Coherent sheaf

Definition

A coherent sheaf is a type of sheaf of modules over a ringed space that satisfies two important properties: it is locally finitely presented and has support that is a closed subset. This concept is crucial for understanding the behavior of algebraic varieties and morphisms between them, particularly in the context of various geometric and topological structures.

5 Must Know Facts For Your Next Test

1. Coherent sheaves can be thought of as generalizations of vector bundles, making them essential in both algebraic geometry and complex geometry.
2. In affine schemes, coherent sheaves correspond to finitely generated ideals in the ring of regular functions, linking algebra and geometry closely.
3. One key property of coherent sheaves is that their stalks are finitely generated over their local rings, which provides a bridge between local and global analysis.
4. Coherent sheaves are stable under taking finite direct sums and tensor products, allowing for rich constructions in geometric contexts.
5. The notion of coherence is essential in the proof of several important results, including the coherence theorem for projective morphisms.

Review Questions

• How do coherent sheaves relate to finitely generated modules in the context of algebraic varieties?
  • Coherent sheaves are intimately connected to finitely generated modules because they are defined as locally finitely presented sheaves. This means that at every point in the variety, the stalk of a coherent sheaf is a finitely generated module over the local ring. This relationship allows for a seamless transition between algebraic properties and geometric structures, making coherent sheaves essential for studying algebraic varieties.
• Discuss how coherent sheaves contribute to our understanding of morphisms between algebraic varieties.
  • Coherent sheaves play a crucial role in understanding morphisms between algebraic varieties since they encapsulate important information about the structure and behavior of these varieties. When studying morphisms, coherent sheaves allow us to analyze how sections behave under pullback and pushforward, which is vital for exploring properties like flatness and properness. These aspects help mathematicians construct relationships between varieties and investigate their geometric features.
• Evaluate the implications of coherent sheaves on the coherence theorem for projective morphisms and how it affects algebraic geometry.
  • The coherence theorem for projective morphisms asserts that the direct image of a coherent sheaf under a proper morphism remains coherent. This has significant implications for algebraic geometry because it ensures that when one studies morphisms from projective varieties, they can rely on the coherence property to maintain control over their geometric structures. This theorem also facilitates working with families of coherent sheaves, thus enhancing our understanding of deformation theory and moduli problems within algebraic geometry.

Related terms

Sheaf:

A sheaf is a mathematical tool that assigns data to open sets of a topological space, allowing for the systematic study of local properties and global behavior.

Finitely generated module: A finitely generated module is an algebraic structure that can be generated by a finite number of elements, playing a key role in the theory of coherent sheaves.

Support of a sheaf: The support of a sheaf refers to the closure of the set where the sheaf does not vanish, which is essential in understanding its global properties.