An Artin algebra is by definition virtually Gorenstein if the class of modules which are right orthogonal (with respect to Ext^*(-,-)) to all Gorenstein projective modules coincides with the class of modules which are left orthogonal to all Gorenstein injective modules. We provide a new characterization in terms of finitely generated modules. In addition, an example of an algebra is presented which is not virtually Gorenstein.
Let $Λ$ be an Auslander's 1-Gorenstein Artinian algebra with global dimension two. If $Λ$ admits a trivial maximal 1-orthogonal subcategory of $\modΛ$, then for any indecomposable module $M \in \mod Λ$, we have that the projective dimension of $M$ is equal to one if and only if so is its injective dimension and that $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $Λ$ is a tilted algebra.
The relation between the $n$-recollements of stable categories of Gorenstein projective modules and the virtual Gorensteinness of algebras are investigated. Let $A,B$, and $C$ be finite dimensional algebras. We prove that if the stable category of Gorenstein projective $A$-modules admits a $n$-recollement relative to that of $B$ and $C$ with $n\geq 2$, then $A$ is virtually Gorenstein if and only if so are $B$ and $C$.
We generalize the notions of $n$-cluster tilting subcategories and $τ$-selfinjective algebras into $n$-precluster tilting subcategories and $τ_n$-selfinjective algebras, where we show that a subcategory naturally associated to $n$-precluster tilting subcategories has a higher Auslander--Reiten theory. Furthermore, we give a bijection between $n$-precluster tilting subcategories and $n$-minimal Auslander--Gorenstein algebras, which is a higher dimensional analog of Auslander--Solberg correspondence (Auslander--Solberg, 1993) as well as a Gorenstein analog of $n$-Auslander correspondence (Iyama,…
Motivated by understanding the Nakayama conjecture which states that algebras of infinite dominant dimension should be self-injective, we study self-orthogonal modules with virtually Gorenstein endomorphism algebras and prove the following result: Given a finitely generated, self-orthogonal module over an Artin algebra with an orthogonal condition on its Nakayama translation, if its endomorphism algebra is virtually Gorenstein, then the module is projective. As a consequence, we re-obtain a recent result: the Nakayama conjecture holds true for the class of strongly Morita, virtually Gorenstein…
We review the theory of Co-Gorenstein algebras, which was introduced by Beligiannis in the article "The Homological Theory of Contravariantly Finite Subcategories: Gorenstein Categories, Auslander-Buchweitz Contexts and (Co-)Stabilization". We show a connection between Co-Gorenstein algebras and the Nakayama and Generalized Nakayama conjecture.
We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which the balanced pair of subcategories gives rise to equivalent homotopy categories of complexes. As an application, we prove that for a left-Gorenstein ring, there exists a triangle-equivalence between the homotopy category of its Gorenstein projective modules and the homotopy category of its Gorenstein injective modules, which restricts to a triangle-equivalence between the homotopy category of projective modules and the homotopy category of injective modules. In t…
We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\C$ of an abelian category $\A$, and prove that the right Gorenstein subcategory $r\mathcal{G}(\mathscr{C})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\C$ is self-orthogonal, we give a characterization for objects in $r\mathcal{G}(\mathscr{C})$, and prove that any object in $\A$ with finite $r\mathcal{G}(\C)$-projective dimension is isomorphic to a kernel (resp. a cokernel) of a morphism from an object in $\A$ with finite $\C$-projective dimension t…
For an $(n-1)$-Auslander algebra $Λ$ with global dimension $n$, we give some necessary conditions for $Λ$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $Λ$-modules with projective dimension $n-1$ or $n$. For an almost hereditary algebra $Λ$ with global dimension 2, we prove that $Λ$ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable $Λ$-module $M$, $M$ is injective is equivalent to that the reduced grade of $M$ is equal to 2. We give a connection between the Gorenstein Symmetric Conjecture and the existence o…
Dualities of resolving subcategories of finitely generated modules over Artin algebras are characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of these dualities to resolving subcategories of finitely generated modules with finite projective or Gorenstein-projective dimensions, Miyashita's duality and Huisgen-Zimmermann's correspondence on tilting modules as well as their Gorenstein version are obtained. Applications include constructing triangle equivalences of derived categories of finitely generated Gorenstein-projective modules and showing the invariance …
