Remark 52.6.15 (Completed tensor product). Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Denote $K \mapsto K^\wedge$ the adjoint of Proposition 52.6.12. Then we set

$K \otimes ^\wedge _\mathcal {O} L = (K \otimes _\mathcal {O}^\mathbf {L} L)^\wedge$

This completed tensor product defines a functor $D_{comp}(\mathcal{O}) \times D_{comp}(\mathcal{O}) \to D_{comp}(\mathcal{O})$ such that we have

$\mathop{\mathrm{Hom}}\nolimits _{D_{comp}(\mathcal{O})}(K, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D_{comp}(\mathcal{O})}(K \otimes _\mathcal {O}^\wedge L, M)$

for $K, L, M \in D_{comp}(\mathcal{O})$. Note that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, M) \in D_{comp}(\mathcal{O})$ by Lemma 52.6.5.

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