Koszul homology of quotients by edge ideals

TL;DR

This paper proves that the Koszul homology algebra of quotients by edge ideals of forests is generated by the lowest linear strand, expanding understanding of Koszul algebra structures.

Contribution

It introduces a method to construct minimal graded free resolutions for such quotients and demonstrates their Koszul homology properties, including a new proof of Betti number results.

Findings

Koszul homology algebra of forest edge ideal quotients is generated by the lowest linear strand

Constructed minimal graded free resolutions using iterated mapping cone

Reproved Betti number results for tree edge ideal quotients

Abstract

We show that the Koszul homology algebra of a quotient by the edge ideal of a forest is generated by the lowest linear strand. This provides a large class of Koszul algebras whose Koszul homology algebras satisfy this property. We obtain this result by constructing the minimal graded free resolution of a quotient by such an edge ideal via the so called iterated mapping cone construction and using the explicit bases of Koszul homology given by Herzog and Maleki. Using these methods we also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients of edge ideals of trees.