8.4 Coherent sheaves

Coherent sheaves are a key concept in algebraic geometry, bridging algebra and geometry. They provide a way to study geometric objects using algebraic tools, satisfying specific finiteness and compatibility conditions that make them well-behaved under various operations.

These sheaves play a crucial role in understanding local and global properties of schemes and their morphisms. They're a subcategory of quasi-coherent sheaves, possessing better properties for applications like intersection theory and duality in algebraic geometry.

Definition of coherent sheaves

• Coherent sheaves are a fundamental concept in algebraic geometry that provide a way to study geometric objects using algebraic tools
• They are a special type of sheaf that satisfies certain finiteness and compatibility conditions
• Coherent sheaves play a crucial role in understanding the local and global properties of schemes and their morphisms

Coherence conditions

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• A sheaf $\mathcal{F}$ on a scheme $X$ is coherent if it satisfies two conditions:
  1. $\mathcal{F}$ is of finite type, meaning locally it is generated by a finite number of sections
  2. For any open set $U \subset X$ and any finite collection of sections $s_1, \ldots, s_n \in \mathcal{F}(U)$, the kernel of the map $\mathcal{O}_U^n \to \mathcal{F}|_U$ is of finite type
• These conditions ensure that coherent sheaves have good finiteness properties and are well-behaved under various operations
• Examples of coherent sheaves include the structure sheaf $\mathcal{O}_X$ and the sheaf of differentials $\Omega_X$ on a scheme $X$

Relationship to quasi-coherent sheaves

• Coherent sheaves are a subcategory of quasi-coherent sheaves, which are sheaves that locally have a presentation as the cokernel of a map between free modules
• Every coherent sheaf is quasi-coherent, but not every quasi-coherent sheaf is coherent (requires additional finiteness conditions)
• Quasi-coherent sheaves are an important tool in algebraic geometry, but coherent sheaves have even better properties and are more suitable for certain applications (intersection theory, duality, etc.)

Properties of coherent sheaves

• Coherent sheaves possess several important properties that make them a central object of study in algebraic geometry
• These properties are related to the finiteness conditions in the definition of coherence and have significant consequences for the geometry of schemes

Finite type

• A sheaf $\mathcal{F}$ on a scheme $X$ is of finite type if for every point $x \in X$, there exists an open neighborhood $U$ of $x$ such that $\mathcal{F}|_U$ is generated by a finite number of sections
• This means that locally, a coherent sheaf can be described using a finite amount of data
• Being of finite type is a necessary condition for a sheaf to be coherent

Finite presentation

• A sheaf $\mathcal{F}$ is of finite presentation if it is of finite type and for every open set $U \subset X$ and surjective morphism $\mathcal{O}_U^n \to \mathcal{F}|_U$, the kernel is also of finite type
• Finite presentation is a stronger condition than finite type and is equivalent to being coherent on a noetherian scheme
• Sheaves of finite presentation are well-behaved under pullbacks and tensor products

Local freeness

• A coherent sheaf $\mathcal{F}$ is locally free of rank $r$ if for every point $x \in X$, there exists an open neighborhood $U$ of $x$ such that $\mathcal{F}|_U \cong \mathcal{O}_U^r$
• Locally free sheaves of rank $r$ correspond to vector bundles of rank $r$ on the scheme $X$
• Examples of locally free sheaves include the tangent sheaf $\mathcal{T}_X$ and the sheaf of differentials $\Omega_X$ on a smooth scheme $X$

Coherent sheaves on noetherian schemes

• Noetherian schemes are a class of schemes that satisfy the ascending chain condition for closed subschemes, which has important consequences for the behavior of coherent sheaves
• On noetherian schemes, coherent sheaves have particularly nice characterizations and properties

Characterization on affine schemes

• On an affine noetherian scheme $X = \operatorname{Spec} A$, a sheaf $\mathcal{F}$ is coherent if and only if it corresponds to a finitely generated $A$-module $M$ under the correspondence between sheaves and modules
• This characterization allows for a purely algebraic description of coherent sheaves on affine noetherian schemes
• As a consequence, many properties of coherent sheaves can be studied using the properties of finitely generated modules over noetherian rings

