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Subprojectivity in Abelian Categories

Abstract

In the last few years, López-Permouth and several collaborators have introduced a new approach in the study of the classical projectivity, injectivity and flatness of modules. This way, they introduced subprojectivity domains of modules as a tool to measure, somehow, the projectivity level of such a module (so not just to determine whether or not the module is projective). In this paper we develop a new treatment of the subprojectivity in any abelian category which shed more light on some of its various important aspects. Namely, in terms of subprojectivity, some classical results are unified and some classical rings are characterized. It is also shown that, in some categories, the subprojectivity measures notions other than the projectivity. Furthermore, this new approach allows, in addition to establishing nice generalizations of known results, to construct various new examples such as the subprojectivity domain of the class of Gorenstein projective objects, the class of semi-projective complexes and particular types of representations of a finite linear quiver. The paper ends with a study showing that the fact that a subprojectivity domain of a class coincides with its first right Ext-orthogonal class can be characterized in terms of the existence of preenvelopes and precovers.

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Acknowledgements

The authors want to express their gratitude to the referee for the very helpful comments and suggestions. Part of this work was done while Amzil and Ouberka were visiting the University of Almería under the Erasmus+ KA107 scholarship grant from November 2019 to July 2020. They would like to express their sincere thanks for the warm hospitality and the excellent working conditions. Also, during the preparation of this paper, Bennis has been invited by the University of Almería. He is always thankful to the department of Mathematics for the warm hospitality during his stay in Almería. The authors Luis Oyonarte and J.R. García Rozas were partially supported by Ministerio de Economía y Competitividad, grant reference 2017MTM2017-86987-P.

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Correspondence to Driss Bennis.

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Amzil, H., Bennis, D., Rozas, J.R.G. et al. Subprojectivity in Abelian Categories. Appl Categor Struct 29, 889–913 (2021). https://doi.org/10.1007/s10485-021-09638-w

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Keywords

  • Subprojectivity
  • Subprojectivity domain
  • Precovers
  • Preenvelopes

Mathematics Subject Classification

  • 18G25