On GK Dimension and Generator Bounds for a Class of Graded Algebras

TL;DR

This paper introduces monotonic algebras, broadening the understanding of GK dimension bounds and generator counts, and proves a parity theorem relating global and GK dimensions for Artin-Schelter regular algebras.

Contribution

It defines monotonic algebras and establishes bounds on GK dimension and generators, along with a parity theorem for Artin-Schelter regular algebras.

Findings

GK dimension is bounded by global dimension in monotonic algebras

Minimal number of generators is similarly bounded

Difference between global and GK dimension is always even for Artin-Schelter regular algebras

Abstract

In this paper, we introduce the concept of \textit{monotonic algebras}, a broad class of algebras that includes all Artin-Schelter regular algebras of dimension at most four, as well as algebras with \textit{pure} resolutions, such as Koszul and piecewise Koszul algebras. We show that the Gelfand-Kirillov (GK) dimension of these algebras is bounded above by their global dimension and establish a similar result for the minimal number of generators. Furthermore, we prove a parity theorem for Artin-Schelter regular algebras, demonstrating that the difference between their global dimension and GK dimension is always an even integer.