Lemma 24.18.1 (0FRU)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 24: Differential Graded Sheaves
Section 24.18: Pull and push for sheaves of differential graded modules
Lemma 24.18.1 (cite)

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Lemma 24.18.1. In the situation above we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{B}, \text{d})}( \mathcal{N}, f_*\mathcal{M}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}( f^*\mathcal{N}, \mathcal{M}) \]

Proof. Omitted. Hints: This is true for the underlying graded categories by Lemma 24.9.1. A calculation shows that these isomorphisms are compatible with differentials. $\square$

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