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Products of flat modules and global dimension relative to $ \mathcal F$-Mittag-Leffler modules


Author: Manuel Cortés-Izurdiaga
Journal: Proc. Amer. Math. Soc. 144 (2016), 4557-4571
MSC (2010): Primary 16D40, 16E10
DOI: https://doi.org/10.1090/proc/13059
Published electronically: July 21, 2016
MathSciNet review: 3544508
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Abstract: Let $ R$ be any ring. We prove that all direct products of flat right $ R$-modules have finite flat dimension if and only if each finitely generated left ideal of $ R$ has finite projective dimension relative to the class of all $ \mathcal F$-Mittag-Leffler left $ R$-modules, where $ \mathcal F$ is the class of all flat right $ R$-modules. In order to prove this theorem, we obtain a general result concerning global relative dimension. Namely, if $ \mathcal X$ is any class of left $ R$-modules closed under filtrations that contains all projective modules, then $ R$ has finite left global projective dimension relative to $ \mathcal X$ if and only if each left ideal of $ R$ has finite projective dimension relative to $ \mathcal X$. This result contains, as particular cases, the well-known results concerning the classical left global, weak and Gorenstein global dimensions.


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Manuel Cortés-Izurdiaga
Affiliation: Departamento de Matemáticas, University of Almeria, E-04071, Almeria, Spain
Email: mizurdia@ual.es

DOI: https://doi.org/10.1090/proc/13059
Keywords: Products of flat modules, global dimension, global dimension relative to Mittag-Leffler modules
Received by editor(s): August 4, 2014
Published electronically: July 21, 2016
Additional Notes: Part of this paper was written while the author was visiting the School of Mathematics at the University of Manchester. The author is very grateful to Mike Prest for his hospitality and for many interesting discussions on the subject
The author was partially supported by research project MTM-2014-54439 and by research group “Categorías, computación y teoría de anillos” (FQM211) of the University of Almería
Communicated by: Harm Derksen
Article copyright: © Copyright 2016 American Mathematical Society