3.3 Local properties of sheaves

Sheaves allow us to study local properties of mathematical structures on topological spaces. By restricting sheaves to open subsets and examining their stalks, we can analyze how they behave at specific points and in small neighborhoods.

Local properties of sheaves are crucial for understanding global behavior. Concepts like sheaf restriction, stalks, and sheafification provide powerful tools for connecting local and global perspectives in algebraic topology and geometry.

Sheaf restriction to open subsets

• Restricting a sheaf to an open subset allows for the study of local properties and behavior of the sheaf
• The restriction process involves considering the presheaf of sections over open subsets of the space
• Gluing axioms ensure that the restricted sheaves maintain the key properties of the original sheaf

Presheaf of sections over open subsets

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• For each open subset $U$ of the topological space $X$, the set of sections of the sheaf $F$ over $U$ forms a presheaf
• The restriction maps between sections over open subsets are induced by the restriction maps of the original sheaf $F$
• The presheaf of sections captures the local behavior of the sheaf over the open subsets of the space

Gluing axiom for restricted sheaves

• The gluing axiom ensures that the restricted sheaves maintain the essential properties of the original sheaf
• Given a family of sections $s_i$ over an open cover $\{U_i\}$ of an open subset $U$, if the sections agree on the overlaps $U_i \cap U_j$, then they can be glued together to form a unique section over $U$
• The gluing axiom guarantees the existence and uniqueness of the glued section, preserving the sheaf structure

Uniqueness of glued sections

• The gluing axiom implies the uniqueness of the glued section over an open subset
• If two sections over an open subset $U$ agree on an open cover of $U$, then they must be equal on the entire subset $U$
• This uniqueness property is crucial for the well-definedness of the restriction process and the consistency of the sheaf structure

Stalks of a sheaf

• Stalks provide a way to study the local behavior of a sheaf at a specific point in the topological space
• The stalk of a sheaf at a point captures the germs of sections near that point
• Stalks play a crucial role in understanding the local properties of a sheaf and its relation to the underlying space

Germs of sections at a point

• A germ of a section at a point $x$ is an equivalence class of sections defined on open neighborhoods of $x$
• Two sections are equivalent if they agree on some open neighborhood of $x$
• The set of all germs of sections at a point $x$ forms the stalk of the sheaf at $x$, denoted as $F_x$

Stalks as colimits of sections

• The stalk of a sheaf at a point $x$ can be constructed as the colimit of the sections over open neighborhoods of $x$
• The colimit construction formalizes the idea of germs as equivalence classes of sections
• The colimit property ensures that the stalks capture the local behavior of the sheaf near each point

Morphisms between stalks

• Morphisms between stalks of sheaves at a point $x$ are induced by morphisms between the sheaves
• A morphism of sheaves $\varphi: F \to G$ induces a morphism between the stalks $\varphi_x: F_x \to G_x$ at each point $x$
• The induced morphisms between stalks preserve the local structure and compatibility of the sheaf morphisms

Stalks vs fibers of a sheaf

• The stalk of a sheaf at a point $x$ is related to, but distinct from, the fiber of the sheaf over $x$
• The fiber of a sheaf $F$ over a point $x$ is the set $F(x)$, which is the value of the sheaf at the point $x$
• While the fiber captures the value of the sheaf at a specific point, the stalk captures the local behavior of the sheaf near that point

Sheafification of a presheaf

• Sheafification is the process of converting a presheaf into a sheaf by enforcing the gluing and uniqueness axioms
• The sheafification construction ensures that the resulting sheaf satisfies the sheaf axioms while preserving the essential structure of the presheaf
• Sheafification is an important tool for studying the relationship between presheaves and sheaves

Sheafification construction

• The sheafification of a presheaf $F$ is denoted as $F^+$ or $\mathcal{F}$
• The sheafification construction involves adding new sections to the presheaf to satisfy the gluing and uniqueness axioms
• The sections of the sheafified presheaf $F^+$ over an open subset $U$ are obtained by considering compatible families of sections over open covers of $U$

Universal property of sheafification

• The sheafification of a presheaf satisfies a universal property
• For any sheaf $G$ and a morphism of presheaves $\varphi: F \to G$, there exists a unique morphism of sheaves $\varphi^+: F^+ \to G$ such that $\varphi = \varphi^+ \circ i$, where $i: F \to F^+$ is the natural map from the presheaf to its sheafification
• The universal property characterizes the sheafification as the "best" way to convert a presheaf into a sheaf

