Stacks of algebras and their homology

TL;DR

This paper constructs finite dimensional algebras with prescribed finitistic dimensions using a stacking technique, enabling precise control over their homological properties and higher syzygies.

Contribution

It introduces a novel stacking method to build algebras with specific finitistic dimension functions, advancing the understanding of homological dimensions.

Findings

Constructed algebras with finitistic dimension functions matching any finite-valued increasing function.

Developed a stacking technique to control higher syzygies of modules.

Provided new examples illustrating the range of finitistic dimensions.

Abstract

For any increasing function $f: {\Bbb N} \rightarrow {\Bbb N}_{\ge 2}$ which takes only finitely many distinct values, a connected finite dimensional algebra $\Lambda$ is constructed, with the property that $\text{fin.dim}_n\, \Lambda = f(n)$ for all $n$; here $\text{fin.dim}_n\, \Lambda$ is the $n$-generated finitistic dimension of $\Lambda$. The stacking technique developed for this construction of homological examples permits strong control over the higher syzygies of $\Lambda$-modules in terms of the algebras serving as layers.