No Arabic abstract
Let $T$ be a tilting module. In this paper, Gorenstein $pi[T]$-projective modules are introduced and some of their basic properties are studied. Moreover, some characterizations of rings over which all modules are Gorenstein $pi[T]$-projective are given. For instance, on the $T$-cocoherent rings, it is proved that the Gorenstein $pi[T]$-projectivity of all $R$-modules is equivalent to the $pi[T]$-projectivity of $sigma[T]$-injective as a module.
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.
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.
Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $Gamma. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulays correspondence characterizing the $R$-submodules of $Gamma $ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein $k$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed.
Let $mathfrak{a}$ be an ideal of a noetherian (not necessarily local) ring $R$ and $M$ an $R$-module with $mathrm{Supp}_RMsubseteqmathrm{V}(mathfrak{a})$. We show that if $mathrm{dim}_RMleq2$, then $M$ is $mathfrak{a}$-cofinite if and only if $mathrm{Ext}^i_R(R/mathfrak{a},M)$ are finitely generated for all $ileq 2$, which generalizes one of the main results in [Algebr. Represent. Theory 18 (2015) 369--379]. Some new results concerning cofiniteness of local cohomology modules $mathrm{H}^i_mathfrak{a}(M)$ for any finitely generated $R$-module $M$ are obtained.
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.