An equivalence of categories for graded modules over monomial algebras and path algebras of quivers

TL;DR

This paper establishes an equivalence between certain quotient categories of graded modules over monomial algebras and path algebras of quivers, using a graph construction and algebra homomorphism.

Contribution

It introduces a new categorical equivalence linking monomial algebra modules to quiver path algebra modules via the Ufnarovskii graph.

Findings

Categories QGr(A) and QGr(kQ) are equivalent for the given algebras.

The construction of the Ufnarovskii graph is key to the equivalence.

An algebra homomorphism from A to kQ is central to the proof.

Abstract

Let A be a finitely generated connected graded k-algebra defined by a finite number of monomial relations. Then there is a finite directed graph, Q, the Ufnarovskii graph of A, for which the categories QGr(A) and QGr(kQ) are equivalent: QGr(A) denotes the quotient category of graded A-modules modulo the subcategory consisting of those that are the sum of their finite dimensional submodules; QGr(kQ) has a similar definition. The proof makes use of an algebra homomorphism A--->kQ that may be of independent interest.