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Let $R$ be an Artin algebra. Under certain Auslander-type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander-type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if $R$ is left quasi Auslander, then $R$ is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if $R$ satisfies the Auslander condition, then $R$ is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander--Reiten's conjecture, which states that $R$ is Gorenstein if $R$ satisfies the Auslander condition., Comment: 16 pages; accepted for publication in Bulletin of the London Mathematical Society

We explore the assignment of norms to $\Lambda$-modules over a finite-dimensional algebra $\Lambda$, resulting in the establishment of normed $\Lambda$-modules. Our primary contribution lies in constructing a new category $\mathscr{N}\!\!or^p$ related to normed modules along with its full subcategory $\mathscr{A}^p$. By examining the objects and morphisms in these categories, we establish a framework for understanding the categorification of integration, series expansions and derivatives. Furthermore, we obtain the Stone--Weierstrass approximation theorem in the sense of $\mathscr{A}^p$., Comment: 36 pages, 1 figures

Let $A=kQ/I$ be a finite dimensional basic algebra over an algebraically closed field $k$ which is a gentle algebra with the marked ribbon surface $(\mathcal{S}_A,\mathcal{M}_A,\Gamma_A)$. It is known that $\mathcal{S}_A$ can be divided into some elementary polygons $\{\Delta_i\mid 1\le i\le d\}$ by $\Gamma_A$ which has exactly one side in the boundary of $\mathcal{S}_A$. Let $\mathfrak{C}(\Delta_i)$ be the number of sides of $\Delta_i$ belonging to $\Gamma_A$ if the unmarked boundary component of $\mathcal{S}_A$ is not a side of $\Delta_i$; otherwise, $\mathfrak{C}(\Delta_i)=\infty$, and let $\mathsf{f}\text{-}\Delta$ be the set of all non-$\infty$-elementary polygons and $\mathcal{F}_A$ (respectively, ${\mathsf{f}\text{-}\mathcal{F}}_A$) the set of all forbidden threads (respectively, of finite length). Then we have \begin{enumerate} \item[{\rm (1)}] The global dimension of $A=\max\limits_{1\leq i\leq d}{\mathfrak{C}(\Delta_i)}-1 =\max\limits_{\mathit{\Pi}\in\mathcal{F}_A} l(\mathit{\Pi})$, where $l(\mathit{\Pi})$ is the length of $\mathit{\Pi}$. \item[{\rm (2)}] The left and right self-injective dimensions of $A=$ \begin{center} $\begin{cases} 0,\ \mbox{\text{if {\it Q} is either a point or an oriented cycle with full relations};}\\ \max\limits_{\Delta_i\in{\mathsf{f}\text{-}\Delta}}\big\{1, {\mathfrak{C}(\Delta_i)}-1 \big\}= \max\limits_{\mathit{\Pi}\in{\mathsf{f}\text{-}\mathcal{F}}_A} l(\mathit{\Pi}),\ \mbox{\text{otherwise}.} \end{cases}$ \end{center} \end{enumerate} As a consequence, we get that the finiteness of the global dimension of gentle algebras is invariant under AG-equivalence. In addition, we get that the number of indecomposable non-projective Gorenstein projective modules over gentle algebras is also invariant under AG-equivalence., Comment: 39 pages, accepted for publication in Science China Mathematics

For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $\Lambda$, we give the upper bounds of the extension dimension of $\Lambda$ in terms of the radical layer length of $\Lambda$ and certain relative projective (or injective) dimension of some simple $\Lambda$-modules, from which some new upper bounds of the derived dimension of $\Lambda$ are induced., Comment: Journal of Algebra, 606 (2022), 243--265

