Gorenstein Projective Objects in Functor categories

Abstract

Let k be a commutative ring, let C be a small, k-linear, Hom-finite, locally bounded category, and let B be a k-linear abelian category. We construct a Frobenius exact subcategory GP(GPP (B C )) of the functor category B C , and we show that it is a subcategory of the Gorenstein projective objects GP(B C ) in B C . Furthermore, we obtain criteria for when GP(GPP (B C )) = GP(B C ). We show in examples that this can be used to compute GP(B C ) explicitly.

A functorial approach to monomorphism categories for species I

“…Furthermore, analogues of the top and socle functor are introduced in the abstract setting. Section 6 is devoted to the proof of Theorem C using the theory of adjunctions with Nakayama functors introduced in [Kva17]. Furthermore, we show that in our setup the category of (relative) Gorenstein projective objects can be described as a generalisation of the monomorphism category.…”

“…Recently, these results by Ringel and Schmidmeier as well as the connections to Gorenstein projective modules have been further explored, see e.g. [Zha11, XZ12, LZ13, XZZ14, RZ17, Kül17, Kva17,Kva18]. Noting that H(A) ∼ = mod(A ⊗ kA 2 ) ∼ = mod ( A A 0 A ), where A 2 denotes the Dynkin quiver A 2 , the generalisations are in two directions: Firstly, they concern monomorphism categories for algebras of the form A ⊗ kQ where Q is a finite quiver, or more generally categories of functors Q → mod A where Q is a locally bounded quiver, and secondly monomorphism subcategories in module categories of algebras of the form ( A X 0 B ), where A and B is an Artin algebra and X is a B-A-bimodule.…”

“…This approach generalises the theory of monomorphism categories to functor categories, similarly to [Kva17]. However, it does not apply to categories of generalised species like modules over the matrix ring ( A X 0 B ).…”

A Bijection theorem for Gorenstein projective τ-tilting modules

“…From then on, Gorenstein homological algebra became fruitful. For the recent work on Gorenstein projective modules over finite dimensional algebras, we refer to [CSZ,HuLXZ,K,RZ1,RZ2].…”