LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Objectwise comparison of two projective resolution data induces an isomorphism on derived objects

Statement

Assume the Axiom of Dependent Choice.

Let $P$ and $Q$ be supplied projective resolution data on the same domain, and let $F : A \to B$ be an additive functor between abelian categories. For every object $A$ in the common domain and every $n \in Z$ , there is an isomorphism $θ_{P, Q} (A) : L_{n}^{P} F (A) \tilde{\to} L_{n}^{Q} F (A)$ induced by a comparison map between the chosen resolutions of $A$ .

Facts & Assumptions

Given: An object $A$ in the common domain and an integer $n$ .

[L1]

The data $P$ and $Q$ supply specific projective resolutions of $A$ (Supplied projective resolution data).

[L2]

Any two projective resolutions of the same object are homotopy equivalent over that object (Projective resolutions of the same object are homotopy equivalent over that object).

[L3]

Chain-homotopic maps induce the same homology map, and homology respects composition (Chain-homotopic maps induce the same map on homology, Homology respects identities and composition).

[L4]

The derived objects are the homology objects of the chosen deleted resolutions after applying $F$ (Left derived objects relative to supplied projective resolution data).

Proof

technique · direct

1.1

By [L1] and [L2], there exist comparison maps $c_{∙} : P_{∙} (A) \to Q_{∙} (A)$ and $d_{∙} : Q_{∙} (A) \to P_{∙} (A)$ whose composites are homotopic to the identity chain maps on the two resolutions.

L1L2givenconstruct

2.1

Apply $F$ degreewise and pass to homology. By [L3], the induced maps $H_{n} (F (c_{∙}))$ and $H_{n} (F (d_{∙}))$ are inverse because their composites equal the homology maps of chain maps homotopic to the identities. Using [L4], this yields the claimed isomorphism $θ_{P, Q} (A) : L_{n}^{P} F (A) \tilde{\to} L_{n}^{Q} F (A)$ .

L3L4step 1.1∎

Depends on

Used by

Dependency tree · two levels

16 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

Romyar Sharifi, Homological Algebra (standard reference, not scraped)