Let $\mathscr{A}$ be an abelian category having enough projective and injective objects, and let $\mathscr{T}$ be an additive subcategory of $\mathscr{A}$ closed under direct summands. A known assertion is that in a short exact sequence in $\mathscr{A}$, the $\mathscr{T}$-projective (respectively, $\mathscr{T}$-injective) dimensions of any two terms can sometimes induce an upper bound of that of the third term by using the same comparison expressions. We show that if $\mathscr{T}$ contains all projective (respectively, injective) objects of $\mathscr{A}$, then the above assertion holds true if and only if $\mathscr{T}$ is resolving (respectively, coresolving). As applications, we get that a left and right Noetherian ring $R$ is $n$-Gorenstein if and only if the Gorenstein projective (respectively, injective, flat) dimension of any left $R$-module is at most $n$. In addition, in several cases, for a subcategory $\mathscr{C}$ of $\mathscr{T}$, we show that the finitistic $\mathscr{C}$-projective and $\mathscr{T}$-projective dimensions of $\mathscr{A}$ are identical., Comment: 28 pages, accepted for publication in Forum Mathematicum

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 $\mathcal{G}\mathcal{F}_C(R)$ is the subcategory of $\Mod R$ consisting of $C$-Gorenstein flat modules and $\mathcal{G}\mathcal{I}_C(R^{op})$ is the subcategory of $\Mod R^{op}$ consisting of $C$-Gorenstein injective modules; we also get that if $S$ is right coherent, then $(\mathcal{A}_C(R^{op}),l\mathcal{G}(\mathcal{F}_C(R)))$ is a coproduct-closed and product-closed duality pair and $\mathcal{A}_C(R^{op})$ is covering and preenveloping in $\Mod R^{op}$, where $\mathcal{A}_C(R^{op})$ is the Auslander class in $\Mod R^{op}$ and $l\mathcal{G}(\mathcal{F}_C(R))$ is the left Gorenstein subcategory of $\Mod R$ relative to $C$-flat modules.

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 to an object in $r\mathcal{G}(\C)$. As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in $\A$ having enough injectives.

Let $(\mathfrak{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear extriangulated category with $k$ a commutative artinian ring. We define an additive subcategory $\mathfrak{C}_r$ (respectively, $\mathfrak{C}_l$) of $\mathfrak{C}$ in terms of the representable functors from the stable category of $\mathfrak{C}$ modulo $\mathfrak{s}$-injectives (respectively, $\mathfrak{s}$-projectives) to $k$-modules, which consists of all $\mathfrak{s}$-projective (respectively, $\mathfrak{s}$-injective) objects and objects isomorphic to direct summands of finite direct sums of all third (respectively, first) terms of almost split $\mathfrak{s}$-triangles. We investigate the subcategories $\mathfrak{C}_r$ and $\mathfrak{C}_l$ in terms of morphisms determined by objects, and then give equivalent characterizations on the existence of almost split $\mathfrak{s}$-triangles., Comment: arXiv admin note: text overlap with arXiv:1612.02591 by other authors

Let $\Lambda,\Gamma$ be rings and $R=\left(\begin{array}{cc}\Lambda & 0 \\ M & \Gamma\end{array}\right)$ the triangular matrix ring with $M$ a $(\Gamma,\Lambda)$-bimodule. Let $X$ be a right $\Lambda$-module and $Y$ a right $\Gamma$-module. We prove that $(X, 0)$$\oplus$$(Y\otimes_\Gamma M, Y)$ is a silting right $R$-module if and only if both $X_{\Lambda}$ and $Y_{\Gamma}$ are silting modules and $Y\otimes_\Gamma M$ is generated by $X$. Furthermore, we prove that if $\Lambda$ and $\Gamma$ are finite dimensional algebras over an algebraically closed field and $X_{\Lambda}$ and $Y_{\Gamma}$ are finitely generated, then $(X, 0)$$\oplus$$(Y\otimes_\Gamma M, Y)$ is a support $\tau$-tilting $R$-module if and only if both $X_{\Lambda}$ and $Y_{\Gamma}$ are support $\tau$-tilting modules, $\Hom_\Lambda(Y\otimes_\Gamma M,\tau X)=0$ and $\Hom_\Lambda(e\Lambda, Y\otimes_\Gamma M)=0$ with $e$ the maximal idempotent such that $\Hom_\Lambda(e\Lambda, X)=0$., Comment: 17 pages, accepted for publication in Taiwanese Journal of Mathematics

