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Gorenstein projective bimodules via monomorphism categories and filtration categories. (English) Zbl 1399.18005

Gorenstein projective modules over tensor products of finite dimensional algebras over a field \(k\) are studied, extending previous work on Gorenstein projectives over \(T_2\)-extensions [Z.-W. Li and P. Zhang, J. Algebra 323, No. 6, 1802–1812 (2010; Zbl 1210.16011)], Gorenstein projective quiver representations over dual numbers \(k[x]/(x^2)\) [C. M. Ringel and P. Zhang, J. Algebra 475, 327–360 (2017; Zbl 1406.16010)], and related work of [X.-H. Luo and P. Zhang, J. Algebra 479, 1–34 (2017; Zbl 1405.16022)], and others.
Of course, all these investigations are closely related to the invariant theory of monomorphisms in an abelian category, first studied by G. Birkhoff [Proc. Lond. Math. Soc. (2) 38, 385–401 (1934; JFM 60.0893.03)] who classifies subgroups of finite abelian groups with respect to a fixed basis. Accordingly, the recent investigations led to an intensified study of monomorphism categories, a class of exact categories which can be handled to a certain extent by Auslander-Reiten theory.
In the paper under review, the authors give a new definition of the monomorphism category over a finite dimensional algebra, extending previous variants based on acyclic quivers. On the other hand, the Gorenstein projectives over a tensor product of Gorenstein algebras are represented as a category of filtrations of tensor products of Gorenstein projectives.

MSC:

18G25 Relative homological algebra, projective classes (category-theoretic aspects)
16D20 Bimodules in associative algebras
16G10 Representations of associative Artinian rings
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References:

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