A strongly compact cardinal yields a left and right coherent ring with $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$

Abstract: For a ring $R$, let $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, and $\mathcal{PGF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We introduce the local Boolean--Roos hypothesis $\textsf{LBR}$: the existence of a strongly compact cardinal implies $\textsf{LBR}$, while $\textsf{LBR}$ implies the existence of a measurable cardinal. Assuming $\textsf{LBR}$, we construct a left and right coherent ring $R$ and a strongly Gorenstein projective left $R$-module $G$ which is not Gorenstein flat; hence $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$.

• The paper proves that, assuming a strongly compact cardinal, there is a left and right coherent ring R with PGF(R) properly contained in GP(R), resolving the equality problem negatively under a large-cardinal hypothesis.
• The authors establish the required Ext-vanishing using Boolean rings, complete ultrafilters, and Roos-complex derived limits, then construct a strongly Gorenstein projective module whose nonzero Tor with an injective module shows it is not Gorenstein flat.
• The result provides the first counterexample in the two-sided coherent setting while leaving open whether PGF(R)=GP(R) is provable, refutable, or independent in ZFC alone.

The problem and the main result

For a ring $R$, the inclusions $PGF(R)\subseteq GP(R)\cap GF(R)$ hold unconditionally, where $GP$, $GF$, and $PGF$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat modules, respectively (2608.17748). Iacob's equivalence $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$ reduces the question of whether every Gorenstein projective module is Gorenstein flat to the equality $GP(R)=PGF(R)$, a problem stated by Šaroch–Šťovíček that remained open in ZFC. The paper resolves this in the negative under a large-cardinal hypothesis. The main theorem asserts: assuming ZFC together with the existence of a strongly compact cardinal, there exist a left and right coherent ring $R$ and a strongly Gorenstein projective left $R$-module $G$ that is not Gorenstein flat; consequently $PGF(R)\subseteq GP(R)\cap GF(R)$0 (2608.17748). The construction is a counterexample over a two-sided coherent ring, which is the strongest ring-theoretic setting in which the question was previously unresolved.

The local Boolean–Roos hypothesis

The set-theoretic input is isolated as the hypothesis $PGF(R)\subseteq GP(R)\cap GF(R)$1 for an uncountable cardinal $PGF(R)\subseteq GP(R)\cap GF(R)$2, requiring (1) a nonprincipal $PGF(R)\subseteq GP(R)\cap GF(R)$3-complete ultrafilter on $PGF(R)\subseteq GP(R)\cap GF(R)$4, i.e., measurability of $PGF(R)\subseteq GP(R)\cap GF(R)$5, and (2) an $PGF(R)\subseteq GP(R)\cap GF(R)$6-complete fine ultrafilter on $PGF(R)\subseteq GP(R)\cap GF(R)$7 (2608.17748). The paper proves the chain

$PGF(R)\subseteq GP(R)\cap GF(R)$8

using a pushforward argument: the map $PGF(R)\subseteq GP(R)\cap GF(R)$9 transports a fine ultrafilter on $GP$0 to a nonprincipal $GP$1-complete ultrafilter on $GP$2 (2608.17748). Thus strong compactness suffices and measurability is necessary; whether the intermediate hypothesis $GP$3 is strictly weaker than strong compactness is not determined.

Ext-vanishing via the Roos complex

The homological core is the computation $GP$4 for all $GP$5 and all sets $GP$6, where $GP$7 is the Boolean ring and $GP$8 is the ultrapower quotient by the maximal ideal induced by the $GP$9-complete ultrafilter (2608.17748). Degrees $GF$0 and $GF$1 use only nonprincipality and $GF$2-completeness of the ultrafilter: singletons lie in the dual ideal, and any countable set does as well, which forces every homomorphism $GF$3 to extend to $GF$4.

