Jiaqun Wei
Inspired by recent works on rings satisfying Auslander's conjecture, we study invariants, which we call Auslander bounds, and prove that they have strong relations to some homological conjectures.
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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 …
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 will study the relationship of quite different object in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, $τ$-categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers.
We prove that a certain eventually homological isomorphism between module categories induces a triangle equivalence between their singularity categories, Gorenstein defect categories and the stable categories of Gorenstein projective modules. Further, we show that Auslander-Reiten conjecture and Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms. Applying the results to arrow removal and vertex removal, we describe the Gorenstein projective modules over some non-monomial algebras, and we verify the Auslander-Reiten conjecture for certain algebras.
In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $Λ$, we show that the right self-injective dimension of $Λ$ is at most $k$ if and only if each $M \in$mod $Λ$ satisfying the condition that Ext$_Λ^i(M, Λ)=0$ for any $1\leq i \leq k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $Λ$ are identical. In addition, we show that if $Λ$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $Λ$ with grade$M$ finite, then Ext$_…
We prove that the generic quantized coordinate ring $\mathcal{O}_q(G)$ is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group $G$. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a new set of positive cases for a question of Brown and the first author.
The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this note is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring $R$, then it holds for $R$.
We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen-Macaulay property from the same perspective of that for the Auslander condtion. A key step of our approach is to establish homological relations between a graded ring $R$, its quotient ring modulo the ideal $\hbar R$ and its localization ring with respect to the Ore set $\{\, \hbar^i\, \}_{i\geq0}$, where $\hbar$ is a homogeneous regular normal…
We provide a framework for part of the homological theory of Z-algebras and their generalizations, directed towards analogues of the Auslander-Gorenstein condition and the associated double Ext spectral sequence that are useful for enveloping algebras of Lie algebras and related rings. As an application, we prove the equidimensionality of the characteristic variety of an irreducible representation of the Z-algebra, and for related representations over quantum symplectic resolutions. In the special case of Cherednik algebras of type A, this answers a question raised by the authors.
In studying Nakayama's 1958 conjecture on rings of infinite dominant dimension, Auslander and Reiten proposed the following generalization: Let Lambda be an Artin algebra and M a Lambda-generator such that Ext^i_Lambda(M,M)=0 for all i \geq 1; then M is projective. This conjecture makes sense for any ring. We establish Auslander and Reiten's conjecture for excellent Cohen--Macaulay normal domains containing the rational numbers, and slightly more generally.
Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain singular equivalences induced by adjoint pairs, which occur often in matrix algebras, recollements and change of rings. Accordingly, several reduction methods are established to study this conjecture.
In this paper we give a sufficient condition of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations. We also introduce property (W$^{k}$). As an application of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations we give a connection between the finitistic dimension conjecture, the Auslander-Reiten conjecture and property (W$^{k}$).
In this note, we prove that the generalized Auslander-Reiten conjecture is preserved under derived equivalences between Artin algebras.
This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita type). As for weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between weakly symmetric algebras o…
Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.
Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture - by a 2003 counterexample due to Jorgensen and Sega - motivates the consideration of the class of rings that do satisfy Auslander's condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander-Reiten Conjecture is proved for AC rings, and Auslander's G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex…
In this paper, we study representations of certain string algebras, which are referred to as of affine type $\widetilde{C}$. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten quivers of the string algebras and $τ$-locally free modules defined by Geiss-Lerclerc-Schröer. As an application, we prove Geiss-Leclerc-Schröer's conjecture on the correspondence between positive roots of type $\widetilde{C}$ and $τ$-locally free modules of the corresponding string algebras.
In this short paper we study for the skew PBW (Poincaré-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the …
Igusa--Todorov Algebras and the Auslander--Reiten Conjecture
本文证明所有 Igusa--Todorov Artin 代数满足 Auslander--Reiten 猜想，并给出两个应用类：表示维数至多 3 的代数及满足特定幂零条件的代数。
The Auslander--Reiten conjecture for quantum complete intersections
本文证明任意域上有限维量子完全交满足 Auslander--Reiten 猜想与 Tachikawa 第二猜想，通过刚性模分级扭转与自扩张判据实现。
We introduce syzygies for derived categories and study their properties. Using these, we prove the derived invariance of the following classes of artin algebras: (1) syzygy-finite algebras, (2) Igusa-Todorov algebras, (3) AC algebras, (4) algebras satisfying the finitistic Auslander conjecture, and (5) algebras satisfying the generalized Auslander-Reiten conjecture. In particular, Gorenstein CM-finite algebras are derived invariants.
