8.2 Sheaves on schemes

Sheaves on schemes bridge the gap between algebraic geometry and topology. They provide a powerful framework for studying geometric properties of algebraic varieties, allowing us to analyze local and global structures simultaneously.

Quasi-coherent and coherent sheaves play central roles in this theory. These objects generalize modules over rings to the scheme setting, enabling us to apply algebraic techniques to geometric problems and vice versa.

Definition of sheaves on schemes

• Sheaves on schemes generalize the notion of sheaves on topological spaces to the algebraic geometry setting
• Schemes are the fundamental objects of study in algebraic geometry, locally modeled on spectra of commutative rings
• Sheaves on schemes allow for the study of geometric properties and constructions in a more abstract and flexible framework

Schemes as locally ringed spaces

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• Schemes are defined as locally ringed spaces $(X, \mathcal{O}_X)$, where $X$ is a topological space and $\mathcal{O}_X$ is a sheaf of rings on $X$
• The stalks $\mathcal{O}_{X,x}$ at each point $x \in X$ are local rings, capturing the local algebraic structure
• Affine schemes, such as $\operatorname{Spec}(R)$ for a commutative ring $R$, serve as building blocks for general schemes

Structure sheaves on schemes

• The structure sheaf $\mathcal{O}_X$ on a scheme $X$ is a fundamental object that encodes the algebraic structure
• For an affine scheme $\operatorname{Spec}(R)$, the structure sheaf is determined by the ring $R$
• On a general scheme, the structure sheaf is obtained by gluing the structure sheaves of affine open subsets

Morphisms of schemes

• Morphisms of schemes are defined as morphisms of locally ringed spaces
• A morphism of schemes $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ consists of a continuous map $f: X \to Y$ and a sheaf morphism $f^{\#}: \mathcal{O}_Y \to f_*\mathcal{O}_X$
• Morphisms of schemes allow for the study of relationships and functoriality between different schemes

Quasi-coherent sheaves

• Quasi-coherent sheaves are a class of sheaves on schemes that generalize the notion of modules over rings
• They provide a natural framework for studying vector bundles and other geometric objects on schemes
• Quasi-coherent sheaves have nice properties and are well-behaved under various operations

Quasi-coherent modules

• A quasi-coherent sheaf $\mathcal{F}$ on a scheme $X$ is a sheaf of $\mathcal{O}_X$-modules that is locally the sheaf associated to a module
• On an affine scheme $\operatorname{Spec}(R)$, quasi-coherent sheaves correspond to $R$-modules
• Quasi-coherent sheaves can be obtained by gluing quasi-coherent sheaves on affine open subsets

Properties of quasi-coherent sheaves

• Quasi-coherent sheaves form an abelian category $\operatorname{QCoh}(X)$ on a scheme $X$
• Operations such as kernels, cokernels, and tensor products of quasi-coherent sheaves are well-defined
• Quasi-coherent sheaves have a nice behavior under pullbacks and pushforwards along morphisms of schemes

Morphisms of quasi-coherent sheaves

• Morphisms between quasi-coherent sheaves are morphisms of $\mathcal{O}_X$-modules
• The category of quasi-coherent sheaves $\operatorname{QCoh}(X)$ is an abelian category
• Morphisms of quasi-coherent sheaves can be studied using tools from homological algebra

Coherent sheaves

• Coherent sheaves are a subclass of quasi-coherent sheaves with additional finiteness conditions
• They play a central role in algebraic geometry and have good properties
• Coherent sheaves are closely related to finitely generated modules over commutative rings

Definition and properties

• A coherent sheaf on a scheme $X$ is a quasi-coherent sheaf that is locally finitely presented
• Coherent sheaves are stable under kernels, cokernels, and extensions in the category of quasi-coherent sheaves
• On a noetherian scheme, a sheaf is coherent if and only if it is locally finitely generated

Finitely presented sheaves

• A sheaf $\mathcal{F}$ on a scheme $X$ is finitely presented if there exists an exact sequence $\mathcal{O}_X^{\oplus n} \to \mathcal{O}_X^{\oplus m} \to \mathcal{F} \to 0$
• Finitely presented sheaves are coherent sheaves
• The notion of finitely presented sheaves is a generalization of finitely presented modules over rings

Coherent sheaves vs quasi-coherent sheaves

• Coherent sheaves form a full subcategory of the category of quasi-coherent sheaves
• On a noetherian scheme, every coherent sheaf is quasi-coherent
• Not every quasi-coherent sheaf is coherent, as they may lack the finiteness conditions

Invertible sheaves and line bundles

• Invertible sheaves, also known as line bundles, are a special class of coherent sheaves of rank one
• They play a fundamental role in the study of divisors, the Picard group, and the geometry of schemes
• Invertible sheaves provide a bridge between algebraic and geometric aspects of schemes