Stalks of sheafified presheaves

• The stalks of a sheafified presheaf $F^+$ at a point $x$ are isomorphic to the stalks of the original presheaf $F$ at $x$
• This property ensures that the sheafification process preserves the local behavior of the presheaf
• The isomorphism between the stalks of $F^+$ and $F$ is a consequence of the universal property of sheafification

Sheafification vs associated sheaf

• The sheafification of a presheaf is also known as the associated sheaf
• The associated sheaf of a presheaf $F$ is the smallest sheaf that contains $F$ as a subpresheaf
• The sheafification and the associated sheaf construction are equivalent ways of converting a presheaf into a sheaf

Local properties and stalks

• Local properties of sheaves can be studied using stalks
• Stalks provide a way to analyze the behavior of a sheaf near a specific point in the topological space
• Many global properties of sheaves can be determined by examining the local properties at the level of stalks

Local isomorphisms of sheaves

• Two sheaves $F$ and $G$ are said to be locally isomorphic if their stalks $F_x$ and $G_x$ are isomorphic for each point $x$ in the topological space
• Local isomorphisms capture the idea that the sheaves have the same local structure, even if they may differ globally
• Local isomorphisms are important for understanding the local behavior and classification of sheaves (examples: locally constant sheaves, locally free sheaves)

Determining global properties from stalks

• Many global properties of sheaves can be determined by examining the properties of the stalks
• For example, a sheaf $F$ is flasque (flabby) if and only if each stalk $F_x$ is an injective module over the base ring
• Similarly, a sheaf $F$ is soft if and only if each stalk $F_x$ is a flat module over the base ring
• These characterizations allow for the study of global properties of sheaves using local information from the stalks

Locally constant sheaves

• A sheaf $F$ is locally constant if for each point $x$ in the topological space, there exists an open neighborhood $U$ of $x$ such that the restriction of $F$ to $U$ is a constant sheaf
• Locally constant sheaves have the property that their stalks $F_x$ are isomorphic to a fixed abelian group or module for all points $x$
• Examples of locally constant sheaves include the constant sheaf and the sheaf of locally constant functions on a topological space

Locally free sheaves of modules

• A sheaf of modules $F$ over a ringed space $(X, \mathcal{O}_X)$ is locally free of rank $n$ if for each point $x \in X$, there exists an open neighborhood $U$ of $x$ such that the restriction of $F$ to $U$ is isomorphic to the free $\mathcal{O}_U$-module of rank $n$
• Locally free sheaves of modules generalize the notion of vector bundles in algebraic geometry
• The stalks of a locally free sheaf of modules are free modules over the stalks of the structure sheaf $\mathcal{O}_X$

Sheaf operations and locality

• Many sheaf operations, such as kernel, cokernel, image, and tensor product, have a local nature
• The local behavior of these operations can be studied using stalks
• Understanding the locality of sheaf operations is crucial for working with sheaves and their associated cohomology theories

Kernel sheaves and locality

• The kernel of a morphism of sheaves $\varphi: F \to G$ is a sheaf denoted by $\ker(\varphi)$
• The stalk of the kernel sheaf at a point $x$ is isomorphic to the kernel of the induced map on the stalks $\varphi_x: F_x \to G_x$
• This local characterization allows for the study of the kernel sheaf using the kernels of the induced maps on the stalks

Cokernel sheaves and locality

• The cokernel of a morphism of sheaves $\varphi: F \to G$ is a sheaf denoted by $\coker(\varphi)$
• The stalk of the cokernel sheaf at a point $x$ is isomorphic to the cokernel of the induced map on the stalks $\varphi_x: F_x \to G_x$
• The local behavior of the cokernel sheaf can be understood by examining the cokernels of the induced maps on the stalks

Image sheaves and locality

• The image of a morphism of sheaves $\varphi: F \to G$ is a sheaf denoted by $\im(\varphi)$
• The stalk of the image sheaf at a point $x$ is isomorphic to the image of the induced map on the stalks $\varphi_x: F_x \to G_x$
• The local structure of the image sheaf is determined by the images of the induced maps on the stalks

Tensor product sheaves and locality

• The tensor product of two sheaves of modules $F$ and $G$ over a ringed space $(X, \mathcal{O}_X)$ is a sheaf denoted by $F \otimes_{\mathcal{O}_X} G$
• The stalk of the tensor product sheaf at a point $x$ is isomorphic to the tensor product of the stalks $F_x \otimes_{\mathcal{O}_{X,x}} G_x$, where $\mathcal{O}_{X,x}$ is the stalk of the structure sheaf at $x$
• The local behavior of the tensor product sheaf can be studied using the tensor products of the stalks of the constituent sheaves