Left-right symmetry of finite finitistic dimension

TL;DR

This paper establishes an equivalence condition for the finitistic dimension conjecture, linking the finiteness of an algebra's finitistic dimension to that of its opposite algebra, and explores related injective generation properties.

Contribution

It proves that the finitistic dimension conjecture holds iff the finiteness of an algebra's finitistic dimension implies the same for its opposite algebra, and discusses injective generation.

Findings

Finitistic dimension conjecture is equivalent to its symmetry between an algebra and its opposite.

Finiteness of an algebra's finitistic dimension implies finiteness for its opposite algebra.

Results are also applicable to injective generation properties.

Abstract

We show that the finitistic dimension conjecture holds for all finite dimensional algebras if and only if, for all finite dimensional algebras, the finitistic dimension of an algebra being finite implies that the finitistic dimension of its opposite algebra is also finite. We also prove the equivalent statement for injective generation.