F-coherent rings and regular rings

Let $R$ be a complete local domain containing an algebraically closed field. If the perfection of $R$ , $R_{perf}$ (union of all $p^{e}$ th roots, in the sense of positive characteristic commutative algebra), is coherent, must $R$ be a purely inseparable extension or a subring of a regular ring?

Progress summary

Open

The converse remains open: coherence of the perfection has not been shown to force the ring to come from a regular ring or a purely inseparable extension.

The problem asks whether an $F$ -coherent complete local domain containing an algebraically closed field must arise through one of the known regularity constructions.

Known results

Regular rings are $F$ -coherent.
Purely inseparable extensions and subrings of regular rings are $F$ -coherent.
In dimension one, $F$ -coherence is equivalent to purely inseparable normalization.
More generally, an $F$ -coherent reduced Noetherian ring with module-finite normalization has purely inseparable normalization.

January 2024 characterization result

A paper answering a question of Patankar constructs a complete local Noetherian normal domain of characteristic $p>0$ whose perfection is a non-coherent GCD domain. This challenges the landscape of known characterizations, but the source does not identify the example as a counterexample to the stated implication, since its perfection is not coherent.

Current status (as of August 2026): The stated implication remains unresolved; the classical sufficient constructions and purely inseparable-normalization results are known, but no proof or counterexample for this exact formulation was found.

Sources

Sources & referencesView supporting material

References

Source. K. Shimomoto, (F)-coherent rings with applications to tight closure theory, J. Algebra 338 (2011), 24–34. See also M. Asgharzadeh and S. Patankar, Remarks on some Homological Problems regarding Infinite Integral Extensions, arXiv:2608.09473, Question 1.7. A. Simpson, The perfection can be a noncoherent GCD domain. J. Commut. Algebra 16 (2024) no. 3, 363-367. JCA

commutative-algebra

Posted by Shravan Patankar 1 week ago · edited 1 week ago