The Stacks project

22.19 Injective modules and differential graded algebras

If $(A, \text{d})$ is a differential graded algebra and $I$ is an object of $\text{Mod}_{(A, \text{d})}$ then we say $I$ is injective as a graded $A$-module or sometimes $I$ is graded injective to mean that $I$ is a injective object of the abelian category $\text{Mod}_ A$ of graded $A$-modules.

Lemma 22.19.1. Let $(A, \text{d})$ be a differential graded algebra. Let $I \to M$ be an injective homomorphism of differential graded $A$-modules. If $I$ is graded injective, then $I \to M$ is an admissible monomorphism.

Proof. This is immediate from the definitions. $\square$

Let $(A, \text{d})$ be a differential graded algebra. If $M$ is a left, resp. right differential graded $A$-module, then

\[ M^\vee = \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , \mathbf{Q}/\mathbf{Z}) \]

with $A$-module structure constructed in Section 22.18 is a right, resp. left differential graded $A$-module by the discussion in Section 22.13. By Remarks 22.13.5 and 22.13.6 there evaluation map of Section 22.18

\[ M \longrightarrow (M^\vee )^\vee \]

is a homomorphism of left, resp. right differential graded $A$-modules

Lemma 22.19.2. Let $(A, \text{d})$ be a differential graded algebra. If $M$ is a left differential graded $A$-module and $N$ is a right differential graded $A$-module, then

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}_{(A, \text{d})}}(N, M^\vee ) & = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}(\mathbf{Z})}(N \otimes _ A M, \mathbf{Q}/\mathbf{Z}) \\ & = \text{DifferentialGradedBilinear}_ A(N \times M, \mathbf{Q}/\mathbf{Z}) \end{align*}

Proof. The first equality is Lemma 22.13.2 and the second equality was shown in Section 22.12. $\square$

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

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


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

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

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

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