Prove that any two split extensions of modules A by C are equivalent.

Name: Problem 9, Chapter 7: An Introduction to Homological Algebra
Uploaded: 2020-10-28T03:24:40-08:00
Duration: 5 min 17 s
Channel: Anthony Ramos
Description: Prove that any two split extensions of modules A by C are equivalent.

step 1 Hello guys, so for this exercise we're going to consider V on the following notation it's going

Prove that any two split extensions of modules A by C are...

Key Concepts

Equivalence of Extensions

Two extensions are considered equivalent if there exists an isomorphism between their middle modules that makes the associated diagram commute with the identity maps on the submodule and quotient. In the setting of split extensions, this equivalence is often demonstrated by the fact that all split extensions of a given pair of modules are isomorphic to the direct sum, thereby ensuring that any two split extensions are equivalent to each other.

Split Exact Sequence

A split exact sequence is a special type of short exact sequence that 'splits', meaning there exists a homomorphism (a splitting homomorphism) that serves as a right inverse to the projection from B to C. This splitting guarantees that the middle module B is isomorphic to the direct sum of A and C, reflecting a trivial extension where no nontrivial information is introduced in the extension.

Splitting Homomorphism

A splitting homomorphism is a map from the quotient module C back into the module B such that composing it with the natural projection yields the identity on C. Its existence characterizes split exact sequences and ensures that the module B can be decomposed as a direct sum, which is pivotal in showing that any two such split extensions are equivalent.

Module Extension

A module extension is an arrangement where one module is 'extended' by another, formalized by a short exact sequence 0 ? A ? B ? C ? 0. This sequence encapsulates how B, the middle module, contains A as a submodule and has C as its quotient, thereby relating the structure of these modules in a cohesive manner.

Short Exact Sequence

A short exact sequence is an exact sequence of the form 0 ? A ? B ? C ? 0 where the image of each homomorphism coincides with the kernel of the subsequent map. In the context of module theory, it provides a framework for understanding extensions and decompositions of modules by precisely describing how one module is embedded into another and how the quotient is formed.

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