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

TL;DR

This paper establishes new results on derived functors of $a$-adic completion, including an adic reduction to the diagonal in certain algebraic settings and a characterization of cofiniteness via derived completion, generalizing classical results.

Contribution

It introduces a generalized adic reduction to the diagonal and links cofiniteness of Ext modules to derived $a$-adic completion, extending Serre's classical results.

Findings

Adic reduction to the diagonal holds under specified conditions.

Cofiniteness of Ext modules is characterized by finitely generated cohomologies of derived completion.

Generalizes Serre's result to broader algebraic contexts.

Abstract

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 $A = K[[x_1,\dots,x_n]]$.