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

Abstract

© 2017 American Mathematical Society. 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 ⊆ 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: (Formula presented). 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 ⊆ 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 ExtiA(A/a, M) is finitely generated for all i ≥ 0 if and only if the derived a-adic completion LΛa(M) has finitely generated cohomologies over Â. 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[[x1,⋯, xn]].