Let $\Gamma$ be a split extension of a finite-dimensional algebra $\Lambda$ by a nilpotent bimodule $_\Lambda E_\Lambda$, and let $(T,P)$ be a pair in $\mod\Lambda$ with $P$ projective. We prove that $(T\otimes_\Lambda \Gamma_\Gamma, P\otimes_\Lambda \Gamma_\Gamma)$ is a support $\tau$-tilting pair in $\mod \Gamma$ if and only if $(T,P)$ is a support $\tau$-tilting pair in $\mod \Lambda$ and $\Hom_\Lambda(T\otimes_\Lambda E,\tau T_\Lambda)=0=\Hom_\Lambda(P,T\otimes_\Lambda E)$. As applications, we obtain a necessary and sufficient condition such that $(T\otimes_\Lambda \Gamma_\Gamma, P\otimes_\Lambda \Gamma_\Gamma)$ is support $\tau$-tilting pair for a cluster-tilted algebra $\Gamma$ corresponding to a tilted algebra $\Lambda$; and we also get that if $T_1,T_2\in\mod\Lambda$ such that $T_1\otimes_\Lambda \Gamma$ and $T_2\otimes_\Lambda \Gamma$ are support $\tau$-tilting $\Gamma$-modules, then $T_1\otimes_\Lambda \Gamma$ is a left mutation of $T_2\otimes_\Lambda \Gamma$ if and only if $T_1$ is a left mutation of $T_2$.

Let $\Lambda$ be an artin algebra. We give an upper bound for the dimension of the bounded derived category of the category $\mod \Lambda$ of finitely generated right $\Lambda$-modules in terms of the projective and injective dimensions of certain class of simple right $\Lambda$-modules as well as the radical layer length of $\Lambda$. In addition, we give an upper bound for the dimension of the singularity category of $\mod \Lambda$ in terms of the radical layer length of $\Lambda$.

Given an additive category $\mathcal{C}$ and an integer $n\geqslant 2$. We form a new additive category $\mathcal{C}[\epsilon]^n$ consisting of objects $X$ in $\mathcal{C}$ equipped with an endomorphism $\epsilon_X$ satisfying ${\epsilon^n_X}=0$. First, using the descriptions of projective and injective objects in $\mathcal{C}[\epsilon]^n$, we not only establish a connection between Gorenstein flat modules over a ring $R$ and $R[t]/(t^n)$, but also prove that an Artinian algebra $R$ satisfies some homological conjectures if and only if so does $R[t]/(t^n)$. Then we show that the corresponding homotopy category $\K(\mathcal{C}[\epsilon]^n)$ is a triangulated category when $\mathcal{C}$ is an idempotent complete exact category. Moreover, under some conditions for an abelian category $\mathcal{A}$, the natural quotient functor $Q$ from $\K(\mathcal{A}[\epsilon]^n)$ to the derived category $\D(\mathcal{A}[\epsilon]^n)$ produces a recollement of triangulated categories. Finally, we prove that if $\mathcal{A}$ is an Ab4-category with a compact projective generator, then $\D(\mathcal{A}[\epsilon]^n)$ is a compactly generated triangulated category., Comment: 30 pages, accepted for publication in Journal of Algebra

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories and $\mathbf{F}:\mathcal{A}\to \mathcal{B}$ an additive and right exact functor which is perfect, and let $(\mathbf{F},\mathcal{B})$ be the left comma category. We give an equivalent characterization of Gorenstein projective objects in $(\mathbf{F},\mathcal{B})$ in terms of Gorenstein projective objects in $\mathcal{B}$ and $\mathcal{A}$. We prove that there exists a left recollement of the stable category of the subcategory of $(\mathbf{F},\mathcal{B})$ consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in $\mathcal{B}$ and $\mathcal{A}$. Moreover, this left recollement can be filled into a recollement when $\mathcal{B}$ is Gorenstein and $\mathbf{F}$ preserves projectives.

Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We study when the double functor $\Tor^S_i(\omega, \Ext^i_{R}(\omega,-))$ preserves epimorphisms and the double functor $\Ext_{R}^i(\omega, \Tor_i^{S}(\omega,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules. Under certain cograde condition of modules, we construct two complete cotorsion pairs. In addition, we establish the relation between some relative finitistic dimensions of rings and the right and left projective dimensions of $\omega$., Comment: 46 pages, accepted for publication in Publications of the Research Institute for Mathematical Sciences

Let $\A$ be an abelian category having enough projective objects and enough injective objects. We prove that if $\A$ admits an additive generating object, then the extension dimension and the weak resolution dimension of $\A$ are identical, and they are at most the representation dimension of $\A$ minus two. By using it, for a right Morita ring $\La$, we establish the relation between the extension dimension of the category $\mod \La$ of finitely generated right $\Lambda$-modules and the representation dimension as well as the right global dimension of $\Lambda$. In particular, we give an upper bound for the extension dimension of $\mod \Lambda$ in terms of the projective dimension of certain class of simple right $\Lambda$-modules and the radical layer length of $\Lambda$. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements., Comment: 21 pages, accepted for publication in Algebras and Representation Theory

Let $R$ and $S$ be any rings and $_RC_S$ a semidualizing bimodule, and let $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ be the Auslander and Bass classes respectively. Then both the pairs $$(\mathcal{A}_C(R^{op}),\mathcal{B}_C(R))\ {\rm and}\ (\mathcal{B}_C(R),\mathcal{A}_C(R^{op}))$$ are coproduct-closed and product-closed duality pairs and both $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if $\mathcal{B}_C(R)$ is enveloping in $\Mod R$, then $\mathcal{A}_C(S)$ is enveloping in $\Mod S$. Then some applications to the Auslander projective dimension of modules are given., Comment: 21 pages

For a recollement $(\mathcal{A},\mathcal{B},\mathcal{C})$ of abelian categories, we show that torsion pairs in $\mathcal{A}$ and $\mathcal{C}$ can induce torsion pairs in $\mathcal{B}$; and the converse holds true under certain conditions., Comment: 15 pages

Let $\mathscr{A}$ be an abelian category and $\mathscr{P}(\mathscr{A})$ the subcategory of $\mathscr{A}$ consisting of projective objects. Let $\mathscr{C}$ be a full, additive and self-orthogonal subcategory of $\mathscr{A}$ with $\mathscr{P}(\mathscr{A})$ a generator, and let $\mathcal{G}(\mathscr{C})$ be the Gorenstein subcategory of $\mathscr{A}$. Then the right 1-orthogonal category ${\mathcal{G}(\mathscr{C})^{\bot_1}}$ of $\mathcal{G}(\mathscr{C})$ is both projectively resolving and injectively coresolving in $\mathscr{A}$. We also get that the subcategory $\spc(\mathcal{G}(\mathscr{C}))$ of $\mathscr{A}$ consisting of objects admitting special $\mathcal{G}(\mathscr{C})$-precovers is closed under extensions and $\mathscr{C}$-stable direct summands (*). Furthermore, if $\mathscr{C}$ is a generator for $\mathcal{G}(\mathscr{C})^{\perp_1}$, then we have that $\spc(\mathcal{G}(\mathscr{C}))$ is the minimal subcategory of $\mathscr{A}$ containing $\mathcal{G}(\mathscr{C})^{\perp_1}\cup \mathcal{G}(\mathscr{C})$ with respect to the property (*), and that $\spc(\mathcal{G}(\mathscr{C}))$ is $\mathscr{C}$-resolving in $\mathscr{A}$ with a $\mathscr{C}$-proper generator $\mathscr{C}$., Comment: 19 pages, accepted for publication in Science China Mathematics

