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

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: $A{\otimes }_{A{\hat{\otimes }}_{K}A}^L(M{\hat{\otimes }}_K^{L}N)\cong M{\otimes }_A^{L}N$. 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 $Ł{\hat{\Lambda }}_a(M)$ 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 $A=K[[x_1,\dots ,x_n]]$.

12 pages. Final version, to appear in Proceedings of the AMS

Cited by in corpus (3)

• Injective DG-modules over non-positive DG-rings
• Completion and torsion over commutative DG rings
• The homotopy theory of complete modules