Covering classes, strongly flat modules, and completions
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Mathematische Zeitschrift
Covering classes, strongly flat modules, and completions Alberto Facchini1 · Zahra Nazemian2 Received: 5 September 2018 / Accepted: 5 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring R that coincides with the R-topology defined by Matlis when R is commutative. (2) We consider the class SF of strongly flat modules when R is a right Ore domain with classical right quotient ring Q. Strongly flat modules are flat. The completion of R in its R-topology is a strongly flat R-module. (3) We prove some results related to the question whether SF a covering class implies SF closed under direct limits. This is a particular case of the so-called Enochs’ Conjecture (whether covering classes are closed under direct limits). Some of our results concern right chain domains. For instance, we show that if the class of strongly flat modules over a right chain domain R is covering, then R is right invariant. In this case, flat R-modules are strongly flat. Keywords Covering class · Strongly flat module · Completion · Cotorsion module · R-topology Mathematics Subject Classification Primary 16E30 · 16W80; Secondary 18G15
1 Introduction The aim of this paper is to highlight some relations between completions, strongly flat modules and perfect rings in the non-commutative case. We explore the connections between some notions of Homological Algebra (cotorsion modules) and topological rings (completions in
The first author was partially supported by Dipartimento di Matematica “Tullio Levi-Civita” of Università di Padova (Project BIRD163492/16 “Categorical homological methods in the study of algebraic structures” and Research program DOR1714214 “Anelli e categorie di moduli”). The second author was supported by a Grant from IPM.
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Alberto Facchini [email protected] Zahra Nazemian [email protected]
1
Dipartimento di Matematica, Università di Padova, 35121 Padua, Italy
2
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
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A. Facchini, Z. Nazemian
some natural topologies). These connections are well known for modules over commutative rings, thanks to Matlis, who proved that the completion in the R-toplogy for an integral domain R is closely related to the cotorsion completion functor Ext1R (K , −). Here Q is the field of fractions of R and K := Q/R [20]. We investigate these connections in the non-commutative case, defining a suitable R-topology on any module over a not-necessarily commutative ring R. This leads us to the study of strongly flat modules, because the completion of R in its R-topology turns out to be a strongly flat R-module (Corollary 5.13). We consider strongly flat modules over non-commutative rings as defined in [13, Sect. 3]. The class of strongly flat modules lies between the class of projective modules and the class of flat modules. In partic
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