Let $\Lambda$ be a finite dimensional string algebra over a field with the quiver $Q$ such that the underlying graph of $Q$ is a tree, and let $|\Det(\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms between indecomposable left $\Lambda$-modules. Then we have $$|\Det(\Lambda)|=2n-p-q-1,$$ where $n$ is the number of vertices in $Q$, $p=|\{i\mid i$ is a source in $Q$ with two neighbours$\}|$ and $q$ is the number of non-zero vertex ideals of $\Lambda$., Comment: 18 pages. arXiv admin note: text overlap with arXiv:1608.07918

In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if $(\C,\E,\s)$ is an extriangulated category with enough injective objects and projective objects, then there exists a bijective correspondence between any two of the following classes: (1) special precovering ideals of $\C$; (2) special preenveloping ideals of $\C$; (3) additive subfunctors of $\E$ having enough special injective morphisms; and (4) additive subfunctors of $\E$ having enough special projective morphisms. Moreover, we show that if $(\C,\E,\s)$ is an extriangulated category with enough injective objects and projective morphisms, then there exists a bijective correspondence between the following two classes: (1) all object-special precovering ideals of $\C$; (2) all additive subfunctors of $\E$ having enough special injective objects., Comment: 29 pages, the introduction is re-organized and examples 2.8 and 3.2 are added

Let $\Lambda$ be a finite dimensional algebra of type ${\mathbb A}_n$ over an algebraically closed field $K$ with the quiver $Q$ and let $|\Det(\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms between indecomposable left $\Lambda$-modules. If $\Lambda$ is a path algebra, then we have $$|\Det(\Lambda)|= 2n-2, &\mbox{if $p=0$; } 2n-p-1, &\mbox{if $p\geq 1$,}$$ where $p=|\{i\mid i$ is a source in $Q$ with $2\leq i\leq n-1\}|$. If $\Lambda$ is a bound quiver algebra, then we have $$ |\Det(\Lambda)|= 2n-2, &\mbox{if $r=1$; } 2n-p-q-1, &\mbox{if $r\geq 2$,} $$ where $q$ is the number of non-zero sink ideals of $\Lambda$ and $r=|\{i\mid i$ is a sink in $Q$ with $1\leq i\leq n\}|$., Comment: 21 pages, final version, accepted for publication in Journal of Algebra

Motivated by $\tau$-tilting theory developed by Adachi, Iyama and Reiten, for a finite-dimensional algebra $\Lambda$ with action by a finite group $G$, we introduce the notion of $G$-stable support $\tau$-tilting modules. Then we establish bijections among $G$-stable support $\tau$-tilting modules over $\Lambda$, $G$-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective $\Lambda$-modules, and $G$-stable functorially finite torsion classes in the category of finitely generated left $\Lambda$-modules. In the case when $\Lambda$ is the endomorphism of a $G$-stable cluster-tilting object $T$ over a Hom-finite 2-Calabi-Yau triangulated category $\mathcal{C}$ with a $G$-action, these are also in bijection with $G$-stable cluster-tilting objects in $\mathcal{C}$. Moreover, we investigate the relationship between stable support $\tau$-tilitng modules over $\Lambda$ and the skew group algebra $\Lambda G$., Comment: 21 pages

We investigate the properties of pure derived categories of module categories, and show that pure derived categories share many nice properties of classical derived categories. In particular, we show that bounded pure derived categories can be realized as certain homotopy categories. We introduce the pure projective (resp. injective) dimension of complexes in pure derived categories, and give some criteria for computing these dimensions in terms of the properties of pure projective (resp. injective) resolutions and pure derived functors. As a consequence, we get some equivalent characterizations for the finiteness of the pure global dimension of rings. Finally, pure projective (resp. injective) resolutions of unbounded complexes are considered., Comment: 20 pages, accepted for publication in Journal of Algebra

