A strongly compact cardinal yields a left and right coherent ring 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 isolate the local ultrafilter hypothesis $\text{LUH}$: the existence of a strongly compact cardinal implies $\text{LUH}$, while $\text{LUH}$ implies the existence of a measurable cardinal. Assuming $\text{LUH}$, we construct a left and right coherent ring $R$ and a strongly Gorenstein projective left $R$-module $G$ which is not Gorenstein flat; hence $\mathcal{P}\mathcal{GF}(R)⊊\mathcal{G}\mathcal{P}(R)$.

Substantially revised. Added historical context and a comparison with Dai--Zhang. Renamed LBR as LUH and removed the redundant uncountability assumption. Corrected module-side conventions, the Roos--Jensen statement, Roos-complex indices, and notation. Moved the speculative positive direction to discussion appendices and replaced conjectures with questions