Adic reduction to the diagonal and a relation between cofiniteness and derived completion

We prove two results about the derived functor of $a$-adic completion: (1) Let $K$ be a commutative noetherian ring, let $A$ be a flat noetherian $K$-algebra which is $a$-adically complete with respect to some ideal $a\subseteq A$, such that $A\mathrm{/}a$ is essentially of finite type over $K$, and let $M,N$ be finitely generated $A$-modules. Then adic reduction to the diagonal holds: $A\otimes^{L}{ A\hat{\otimes}{K} A } ( M\hat{\otimes}^{L}_{K} N ) \cong M \otimes^{L}_A N$\mathrm{.}Asimilarresultisgiveninthecasewhere$M,N$ are not necessarily finitely generated. (2) Let $A$ be a commutative ring, let $a\subseteq A$ be a weakly proregular ideal, let $M$ be an $A$-module, and assume that the $a$-adic completion of $A$ is noetherian (if $A$ is noetherian, all these conditions are always satisfied). Then $Ex{t}_A^i(A\mathrm{/}a,M)$ is finitely generated for all $i\ge 0$ if and only if the derived $a$-adic completion $\text{L}{\widehat{\mathrm{\Lambda }}}_a(M)$ has finitely generated cohomologies over $\widehat{A}$. The first result is a far reaching generalization of a result of Serre, who proved this in case $K$ is a field or a discrete valuation ring and $A=K[[x_1,\dots ,x_n]]$.