Year: 2020
Type: preprint
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 $A \cong Ω^{n}(\text{Tr}(Ω^{n-2}(V)))$ 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.
Source: arXiv (Cornell University)
Author: Marczinzik, Rene
Institution: University of Stuttgart
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W3025868439