an open set $V$ containing $U$, an extension of $s$ to $V$ is a section $t \in \mathcal{F}(V)$ such that $t|_U = s$
• Extensions allow us to study how local sections can be "glued" together to form sections over larger open sets
• The existence and uniqueness of extensions are closely related to the sheaf axioms

Uniqueness of extensions

• If a sheaf $\mathcal{F}$ satisfies the sheaf axioms, then extensions of sections are unique when they exist
• More precisely, if $s \in \mathcal{F}(U)$ and $t_1, t_2 \in \mathcal{F}(V)$ are two extensions of $s$ to an open set $V$ containing $U$, then $t_1 = t_2$
• This uniqueness property is crucial for the consistency and well-definedness of sheaves

Relationship between sections and sheaves

• Sections and sheaves are intimately connected, with sections providing a way to study sheaves and sheaves imposing conditions on sections
• The interplay between sections and sheaves is at the heart of sheaf theory

Sheaf conditions for sections

• The sheaf axioms impose conditions on the behavior of sections
• The local identity axiom states that for any open set $U$ and any covering $\{U_i\}$ of $U$, if $s, t \in \mathcal{F}(U)$ satisfy $s|_{U_i} = t|_{U_i}$ for all $i$, then $s = t$
• The gluing axiom states that for any open set $U$ and any covering $\{U_i\}$ of $U$, if we have sections $s_i \in \mathcal{F}(U_i)$ such that $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$ for all $i, j$, then there exists a unique section $s \in \mathcal{F}(U)$ such that $s|_{U_i} = s_i$ for all $i$

Gluing sections

• The gluing axiom allows us to construct global sections from compatible local sections
• If we have a collection of local sections $s_i \in \mathcal{F}(U_i)$ that agree on the overlaps $U_i \cap U_j$, we can "glue" them together to obtain a global section $s \in \mathcal{F}(U)$
• Gluing is a powerful technique for constructing sections and studying the global behavior of sheaves

Sheafification using sections

• Given a presheaf $\mathcal{F}$ (a functor that may not satisfy the sheaf axioms), we can construct a sheaf $\mathcal{F}^+$ called the sheafification of $\mathcal{F}$
• The sheafification process involves adding new sections to $\mathcal{F}$ to ensure that the sheaf axioms are satisfied
• The sections of the sheafification $\mathcal{F}^+$ are obtained by gluing together compatible local sections of $\mathcal{F}$

Operations on sections

• Sections can be manipulated and combined using various operations
• These operations often reflect the underlying algebraic or geometric structure of the sheaf

Addition of sections

• If $\mathcal{F}$ is a sheaf of abelian groups (or more generally, a sheaf of modules over a ring), we can add sections pointwise
• Given sections $s, t \in \mathcal{F}(U)$, their sum $s + t \in \mathcal{F}(U)$ is defined by $(s + t)(x) = s(x) + t(x)$ for all $x \in U$
• Addition of sections is compatible with restriction, making $\mathcal{F}(U)$ an abelian group (or module) for each open set $U$

Multiplication of sections by functions

• If $\mathcal{F}$ is a sheaf of modules over a ring $R$, we can multiply sections by functions (elements of $R$)
• Given a section $s \in \mathcal{F}(U)$ and a function $f \in R(U)$, their product $fs \in \mathcal{F}(U)$ is defined by $(fs)(x) = f(x)s(x)$ for all $x \in U$
• Multiplication by functions is compatible with restriction and addition, making $\mathcal{F}(U)$ an $R(U)$-module for each open set $U$

Algebraic structure of sections

• The operations of addition and multiplication by functions endow the sections of a sheaf with a rich algebraic structure
• For each open set $U$, the sections $\mathcal{F}(U)$ form an algebraic object (e.g., a group, ring, or module) that reflects the nature of the sheaf
• This algebraic structure is compatible with restriction and can be used to study the sheaf using tools from algebra

