  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 $A/a$ is essentially of finite type over $K$, and let
$M,N$ be finitely generated $A$-modules. Then adic reduction to the diagonal
holds: $A\otimes^{L}_{ A\hat{\otimes}_{K} A } ( M\hat{\otimes}^{L}_{K} N )
\cong M \otimes^{L}_A N$. A similar result is given in the case where $M,N$ 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 $\mbox{Ext}^i_A(A/a,M)$ is
finitely generated for all $i\ge 0$ if and only if the derived $a$-adic
completion $\L\hat{\Lambda}_{a}(M)$ has finitely generated cohomologies over
$\hat{A}$. 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
$A = K[[x_1,\dots,x_n]]$.
