Finitistic Dimension of Faithfully Flat Weak Hopf-Galois Extension

TL;DR

This paper investigates the relationship between the finitistic dimensions of algebras in weak Hopf-Galois extensions, establishing inequalities and equalities under certain conditions, thus advancing understanding in algebraic homological dimensions.

Contribution

It proves that finitistic dimension of the base algebra is bounded by that of the extension algebra in weak Hopf-Galois extensions, and under semisimplicity, they are equal if the conjecture holds.

Findings

Finitistic dimension of B is less than or equal to that of A.

Equality of finitistic dimensions holds if H is semisimple and the conjecture is true.

Provides new insights into homological properties of weak Hopf-Galois extensions.

Abstract

Let $H$ be a finite-dimensional weak Hopf algebra over a field $k$ and $A/B$ be a right faithfully flat weak $H$-Galois extension. We prove that if the finitistic dimension of $B$ is finite, then it is less than or equal to that of $A$. Moreover, suppose that $H$ is semisimple. If the finitistic dimension conjecture holds, then the finitistic dimension of $B$ is equal to that of $A$.