Sections and presheaves

• Sections are closely related to the notion of presheaves, which are a generalization of sheaves
• Presheaves capture the idea of a "consistent assignment" of data to open sets, without requiring the gluing axiom

Presheaf of sections

• Given a sheaf $\mathcal{F}$, we can construct a presheaf $\Gamma(\mathcal{F})$ called the presheaf of sections of $\mathcal{F}$
• The presheaf of sections assigns to each open set $U$ the set of sections $\mathcal{F}(U)$ and to each inclusion $V \subset U$ the restriction map $\rho_{U, V}: \mathcal{F}(U) \to \mathcal{F}(V)$
• The presheaf of sections captures the local behavior of the sheaf and is a useful tool for studying sheaves

Sheafification of presheaf of sections

• The presheaf of sections $\Gamma(\mathcal{F})$ is not always a sheaf, as it may not satisfy the gluing axiom
• However, we can always construct a sheaf from a presheaf by applying the sheafification process
• The sheafification of the presheaf of sections $\Gamma(\mathcal{F})^+$ is canonically isomorphic to the original sheaf $\mathcal{F}$, providing a way to recover the sheaf from its sections

Sections in specific sheaves

• Sections behave differently in various types of sheaves, reflecting the specific structure and properties of each sheaf
• Studying sections in specific sheaves can provide insights into the underlying geometric or algebraic objects

Sections of constant sheaves

• A constant sheaf $\underline{A}$ on a topological space $X$ assigns the same abelian group (or another algebraic object) $A$ to each open set $U \subset X$
• The sections of a constant sheaf over an open set $U$ are simply the constant functions from $U$ to $A$
• Global sections of a constant sheaf correspond to elements of $A$, making constant sheaves a simple but important example

Sections of locally constant sheaves

• A locally constant sheaf $\mathcal{F}$ on a topological space $X$ assigns an abelian group (or another algebraic object) $\mathcal{F}_x$ to each point $x \in X$, such that for each $x$ there is an open neighborhood $U$ of $x$ where $\mathcal{F}$ is constant
• Sections of a locally constant sheaf are locally constant functions, i.e., functions that are constant on each connected component of their domain
• Locally constant sheaves are important in algebraic topology and the study of covering spaces

Sections of skyscraper sheaves

• A skyscraper sheaf $i_{x*}(A)$ at a point $x \in X$ with value $A$ assigns the abelian group (or another algebraic object) $A$ to open sets containing $x$ and the zero object to open sets not containing $x$
• Sections of a skyscraper sheaf over an open set $U$ are zero unless $U$ contains $x$, in which case they correspond to elements of $A$
• Skyscraper sheaves are useful for studying the local behavior of sheaves and for constructing sheaf cohomology

Applications and examples

• Sections of sheaves find applications in various areas of mathematics, providing a unified language for studying geometric and algebraic objects
• Examples from different fields illustrate the power and versatility of sheaf theory

Sections in geometry and topology

• In differential geometry, sections of the tangent sheaf correspond to vector fields on a manifold
• Sections of the sheaf of differential forms represent differential forms on a manifold, which are crucial for studying integration and cohomology
• In algebraic topology, sections of the locally constant sheaf associated with a covering space correspond to lifts of maps into the base space

Sections in complex analysis

• In complex analysis, sections of the sheaf of holomorphic functions on a complex manifold are holomorphic functions
• The sheaf of holomorphic functions is a fundamental object in complex analysis and geometry, and its sections encode analytic and geometric properties of the manifold
• Sheaf cohomology groups computed using sections of the sheaf of holomorphic functions provide invariants of complex manifolds and vector bundles

Sections in algebraic geometry

• In algebraic geometry, sections of the structure sheaf of an affine scheme correspond to regular functions on the scheme
• Sections of the sheaf of regular functions on a variety encode algebraic and geometric properties of the variety
• Sheaf cohomology groups computed using sections of coherent sheaves (such as the structure sheaf) are important invariants in algebraic geometry and commutative algebra