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

Abstract:

We prove two results about the derived functor of $\mathfrak {a}$-adic completion: (1) Let $\mathbb {K}$ be a commutative noetherian ring, let $A$ be a flat noetherian $\mathbb {K}$-algebra which is $\mathfrak {a}$-adically complete with respect to some ideal $\mathfrak {a}\subseteq A$, such that $A/\mathfrak {a}$ is essentially of finite type over $\mathbb {K}$, and let $M,N$ be finitely generated $A$-modules. Then adic reduction to the diagonal holds: $A\otimes ^{\mathrm {L}}_{ A\widehat {\otimes }_{\mathbb {K}} A } (M\widehat {\otimes }^{\mathrm {L}}_{\mathbb {K}} N ) \cong M \otimes ^{\mathrm {L}}_AN$. A similar result is given in the case where $M,N$ are not necessarily finitely generated. (2) Let $A$ be a commutative ring, let $\mathfrak {a}\subseteq A$ be a weakly proregular ideal, let $M$ be an $A$-module, and assume that the $\mathfrak {a}$-adic completion of $A$ is noetherian (if $A$ is noetherian, all these conditions are always satisfied). Then $\textrm {Ext}^i_A(A/\mathfrak {a},M)$ is finitely generated for all $i\ge 0$ if and only if the derived $\mathfrak {a}$-adic completion $\mathrm {L}\widehat {\Lambda }_{\mathfrak {a}}(M)$ 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 $\mathbb {K}$ is a field or a discrete valuation ring and $A = \mathbb {K}[[x_1,\dots ,x_n]]$.