Lemma 22.26.10 (09LE)—The Stacks project

Lemma 22.26.10. Let $R$ be a ring. Let $\mathcal{A}$ be a differential graded category over $R$ . Let $x$ be an object of $\mathcal{A}$ . Let

$(E, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, x)$

be the differential graded $R$ -algebra of endomorphisms of $x$ . We obtain a functor

$\mathcal{A} \longrightarrow \text{Mod}^{dg}_{(E, \text{d})},\quad y \longmapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y)$

of differential graded categories by letting $E$ act on $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y)$ via composition in $\mathcal{A}$ . This functor induces functors

$\text{Comp}(\mathcal{A}) \to \text{Mod}_{(A, \text{d})} \quad \text{and}\quad K(\mathcal{A}) \to K(\text{Mod}_{(A, \text{d})})$

by an application of Lemma 22.26.5.

Proof. This lemma proves itself. $\square$

The Stacks project

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