1.2 Sheaves

Sheaves are mathematical structures that assign data to open sets of a topological space, allowing for the study of local-to-global properties. They provide a framework for gluing local information together, making them essential in various areas of mathematics.

This chapter covers the definition of sheaves, their properties, and operations. It explores morphisms between sheaves, cohomology theories, and applications in algebraic geometry. Understanding sheaves is crucial for grasping modern developments in topology and geometry.

Definition of sheaves

• Sheaves are mathematical objects that assign data to open sets of a topological space in a way that is compatible with restrictions
• Sheaves provide a framework for studying local-to-global properties and for gluing local data together to obtain global information

Presheaves vs sheaves

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• Presheaves assign data to open sets but may not satisfy the gluing conditions required for sheaves
• Sheaves satisfy two additional conditions: identity axiom (sections over an open set are uniquely determined by their values on a cover) and gluing axiom (compatible sections on a cover can be glued together to obtain a section over the union)
• Every sheaf is a presheaf, but not every presheaf is a sheaf

Sheaf conditions

• Identity axiom: if two sections of a sheaf agree on an open cover of an open set, then they are equal on the entire open set
• Gluing axiom: if there are compatible sections on an open cover of an open set, then there exists a unique section on the entire open set that restricts to the given sections on the cover
• These conditions ensure that local data can be glued together consistently to obtain global information

Sheaves on topological spaces

• Sheaves can be defined on any topological space, where the open sets form the basis for the sheaf structure
• The data assigned to each open set is typically a set, group, ring, or module, depending on the type of sheaf
• Restriction maps between open sets are required to be compatible with the algebraic structure of the data

Sheaves of abelian groups

• A sheaf of abelian groups assigns an abelian group to each open set, with restriction maps being group homomorphisms
• The sheaf conditions ensure that the group structure is preserved when gluing sections together
• Examples include the constant sheaf (assigns the same abelian group to every open set) and the sheaf of continuous functions (assigns the group of continuous functions on each open set)

Sheaves of rings

• A sheaf of rings assigns a ring to each open set, with restriction maps being ring homomorphisms
• The sheaf conditions ensure that the ring structure is preserved when gluing sections together
• Examples include the sheaf of continuous real-valued functions and the sheaf of holomorphic functions on a complex manifold

Sheaves of modules

• A sheaf of modules assigns a module over a fixed ring to each open set, with restriction maps being module homomorphisms
• The sheaf conditions ensure that the module structure is preserved when gluing sections together
• Examples include the sheaf of sections of a vector bundle and the sheaf of differential forms on a smooth manifold

Morphisms of sheaves

• Morphisms of sheaves are the natural notion of maps between sheaves that preserve the sheaf structure
• Morphisms allow for the comparison and relation of different sheaves on the same topological space

Definition of morphisms

• A morphism of sheaves $f: \mathcal{F} \to \mathcal{G}$ consists of a collection of maps $f(U): \mathcal{F}(U) \to \mathcal{G}(U)$ for each open set $U$, commuting with the restriction maps of $\mathcal{F}$ and $\mathcal{G}$
• The commutativity condition ensures that the morphism respects the local nature of the sheaves
• Morphisms can be defined for sheaves of sets, groups, rings, or modules, with the maps $f(U)$ being the appropriate type of homomorphism

Composition of morphisms

• Morphisms of sheaves can be composed, with the composition defined open set-wise: $(g \circ f)(U) = g(U) \circ f(U)$
• The composition of two morphisms is again a morphism, making the collection of sheaves on a topological space into a category

Isomorphisms of sheaves

• An isomorphism of sheaves is a morphism $f: \mathcal{F} \to \mathcal{G}$ that has an inverse morphism $g: \mathcal{G} \to \mathcal{F}$, such that $g \circ f = id_{\mathcal{F}}$ and $f \circ g = id_{\mathcal{G}}$
• Isomorphic sheaves can be thought of as essentially the same, with the isomorphism providing a way to transfer information between them

Kernel and cokernel of morphisms

• The kernel of a morphism $f: \mathcal{F} \to \mathcal{G}$ is the sheaf $\ker(f)$ defined by $\ker(f)(U) = \ker(f(U))$ for each open set $U$
• The cokernel of a morphism $f: \mathcal{F} \to \mathcal{G}$ is the sheaf $\operatorname{coker}(f)$ defined by $\operatorname{coker}(f)(U) = \operatorname{coker}(f(U))$ for each open set $U$
• Kernels and cokernels allow for the study of the exactness properties of morphisms and sequences of sheaves

