Exact Sequences

Definition

An exact sequence is a sequence of algebraic objects (like modules or groups) and morphisms between them such that the image of one morphism is equal to the kernel of the next. This property provides crucial information about how these objects relate to each other and is fundamental in exploring properties of flat modules and their applications in various algebraic contexts.

5 Must Know Facts For Your Next Test

1. Exact sequences can be used to define various properties of modules, such as projectivity and injectivity, which are essential for understanding flatness.
2. In a short exact sequence, if we have a module $A$ mapping into $B$ and then into $C$, exactness ensures that the image of $A$ in $B$ matches precisely with the kernel of the map from $B$ to $C$.
3. Exact sequences provide a way to understand when a property holds for a quotient module based on its relation to another module in the sequence.
4. Exactness can help demonstrate that certain sequences of flat modules maintain flatness when tensored with another module.
5. Studying exact sequences reveals how changes in one module affect others, which is particularly useful in applications like localization and dimension theory.

Review Questions

• How do exact sequences help define important properties like projectivity and injectivity in modules?
  • Exact sequences serve as a foundation for understanding projectivity and injectivity by showing how modules relate through morphisms. In particular, if you have an exact sequence involving a projective module, this indicates that every morphism into this module can be lifted. Similarly, for injective modules, exactness guarantees that any map from another module can be extended. These properties are vital in determining flatness and the behavior of modules under various operations.
• Discuss how short exact sequences can be applied to explore flatness criteria among modules.
  • Short exact sequences are crucial when examining flatness because they allow us to analyze the behavior of modules under tensor products. If you have a short exact sequence involving flat modules, you can show that their tensor product with another module preserves exactness. This property helps establish conditions under which certain modules maintain their flatness through transformations or mappings, making it easier to identify flat modules in broader algebraic structures.
• Evaluate how understanding exact sequences impacts our comprehension of localization in commutative algebra.
  • Understanding exact sequences significantly enhances our grasp of localization because they illustrate how modules behave when localized at a prime ideal. When you have an exact sequence and localize it, you can see how properties like flatness persist or change within the localized context. This deep insight enables mathematicians to determine when certain sequences remain exact after localization, which is critical for applying results related to flat modules in various commutative algebra scenarios.

Related terms

Kernel:

The kernel of a morphism is the set of elements that are mapped to the zero element in the target object, capturing the idea of what gets 'collapsed' by the morphism.

Cokernel: The cokernel of a morphism is the quotient of the target object by the image of the morphism, measuring how far the morphism fails to be surjective.

Short Exact Sequence: A short exact sequence is an exact sequence that consists of three objects and two morphisms, often used to study extensions and properties of modules.