Homological properties of extensions of algebras

TL;DR

This paper studies a special class of algebra extensions called strongly proj-bounded extensions, showing they preserve important homological properties like global dimension and Hochschild homology, with new examples of such extensions.

Contribution

It introduces strongly proj-bounded extensions and proves they preserve key homological invariants, extending previous results and providing new algebraic examples.

Findings

Finiteness of global dimension is preserved

Finiteness of Hochschild homology support is preserved

New class of finite relative global dimension extensions described

Abstract

We consider a class of extensions of associative algebras, which we refer to as ``strongly proj-bounded extensions''. We prove that the finiteness of the left global dimension and the support of the Hochschild homology is preserved by strongly proj-bounded extensions, generalizing results of Cibils, Lanzillota, Marcos and Solotar. Moreover, we show that the finiteness of the big left finitistic dimension is preserved by strongly proj-bounded extensions. In order to construct examples, we describe a new class of extensions of algebras of finite relative global dimension, which may be of independent interest. The results apply both for abstract (meaning no topology) and pseudocompact algebras.