Auslander–Gorenstein resolution

We introduce the notion of Auslander–Gorenstein resolution and show that a Noetherian ring is an Auslander–Gorenstein ring if it admits an Auslander–Gorenstein resolution over another Auslander–Gorenstein ring.     

Auslander–Bridger Modules

Classically, the Auslander–Bridger transpose finds its best applications in the well-known setting of finitely presented modules over a semiperfect ring. We introduce a class of modules over an arbitrary ring R, which we call Auslander–Bridger modules, with the property that the Auslander–Bridger transpose induces a well-behaved bijection between isomorphism classes of Auslander–Bridger right R-modules and isomorphism classes of Auslander–Bridger left R-modules. Thus we generalize what happens for finitely presented modules over a semiperfect ring. Auslander–Bridger modules are characterized by two invariants (epi-isomorphism class and lower-isomorphism class), which are interchanged by the transpose. Via a suitable duality, we find that kernels of morphisms between injective modules of finite Goldie dimension are also characterized by two invariants (mono-isomorphism class and upper-isomorphism class).     

Auslander–Reiten Components Containing Cones

Let A be a locally finite Abelian R-category with Auslander–Reiten sequences and with Auslander–Reiten quiver (A). We give a criterion for Auslander–Reiten components to contain a cone and apply this result to various categories.     

Auslander–Reiten sequences on schemes

Auslander–Reiten sequences are the central item of Auslander–Reiten theory, which is one of the most important techniques for the investigation of the structure of abelian categories. This note considers X, a smooth projective scheme of dimension at least 1 over the field k, and , an indecomposable coherent sheaf on X. It is proved that in the category of quasi-coherent sheaves on X, there is an Auslander–Reiten sequence ending in .     

Covering techniques in Auslander–Reiten theory

Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander–Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.     

Tilting complexes and Auslander–Reiten conjecture

We studied the properties of tilting complexes and proved that derived equivalences preserve the validity of the Auslander–Reiten conjecture.     

Auslander–Reiten Triangles in Homotopy Categories

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 [14 Happel, D. (1988). Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar]]; (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 improve a main result in [12 Nan, G. (2012). Auslander-Reiten triangles on Gorenstein derived categories. Comm. Algebra 40:3912–3919.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

Derived equivalences for Cohen–Macaulay Auslander algebras

Let $A$ and $B$ be Gorenstein Artin algebras of finite Cohen–Macaulay type. We prove that, if $A$ and $B$ are derived equivalent, then their Cohen–Macaulay Auslander algebras are also derived equivalent.     

Homological Behavior of Auslander’s k-Gorenstein Rings

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?∈ mod Λ satisfying the condition that Ext $_{\Lambda}^i(M, \Lambda)=0$ for any 1?≤?i?≤?k is reflexive. For an ∞-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 ∞-Gorenstein ring and M?∈ mod Λ with gradeM finite, then Ext $_{\Lambda}^i($ Ext $_{\Lambda ^{op}}^i($ Ext $_{\Lambda}^{{\rm grade}M}(M, \Lambda), \Lambda), \Lambda)=0$ if and only if i?≠gradeM. For a 2-Gorenstein ring Λ, we show that a non-zero proper left ideal I of Λ is reflexive if and only if Λ/I has no non-zero pseudo-null submodule.     

Auslander–Buchweitz Approximation Theory for Triangulated Categories

We introduce and develop an analogous of the Auslander–Buchweitz approximation theory (see Auslander and Buchweitz, Societe Mathematique de France 38:5–37, 1989) in the context of triangulated categories, by using a version of relative homology in this setting. We also prove several results concerning relative homological algebra in a triangulated category $\mathcal{T},$ which are based on the behavior of certain subcategories under finiteness of resolutions and vanishing of Hom-spaces. For example: we establish the existence of preenvelopes (and precovers) in certain triangulated subcategories of $\mathcal{T}.$ The results resemble various constructions and results of Auslander and Buchweitz, and are concentrated in exploring the structure of a triangulated category $\mathcal{T}$ equipped with a pair $(\mathcal{X},\omega),$ where $\mathcal{X}$ is closed under extensions and ω is a weak-cogenerator in $\mathcal{X},$ usually under additional conditions. This reduces, among other things, to the existence of distinguished triangles enjoying special properties, and the behavior of (suitably defined) (co)resolutions, projective or injective dimension of objects of $\mathcal{T}$ and the formation of orthogonal subcategories. Finally, some relationships with the Rouquier’s dimension in triangulated categories is discussed.     

Cohen–Macaulay Auslander algebras of skewed-gentle algebras

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras.     

Auslander–Reiten Quivers and the Coxeter Complex

Let Q be a quiver of type ADE. We construct the corresponding Auslander–Reiten quiver as a topological complex inside the Coxeter complex associated with the underlying Dynkin diagram. In A_n case, we recover special wiring diagrams. Presented by R. RentschlerMathematics Subject Classifications (2000) 16G70, 17B10, 20F55.     

Auslander–Reiten components with bounded short cycles

We study Auslander–Reiten components of an artin algebra with bounded short cycles, namely, there exists a bound for the depths of maps appearing on short cycles of non-zero non-invertible maps between modules in the given component. First, we give a number of combinatorial characterizations of almost acyclic Auslander–Reiten components. Then, we shall show that an Auslander–Reiten component with bounded short cycles is obtained, roughly speaking, by gluing the connecting components of finitely many tilted quotient algebras. In particular, the number of such components is finite and each of them is almost acyclic with only finitely many DTr-orbits. As an application, we show that an artin algebra is representation-finite if and only if its module category has bounded short cycles. This includes a well known result of Ringel’s, saying that a representation-directed algebra is representation-finite.     

Auslander－Gorenstein滤环上的全律模

本文给出了Ａｕｓｌａｎｄｅｒ－Ｇｏｒｅｎｓｔｅｉｎ滤环上全律模（ｈｏｌｏｎｏｍｉｃｍｏｄｕｌｅｓ）的一个特征簇刻划，由此得到了通过特征理想的极小素因子（有限个）计算余维数的公式。     

Auslander－Gorenstein滤环上的全律模

本文给出了Ａｕｓｌａｎｄｅｒ－Ｇｏｒｅｎｓｔｅｉｎ滤环上全律模（ｈｏｌｏｎｏｍｉｃｍｏｄｕｌｅｓ）的一个特征簇刻划，由此得到了通过特征理想的极小素因子（有限个）计算余维数的公式。