Characterization on projective schemes

• On a projective scheme $X$ over a noetherian ring $A$, a sheaf $\mathcal{F}$ is coherent if and only if it is of finite type
• This characterization is a consequence of the fact that the twisting sheaves $\mathcal{O}_X(n)$ are invertible sheaves (locally free of rank 1) on projective schemes
• Coherent sheaves on projective schemes can be studied using graded modules over the homogeneous coordinate ring of $X$

Serre's theorem

• Serre's theorem states that on a noetherian scheme $X$, a sheaf $\mathcal{F}$ is coherent if and only if it is of finite type and for every affine open subset $U = \operatorname{Spec} A \subset X$, the corresponding $A$-module $\mathcal{F}(U)$ is finitely generated
• This theorem provides a global characterization of coherent sheaves on noetherian schemes in terms of local data
• Serre's theorem is a powerful tool for studying coherent sheaves and their cohomology on noetherian schemes

Coherent sheaves and modules

• There is a deep connection between coherent sheaves and modules over rings, which allows for the application of algebraic techniques to the study of coherent sheaves
• This correspondence is particularly useful on affine and projective schemes, where it provides a bridge between geometric and algebraic properties

Correspondence with finitely generated modules

• On an affine scheme $X = \operatorname{Spec} A$, there is a one-to-one correspondence between coherent sheaves on $X$ and finitely generated $A$-modules
• Given a finitely generated $A$-module $M$, one can construct a coherent sheaf $\widetilde{M}$ on $X$ by localizing $M$ at each prime ideal of $A$
• Conversely, given a coherent sheaf $\mathcal{F}$ on $X$, the global sections $\mathcal{F}(X)$ form a finitely generated $A$-module
• This correspondence allows for the study of coherent sheaves using the rich theory of finitely generated modules over rings

Sheaf of modules associated to a module

• Given an $A$-module $M$, one can construct a sheaf of $\mathcal{O}_X$-modules $\widetilde{M}$ on $X = \operatorname{Spec} A$ by setting $\widetilde{M}(U) = M_f$ for each open subset $U = D(f) \subset X$
• The sheaf $\widetilde{M}$ is called the sheaf of modules associated to $M$ and is always quasi-coherent
• If $M$ is finitely generated, then $\widetilde{M}$ is a coherent sheaf
• This construction provides a way to study modules using the language of sheaves and algebraic geometry

Module associated to a coherent sheaf

• Given a coherent sheaf $\mathcal{F}$ on an affine scheme $X = \operatorname{Spec} A$, one can associate to it a finitely generated $A$-module $\Gamma(X, \mathcal{F})$, which is the $A$-module of global sections of $\mathcal{F}$
• The functor $\Gamma(X, -)$ establishes an equivalence between the category of coherent sheaves on $X$ and the category of finitely generated $A$-modules
• This correspondence allows for the application of algebraic techniques, such as homological algebra, to the study of coherent sheaves

Cohomology of coherent sheaves

• Cohomology is a powerful tool for studying the global properties of coherent sheaves and extracting invariants that provide insight into the geometry of schemes
• The cohomology of coherent sheaves is particularly well-behaved on projective schemes and is the subject of several important theorems and conjectures

Čech cohomology

• Čech cohomology is a cohomology theory that can be used to compute the cohomology of coherent sheaves on schemes
• Given a coherent sheaf $\mathcal{F}$ on a scheme $X$ and an open cover $\mathcal{U} = \{U_i\}$ of $X$, the Čech cohomology groups $\check{H}^p(\mathcal{U}, \mathcal{F})$ are defined as the cohomology of the Čech complex associated to $\mathcal{F}$ and $\mathcal{U}$
• Čech cohomology is particularly useful for computing the cohomology of coherent sheaves on projective schemes, where it agrees with the sheaf cohomology
• Čech cohomology can be used to prove vanishing theorems and study the relationship between the cohomology of coherent sheaves and the geometry of the underlying scheme

Serre's finiteness theorem

• Serre's finiteness theorem states that on a projective scheme $X$ over a noetherian ring $A$, the cohomology groups $H^i(X, \mathcal{F})$ of a coherent sheaf $\mathcal{F}$ are finitely generated $A$-modules for all $i \geq 0$
• This theorem is a consequence of the fact that coherent sheaves on projective schemes are of finite type and the higher direct images of coherent sheaves under proper morphisms are coherent
• Serre's finiteness theorem has important applications in the study of the cohomology of coherent sheaves and the geometry of projective schemes