For degrees $GF$5, the paper identifies $GF$6 with the derived inverse limit $GF$7 over the ideal $GF$8, computed via the normalized Roos complex of the inverse system (2608.17748). The vanishing argument proceeds in two steps: a cocycle $GF$9 is solved on fewer than $PGF$0 selected chain components by adjoining a strict upper bound in $PGF$1 (using $PGF$2-completeness of $PGF$3); then the coordinatewise ultralimit along the fine $PGF$4-complete ultrafilter $PGF$5 on $PGF$6 combines these partial solutions into a global cochain $PGF$7 with $PGF$8. Fineness of $PGF$9 preserves each selected component equation, and its $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$0-completeness preserves finite $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$1-support of the ultralimit. Notably, no normality or $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$2-completeness of $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$3 is required (2608.17748).

Construction of the counterexample

The proof proceeds in three steps (2608.17748).

Step 1. A character $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$4 factoring through $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$5 yields a nonzero class in $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$6 for a deleted free resolution $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$7 of $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$8; nonvanishing follows because $GP(R)\subseteq GF(R)\iff GP(R)=PGF(R)$9 while $GP(R)=PGF(R)$0 annihilates the ideal generated by any element of $GP(R)=PGF(R)$1.

Step 2. The resolution is periodized over the dual-number ring $GP(R)=PGF(R)$2 via the differential $GP(R)=PGF(R)$3. Characteristic $GP(R)=PGF(R)$4 is essential: it makes $GP(R)=PGF(R)$5 and collapses the sign in the Hom differential, so that $GP(R)=PGF(R)$6 is a one-periodic totally acyclic complex of free left $GP(R)=PGF(R)$7-modules. Moreover, $GP(R)=PGF(R)$8, while $GP(R)=PGF(R)$9 is exact for the quotient module $R$0.

Step 3. The lower triangular matrix ring $R$1 is formed. Since $R$2 is von Neumann regular and $R$3 is free of rank $R$4 as a central $R$5-algebra, Harris's coherence transfer shows $R$6 is left and right coherent (2608.17748). The complex $R$7 is one-periodic totally acyclic over $R$8, so its cocycle $R$9 is strongly Gorenstein projective. The injective right $R$0-module $R$1 detects the failure of flatness: a chain of isomorphisms identifies $R$2, hence $R$3, contradicting the Tor-vanishing that Gorenstein flat modules satisfy against injective modules (2608.17748). Therefore $R$4 and $R$5.

The implication is that over a two-sided coherent ring, Gorenstein projective modules need not be Gorenstein flat, and the equality $R$6 fails — the first such example, conditional on a large cardinal.

A conditional positive direction

Two appendices record a speculative converse. Under the auxiliary hypothesis $R$7 (periodic dual-number stationarity) and the absence of monotone $R$8-sequential cardinals — which follows from $R$9 via Choodnovsky's theorem on sequential continuity — the paper proves $G$0 for every ring $G$1 (2608.17748). The argument shows that a strongly Gorenstein projective module with the relevant Ext-vanishing must be strict $G$2-stationary, hence Gorenstein flat by Wang–Liang. The paper is explicit that $G$3 is not a standard axiom, that its consistency relative to ZFC is unknown, and that no familiar hypothesis is known to imply it; no consistency claim is made for $G$4.

Limitations and open questions

The main result is conditional: it requires the existence of a strongly compact cardinal, and the paper does not establish the consistency of the failure $G$5 relative to $G$6 alone. Two conjectures are stated: that $G$7 is consistent relative to ZFC, and that $G$8 is consistent relative to ZFC (2608.17748). If both hold, the universal equality $G$9 is independent of ZFC. Whether $PGF(R)\subseteq GP(R)\cap GF(R)$00 is strictly weaker than strong compactness, and whether the counterexample can be constructed in ZFC or under weaker hypotheses, remain open.

Conclusion

The paper constructs, assuming a strongly compact cardinal, a left and right coherent ring over which a strongly Gorenstein projective module fails to be Gorenstein flat, thereby separating $PGF(R)\subseteq GP(R)\cap GF(R)$01 from $PGF(R)\subseteq GP(R)\cap GF(R)$02 and answering the Šaroch–Šťovíček question negatively under large-cardinal assumptions. The proof combines ultrafilter combinatorics with Roos's derived-limit machinery, and the set-theoretic hypotheses are isolated in the intermediate axiom $PGF(R)\subseteq GP(R)\cap GF(R)$03. The remaining question — whether the failure, or the equality, holds in ZFC itself — is left as an explicit consistency problem.