Some generalizations of Preprojective algebras and their properties

TL;DR

This paper introduces a generalized construction of preprojective algebras associated with relative Frobenius pairs of rings, analyzing their algebraic properties and conditions for finiteness and regularity.

Contribution

It defines a new class of graded algebras linked to Frobenius pairs and establishes their noetherianity, finite generation, and global dimension under specific conditions.

Findings

The algebra $l_R(S)$ coincides with classical preprojective algebras in a special case.

When $S/R$ has rank 4 and $R$ is noetherian, $l_R(S)$ is noetherian and finite over its center.

If $R$ and $S$ are regular, $l_R(S)$ has finite global dimension.

Abstract

In this note we consider a notion of relative Frobenius pairs of commutative rings $S/R$. To such a pair, we associate an $\mathbb{N}$-graded $R$-algebra $\Pi_R(S)$ which has a simple description and coincides with the preprojective algebra of a quiver with a single central node and several outgoing edges in the split case. If the rank of $S$ over $R$ is 4 and $R$ is noetherian, we prove that $\Pi_R(S)$ is itself noetherian and finite over its center and that each $\Pi_R(S)_d$ is finitely generated projective. We also prove that $\Pi_R(S)$ is of finite global dimension if $R$ and $S$ are regular.