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Register a new userStability of Gorenstein flat categories with respect to a semidualizing module
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In this paper, we first introduce $mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: $xymatrix@C=80pt{mathcal {G}(mathcal {F})cap mathcal {A}_C(R) ar@<0.5ex>[r]^{Cotimes_R-} & mathcal {G}(mathcal {W}_F) ar@<0.5ex>[l]^{textrm{Hom}_R(C,-)}} $ where $mathcal {G}(mathcal {F})$, $mathcal {A}_C(R) $ and $mathcal {G}(mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $mathcal {W}_F$-Gorenstein modules respectively. Then, we investigate two-degree $mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $mathcal {W}_F$-Gorenstein if there exists an exact sequence $mathbb{G}_bullet=indent ...longrightarrow G_1longrightarrow G_0longrightarrow G^0longrightarrow G^1longrightarrow...$ in $mathcal {G}(mathcal {W}_F)$ such that $M cong$ $im(G_0rightarrow G^0) $ and that $mathbb{G}_bullet$ is Hom$_R(mathcal {G}(mathcal {W}_F),-)$ and $mathcal {G}(mathcal {W}_F)^+otimes_R-$ exact. We show that two notions of the two-degree $mathcal {W}_F$-Gorenstein and the $mathcal {W}_F$-Gorenstein modules coincide when R is a commutative GF-closed ring.
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We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring which is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo-Gillespie-Hovey. It is also compatible with the notion of $n$-coherent rings introduced by Bravo-Perez: So a $0$-coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$-coherent AC-Gorenstein ring is precisely a Ding-Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects the Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects the Gorenstein AC-injectives.
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We generalize the definition of an exact sequence of tensor categories due to Brugui`eres and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this notion and show their equivalence. In particular, the Deligne tensor product of tensor categories gives rise to an exact sequence in our sense. We also show that the dual to an exact sequence in our sense is again an exact sequence. This generalizes the corresponding statement for exact sequences of Hopf algebras. Finally, we show that the middle term of an exact sequence is semisimple if so are the other two terms.
In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs along the recollement $(mathcal{A},mathcal{C},mathcal{B})$ of abelian categories with enough projective and injective objects. As a consequence, we investigate how to establish recollements of triangulated categories from recollements of abelian categories by using the theory of exact model structures. Finally, we give applications to contraderived categories, projective stable derived categories and stable categories of Gorenstein injective modules over an upper triangular matrix ring.
A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring $R$, then $R$ is Gorenstein. In this paper we investigate some homological dimensions involving a semidualizing complex and improve on Foxbys result by answering a question of Takahashi and White. In particular, we prove for a semidualizing complex $C$, if there exists a complex with finite depth, finite $mathcal{F}_C$-projective dimension, and finite $mathcal{I}_C$-injective dimension over a local ring $R$, then $R$ is Gorenstein.
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