Weak proregularity, weak stability, and the noncommutative MGM equivalence

JOURNAL OF ALGEBRA

卷:513

Article

Vyas, Rishi¹  Yekutieli, Amnon¹ 

[1] Ben Gurion Univ Negev, Dept Math, IL-84105 Beer Sheva, Israel

关键词: Torsion classes;    Derived torsion;    Derived completion;   

DOI  :  10.1016/j.jalgebra.2018.07.023

来源: Elsevier

【 摘 要 】

Let A be a commutative ring, and let a be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived alpha-torsion and alpha-adic completion functors to be nicely behaved is the weak proregularity of alpha. In particular, the MGM Equivalence holds. Because weak proregularity is defined in terms of elements of the ring (it involves limits of Koszul complexes), it is not suitable for noncommutative ring theory. In this paper we introduce a new condition on a torsion class T in a module category: weak stability. Our first main theorem says that in the commutative case, the ideal a is weakly proregular if and only if the corresponding torsion class T is weakly stable. We then study weak stability of torsion classes in module categories over noncommutative rings. There are three main theorems in this context: (sic) For a torsion class T that is weakly stable, quasi-compact and finite dimensional, the right derived torsion functor is isomorphic to a left derived tensor functor. (sic) The Noncommutative MGM Equivalence, that holds under the same assumptions on T. (sic) A theorem about symmetric derived torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and corrects an error in a paper of Yekutieli & Zhang from 2003. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free