We study homological behavior of modules satisfying the Auslander condition. Assume that $\mathcal{AC}$ is the class of left $R$-modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in $\mathcal{AC}$ belongs to $\mathcal{AC}$ for any ring $R$. As a consequence, we show that for any left Noetherian ring $R$, $\mathcal{AC}$ is a resolving subcategory of the category of left $R$-modules if and only if $_RR$ satisfies the Auslander condition if and only if each Gorenstein projective left $R$-module belongs to $\mathcal{AC}$. As an application, we prove that, for an Artinian algebra $R$ satisfying the Auslander condition, $R$ is Gorenstein if and only if $\mathcal{AC}$ coincides with the class of Gorenstein projective left $R$-modules if and only if $({\mathcal{AC}^{< \infty}},(\mathcal{AC}^{<\infty})^\bot)$ is a tilting-like cotorsion pair if and only if (${\mathcal{AC}^{< \infty}},\mathcal{I}$) is a tilting-like cotorsion pair, where $\mathcal{AC}^{<\infty}$ is the class of left $R$-modules with finite $\mathcal{AC}$-dimension and $\mathcal{I}$ is the class of injective left $R$-modules. This leads to some criteria for the validity of the Auslander and Reiten conjecture which says that an Artinian algebra satisfying the Auslander condition is Gorenstein.
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.
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.
In this paper, we study when the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module. Applications go in three directions. The first is to characterize when the little finitistic dimension is finite. The second is to obtain equivalent formulations for a Wakamatsu tilting module to be a tilting module. The third is to give some new characterizations of Gorenstein rings.
Motivated by its links to $τ$-tilting theory, we introduce a generalization of cotorsion pairs in module categories. Such pairs are also linked to co-t-structures in corresponding triangulated categories, and to cotorsion pairs in certain extension closed (and hence extriangulated) subcategories, which we call two-term categories.
We construct an isomorphism between the partially ordered set of tilting modules for the Auslander algebra of $K[x]/(x^n)$ and the interval of rational permutation braids in the braid group on $n$ strands. Hence, there are only finitely many tilting modules.
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,…
We give a characterization of $τ$-rigid modules over Auslander algebras in terms of projective dimension of modules. Moreover, we show that for an Auslander algebra $Λ$ admitting finite number of non-isomorphic basic tilting $Λ$-modules and tilting $Λ^{op}$-modules, if all indecomposable $τ$-rigid $Λ$-modules of projective dimension $2$ are of grade $2$, then $Λ$ is $τ$-tilting finite.
We give a bijection between the tilting complexes in the bounded homotopy category of the Auslander algebra of the truncated polynomial ring and ZxB where B is the Artin braid goup of type A with n-1 generators. The tilting complexes have mutation components parametrized by Z and each component has a natural faithful and transitive operation of B. This also implies that the derived Picard group of this algebra is isomorphic to the direct product of its outer isomorphism group and ZxB. This work is to be seen as a continuation of the work of Geuenich and an application of the work of Aihara and…
Let $Λ$ be a radical square zero Nakayama algebra with $n$ simple modules and let $Γ$ be the Auslander algebra of $Λ$. Then every indecomposable direct summand of a tilting $Γ$-module is either simple or projective. Moreover, if $Λ$ is self-injective, then the number of tilting $Γ$-modules is $2^n$; otherwise, the number of tilting $Γ$-modules is $2^{n-1}$.
We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.
Let $Λ$ be a radical square zero algebra of a Dynkin quiver and let $Γ$ be the Auslander algebra of $Λ$. Then the number of tilting right $Γ$-modules is $2^{m-1}$ if $Λ$ is of $A_{m}$ type for $m\geq 1$. Otherwise, the number of tilting right $Γ$-modules is $2^{m-3}\times14$ if $Λ$ is either of $D_{m}$ type for $m\geq 4$ or of $E_{m}$ type for $m=6,7,8$.