For a ring $R$ and an additive subcategory $\C$ of the category $\Mod R$ of left $R$-modules, under some conditions we prove that the right Gorenstein subcategory of $\Mod R$ and the left Gorenstein subcategory of $\Mod R^{op}$ relative to $\C$ form a coproduct-closed duality pair. Let $R,S$ be rings and $C$ a semidualizing ($R,S$)-bimodule. As applications of the above result, we get that if $S$ is right coherent and $C$ is faithfully semidualizing, then $(\mathcal{GF}_C(R),\mathcal{GI}_C(R^{op}))$ is a coproduct-closed duality pair and $\mathcal{GF}_C(R)$ is covering in $\Mod R$, where $\mat…
In this paper, by using functor rings and functor categories, we study finiteness and purity of subcategories of the module categories. We give a characterisation of contravariantly finite resolving subcategories of the module category of finite representation type in terms of their functor rings. We also characterize contravariantly finite resolving subcategories of the module category $Λ${\rm -mod} that contain the Jacobson radical of $Λ$ of finite type, by their functor categories. We study the pure semisimplicity conjecture for a locally finitely presented category ${\underrightarrow{\lim}…
Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite $p$-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of Cohen-Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of Cohen-Macau…
We show that an iteration of the procedure used to define the Gorenstein projective modules over a commutative ring $R$ yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective $R$-modules $G=...\xra{\partial^G_2}G_1\xra{\partial^G_1}G_0\xra{\partial^G_0} ...$ such that the complexes $\Hom_R(G,H)$ and $\Hom_R(H,G)$ are exact for each Gorenstein projective $R$-module $H$, the module $\coker(\partial^G_1)$ is Gorenstein projective. The proof of this result hinges upon our analysis of Gorenstein subcategories of abelian categories.
We study the properties of rings satisfying Auslander-type conditions. If an artin algebra $Λ$ satisfies the Auslander condition (that is, $Λ$ is an $\infty$-Gorenstein artin algebra), then we construct two kinds of subcategories which form functorially finite cotorsion pairs.
Gorenstein homological algebra is a kind of relative homological algebra which has been developed to a high level since more than four decades. In this report we review the basic theory of Gorenstein homological algebra of artin algebras. It is hoped that such a theory will help to understand the famous Gorenstein symmetric conjecture of artin algebras. With only few exceptions all the results in this report are contained in the existing literature. We have tried to keep the exposition as self-contained as possible. This report can be viewed as a preparation for learning the newly developed th…
We study Artin algebras $A$ and commutative Noetherian complete local rings $R$ in connection with the following decomposition property of Gorenstein-projective modules: $(*)$ any Gorenstein-projective module is a direct sum of finitely generated modules. We show that this direct decomposition property is related to the property of the algebra $A$, or the ring $R$, being (virtually) Gorenstein of finite Cohen-Macaquly type. Along the way we generalize classical results of Auslander and Ringel-Tachikawa from the early seventies, and results of Chen and Yoshino on the structure of Gorenstein-pro…
It is a well-known result of Auslander and Reiten that contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander--Reiten condition, namely contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ (of finitely gen…
We study Morita rings $Λ_{(ϕ,ψ)}=\bigl({smallmatrix} A &_AN_B_BM_A & B {smallmatrix}\bigr)$ in the context of Artin algebras from various perspectives. First we study covariant finite, contravariant finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms $ϕ$ and $ψ$ are zero. Further we give bounds for the global dimension of a Morita ring $Λ_{(0,0)}$, regarded as an Artin algebra, in terms of the global dimensions of $A$ and $B$ in the case when both $ϕ$ and $ψ$ are zero. We illustrate our bounds with some examples. Finally we inve…
Let $Q$ be a finite quiver and $Λ$ be the radical square zero algebra of $Q$ over a field. We give a full and dense functor from the category of reduced differential projective modules over $Λ$ to the category of representations of the opposite of $Q$. If moreover $Q$ has oriented cycles and $Q$ is not a basic cycle, we prove that the algebra of dual numbers over $Λ$ is not virtually Gorenstein.
We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations. Our applications focus on displaying several recollements that glue together various full subcategories of K(R), the homotopy category of chain complexes of modules over a general ring R. When R is (left) Noetherian ring, these recollements involve complexes built from the Gorenstein injective modules. When R is a (left) coherent ring for which all flat modu…
