Prove that [ξ]=[0 →A i⟶ B →C →0] ∈e(C, A) has finite order if and only if there are a nonzero m

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Prove that [ξ]=[0 →A i⟶ B →C →0] ∈e(C, A) has finite order...

Prove that $[\xi]=[0 \rightarrow A \stackrel{i}{\longrightarrow} B \rightarrow C \rightarrow 0] \in e(C, A)$ has finite order if and only if there are a nonzero $m \in \mathbb{Z}$ and a map $s: B \rightarrow A$ with $s i=m \cdot 1_{A}$.

Key Concepts

Extension Classes in Homological Algebra

In homological algebra, a short exact sequence such as 0 ? A ? B ? C ? 0 represents an extension class in the Ext group (often denoted as Ext¹(C, A)). These classes capture the various ways one module (or abelian group) can be 'extended' by another, thereby providing a measure of the nontriviality of the relationship between A, B, and C. The study of these classes is central to understanding how algebraic structures combine and split, and they form an abelian group under the Baer sum operation.

Baer Sum and Torsion in Ext Groups

The Baer sum is the operation used to 'add' two extensions, turning the collection of extension classes into an abelian group. An extension [?] having finite order means that there exists a nonzero integer m such that the m-fold Baer sum of [?] with itself is the trivial extension (i.e., it splits). This characterization of finite order in the Ext group is important because it links algebraic conditions (like the existence of an integer m nullifying the extension) with the geometric or structural properties of the sequence.

Splitting of Exact Sequences

A splitting of a short exact sequence is a map (often called a retraction) that provides a right inverse to the inclusion map, thus making the sequence split or 'trivial'. In the context of the problem, the existence of a map s: B ? A satisfying s ? i = m · 1? (with m a nonzero integer) is a weakened form of splitting, where the splitting occurs up to multiplication by m. This condition is equivalent to the extension class being m-torsion in the Ext group and is key to proving the finite order of the extension.

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