Relative Higher Homology and Representation Theory
本文在阿贝尔范畴的 n-簇倾斜子范畴中，基于 Ext^n 的加法子双函子建立相对高阶同调代数，证明相对 n-Auslander-Reiten 对偶公式并探索相对 Grothendieck 群，给出有限型的充要条件。
Singular equivalences of $n$-adjoint type and standard eventually homological isomorphisms
本文修改最终同调同构定义，引入n-伴随型奇异等价，发展相关理论并应用于同调猜想简化与性质保持，同时建立其与三角函子的对应关系，纳入3-伴随型奇异等价框架。
Singular equivalences and homological conjectures
通过中心化子矩阵代数的奇异范畴和奇异等价，验证了Auslander-Reiten和Cartan行列式猜想，进而证明所有经典同调猜想对域上中心化子矩阵代数成立。
Auslander conditions and tilting-like cotorsion pairs
研究满足Auslander条件的模类的同调性质，证明其预解性与Gorenstein投射模的包含关系，并给出Artinian代数下类倾斜余挠对与Gorenstein性质的等价判据。
n-recollements of triangulated categories and n-derived-simple algebras are introduced. The relations between the n-recollements of derived categories of algebras and the Cartan determinants, homological smoothness and Gorensteinness of algebras respectively are clarified. As applications, the Cartan determinant conjecture is reduced to 1-derived-simple algebras, and the Gorenstein symmetry conjecture is reduced to 2-derived-simple algebras.
On the homological conjectures for Artin algebras
本文证明Tachikawa第二猜想蕴含Artin代数的Auslander-Reiten猜想，并通过二重平凡扩张及已知关系，揭示多个重要猜想等价，且第二猜想蕴含第一猜想。
We consider the converse of the Butler, Auslander-Reiten's Theorem which is on the relations for Grothendieck groups. We show that a Gorenstein ring is of finite representation type if the Auslander-Reiten sequences generate the relations for Grothendieck groups. This gives an affirmative answer of the conjecture due to Auslander.
We show that recollements of module categories give rise to homomorphisms between the associated Hochschild cohomology algebras which preserve the strict Gerstenhaber structure, i.e., the cup product, the graded Lie bracket and the squaring map. We review various long exact sequences in Hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict Gerstenhaber structure as well. As a byproduct, we generalise a known long exact cohomology sequence of Koenig-Nagase to arbitrary surjective homological epimorphisms. We use our observations to motivate …
Categorical properties and homological conjectures for bounded extensions of algebras
本文定义了代数的有界扩张，证明其满足带有级别的 Morita 型奇异等价及相关稳定范畴等价，并探讨了多种同调猜想，给出了实用函子准则及在平凡扩张和三角矩阵代数上的应用。
In this paper, various Homological Conjectures are studied for local rings which are locally finitely generated over a discrete valuation ring $V$ of mixed characteristic. Typically, we can only conclude that a particular Conjecture holds for such a ring provided the residual characteristic of $V$ is sufficiently large in terms of the complexity of the data, where the complexity is primarily given in terms of the degrees of the polynomials over $V$ that define the data, but possibly also by some additional invariants such as (homological) multiplicity. Thus asymptotic versions of the Improved …
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 Na…
In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely related to a longstanding conjecture, the finitistic dimension conjecture, in representation theory and homological algebra. Further, we apply our results to a series of situations of particular interest: exact contexts, ring extensions, trivial extensions, pullbacks of rings, and algebras induced from Auslander-Reiten sequences. In particular, we not only extend …
Homological theory of orthogonal modules
本研究从稳定模范畴视角研究自入射Artin代数上的有限生成正交生成元，建立M-相对稳定范畴的粘合、刻画紧对象，给出Tachikawa第二猜想的等价刻画，还引入Gorenstein-Morita代数并证明Nakayama猜想对其成立。
The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, without the use of the powerful Lefschetz theorem, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an applic…
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