Exact sequences of sheaves

• A sequence of sheaves and morphisms $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ is called exact if the kernel of each morphism is equal to the image of the previous one
• Exact sequences provide a way to relate different sheaves and study their interdependencies
• The sheaf-theoretic analogue of the snake lemma allows for the construction of long exact sequences from short ones

Operations on sheaves

• Various operations can be performed on sheaves, allowing for the construction of new sheaves from existing ones
• These operations often have analogues in other algebraic contexts (modules, vector spaces) and provide a way to study the relationships between different sheaves

Direct sum and product of sheaves

• The direct sum of two sheaves $\mathcal{F} \oplus \mathcal{G}$ is defined by $(\mathcal{F} \oplus \mathcal{G})(U) = \mathcal{F}(U) \oplus \mathcal{G}(U)$ for each open set $U$
• The direct product of two sheaves $\mathcal{F} \times \mathcal{G}$ is defined by $(\mathcal{F} \times \mathcal{G})(U) = \mathcal{F}(U) \times \mathcal{G}(U)$ for each open set $U$
• These constructions allow for the study of sheaves componentwise and the expression of certain sheaves as a combination of simpler ones

Tensor product of sheaves

• The tensor product of two sheaves $\mathcal{F} \otimes \mathcal{G}$ is defined by $(\mathcal{F} \otimes \mathcal{G})(U) = \mathcal{F}(U) \otimes \mathcal{G}(U)$ for each open set $U$
• The tensor product is a way to combine two sheaves into a new one, often encoding a form of multiplication or interaction between the original sheaves
• The tensor product is a key operation in the study of sheaves of modules and their associated constructions (sheaf cohomology, sheaf Ext)

Inverse image sheaf

• Given a continuous map $f: X \to Y$ and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^*\mathcal{F}$ on $X$ is defined by $f^*\mathcal{F}(U) = \mathcal{F}(f(U))$ for each open set $U$ in $X$
• The inverse image sheaf allows for the pullback of sheaves along continuous maps, providing a way to relate sheaves on different spaces
• The inverse image sheaf is a contravariant functor from the category of sheaves on $Y$ to the category of sheaves on $X$

Direct image sheaf

• Given a continuous map $f: X \to Y$ and a sheaf $\mathcal{F}$ on $X$, the direct image sheaf $f_*\mathcal{F}$ on $Y$ is defined by $f_*\mathcal{F}(V) = \mathcal{F}(f^{-1}(V))$ for each open set $V$ in $Y$
• The direct image sheaf allows for the pushforward of sheaves along continuous maps, providing a way to relate sheaves on different spaces
• The direct image sheaf is a covariant functor from the category of sheaves on $X$ to the category of sheaves on $Y$
• The direct image sheaf is left adjoint to the inverse image sheaf, a key property in the study of sheaf cohomology and sheaf theoretic constructions in algebraic geometry

Sheaf Hom and sheaf Ext

• Given two sheaves $\mathcal{F}$ and $\mathcal{G}$ on a topological space $X$, the sheaf Hom $\mathcal{H}om(\mathcal{F}, \mathcal{G})$ is defined by $\mathcal{H}om(\mathcal{F}, \mathcal{G})(U) = Hom(\mathcal{F}|_U, \mathcal{G}|_U)$ for each open set $U$
• The sheaf Hom encodes the local homomorphisms between two sheaves, providing a way to study the relationships between sheaves
• The sheaf Ext $\mathcal{E}xt^i(\mathcal{F}, \mathcal{G})$ is a derived version of sheaf Hom, defined using the derived functors of the Hom functor
• Sheaf Ext measures the higher obstruction to the existence of sheaf homomorphisms and plays a key role in the study of sheaf cohomology and the classification of sheaves

Sheaf cohomology

• Sheaf cohomology is a powerful tool for studying the global properties of sheaves and the underlying topological space
• It provides a way to measure the obstruction to the existence of global sections and the deviation from the local-to-global principle

Čech cohomology of sheaves

• Čech cohomology is a concrete approach to sheaf cohomology, based on the construction of a cochain complex from an open cover of the topological space
• Given a sheaf $\mathcal{F}$ and an open cover $\mathcal{U} = \{U_i\}$ of $X$, the Čech cochain complex $C^\bullet(\mathcal{U}, \mathcal{F})$ is defined using the sections of $\mathcal{F}$ on finite intersections of open sets in the cover
• The Čech cohomology groups $\check{H}^i(\mathcal{U}, \mathcal{F})$ are the cohomology groups of this cochain complex
• Čech cohomology is a useful computational tool and can be shown to be isomorphic to the derived functor cohomology under certain conditions (e.g., for fine sheaves on paracompact Hausdorff spaces)

