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 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,OX), where X is a topological space and OX is a of rings on X
The stalks OX,x at each point x∈X are local rings, capturing the local algebraic structure
Affine schemes, such as Spec(R) for a commutative ring R, serve as building blocks for general schemes
Structure sheaves on schemes
The structure sheaf OX on a scheme X is a fundamental object that encodes the algebraic structure
For an 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 of schemes f:(X,OX)→(Y,OY) consists of a continuous map f:X→Y and a sheaf morphism f#:OY→f∗OX
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- F on a scheme X is a sheaf of OX-modules that is locally the sheaf associated to a module
On an affine scheme 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 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 OX-modules
The category of quasi-coherent sheaves 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 F on a scheme X is finitely presented if there exists an exact sequence OX⊕n→OX⊕m→F→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 L such that for every point x∈X, the stalk Lx is a free OX,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 Pic(X) of a scheme X is the group of isomorphism classes of invertible sheaves on X
The group operation in 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 Cl(X) is isomorphic to the Picard group 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 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 Γ give rise to the sheaf cohomology groups Hi(X,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 F to the cohomology of its dual sheaf F∨⊗ωX, where ωX 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 I⊂OX
The structure sheaf of a closed subscheme Y is given by OY=OX/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
Key Terms to Review (19)
Affine scheme: An affine scheme is a basic building block in algebraic geometry, defined as the spectrum of a commutative ring. It captures the notion of algebraic varieties and serves as a fundamental example of a scheme, where each affine scheme corresponds to a ring, allowing us to study geometric properties through algebraic techniques.
Alexander Grothendieck: Alexander Grothendieck was a French mathematician who made groundbreaking contributions to algebraic geometry, particularly through the development of sheaf theory and the concept of schemes. His work revolutionized the field by providing a unifying framework that connected various areas of mathematics, allowing for deeper insights into algebraic varieties and their cohomological properties.
Coherent Sheaf: A coherent sheaf is a type of sheaf that has properties similar to those of finitely generated modules over a ring, particularly in terms of their local behavior. Coherent sheaves are significant in algebraic geometry and other areas because they ensure that certain algebraic structures behave nicely under localization and restriction, which connects them with various topological and algebraic concepts.
Constant Sheaf: A constant sheaf is a type of sheaf that assigns the same set, usually a fixed set of elements, to every open set in a topological space. This notion is crucial because it provides a simple way to study sheaves by associating them with constant functions over various open sets, making them foundational in understanding more complex sheaves and their properties.
Direct Image Theorem: The Direct Image Theorem states that for a morphism of schemes, a sheaf can be pushed forward to the target scheme via this morphism. This theorem helps in understanding how sheaves behave under continuous maps, allowing us to analyze the properties of sheaves in the context of different spaces. It provides a crucial connection between the geometry of the source and target schemes and the algebraic structures defined by the sheaves.
Divisor Sheaf: A divisor sheaf is a specific type of sheaf on a scheme that encodes information about effective divisors and their associated properties, such as local functions that have poles or zeros along a divisor. It allows for the study of algebraic properties in a geometric setting by associating to each open set of the scheme a ring of functions that exhibit behavior related to the divisor. This concept is crucial in understanding how divisors can be interpreted within the language of sheaves and schemes.
Gluing Property: The gluing property is a fundamental aspect of sheaf theory that allows one to construct global sections from local data. Specifically, it states that if you have a collection of local sections defined on open sets of a topological space that agree on overlaps, then there exists a unique global section on the entire space that corresponds to these local sections. This concept is crucial for understanding how local behaviors can be stitched together into a cohesive global structure.
Jean-Pierre Serre: Jean-Pierre Serre is a renowned French mathematician known for his foundational contributions to algebraic geometry, topology, and number theory. His work laid the groundwork for many important concepts and theorems in modern mathematics, influencing areas such as sheaf theory, cohomology, and the study of schemes.
Locality: Locality refers to the property of sheaves that allows them to capture local data about spaces, making them useful for studying properties that can be understood through local neighborhoods. This concept connects various aspects of sheaf theory, particularly in how information can be restricted to smaller sets and still retain significant meaning in broader contexts.
Locally ringed sheaf: A locally ringed sheaf is a type of sheaf that assigns to each open set in a topological space a ring of sections, and at each point, the stalk of the sheaf is a local ring. This structure allows for local algebraic properties to be studied in relation to the topological space, making it particularly useful in algebraic geometry and the study of schemes.
Morphism: A morphism is a structure-preserving map between two mathematical objects that allows the transfer of properties and relationships. It plays a critical role in category theory, providing a way to relate different objects like sheaves or schemes while maintaining their essential characteristics. In the context of sheaves and schemes, morphisms allow for the interaction between sheaves on different topological spaces and establish how schemes can be transformed or mapped to one another.
Open Cover: An open cover is a collection of open sets in a topological space that together cover the entire space, meaning every point in the space is contained within at least one of the open sets in the collection. This concept plays a crucial role in various mathematical contexts, such as ensuring that certain properties hold locally or globally, as well as being integral to the construction of sheaves, the formulation of Čech cohomology, and the study of locally ringed spaces and manifolds.
Pullback: A pullback is a construction in category theory that allows us to take a pair of morphisms and create a new object that effectively combines their information. It relates two objects through their mappings, providing a way to 'pull back' data along these morphisms, which is crucial in many areas including sheaf theory, coherent sheaves, and the study of logical structures.
Pushforward: Pushforward refers to a way of transferring structures, such as sheaves or morphisms, from one space to another via a continuous map. This concept plays a crucial role in connecting different spaces in sheaf theory, allowing us to understand how properties and information propagate through maps, particularly when working with sheaves and morphisms in various contexts.
Scheme: A scheme is a mathematical structure that generalizes the concept of algebraic varieties, providing a framework for studying solutions to polynomial equations in a more flexible way. Schemes consist of a topological space along with a sheaf of rings, which allows for local data to be combined with global geometric properties. This structure connects algebra, geometry, and topology, making it a powerful tool in modern mathematics.
Sheaf: A sheaf is a mathematical structure that captures local data attached to the open sets of a topological space, enabling the coherent gluing of these local pieces into global sections. This concept bridges several areas of mathematics by allowing the study of functions, algebraic structures, or more complex entities that vary across a space while maintaining consistency in how they relate to each other.
Sheafification Theorem: The Sheafification Theorem states that for any presheaf, there exists a unique sheaf associated with it, which captures its local properties while maintaining the original global data. This process effectively 'corrects' a presheaf into a sheaf, ensuring that it satisfies the sheaf axioms, like locality and gluing. The theorem highlights how one can derive sheaves from presheaves and emphasizes the importance of local information in defining sheaf properties.
Structural Sheaf: A structural sheaf is a fundamental concept in algebraic geometry that assigns a set of functions to open subsets of a topological space or a scheme, providing a way to locally study algebraic properties. It acts as a bridge between the geometric and algebraic aspects of a space, allowing one to define functions, sections, and operations on the scheme. The structural sheaf plays a crucial role in defining morphisms between schemes and understanding the properties of varieties.
Vector Sheaf: A vector sheaf is a type of sheaf that associates to each open set of a topological space a vector space, allowing for the study of sections that can be added together and multiplied by scalars. This concept becomes particularly important in algebraic geometry and scheme theory, as it provides a way to understand how vector bundles can be constructed over various spaces. Vector sheaves can describe geometric objects and their properties, reflecting the structure of the underlying space through linear algebra.