Invertible sheaves on schemes

• An invertible sheaf on a scheme $X$ is a coherent sheaf $\mathcal{L}$ such that for every point $x \in X$, the stalk $\mathcal{L}_x$ is a free $\mathcal{O}_{X,x}$-module of rank one
• Invertible sheaves correspond to line bundles, which are locally trivial rank one vector bundles
• The tensor product of two invertible sheaves is again an invertible sheaf

Picard group of a scheme

• The Picard group $\operatorname{Pic}(X)$ of a scheme $X$ is the group of isomorphism classes of invertible sheaves on $X$
• The group operation in $\operatorname{Pic}(X)$ is given by the tensor product of invertible sheaves
• The Picard group measures the obstruction to the existence of global sections of invertible sheaves

Line bundles and divisors

• On a smooth projective variety $X$, there is a one-to-one correspondence between invertible sheaves and divisors modulo linear equivalence
• The divisor class group $\operatorname{Cl}(X)$ is isomorphic to the Picard group $\operatorname{Pic}(X)$
• This correspondence allows for the study of the geometry of $X$ using the language of line bundles and divisors

Sheaf cohomology on schemes

• Sheaf cohomology is a powerful tool for studying global properties of sheaves on schemes
• It generalizes the notion of cohomology of sheaves on topological spaces to the algebraic geometry setting
• Sheaf cohomology provides invariants that capture important geometric information about schemes

Čech cohomology for schemes

• Čech cohomology is a computational tool for calculating sheaf cohomology on schemes
• It is defined using Čech cocycles and coboundaries with respect to an open cover of the scheme
• Čech cohomology is particularly useful for computing cohomology of quasi-coherent sheaves on affine schemes

Derived functors and sheaf cohomology

• Sheaf cohomology can be defined using the language of derived functors in homological algebra
• The right derived functors of the global sections functor $\Gamma$ give rise to the sheaf cohomology groups $H^i(X, \mathcal{F})$
• Derived functor approach provides a systematic way to study sheaf cohomology and its properties

Serre duality for schemes

• Serre duality is a fundamental duality theorem in the theory of sheaf cohomology on schemes
• It relates the cohomology of a coherent sheaf $\mathcal{F}$ to the cohomology of its dual sheaf $\mathcal{F}^{\vee} \otimes \omega_X$, where $\omega_X$ is the dualizing sheaf
• Serre duality has important applications in the study of the geometry of schemes and the Riemann-Roch theorem

Sheaves and geometric constructions

• Sheaves on schemes provide a framework for studying various geometric constructions and their properties
• Many geometric objects and operations can be naturally described and studied using the language of sheaves
• Sheaves allow for a unified treatment of local and global aspects of schemes

Sheaves and closed subschemes

• Closed subschemes of a scheme $X$ correspond to quasi-coherent sheaves of ideals $\mathcal{I} \subset \mathcal{O}_X$
• The structure sheaf of a closed subscheme $Y$ is given by $\mathcal{O}_Y = \mathcal{O}_X / \mathcal{I}$
• Sheaf-theoretic operations, such as restriction and extension, can be used to study properties of closed subschemes

Sheaves on projective schemes

• Projective schemes, such as projective varieties, are important objects in algebraic geometry
• The study of sheaves on projective schemes is particularly rich and has connections to representation theory and physics
• Concepts like twisting sheaves and the Serre correspondence play a crucial role in the study of sheaves on projective schemes

Sheaves and blowups

• Blowups are a fundamental construction in algebraic geometry that allows for the resolution of singularities
• Sheaves can be used to describe the behavior of blowups and the exceptional divisor
• The study of sheaves on blowups provides insights into the birational geometry of schemes

Applications and examples

• Sheaves on schemes have numerous applications in various areas of mathematics and beyond
• They provide a unifying language and powerful tools for solving problems and understanding geometric structures
• Examples demonstrate the wide range of contexts in which sheaves on schemes naturally arise

Sheaves in algebraic geometry

• Sheaves are ubiquitous in algebraic geometry and play a central role in the study of schemes
• They provide a way to encode geometric information and study properties such as coherence, cohomology, and duality
• Sheaves are used in the study of moduli spaces, intersection theory, and the geometry of algebraic varieties

Sheaves and intersection theory

• Intersection theory is a branch of algebraic geometry that studies the intersection properties of subvarieties
• Sheaves, particularly coherent sheaves, are used to formulate and study intersection-theoretic problems
• Chern classes, which are cohomology classes associated to vector bundles, are important tools in intersection theory

Sheaves in moduli problems

• Moduli problems concern the classification and parameterization of geometric objects, such as curves or vector bundles
• Sheaves on schemes provide a language for formulating and studying moduli problems
• Moduli spaces, such as the moduli space of curves or the moduli space of vector bundles, can be constructed and studied using sheaf-theoretic techniques