Related papers: Auslander regular algebras and Coxeter matrices

Co-Gorenstein algebras were introduced by A. Beligiannis in \cite{B}. In \cite{KM}, the authors propose the following conjecture (Co-GC): if $\Omega^n (\mod A)$ is extension closed for all $n \leq 1$, then $A$ is right Co-Gorenstein, and…

We provide a categorification of Oh and Suh's combinatorial Auslander-Reiten quivers in the simply laced case. We work within the perfectly valued derived category $\mathrm{pvd}(\Pi_Q)$ of the 2-dimensional Ginzburg dg algebra of a Dynkin…

In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it.…

In this paper we show that for a simply-laced root system a choice of $C$ gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to $C$-orbits in $R$; moreover, it gives an identification of…

Regular and higher regular graded algebras (in simplest case satisfying Von Neumann regularity $\Theta_{1}\Theta_{2}\Theta_{1}=\Theta_{1}$ instead of anticommutativity) are introduced and their properties are studied. They are described in…

We generalize the "facial weak order" of a finite Coxeter group to a partial order on a set of intervals in a complete lattice. We apply our construction to the lattice of torsion classes of a finite-dimensional algebra and consider its…

We show that the Auslander-Reiten Formula for a finite dimensional hereditary algebra is invariant under the Auslander-Reiten translate.

We deduce a necessary condition for Auslander-Reiten components of the bounded derived category of a finite dimensional algebra to have Euclidean tree class by classifying certain types of irreducible maps in the category of complexes. This…

In 1996, Doty, Nakano and Peters defined infinitesimal Schur algebras, combining the approach via polynomial representations with the approach via $G_r T$-modules to representations of the algebraic group $G = \mathrm{GL}_n$. We study…

In this paper, we construct a version of Auslander-Reiten sequences for the $K(n)$-local stable homotopy category. In particular, the role of the Auslander-Reiten translation is played by the local Brown-Comenetz duality functor. As an…

We study the relationship between Serre duality and the Mukai pairing for smooth and proper dg-algebras. We introduce an alternative definition of the Mukai pairing and prove that it coincides with the Mukai pairings defined by…

We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets,…

Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$\omega$-cotorsionfree modules and a subclass of the class of $\omega$-adstatic modules. Also…

We describe the structure of semi-regular Auslander-Reiten components of artin algebras without external short paths in the module category. As an application we give a complete description of self-injective artin algebras whose…

We consider the Kronecker algebra $A=\mathcal{O}[X, Y]/(X^2, Y^2)$, where $\mathcal{O}$ is a complete discrete valuation ring. Since $A \otimes\kappa$ is a special biserial algebra, where $\kappa$ is the residue field of $\mathcal{O}$, one…

We introduce new partial orders on the sequence positive roots and study the statistics of the poset by using Auslander-Reiten quivers for finite type ADE. Then we can prove that the statistics provide interesting information on the…

In recent work, the second author introduced the concept of Coxeter quivers, generalizing several previous notions of a quiver representation. Finite type Coxeter quivers were classified, and their indecomposable objects were shown to be in…

We calculate explicitly cohomologies of the lattices over the Kleinian 4-group belonging to the regular components of the Auslander-Reiten quiver as well as of their dual modules. The result is applied to the classification of some…

Let $R$ be a ring and $\mathcal{Q}$ be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category $\rm{Rep} (\mathcal{Q} ,R)$ of representations of $\mathcal{Q}$ by left…

Auslander-Reiten theory is fundamental to study categories which appear in representation theory, for example, modules over artin algebras, Cohen-Macaulay modules over Cohen-Macaulay rings, lattices over orders, and coherent sheaves on…