Let $R$ and $S$ be rings and $_Rω_S$ a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$ω$-cotorsionfree modules and a subclass of the class of $ω$-adstatic modules. Also we establish the relation between the relative homological dimensions of a module $M$ and the corresponding standard homological dimensions of ${\rm Hom}(ω,M)$. 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 …
We build a bijection between the set $\sttiltΛ$ of isomorphism classes of basic support $τ$-tilting modules over the Auslander algebra $Λ$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\sttiltΛ$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tiltΛ$ of isomorphism classes of basic tilting $Λ$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $Γ$ of Dynkin type $A…
We study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules. Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension $2$, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least $2$, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which ha…
The main theme of this paper is to study $τ$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $τ$-cotorsion torsion triples and show a bijection between the collection of $τ$-cotorsion torsion triples in $\mathscr{A}$ and the collection of $τ$-tilting subcategories of $\mathscr{A}$, generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr{A}$. General definitions and results are exemplified using persistent mod…
For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, it is proved that the decomposition of $\mathcal{C}$ is determined by the special decomposition of $\mathcal{T}$, namely, $\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$, where $\mathcal{C}_i, i\in I$ are triangulated subcategories, if and only if $\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$ where $\mathcal{T}_i, i\in I$ are subcategories with $Hom_{\mathcal{C}}(\mathcal{T}_i[t],\mathcal{T}_j)=0, \forall 1\leq t\leq d-2$ and $i\not= j.$ This induces that the Gabriel …
We construct two functors from the submodule category of a self-injective representation-finite algebra $Λ$ to the module category of the stable Auslander algebra of $Λ$. These functors factor through the module category of the Auslander algebra of $Λ$. Moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism. Their construction uses a recollement of the module category of the Auslander algebra induced by an idempotent and this recollement determines a ch…
In this paper we investigate the endomorphism algebras of standard cluster tilting objects in the stably 2-Calabi-Yau categories $\Sub{Λ_w}$ with elements $w$ in Coxeter groups in \cite{BIRSc}. They are examples of the 2-Auslander algebras introduced in \cite{I1}. Generalizing work in \cite{GLS1} we show that they are quasihereditary, even strongly quasihereditary in the sense of \cite{R}. We also describe the cluster tilting object giving rise to the Ringel dual, and prove that there is a duality between $\Sub{Λ_w}$ and the category $\mathcal{F}(Δ)$ of good modules over the quasihereditary al…
We revisit faithfully balanced modules. These are faithful modules having the double centralizer property. For finite-dimensional algebras our main tool is the category ${\rm cogen}^1(M)$ of modules with a copresentation by summands of finite sums of $M$ on which ${\rm Hom}(-,M)$ is exact. For a faithfully balanced module $M$ the functor ${\rm Hom}(-,M)$ is a duality on these categories - for cotilting modules this is the Brenner-Butler theorem. We also study new classes of faithfully balanced modules combining cogenerators and cotilting modules. Then we turn to relative homological algebra in…
In the 1980s, Harada introduced a new class of algebras now called Harada algebras. Harada algebras provides us with a rich source of Auslander's 1-Gorenstein algebras. In this paper, we have two main results about Harada algebras. The first is the classification of modules over Harada algebras whose projective dimension is at most one. The second is the classification of tilting modules over Harada algebras, which is shown by giving a bijection between tilting modules over Harada algebras and tilting modules over direct products of upper triangular matrix algebras over $K$. A combinatorial de…
Let $R$ be a commutative artinian ring. We extend higher Auslander correspondence from Artin $R$-algebras of finite representation type to dualizing $R$-varieties. More precisely, for a positive integer $d$, we show that a dualizing $R$-variety is $d$-abelian if and only if it is a $d$-Auslander dualizing $R$-variety if and only if it is equivalent to a $d$-cluster-tilting subcategory of the category of finitely presented modules over a dualizing $R$-variety.
Let $R$ be a left noetherian ring, $S$ a right noetherian ring and $_RU$ a generalized tilting module with $S={\rm End}(_RU)$. The injective dimensions of $_RU$ and $U_S$ are identical provided both of them are finite. Under the assumption that the injective dimensions of $_RU$ and $U_S$ are finite, we describe when the subcategory $\{{\rm Ext}_S^n(N, U)|N$ is a finitely generated right $S$-module$\}$ is closed under submodules. As a consequence, we obtain a negative answer to a question posed by Auslander in 1969. Finally, some partial answers to Wakamatsu Tilting Conjecture are given.
