An analogue of Hilbert's Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra

TL;DR

This paper proves an analogue of Hilbert's Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra, establishing formulas for global and weak dimensions over arbitrary rings.

Contribution

It introduces a noncommutative algebra of one-sided inverses and generalizes the Syzygy Theorem to this setting, including the projective dimension.

Findings

Global dimension of the algebra equals that of the base ring plus n.

Weak dimension increases by n for Noetherian base rings.

The algebra is noncommutative, non-Noetherian, and not a domain.

Abstract

An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra $\mS_n(A)$ is noncommutative, neither left nor right Noetherian and not a domain. The proof is based on a generalization of the Theorem of Kaplansky (on the projective dimension) obtained in the paper. As a consequence it is proved that for a left or right Noetherian algebra $A$: $$ \wdim (\mS_n(A))= \wdim (A) +n.$$