Constructing Endomorphism Rings of Large Finite Global Dimension

TL;DR

This paper explores methods to construct endomorphism rings with varying global dimensions over rings linked to numerical semigroups, revealing the potential for arbitrarily large or fixed global dimensions through two distinct constructions.

Contribution

It introduces lazy and greedy constructions for endomorphism rings, demonstrating their capabilities to produce rings with large or fixed global dimensions, respectively.

Findings

Lazy construction yields arbitrarily large global dimension.

Greedy construction always results in global dimension two.

Difference between maximum and minimum global dimension can be arbitrarily large.

Abstract

In this paper we study endomorphism rings of finite global dimension over a ring associated to a numerical semigroup. We construct these endomorphism rings in two ways, called the lazy and greedy construction. The first main result of this paper shows that the lazy construction enables us to obtain endomorphism rings of arbitrarily large global dimension. The second main result of this paper shows that the greedy construction gives us endomorphism rings which always have global dimension two. As a consequence, for a fixed numerical semigroup, the difference of the maximal possible value and the minimal possible value of the global dimension of an endomorphism ring over that numerical semigroup can be arbitrarily large.