Global Dimension of Polynomial Rings in Partially Commuting Variables

TL;DR

This paper extends the understanding of polynomial rings with partially commuting variables by computing their global dimension and generalizing Hilbert's Syzygy Theorem within this context.

Contribution

It introduces a method to compute the global dimension of categories of objects over free partially commutative monoids and generalizes a classical theorem to this setting.

Findings

Computed global dimension for categories over partially commutative monoids

Generalized Hilbert's Syzygy Theorem to partially commuting variables

Provides new tools for homological analysis in non-commutative polynomial rings

Abstract

For any free partially commutative monoid $M(E,I)$, we compute the global dimension of the category of $M(E,I)$-objects in an Abelian category with exact coproducts. As a corollary, we generalize Hilbert's Syzygy Theorem to polynomial rings in partially commuting variables.