Lemma 15.87.9. Let $(A_ n)$ be an inverse system of rings. Given $K, L \in D(\textit{Mod}(\mathbf{N}, (A_ n)))$ there is a canonical derived tensor product $K \otimes ^\mathbf {L} L$ in $D(\mathbf{N}, (A_ n))$ compatible with the maps to $D(A_ n)$. The construction is symmetric in $K$ and $L$ and an exact functor of triangulated categories in each variable.
Proof. Choose a representative $(K_ n^\bullet )$ for $K$ such that each $K_ n^\bullet $ is a K-flat complex (Lemma 15.87.8). Then you can define $K \otimes ^\mathbf {L} L$ as the object represented by the system of complexes
\[ (\text{Tot}(K_ n^\bullet \otimes _{A_ n} L_ n^\bullet )) \]
for any choice of representative $(L_ n^\bullet )$ for $L$. This is well defined in both variables by Lemmas 15.59.2 and 15.59.12. Compatibility with the map to $D(A_ n)$ is clear. Exactness follows exactly as in Lemma 15.58.4. $\square$
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