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Syzygies from two projective resolutions are stably isomorphic

Statement

Syzygies arising from two projective resolutions of the same object are stably isomorphic. In particular, the first syzygies are related by Schanuel's lemma, and higher displayed syzygies inherit the same stable-comparison pattern after truncation.

Facts & Assumptions

Given: Two projective resolutions of the same object $A$ .

[L1]

Syzygies are the kernels selected from a displayed resolution (Syzygies and cosyzygies relative to a chosen resolution).

[L2]

Schanuel's lemma identifies the stable class of two projective presentations of the same object (Schanuel's lemma in an abelian category).

Proof

technique · direct

1.1L1L2

The first syzygies of the two resolutions are the kernels of two projective presentations of $A$ by [L1]. Therefore [L2] gives a stable isomorphism between them.

2.1L1L2step 1.1induction∎

Proceed by induction. Suppose $Ω_{P}^{n - 1} (A) \oplus E ≅ Ω_{Q}^{n - 1} (A) \oplus F$ for projective objects $E, F$ . After identifying these sums with a common object, the two exact rows $0 \to Ω_{P}^{n} (A) \to P_{n - 1} \oplus E \to Ω_{P}^{n - 1} (A) \oplus E \to 0$ and $0 \to Ω_{Q}^{n} (A) \to Q_{n - 1} \oplus F \to Ω_{Q}^{n - 1} (A) \oplus F \to 0$ are projective presentations of that common object. Applying [L2] gives $Ω_{P}^{n} (A) \oplus Q_{n - 1} \oplus F ≅ Ω_{Q}^{n} (A) \oplus P_{n - 1} \oplus E,$ so the $n$ th syzygies are stably isomorphic. Together with step 1.1 this proves the claim in every degree.

Depends on

Used by

Two projective resolutions with nonisomorphic first syzygies Counterexample
FALSE: two syzygies of an object are canonically isomorphic False statement

Dependency tree · two levels

10 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)