Covering classes, strongly flat modules, and completions

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 \( \mathcal {SF}\) 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 \( \mathcal {SF}\) a covering class implies \( \mathcal {SF}\) 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.

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Correspondence to Alberto Facchini.

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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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Facchini, A., Nazemian, Z. Covering classes, strongly flat modules, and completions. Math. Z. 296, 239–259 (2020). https://doi.org/10.1007/s00209-019-02417-3

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Keywords

  • Covering class
  • Strongly flat module
  • Completion
  • Cotorsion module
  • R-topology

Mathematics Subject Classification

  • Primary 16E30
  • 16W80
  • Secondary 18G15