Related papers: Noncommutative Auslander theorem
We study a class of noncommutative surfaces and their higher dimensional analogues which provide answers to several open questions in noncommutative projective geometry. Specifically, we give the first known graded algebras which are…
For a hereditary, finite-dimensional algebra $A$ the Coxeter transformation extends the action of the Auslander--Reiten translation on the non-projective indecomposable modules to a linear endomorphism of the Grothendieck group of the…
Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups $\Gamma$ in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural…
The homological theory of Auslander-Platzeck-Todorov on idempotent ideals laid much of the groundwork for higher Auslander-Reiten theory, providing the key technical lemmas for both higher Auslander correspondence as well as the…
Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for…
We provide a new non-commutative generalisation of the Auslander-Buchsbaum formula for higher Auslander algebras and use this to show that the class of tilted Auslander algebras, studied recently by Zito, and QF-1 algebras of global…
We show that every higher Auslander algebra $A_{n+1}^d$ of type $\mathbb{A}$ such that $\gcd(n,d)=1$ is derived equivalent to a certain replicated algebra $B=B_0^{(n+d)}$. Moreover ${\rm{gldim}} B = nd$ and $B$ admits an $nd$-cluster…
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…
This article concerns commutative algebras over a field $k$ of characteristic zero which are finite dimensional as vectorspaces, and particularly those of such algebras which are graded. Here the term graded is applied to non-negatively…
We discuss the noncommutative generalizations of polynomial algebras which after appropriate completions can be used as coordinate algebras in various noncommutative settings, (noncommutative differential geometry, noncommutative algebraic…
Let $\Lambda$ be a finite dimensional Auslander algebra. For a $\Lambda$-module $M$, we prove that the projective dimension of $M$ is at most one if and only if the projective dimension of its socle soc\,$M$ is at most one. As an…
A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria…
According to the Auslander's formula one way of studying an abelian category ${\mathcal{C}}$ is to study ${\rm mod}\mbox{-}{\mathcal{C}}$, that has nicer homological properties than ${\mathcal{C}}$, and then translate the results back to…
This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn…
The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The $n$-Auslander-Reiten translation functor $\tau_n$ plays an important role in the…
We find a class of algebras A satisfying the following property: for every nontrivial noncommutative polynomial, the linear span of all its values in A equals A. This class includes the algebras of all bounded and all compact operators on…
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…
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.
We define noncommutative binary forms. Using the typical representation of Hermite we prove the fundamental theorem of algebra and we derive a noncommutative Cardano formula for cubic forms. We define quantized elliptic and hyperelliptic…
Given an algebra $R$ and $G$ a finite group of automorphisms of $R$, there is a natural map $\eta_{R,G}:R\#G \to \mathrm{End}_{R^G} R$, called the Auslander map. A theorem of Auslander shows that $\eta_{R,G}$ is an isomorphism when…