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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Schanuel's lemma for two presentations of a module

Example

For $Z / 2 Z$ , compare the two short exact sequences $0 \to 2 Z \to Z \to Z / 2 Z \to 0$ and $0 \to 2 Z \oplus Z \to Z \oplus Z \to Z / 2 Z \to 0,$ where the second surjection sends $(a, b)$ to $\overset{ˉ}{a}$ . Schanuel's lemma predicts $2 Z \oplus (Z \oplus Z) ≅ (2 Z \oplus Z) \oplus Z .$

Facts & Assumptions

Given: The two displayed short exact sequences.

[L1]

Schanuel's lemma gives a stable isomorphism between the two kernels (Schanuel's lemma in an abelian category).

[L2]

Short exact sequences of modules are exact module-theoretic rows (Exact sequences and short exact sequences of modules).

Verification

technique · direct

1.1L2givenalgebra

The kernel of $Z \to Z / 2 Z$ is $2 Z$ , and the kernel of $(a, b) \mapsto \overset{ˉ}{a}$ is $2 Z \oplus Z$ . Thus the two displayed rows are short exact in the sense of [L2].

2.1L1step 1.1algebra∎

Since $2 Z ≅ Z$ , both sides of Schanuel's conclusion are free abelian groups of rank three. Hence they are isomorphic, exactly as [L1] predicts.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 10.109: Rings of finite global dimension (standard reference, not scraped)