ADIC REDUCTION TO THE DIAGONAL AND A RELATION BETWEEN COFINITENESS AND DERIVED COMPLETION

作者：Shaul Liran^*

来源：PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2017, 145(12): 5131-5143.

DOI：10.1090/proc/13662

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 subset of 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 circle times(AL)(A (circle times) over capK) (M (circle times) over cap (L)(K) N) similar or equal to M circle times(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 subset of 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 Exti A (A/a, M) is finitely generated for all i >= 0 if and only if the derived a-adic completion L (Lambda) over cap (a)(M) has finitely generated cohomologies over (Lambda) over cap. 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),..., x(n)]].

• 出版日期2017-12