Lemma 22.19.3 (09K4)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.19: Injective modules and differential graded algebras
Lemma 22.19.3 (cite)

previous lemma

Lemma 22.19.3. Let $(A, \text{d})$ be a differential graded algebra. Then we have

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

and

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

as functors in the differential graded $A$-module $M$.

Proof. This is clear from the discussion above. $\square$

Comments (2)

Comment #535 by m.o. on April 07, 2014 at 08:52

Just typos: $\textrm{Hom}_{K(\text{Mod}_{(A, \text{d})})}(A[k], M) = H^k(M^\vee)$ should be $\textrm{Hom}_{K(\text{Mod}_{(A, \text{d})})}(M, A^\vee[k]) = H^k(M^\vee)$

Comment #547 by Johan on April 09, 2014 at 00:30

Thanks, fixed here.

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