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

TL;DR

The paper addresses whether the Gelfand-Kirillov dimension can be bounded by the global dimension and how many generators are required for a broad class of graded algebras extending Artin-Schelter regular algebras. It introduces monotonic algebras, leveraging the Hilbert series form $h_A(t)=\frac{1}{p(t)}$ and the reciprocity relation $t^l p(t^{-1}) = (-1)^d p(t)$ to derive a parity theorem $GKdim(A) \equiv d \pmod{2}$ for AS-regular algebras, and shows that $GKdim$ is bounded above by the global dimension $d$ for monotonic algebras with finite GK-dimension. Under the weighted polynomial Hilbert-series form $h_A(t)=\frac{1}{\prod_{i=1}^m (1-t^i)^{n_i}}$, the paper also proves a generator bound: $A$ is generated by at most $d$ elements. These results unify several known cases (pure resolutions, Koszul, small-dimension AS-regular algebras) and provide a framework for bounding invariants in noncommutative graded algebras, supporting the broader conjecture that GK-dimension often coincides with the global dimension in practice.

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.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• proof
• Corollary 2.5