PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Cohomology of derived hom is ext

Statement

In the mixed bounded range of derived Hom, $H^{n} R Hom (M, N) ≅ {Hom}_{D (A)} (M, N [n])$ for every integer $n$ . For objects $M, N$ in degree zero and $n \geq 0$ this is classical ${Ext}^{n} (M, N)$ under the supplied one-sided resolution hypothesis.

Facts & Assumptions

Given: In the mixed bounded range of derived Hom, $H^{n} R Hom (M, N) ≅ {Hom}_{D (A)} (M, N [n])$ for every integer $n$ . For objects $M, N$ in degree zero and $n \geq 0$ this is classical ${Ext}^{n} (M, N)$ under the supplied one-sided resolution hypothesis.

[F1]

Derived Hom in the mixed bounded range uses a projective source or injective target, with no-roof cohomology comparisons and a mixed comparison zigzag (Derived hom in the bounded setting).

[F2]

Classical Ext identifies with derived Hom from $M [0]$ to $N [n]$ for $n \geq 0$ (Ext is hom in the derived category).

Proof

1.1

In the projective construction, cycles of degree $n$ in $\underset{}{Hom} (P_{M}, N)$ are chain maps $P_{M} \to N [n]$ . Boundaries are their nullhomotopies, since multiplying a homotopy by $(- 1)^{n}$ converts the shifted homotopy formula into $d u = d_{N} u - (- 1)^{n - 1} u d_{P}$ . Hence the cohomology is ${Hom}_{K} (P_{M}, N [n])$ . The no-roof comparison built into the derived Hom construction identifies it with ${Hom}_{D} (M, N [n])$ . The same calculation for $\underset{}{Hom} (M, I_{N})$ uses the injective no-roof comparison. Zero objects and every integer $n$ are allowed.

F1algebra

2.1

For degree-zero inputs, the classical Ext comparison identifies the last Hom group with the supplied resolution Ext for $n \geq 0$ . The identifications use the same cocycles and comparison maps, so they are natural in both variables. The mixed Hom zigzag makes the two one-sided descriptions agree when both are available.

F2step 1.1∎

Depends on

Used by

ex-derived-hom-of-cyclic-abelian-groups.md Example

Dependency tree · two levels

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Sources

10.7.2–10.7.5 and Exercise 10.7.1, pp. 399–400 (standard reference, not scraped)