11.1 Introduction to sheaf theory

Sheaves are like secret agents, attaching algebraic data to open sets of spaces. They follow strict rules of locality and gluing, allowing us to study global properties by looking at local behavior. It's like piecing together a puzzle!

Sheaves are crucial in algebraic geometry, helping us understand varieties and schemes. They're the backbone of cohomology theories and provide a unified language for describing local and global properties. It's like having a universal translator for geometry!

Sheaves in Algebraic Geometry

Definition and Basic Properties

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• Define sheaves as tools that attach algebraic data to open sets of a topological space compatibly with restrictions
• Specify that sheaves consist of a topological space X, a target category D (often abelian groups, rings, or modules), and a contravariant functor F from the category of open sets of X to D
• State the two key properties satisfied by sheaves:
  • Locality axiom: compatible sections on overlapping open sets agree on the intersection
  • Gluing axiom: compatible sections on an open cover can be uniquely glued to a section on the union
• Explain that sheaves capture local-to-global phenomena, allowing the study of global properties by analyzing local behavior
• Provide important examples of sheaves:
  • Sheaf of regular functions on an algebraic variety
  • Sheaf of continuous functions on a topological space
  • Sheaf of sections of a vector bundle (e.g., the tangent bundle or cotangent bundle)

Applications and Motivation

• Discuss how sheaves are used to study geometric properties of algebraic varieties by examining algebraic data attached to open sets
• Explain that sheaves provide a language to describe local and global properties of spaces in a unified manner
• Highlight the role of sheaves in cohomology theories, such as sheaf cohomology and Čech cohomology, which measure global properties of spaces
• Mention that sheaves are essential in the construction of schemes, the foundational objects of modern algebraic geometry
• Provide an example of how sheaves can be used to study the behavior of functions or sections near a point or along a subvariety

Stalks and Sheafification

Stalks and Local Properties

• Define the stalk of a sheaf F at a point x as the direct limit (colimit) of the sections of F over all open sets containing x, denoted Fx
• Explain that the stalk represents the germs of sections near x, capturing the local behavior of the sheaf
• Describe the stalk functor as a left adjoint to the functor that assigns to each sheaf its underlying presheaf, providing a way to recover the sheaf from its stalks
• Discuss how stalks allow the study of local properties of a sheaf at a specific point, such as:
  • The dimension of the stalk as a vector space
  • The maximal ideal of the stalk as a local ring
• Provide an example of computing the stalk of a sheaf at a point and interpreting its properties

Sheafification Process

• Explain the sheafification process as a way to construct a sheaf from a presheaf by ensuring that the locality and gluing axioms are satisfied
• Describe the two-step process of sheafification:
  1. Take the presheaf of stalks
  2. Sheafify the resulting presheaf
• State that the sheafification functor is left adjoint to the forgetful functor from the category of sheaves to the category of presheaves
• Emphasize that sheafification provides a universal way to turn presheaves into sheaves
• Give an example of a presheaf that is not a sheaf and demonstrate the sheafification process

Constructing and Manipulating Sheaves

Constructing Sheaves from Presheaves

• Define a presheaf as a contravariant functor from the category of open sets of a topological space to a target category D, without necessarily satisfying the locality and gluing axioms
• Explain that the sheafification functor can be applied to a presheaf to obtain an associated sheaf, which is the "closest" sheaf to the given presheaf in a universal sense
• Provide an example of constructing a sheaf from a presheaf using sheafification, such as the sheaf of continuous functions from the presheaf of locally constant functions

Sheaf Operations

• Introduce sheaf operations, such as direct sums, tensor products, and sheaf Hom, as ways to construct new sheaves from existing ones
• Describe the direct sum of sheaves as the sheaf obtained by taking the direct sum of the sections on each open set
• Explain the tensor product of sheaves as the sheaf obtained by tensoring the sections on each open set
• Define the sheaf Hom as the sheaf that assigns to each open set the set of morphisms between the restrictions of two sheaves to that open set
• Discuss how sheaf Hom provides a way to study sheaf morphisms locally
• Give examples of computing sheaf operations, such as the direct sum or tensor product of the sheaf of regular functions and the sheaf of differential forms on an algebraic variety

Sheaves and Locally Ringed Spaces

Ringed Spaces and Locally Ringed Spaces

• Define a ringed space as a pair (X, OX) consisting of a topological space X and a sheaf of rings OX on X, where OX is called the structure sheaf
• Introduce locally ringed spaces as ringed spaces (X, OX) where the stalks OX,x are local rings for all points x in X
• Explain that locally ringed spaces provide a natural setting for studying geometric spaces with algebraic structure
• Give examples of locally ringed spaces, such as:
  • The prime spectrum of a ring A, denoted Spec(A), where X is the set of prime ideals of A and OX is the sheaf of localizations of A at each prime ideal
  • An algebraic variety with its sheaf of regular functions

Affine Schemes and Morphisms

• Define affine schemes as the prime spectra of commutative rings, which serve as the building blocks of algebraic geometry
• Explain that the category of affine schemes is equivalent to the opposite category of commutative rings, providing a bridge between algebraic geometry and commutative algebra via the language of sheaves and locally ringed spaces
• Introduce morphisms of locally ringed spaces as pairs consisting of:
  • A continuous map between the underlying topological spaces
  • A morphism of sheaves of rings, compatible with the local ring structure of the stalks
• Provide an example of a morphism between affine schemes induced by a ring homomorphism, and describe its properties in terms of the associated sheaf morphism