A bimodule approach to dominant dimension

TL;DR

This paper characterizes finite dimensional algebras with high dominant dimension using bimodule torsionfreeness and provides new formulas for Hochschild (co)homology, linking several conjectures in algebra.

Contribution

It introduces a bimodule approach to dominant dimension, establishing equivalences and new formulas, and connects dominant dimension to Hochschild (co)homology and major conjectures.

Findings

Characterization of dominant dimension via bimodule torsionfreeness

New formulas for Hochschild homology and cohomology

Established relations between conjectures in algebra

Abstract

We show that a finite dimensional algebra $A$ has dominant dimension at least $n \geq 2$ if and only if the regular bimodule $A$ is $n$-torsionfree if and only if $A \cong \Omega^{n}(\text{Tr}(\Omega^{n-2}(V)))$ as $A$-bimodules, where $V=\text{Hom}_A(D(A),A)$ is the canonical $A$-bimodule in the sense of \cite{FKY}. We apply this to give new formulas for the Hochschild homology and cohomology for algebras with dominant dimension at least two and show a new relation between the first Tachikawa conjecture, the Nakayama conjecture and Gorenstein homological algebra.