Cellular Noetherian algebras with finite global dimension are split quasi-hereditary

TL;DR

This paper proves that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over certain rings, establishing their structure, uniqueness, and providing formulas and bounds for their global and finitistic dimensions.

Contribution

It demonstrates that such algebras are split quasi-hereditary with unique structure and derives formulas and bounds for their global and finitistic dimensions.

Findings

Cellular Noetherian algebras with finite global dimension are split quasi-hereditary.

The quasi-hereditary structure of these algebras is unique up to equivalence.

Provides formulas and bounds for global and finitistic dimensions of these algebras.

Abstract

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.