Coherent sheaf

Definition

A coherent sheaf is a type of sheaf that behaves nicely with respect to the algebraic structure of a scheme, particularly in the context of the sheaf's ability to be represented by finitely generated modules over rings. This notion is crucial for understanding properties like the support of a sheaf and its relationships with morphisms in algebraic geometry. Coherent sheaves allow us to study varieties and schemes through algebraic means, providing a bridge between geometry and algebra.

5 Must Know Facts For Your Next Test

1. Coherent sheaves are defined on schemes and are crucial for understanding their geometric properties through algebraic means.
2. For a sheaf to be coherent, it must be both finitely generated and satisfy the condition that its stalks are finitely presented as modules over the local ring.
3. The concept of coherence is closely tied to the notion of projective varieties, as coherent sheaves can be associated with the algebraic functions defined on these varieties.
4. Coherent sheaves can represent various geometrical constructs, such as ideals of functions vanishing on subvarieties, making them essential for studying schemes.
5. The category of coherent sheaves has nice properties, such as being closed under taking kernels and cokernels, which helps in various constructions in algebraic geometry.

Review Questions

• How do coherent sheaves relate to finitely generated modules over rings in the context of schemes?
  • Coherent sheaves are essentially represented by finitely generated modules over the rings associated with the schemes they are defined on. This relationship allows us to study the local properties of schemes using algebraic techniques. The condition that they are finitely generated ensures that these sheaves maintain a manageable structure, making them applicable in various geometric contexts.
• Discuss the implications of having locally free sheaves versus coherent sheaves when analyzing schemes.
  • Locally free sheaves can be seen as coherent sheaves that behave like vector bundles, offering more flexibility in terms of local trivialization. While coherent sheaves provide useful information about the structure and function ideals on a scheme, locally free sheaves extend this idea by allowing for rank and dimensional considerations. This distinction is important when classifying geometrical objects and understanding their properties under morphisms.
• Evaluate how coherent sheaves contribute to our understanding of projective varieties and their associated functions.
  • Coherent sheaves play a vital role in studying projective varieties since they correspond to algebraic functions that vanish on these varieties. This connection enables us to translate geometric questions into algebraic ones, which are often easier to handle. By associating coherent sheaves with ideals of functions defining subvarieties, we can derive important properties about these varieties, such as dimension and singularity analysis, linking geometry directly to algebra.

Related terms

Sheaf: A mathematical structure that associates data to open sets of a topological space, allowing for local-to-global principles.

Finitely Generated Module: A module over a ring that can be generated by a finite number of elements, which plays a significant role in the study of coherent sheaves.

Locally Free Sheaf: A sheaf that looks like a free module in small neighborhoods, often used in the context of vector bundles on schemes.