One characteristical property of semilattices

The operations of node deletion and insertion in a finite dimensional quiver algebra were introduced in Martínez-Villa (1980) as an abstraction of the operations used in earlier works (Auslander and Reiten, 1973; Bongartz and Riedtmann, 1979; Platzeck, 1978), such constructions are the easiest way to produce stably equivalent algebras.

In general, it is not easy to decide whether or not a given quadratic algebra is Koszul, then it is of interest to construct new Koszul algebras from given ones. The aim of the article is to prove that node deletion and insertion generalizes to graded quiver algebras producing, as in the finite dimensional case, stably equivalent algebras and, in this Situation, either both or neither of the two algebras are Koszul.