Related papers: On a generalized Auslander-Reiten conjecture

Let $R$ be an isolated Gorenstein singularity with a non-commutative resolution $A=End_R(R\oplus M)$. In this paper, we show that the relative singularity category $\Delta_R(A)$ of $A$ has a number of pleasant properties, such as being…

We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here. Stable homology is a…

We define a congruence module $\Psi_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $\lambda\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra…

Let $R$ be a ring and $\mathsf S$ be a class of strongly finitely presented (FP${}_\infty$) $R$-modules closed under extensions, direct summands, and syzygies. Let $(\mathsf A,\mathsf B)$ be the (hereditary complete) cotorsion pair…

We prove a versions of amplitude inequalities of Iversen, Foxby and Iyengar, and Frankild and Sather-Wagstaff that replace finite generation conditions with adic finiteness conditions. As an application, we prove that a local ring $R$ of…

We study Artin algebras $A$ and commutative Noetherian complete local rings $R$ in connection with the following decomposition property of Gorenstein-projective modules: $(*)$ any Gorenstein-projective module is a direct sum of finitely…

We say an excellent local domain $(S,n)$ satisfies the vanishing conditions for maps of Tor, if for every $A\to R\to S$ with $A$ regular and $A\to R$ module-finite torsion-free extension, and every $A$-module $M$, the map $Tor^A_i(M, R)\to…

We consider a version of Palais' Principle of Symmetric Criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. PSC asserts that, given a group action, for any group-invariant Lagrangian the…

A 2003 counterexample to a conjecture of Auslander brought attention to a family of rings - colloquially called AC rings - that satisfy a natural condition on vanishing of cohomology. Several results attest to the remarkable homological…

One of the most stunning results in the representation theory of Cohen-Macaulay rings is Auslander's well known theorem which states a CM local ring of finite CM type can have at most an isolated singularity. There have been some…

P. Aluffi introduced in [1] a new graded algebra in order to conveniently express characteristic cycles in the theory of singular varieties. This algebra is attached to a surjective ring homomorphism $A\surjects B$ by taking a suitable…

In this thesis we consider two constructions generalizing the classical Arf invariant. In the first construction an $\epsilon$-symmetric quadratic form over a ring with involution $R$ is lifted to an $\epsilon(1+T)$-symmetric quadratic form…

We provide a general method to study representations of quivers over abstract stable homotopy theories (e.g. arbitrary rings, schemes, dg algebras, or ring spectra) in terms of Auslander-Reiten diagrams. For a finite acyclic quiver $Q$ and…

Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity,…

In this paper we explore which part of the ideal lattice of a general ring is parametrized by its Cuntz semigroup $\mathrm{S}(R)$ and its ambient semigroup $\Lambda(R)$. We identify these classes of ideals as the quasipure ideals (a…

Inspired by the construction of the Cuntz semigroup for a C*-algebra, we introduce the matrix Malcolmson semigroup and the finitely presented module Malcolmson semigroup for a unital ring. These two semigroups are shown to have isomorphic…

We study cyclotomic association schemes over a finite commutative ring $R$ with identity. The main interest for us is to identify the normal cyclotomic schemes $C$, i.e. those for which $Aut(C)$ is a subgroup of the one-dimensional affine…

We construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC).…

Let $R$ be a commutative noetherian ring, let $\frak a$ and $\frak b$ be two ideals of $R$; and let $\Ss$ be a Serre subcategory of $R$-modules. We give a necessary and sufficient condition by which $\Ss$ satisfies $C_{\frak a}$ and…

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…