Coherent sheaves

Definition

Coherent sheaves are a type of sheaf in algebraic geometry that encapsulate the idea of algebraic data being locally finitely presented. They are important for studying properties of varieties and schemes because they allow for a well-behaved notion of local finiteness, making them essential in both the algebraic and geometric contexts. Coherent sheaves provide a bridge between the algebraic structure of rings and the topological structure of spaces.

5 Must Know Facts For Your Next Test

1. Coherent sheaves are defined using locally finitely generated modules over the structure sheaf of a scheme or variety.
2. A coherent sheaf can be thought of as being 'locally free' in a certain sense, meaning it behaves like a vector bundle over small neighborhoods.
3. They play a crucial role in the formulation of important concepts such as projective varieties and morphisms between schemes.
4. Coherent sheaves are closed under taking kernels and cokernels, making them a well-behaved category in the context of homological algebra.
5. The category of coherent sheaves on a scheme is an abelian category, which allows for the application of various tools from homological algebra.

Review Questions

• How do coherent sheaves relate to the notion of local finiteness in algebraic geometry?
  • Coherent sheaves embody the concept of local finiteness by ensuring that over every open subset of a scheme, the sheaf can be generated by finitely many sections. This means that while we may have complex global behavior, locally, we can work with manageable data. This relationship is vital in understanding how local properties contribute to the global structure of varieties and schemes.
• What are some key properties that make coherent sheaves suitable for use in homological algebra?
  • Coherent sheaves have several key properties that make them suitable for homological algebra. They are closed under taking kernels and cokernels, which is crucial for defining exact sequences. Additionally, the category of coherent sheaves is an abelian category, allowing one to apply tools like derived functors and spectral sequences effectively. These features enable deep insights into their structure and relationships.
• Evaluate how coherent sheaves enhance our understanding of morphisms between schemes.
  • Coherent sheaves significantly enhance our understanding of morphisms between schemes by providing a framework to study how these morphisms behave with respect to local properties. When considering morphisms, coherent sheaves allow us to track how sections transform under these mappings, revealing insights into geometric aspects such as dimension and singularity. This connection is crucial for understanding the broader implications of morphisms in algebraic geometry, especially in studying moduli spaces and deformation theory.

Related terms

Sheaf:

A sheaf is a mathematical tool that allows for the systematic way to keep track of local data attached to the open sets of a topological space.

Locally finitely presented: This refers to an object that can be represented by finitely generated data over every open set, facilitating the study of local properties in algebraic geometry.

Scheme:

A scheme is a fundamental concept in algebraic geometry that generalizes varieties and allows for a more flexible treatment of geometric objects.