Lemma 22.16.2 (09K1)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.16: Projective modules and differential graded algebras
Lemma 22.16.2 (cite)

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Lemma 22.16.2. Let $(A, d)$ be a differential graded algebra. Then we have

$\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}_{(A, \text{d})}}(A[k], M) = \mathop{\mathrm{Ker}}(\text{d} : M^{-k} \to M^{-k + 1})$

and

$\mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(A[k], M) = H^{-k}(M)$

for any differential graded $A$ -module $M$ .

Proof. Immediate from the definitions. $\square$

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