Vanishing theorems

• Vanishing theorems are results that provide conditions under which the higher cohomology groups of coherent sheaves vanish
• One of the most important vanishing theorems is the Kodaira vanishing theorem, which states that on a smooth projective variety $X$ over a field of characteristic 0, the cohomology groups $H^i(X, \mathcal{L}^{-1})$ vanish for $i < \dim X$ and any ample line bundle $\mathcal{L}$
• Other important vanishing theorems include the Kawamata-Viehweg vanishing theorem and the Nakano vanishing theorem
• Vanishing theorems are powerful tools for studying the cohomology of coherent sheaves and have applications in the classification of algebraic varieties and the study of linear series

Operations on coherent sheaves

• Coherent sheaves are closed under several important operations that allow for the construction of new coherent sheaves from existing ones
• These operations are functorial and have good properties with respect to the cohomology of coherent sheaves

Tensor product

• Given two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on a scheme $X$, their tensor product $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ is also a coherent sheaf
• The tensor product of coherent sheaves corresponds to the tensor product of the associated modules under the correspondence between sheaves and modules on affine schemes
• The tensor product is associative, commutative, and distributive with respect to direct sums
• The tensor product of locally free sheaves corresponds to the tensor product of vector bundles

Hom sheaf

• Given two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on a scheme $X$, the sheaf of homomorphisms $\mathcal{H}om_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is also a coherent sheaf
• The Hom sheaf corresponds to the sheaf associated to the module of homomorphisms between the associated modules under the correspondence between sheaves and modules on affine schemes
• The Hom sheaf is contravariant in the first argument and covariant in the second argument
• The Hom sheaf of locally free sheaves corresponds to the sheaf of morphisms between vector bundles

Pullback and pushforward

• Given a morphism $f: X \to Y$ of schemes and a coherent sheaf $\mathcal{F}$ on $Y$, the pullback $f^*\mathcal{F}$ is a coherent sheaf on $X$
• The pullback is functorial and preserves tensor products and Hom sheaves
• Given a morphism $f: X \to Y$ of noetherian schemes and a coherent sheaf $\mathcal{F}$ on $X$, the pushforward $f_*\mathcal{F}$ is a coherent sheaf on $Y$
• The pushforward is functorial and has a right adjoint given by the pullback functor
• The higher direct images $R^if_*\mathcal{F}$ are also coherent sheaves on $Y$ and play a crucial role in the study of the cohomology of coherent sheaves

Applications of coherent sheaves

• Coherent sheaves have numerous applications in various branches of mathematics, including algebraic geometry, complex analytic geometry, and representation theory
• These applications demonstrate the power and versatility of the theory of coherent sheaves and its connections to other areas of mathematics

In algebraic geometry

• Coherent sheaves are a fundamental tool in algebraic geometry and are used to study the geometry of schemes and their morphisms
• The cohomology of coherent sheaves provides invariants that capture important geometric information, such as the genus of a curve or the Euler characteristic of a variety
• Coherent sheaves are used in the construction of moduli spaces, which parametrize geometric objects such as curves, surfaces, or vector bundles
• The theory of coherent sheaves plays a crucial role in the study of birational geometry, the minimal model program, and the classification of algebraic varieties

In complex analytic geometry

• Coherent sheaves also play an important role in complex analytic geometry, where they are used to study the geometry of complex analytic spaces
• The theory of coherent sheaves on complex analytic spaces is closely related to the theory of coherent sheaves on schemes, with many results and techniques carrying over to the analytic setting
• Coherent sheaves are used in the study of complex analytic vector bundles, the Hodge theory of complex manifolds, and the theory of $\mathcal{D}$-modules
• The Oka-Cartan theory of coherent sheaves on Stein spaces is a powerful tool for studying the cohomology and global properties of coherent sheaves in complex analytic geometry

In representation theory

• Coherent sheaves have applications in representation theory, where they are used to study the geometry of representation spaces and the structure of algebraic groups
• The Beilinson-Bernstein localization theorem establishes a correspondence between certain categories of representations of Lie algebras and categories of coherent sheaves on flag varieties
• Coherent sheaves on homogeneous spaces and their equivariant analogues play a central role in the geometric representation theory of algebraic groups
• The theory of perverse sheaves, which is based on the theory of coherent sheaves, has important applications in the representation theory of finite groups and the study of character sheaves on algebraic groups