The Stacks project

Lemma 24.19.1. In the situation above we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}( j_!\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}_ U, \text{d})}( \mathcal{M}, j^*\mathcal{N}) \]

Proof. Omitted. Hint: We have seen in Lemma 24.10.1 that the lemma is true on graded level. Thus all that needs to be checked is that the resulting isomorphism is compatible with differentials. $\square$

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