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

Author: Liran Shaul
Journal: Proc. Amer. Math. Soc. 145 (2017), 5131-5143
MSC (2010): Primary 13B35, 13C12, 13H15
Published electronically: June 8, 2017
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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]]$.

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Liran Shaul
Affiliation: Departement Wiskunde-Informatica, Universiteit Antwerpen, Middelheim Campus, Middelheimlaan 1, 2020 Antwerp, Belgium
Address at time of publication: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

Received by editor(s): February 23, 2016
Received by editor(s) in revised form: October 24, 2016, December 6, 2016, and January 10, 2017
Published electronically: June 8, 2017
Additional Notes: The author acknowledges the support of the European Union via ERC grant No. 257004-HHNcdMir.
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2017 American Mathematical Society

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