Let $A$ be an artin algebra. We show that the bounded homotopy category of finitely generated right $A$-modules has Auslander-Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [H2]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules admits Auslander-Reiten triangles, which improves a main result in [G]., Comment: 10 pages, accepted for publication in Communications in Algebra

Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$\omega$-cotorsionfree modules and a subclass of the class of $\omega$-adstatic modules. Also we establish the relation between the relative homological dimensions of a module $M$ and the corresponding standard homological dimensions of ${\rm Hom}(\omega,M)$. By investigating the properties of the Bass injective dimension of modules (resp. complexes), we get some equivalent characterizations of semi-tilting modules (resp. Gorenstein artin algebras). Finally we obtain a dual version of the Auslander-Bridger's approximation theorem. As a consequence, we get some equivalent characterizations of Auslander $n$-Gorenstein artin algebras., Comment: 37 pages, accepted for publication in Kyoto Journal of Mathematics

In an abelian category $\mathscr{A}$ with small ${\rm Ext}$ groups, we show that there exists a one-to-one correspondence between any two of the following: balanced pairs, subfunctors $\mathcal{F}$ of ${\rm Ext}^{1}_{\mathscr{A}}(-,-)$ such that $\mathscr{A}$ has enough $\mathcal{F}$-projectives and enough $\mathcal{F}$-injectives and Quillen exact structures $\mathcal{E}$ with enough $\mathcal{E}$-projectives and enough $\mathcal{E}$-injectives. In this case, we get a strengthened version of the translation of the Wakamatsu lemma to the exact context, and also prove that subcategories which are $\mathcal{E}$-resolving and epimorphic precovering with kernels in their right $\mathcal{E}$-orthogonal class and subcategories which are $\mathcal{E}$-coresolving and monomorphic preenveloping with cokernels in their left $\mathcal{E}$-orthogonal class are determined by each other. Then we apply these results to construct some (pre)enveloping and (pre)covering classes and complete hereditary $\mathcal{E}$-cotorsion pairs in the module category., Comment: 15 pages, accepted for publication in Publicationes Mathematicae Debrecen

Let $(\mathscr{X}$, $\mathscr{Y})$ be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to $(\mathscr{X}$, $\mathscr{Y})$, and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the $\mathscr{X}$-resolution dimension of $\mathscr{Y}$ (resp. $\mathscr{Y}$-coresolution dimension of $\mathscr{X}$) is finite, then the bounded homotopy category of $\mathscr{Y}$ (resp. $\mathscr{X}$) is contained in that of $\mathscr{X}$ (resp. $\mathscr{Y}$). As a consequence, we get that the right $\mathscr{X}$-singularity category coincides with the left $\mathscr{Y}$-singularity category if the $\mathscr{X}$-resolution dimension of $\mathscr{Y}$ and the $\mathscr{Y}$-coresolution dimension of $\mathscr{X}$ are finite., Comment: 17 pages, accepted for publication in Science China Mathematics

As a dual of the Auslander transpose of modules, we introduce and study the cotranspose of modules with respect to a semidualizing module $C$. Then using it we introduce $n$-$C$-cotorsionfree modules, and show that $n$-$C$-cotorsionfree modules possess many dual properties of $n$-torsionfree modules. In particular, we show that $n$-$C$-cotorsionfree modules are useful in characterizing the Bass class and investigating the approximation theory for modules. Moreover, we study $n$-cotorsionfree modules over artin algebras and answer negatively an open question of Huang and Huang posed in 2012., Comment: 24 pages, to appear in Forum Mathematicum