From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $Λ$ be an Artin algebra and $\mathcal{M}$ be an $n$-cluster tilting subcategory of $Mod$-$Λ$. We show that $\mathcal{M}$ is pure semisimple if and only if each module in $\mathcal{M}$ is a direct sum of finitely generated modules. Let $\mathfrak{m}$ be an $n$-cluster tilting subcategory of $mod$-$Λ$. We show that $Add(\mathfrak{m})$ is an $n$-cluster tilting subcategory of $Mod$-$Λ$ if and only if $\mathfrak{m}$ has an additive generator…
We consider an arbitrary Abelian category $\mathcal{A}$ and a subcategory $\mathcal{T}$ closed under extensions and direct summands, and characterize those $\mathcal{T}$ that are (semi-)special preenveloping in $\mathcal{A}$; as a byproduct, we generalize to this setting several classical results for categories of modules. For instance, we get that the special preenveloping subcategories $\mathcal{T}$ of $\mathcal{A}$ closed under extensions and direct summands are precisely those for which $(_{}^{\perp_1}\mathcal{T},\mathcal{T})$ is a right complete cotorsion pair, where $_{}^{\perp_1}\mathca…
A classic result by Bass says that the class of all projective modules is covering, if and only if it is closed under direct limits. Enochs extended the if-part by showing that every class of modules $\mathcal C$, which is precovering and closed under direct limits, is covering, and asked whether the converse is true. We employ the tools developed in [18] and give a positive answer when $\mathcal C = \mathcal A$, or $\mathcal C$ is the class of all locally $\mathcal A ^{\leq ω}$-free modules, where $\mathcal A$ is any class of modules fitting in a cotorsion pair $(\mathcal A, \mathcal B)$ such…
For any truncated path algebra $Λ$, we give a structural description of the modules in the categories ${\cal P}^{<\infty}(Λ\text{-mod})$ and ${\cal P}^{<\infty}(Λ\text{-Mod})$, consisting of the finitely generated (resp. arbitrary) $Λ$-modules of finite projective dimension. We deduce that these categories are contravariantly finite in $Λ\text{-mod}$ and $Λ\text{-Mod}$, respectively, and determine the corresponding minimal ${\cal P}^{<\infty}$-approximation of an arbitrary $Λ$-module from a projective presentation. In particular, we explicitly construct - based on the Gabriel quiver $Q$ and th…
Let $R$ be a perfect ring and $T$ be an $R$-module. We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. It is proved that $T$ is sincere silting if and only if $T$ is presilting satisfing the vanishing condition $\mathrm{KerExt}^{0\le i\le 1}_R(T,-)=0$, and that $T$ is tilting if and only if $\mathrm{Ker}\mathrm{Ext}^{0\leqslant i\leqslant 1}_{R}(T,-)=0$ and $\mathrm{Gen}T\subseteq \mathrm{Ker}\mathrm{Ext}^{1\leqslant i\leqslant 2}_{R}(T,-)$. As an application, we prove that a sincere silting $R$-module $T$ of fin…
Let $\mathcal{A}$ be an abelian category. For a pair $(\mathcal{X},\mathcal{Y}$ of classes of objects in $\mathcal{A},$ we define the weak and the $(\mathcal{X},\mathcal{Y})$-Gorenstein relative projective objects in $\mathcal{A}$. We point out that such objects generalize the usual Gorenstein projective objects and others generalizations appearing in the literature as Ding-projective, Ding-injective, $\mathcal{X}$-Gorenstein projective, Gorenstein AC-projective and $G_C$-projective modules and Cohen-Macaulay objects in abelian categories. We show that the principal results on Gorenstein proje…
Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [18], [12], [17]. If $R$ is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [14], and it does not provide for approximations when $R$ has cardinality $\leq \aleph_0$, [6]. We remove the cardinality restriction on $R$ in the latter result. We also prove an extension of the Countable Telescope Conjecture [21]: a cotorsion pair $(\mathcal A,\mathcal B)$ is of countable type when…
