Abstract
Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .
Highlights
Auslander and Bridger [1] generalized finitely generated projective modules to modules of Gorenstein dimension zero over two-sided Noetherian rings
We describe all complete projective resolutions over an upper triangular
Enochs and Jenda [2] generalized it to an arbitrary ring and called it Gorenstein-projective modules
Summary
Auslander and Bridger [1] generalized finitely generated projective modules to modules of Gorenstein dimension zero over two-sided Noetherian rings. Enochs and Jenda [2] generalized it to an arbitrary ring and called it Gorenstein-projective modules. After that, this class of modules got special attention and has been studied by many authors (see, e.g., [3–15]). Is gives us a strong motivation to study Gorenstein-projective modules over upper triangular matrix Artin algebras. E main aim of this paper is to describe all the complete projective resolutions and all finitely generated Gorensteinprojective modules over an upper triangular matrix Artin algebra. Roughout the paper, A − Mod denotes the category of left A-modules and A − mod denotes the category of finitely generated left A-modules for a ring A
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