We study the properties of the relative derived category $D_{\mathscr{C}}^{b}$($\mathscr{A}$) of an abelian category $\mathscr{A}$ relative to a full and additive subcategory $\mathscr{C}$. In particular, when $\mathscr{A}=A{\text -}\mod$ for a finite-dimensional algebra $A$ over a field and $\mathscr{C}$ is a contravariantly finite subcategory of $A$-$\mod$ which is admissible and closed under direct summands, the $\mathscr{C}$-singularity category $D_{\mathscr{C}{\text sg}}$($\mathscr{A}$)=$D_{\mathscr{C}}^{b}$($\mathscr{A}$)/$K^{b}(\mathscr{C})$ is studied. We give a sufficient condition when this category is triangulated equivalent to the stable category of the Gorenstein category $\mathscr{G}(\mathscr{C})$ of $\mathscr{C}$., Comment: 17 pages, to appear in Journal of Pure and Applied Algebra

We introduce relative preresolving subcategories and precoresolving subcategories of an abelian category and define homological dimensions and codimensions relative to these subcategories respectively. We study the properties of these homological dimensions and codimensions and unify some important properties possessed by some known homological dimensions. Then we apply the obtained properties to special subcategories and in particular to module categories. Finally we propose some open questions and conjectures, which are closely related to the generalized Nakayama conjecture and the strong Nakayama conjecture., Comment: 31 pages, Kyoto Journal of Mathematics, 54(4) (2014), 727--757

Let $R$ be an arbitrary ring and $(-)^+=\Hom_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers, and let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that $X^+\in \mathcal{D}$ for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ of left $R$-modules with $C\in \mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided $f^+: C^+\to A^+$ is a $\mathcal{D}$-(pre)cover of $A^+$. Some applications of this result are given., Comment: 9 pages, to appear in Canadian Mathematical Bulletin

Let $\mathscr{A}$ be an abelian category and $\mathscr{C}$ an additive full subcategory of $\mathscr{A}$. We provide a method to construct a proper $\mathscr{C}$-resolution (resp. coproper $\mathscr{C}$-coresolution) of one term in a short exact sequence in $\mathscr{A}$ from that of the other two terms. By using these constructions, we answer affirmatively an open question on the stability of the Gorenstein category $\mathcal{G}(\mathscr{C})$ posed by Sather-Wagstaff, Sharif and White; and also prove that $\mathcal{G}(\mathscr{C})$ is closed under direct summands. In addition, we obtain some criteria for computing the $\mathscr{C}$-dimension and the $\mathcal{G}(\mathscr{C)}$-dimension of an object in $\mathscr{A}$., Comment: 35 pages. arXiv admin note: substantial text overlap with arXiv:1012.1703

We prove that for a left and right Noetherian ring $R$, $_RR$ satisfies the Auslander condition if and only if so does every flat left $R$-module, if and only if the injective dimension of the $i$th term in a minimal flat resolution of any injective left $R$-module is at most $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of the $i$th term in a minimal injective coresolution (resp. flat resolution) of any left $R$-module $M$ is at most the flat (resp. injective) dimension of $M$ plus $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of the injective envelope (resp. flat cover) of any left $R$-module $M$ is at most the flat (resp. injective) dimension of $M$, and if and only if any of the opposite versions of the above conditions hold true. Furthermore, we prove that for an Artinian algebra $R$ satisfying the Auslander condition, $R$ is Gorenstein if and only if the subcategory consisting of finitely generated modules satisfying the Auslander condition is contravariantly finite. As applications, we get some equivalent characterizations of Auslander-Gorenstein rings and Auslander-regular rings., Comment: 29 pages. The original title is "Proper Resolutions and Auslander-Type Conditions of Modules". All the results in Section 3 in the original version has been extended completely to a much more general setting in Section 3 in "Proper Resolutions and Gorenstein Categories" (arXiv:1203.4110), so this section is removed and the title of the paper is changed

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the principle. Over commutative, noetherian rings injective modules have direct sum decompositions into indecomposable modules. We will show that Gorenstein injective modules over Gorenstein rings of finite Krull dimension have filtrations analogous to those provided by these decompositions. This result will then provide us with the tools to prove that all tensor products of Gorenstein injective modules over these rings are Gorenstein injective., Comment: 9 pages; It has been accepted for publication in Proceedings of the American Mathematical Society

