Two-sided coherent algebras over any field with $\mathcal{P}\mathcal{GF}(R)⊊\mathcal{G}\mathcal{P}(R)$

Abstract

For a ring $R$, let $\mathcal{G}\mathcal{P}(R)$, $\mathcal{GF}(R)$, and $\mathcal{P}\mathcal{GF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We answer negatively the question whether $\mathcal{G}\mathcal{P}(R)=\mathcal{P}\mathcal{GF}(R)$ for every ring. More precisely, over every field $k$ we construct a left and right coherent central $k$-algebra $T$ and a strongly Gorenstein projective left $T$-module which is not Gorenstein flat; hence $\mathcal{P}\mathcal{GF}(T)⊊\mathcal{G}\mathcal{P}(T)$.

The construction is extended from fields of characteristic (2) to arbitrary fields. A new corollary section collects consequences of the main theorem obtained from existing results in Gorenstein homological algebra