Gorenstein projective modules and Gorenstein flatness

Determine whether every Gorenstein projective module over an arbitrary ring is Gorenstein flat.

Background

Gorenstein projective and Gorenstein flat modules are central classes in Gorenstein homological algebra. The paper establishes positive partial results: every ℵ₁-generated strongly Gorenstein projective module is Gorenstein flat, and every strongly ℵ₁-presented Gorenstein projective module is Gorenstein flat. These size-restricted results do not resolve the general problem posed by Holm for arbitrary Gorenstein projective modules.

References

One central open problem, posed by Holm , asks whether every Gorenstein projective module is Gorenstein flat.

— Mittag-Leffler Conditions, Gorenstein Modules and Homological Invariants  (2608.13898 - Dai et al., 14 Aug 2026) in Introduction; Section 3, immediately before Proposition 3.1

Hence the following questions are equivalent: \begin{itemize} \item Does $GP(R)\subseteqGF(R)$ hold for every ring $R$? \item Does $GP(R)=PGF(R)$ hold for every ring $R$? \end{itemize} Šaroch--Šťovíček stated this problem Remark following Example~3.10; it remains open in $\textsf{ZFC}$.

— A strongly compact cardinal yields a left and right coherent ring with $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$  (2608.17748 - Zhang, 18 Aug 2026) in Section 1, Introduction

It is not known whether the above inclusion is an equality, nor whether $PGF(R)=GProj(R)$ in general; as observed in Section~4, the equality $PGF(R)=GProj(R)$ holds if and only if every Gorenstein projective module is Gorenstein flat, which is a long-standing open problem.

— $\aleph_m$-presented Gorenstein flat modules and Gorenstein dimensions of groups  (2609.24512 - Stergiopoulou, 21 Sep 2026) in Section 2, Subsection 2.1, immediately after Definition 2.2; also noted in Section 1, Introduction