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring $R$, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left $R$-module is at most the flat dimension of the injective envelope of $_RR$. Then we get that the injective envelope of $_RR$ is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left $R$-module is (Gorenstein) flat, if and only if the injective envelope of every flat left $R$-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left $R$-module is injective, and if and only if the opposite version of one of these conditions is satisfied., Comment: 17 pages. It has been accepted for publication in Algebras and Representation Theory

Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n\geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and only if every finitely generated left $R$-module and every finitely generated right $R$-module have torsionfree dimension at most $n$, if and only if every finitely generated left (or right) $R$-module has Gorenstein dimension at most $n$. For any $n \geq 1$, we study the properties of the finitely generated $R$-modules $M$ with $\Ext_R^i(M, R)=0$ for any $1\leq i \leq n$. Then we investigate the relation between these properties and the self-injective dimension of $R$., Comment: 16 pages. The proof of Lemma 3.8 is modified. It has been accepted for publication in Osaka Journal of Mathematics

In this paper, we study the relation between $m$-strongly Gorenstein projective (resp. injective) modules and $n$-strongly Gorenstein projective (resp. injective) modules whenever $m \neq n$, and the homological behavior of $n$-strongly Gorenstein projective (resp. injective) modules. We introduce the notion of $n$-strongly Gorenstein flat modules. Then we study the homological behavior of $n$-strongly Gorenstein flat modules, and the relation between these modules and $n$-strongly Gorenstein projective (resp. injective) modules., Comment: 20 pages. It has been accepted for publication in Communications in Algebra. Question 3.19 in the first version has been answered negatively in Proposition 3.15(1) in the current version. To make the paper more compact, all the dual parts on the $n$-strongly Gorenstein injective modules in Section 3 have been omitted

Let $R$ be a left and right Noetherian ring and $n,k$ any non-negative integers. $R$ is said to satisfy the Auslander-type condition $G_n(k)$ if the right flat dimension of the $(i+1)$-st term in a minimal injective resolution of $R_R$ is at most $i+k$ for any $0 \leq i \leq n-1$. In this paper, we prove that $R$ is $G_n(k)$ if and only if so is a lower triangular matrix ring of any degree $t$ over $R$., Comment: 13 pages

For any ring $R$ and any positive integer $n$, we prove that a left $R$-module is a Gorenstein $n$-syzygy if and only if it is an $n$-syzygy. Over a left and right Noetherian ring, we introduce the notion of the Gorenstein transpose of finitely generated modules. We prove that a module $M\in \mod R^{op}$ is a Gorenstein transpose of a module $A\in \mod R$ if and only if $M$ can be embedded into a transpose of $A$ with the cokernel Gorenstein projective. Some applications of this result are given., Comment: 15 pages. To appear in Journal of Algebra

Let $\Lambda$ be an Auslander's 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda$ admits a trivial maximal 1-orthogonal subcategory of $\mod\Lambda$, then for any indecomposable module $M \in \mod \Lambda$, 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 $\Lambda$ is a tilted algebra., Comment: 13 pages; The main result in the original version of this paper is extended

For an $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n$, we give some necessary conditions for $\Lambda$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $\Lambda$-modules with projective dimension $n-1$ or $n$. For an almost hereditary algebra $\Lambda$ with global dimension 2, we prove that $\Lambda$ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable $\Lambda$-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 of maximal $n$-orthogonal subcategories of $^{\bot}T$ for a cotilting module $T$. For a Gorenstein algebra, we prove that all non-projective direct summands of a maximal $n$-orthogonal module are $\Omega ^n\tau$-periodic. In addition, we study the relation between the complexity of modules and the existence of maximal $n$-orthogonal subcategories for the tensor product of two finite-dimensional algebras., Comment: 19 pages, to appear in Journal of Algebra

This paper has been withdrawn by the authors because it has been combined with "Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories" (arXiv:0903.0761) together. Please see the new version of the latter paper for the details., Comment: This paper has been withdrawn