Finitely Presented Module

Definition

A finitely presented module is a module over a ring that can be described by a finite number of generators and relations. Specifically, it is represented as a quotient of a free module with a finite number of generators by the image of a finitely generated submodule. This concept is crucial for understanding flatness since finitely presented modules exhibit nice properties related to homological dimensions and can often be analyzed using flat modules.

5 Must Know Facts For Your Next Test

1. Finitely presented modules are important because they often allow for easier computation of properties such as projective dimension and flatness.
2. Every finitely generated module over a Noetherian ring is a quotient of a finitely presented module, linking the two concepts closely.
3. Flatness criteria often involve finitely presented modules, particularly in terms of preserving exact sequences when tensoring.
4. The property of being finitely presented helps in various applications, such as in algebraic geometry, where it corresponds to coherent sheaves.
5. In homological algebra, finitely presented modules lead to significant results regarding their resolutions and the behavior under functors.

Review Questions

• How does the concept of finitely presented modules relate to free modules and their generators?
  • Finitely presented modules can be thought of as quotients of free modules, where these free modules have a finite number of generators. A finitely presented module can be expressed with both a finite set of generators and relations among those generators. This structure allows for the analysis of finitely presented modules through the properties of free modules, making it easier to study their homological characteristics.
• In what ways do finitely presented modules interact with exact sequences in commutative algebra?
  • Finitely presented modules are significant in the study of exact sequences because they preserve exactness when tensoring with flat modules. If we have an exact sequence of finitely presented modules, tensoring this sequence with a flat module will yield another exact sequence. This interaction is essential in understanding how properties such as flatness and projectivity behave under these conditions and provides valuable insights into the structure of modules.
• Evaluate the role of finitely presented modules in applications within algebraic geometry and their impact on coherent sheaves.
  • Finitely presented modules play a crucial role in algebraic geometry as they correspond to coherent sheaves, which are fundamental objects in the study of schemes. Coherent sheaves are those that locally look like the sheaf associated with finitely generated algebras; thus, understanding finitely presented modules gives insight into the behavior of coherent sheaves. This relationship impacts how one studies properties such as morphisms between varieties and the geometric interpretation of algebraic concepts.

Related terms

Free Module: A free module is a module that has a basis, meaning it is isomorphic to a direct sum of copies of the ring, allowing for linear combinations without any relations.

Exact Sequence: An exact sequence is a sequence of module homomorphisms between modules such that the image of one homomorphism equals the kernel of the next, reflecting how modules relate to each other.

Projective Module: A projective module is a module that satisfies the property that every surjective homomorphism onto it can be lifted to a homomorphism from its source, which implies it behaves like a direct summand of a free module.