Derived functors and sheaf cohomology

• Sheaf cohomology can be defined using the derived functors of the global sections functor $\Gamma(X, -)$
• The right derived functors $R^i\Gamma(X, \mathcal{F})$ measure the higher obstructions to the existence of global sections of the sheaf $\mathcal{F}$
• The sheaf cohomology groups $H^i(X, \mathcal{F})$ are defined as $H^i(X, \mathcal{F}) = R^i\Gamma(X, \mathcal{F})$
• Derived functor cohomology provides a more intrinsic definition of sheaf cohomology and allows for the development of a rich theory, including the study of long exact sequences and the Leray spectral sequence

Cohomology with supports

• Cohomology with supports is a variant of sheaf cohomology that takes into account a closed subset of the topological space
• Given a sheaf $\mathcal{F}$ and a closed subset $Z \subset X$, the cohomology groups with support in $Z$ are defined as $H^i_Z(X, \mathcal{F}) = R^i\Gamma_Z(X, \mathcal{F})$, where $\Gamma_Z(X, \mathcal{F})$ denotes the sections of $\mathcal{F}$ with support in $Z$
• Cohomology with supports allows for the study of local cohomological phenomena and the definition of local cohomology
• The excision theorem for cohomology with supports is a key tool in the study of sheaf cohomology and its applications

Cohomological dimension

• The cohomological dimension of a topological space $X$ with respect to a sheaf $\mathcal{F}$ is the smallest integer $n$ such that $H^i(X, \mathcal{F}) = 0$ for all $i > n$ and all sheaves $\mathcal{F}$
• The cohomological dimension measures the complexity of the space from the perspective of sheaf cohomology
• Spaces with finite cohomological dimension (e.g., compact manifolds, algebraic varieties) are particularly amenable to study using sheaf-theoretic methods

Applications of sheaf cohomology

• Sheaf cohomology has numerous applications in various fields of mathematics, including:
  • Algebraic topology: classification of vector bundles, characteristic classes, obstruction theory
  • Complex analysis: Dolbeault cohomology, Hodge theory, Serre duality
  • Algebraic geometry: coherent sheaf cohomology, Serre duality, Riemann-Roch theorem, Hodge theory
• Sheaf cohomology provides a unifying language for the study of various cohomology theories and allows for the transfer of techniques between different fields

Sheaves in algebraic geometry

• Sheaves play a central role in modern algebraic geometry, providing a framework for the study of algebraic varieties and their properties
• The use of sheaves allows for the development of a rich theory that combines algebraic, geometric, and topological ideas

Sheaves on schemes

• In algebraic geometry, sheaves are defined on schemes, which are a generalization of algebraic varieties that allow for the inclusion of nilpotent elements and non-reduced structures
• Sheaves on schemes can be defined using the Zariski topology or the étale topology, with the latter providing a more flexible and powerful framework
• The structure sheaf $\mathcal{O}_X$ of a scheme $X$ is a fundamental object that encodes the algebraic structure of the scheme

Quasi-coherent sheaves

• A sheaf $\mathcal{F}$ on a scheme $X$ is called quasi-coherent if it is locally the cokernel of a morphism between free $\mathcal{O}_X$-modules
• Quasi-coherent sheaves form an abelian category $QCoh(X)$ that is closed under various operations (tensor products, Hom, direct limits)
• The category of quasi-coherent sheaves is a natural generalization of the category of modules over a ring and shares many of its properties

Coherent sheaves

• A sheaf $\mathcal{F}$ on a scheme $X$ is called coherent if it is quasi-coherent and locally finitely presented (i.e., locally the cokernel of a morphism between finite rank free $\mathcal{O}_X$-modules)
• Coherent sheaves form an abelian category $Coh(X)$ that is a full subcategory of $QCoh(X)$
• Coherent sheaves are particularly well-behaved and have good finiteness properties, making them a central object of study in algebraic geometry

Locally free sheaves and vector bundles

• A sheaf $\mathcal{F}$ on a scheme $X$ is called locally free if it is locally isomorphic to a direct sum of copies of the structure sheaf $\mathcal{O}_X$
• Locally free sheaves of rank $n$ correspond to algebraic vector bundles of rank $n$ on the scheme $X$
• The study of vector bundles and their properties (e.g., Chern classes, Chow rings) is a key aspect of algebraic geometry and relates to the study of algebraic cycles and intersection theory

Sheaves of differentials

• The sheaf of differentials $\Omega^1_X$ on a scheme $X$ is a coherent sheaf that encodes the algebraic differential forms on $X$
• Higher exterior powers of the sheaf of differentials $\Omega^p_X$ provide a theory of algebraic differential forms and the de Rham complex
• The study of the sheaf of differentials and its cohomology relates to questions in algebraic geometry, such as the smoothness and regularity of schemes, and the theory of algebraic