F-coherent rings — Noetherian rings of prime characteristic p>0p>0 whose perfect closure R∞R^{\infty} is coherent — were introduced by Shimomoto to study tight closure through perfect rings: in such rings tight closure coincides with Frobenius closure and commutes with localization. While F-coherence is known to descend along faithfully flat maps and to be preserved by localization, purely inseparable extension, and étale and henselian extensions, its ascent behavior is open. It is unknown whether F-coherence ascends to polynomial rings R[x1,…,xn]R[x_1,\dots,x_n] — equivalently, along smooth maps, since stability under localization and étale maps is already known — or under any natural fiber condition on a faithfully flat map R→SR\to S, and whether F-coherence of a hypersurface quotient R/tRR/tR by a non-zero-divisor tt implies that of RR. The only known partial results are polynomial ascent for one-dimensional local rings and the fact that RR is at least a domain when R/tRR/tR is reduced and F-coherent. The problem asks for the ascent and base-change permanence of F-coherence in the generality posed by the source — the algebraic counterpart of the still-open question of whether tight closure localizes at single elements.

Background

Let RR be a Noetherian ring of prime characteristic p>0p>0. The Frobenius map F:R→RF:R\to R sends r↦rpr\mapsto r^p, and its ee-th iterate sends r↦rper\mapsto r^{p^e}. The perfect closure (or perfection) R∞R^{\infty} is the smallest perfect overring of RR, where perfect means that Frobenius is bijective; for a reduced ring it is constructed inside the total ring of fractions as the union ⋃e≥1Rp−e\bigcup_{e\geq 1}R^{p^{-e}} of rings of pep^e-th roots. A theorem of Kunz says that the ee-th Frobenius iterate is flat exactly when RR is regular, and this is the main computational handle on perfect closures. A ring is coherent when every finitely generated ideal is finitely presented; this is the standard weakening of Noetherianity for non-Noetherian rings such as R∞R^{\infty}. Coherence is stable under localization and under flat colimits, but it does not behave like Noetherianity in other respects: a polynomial ring over a coherent ring need not be coherent — coherence of A[x]A[x] is guaranteed when the global dimension of AA (the supremum of the projective dimensions of AA-modules) is at most 2 and can fail otherwise.

Shimomoto (2011) introduced the class of F-coherent rings: RR is F-coherent when R∞R^{\infty} is coherent. Regular rings are F-coherent by Kunz's theorem and the stability of coherence under flat colimits, and so is any purely inseparable extension or subextension of a regular ring (the Frobenius twist construction identifies the two perfections); this yields non-regular examples such as affine semigroup subrings like k[x4,x3y,xy3,y4]k[x^4,x^3y,xy^3,y^4] in characteristic 2. Shimomoto established the known permanence properties: F-coherence is stable under localization and under purely inseparable extension, descends along faithfully flat maps (if SS is F-coherent and R→SR\to S is faithfully flat, so is RR), and is compatible with henselization (a filtered colimit construction of étale local extensions of a local ring, inside which Hensel's lemma holds) and with étale maps: any standard étale extension of an F-coherent ring is F-coherent, hence so is any étale extension, since an étale map is Zariski-locally standard étale and F-coherence is a local property. He also proved structural consequences: F-coherent rings have purely inseparable normalizations, the perfect closure of a local F-coherent ring is a big Cohen-Macaulay algebra (a not-necessarily-finite module on which a system of parameters — a sequence of length equal to the dimension of the ring generating an ideal primary to the maximal ideal — is a regular sequence), and tight closure coincides with Frobenius closure, where the Frobenius closure IFI^{F} of an ideal II consists of the elements x∈Rx\in R with xpe∈I[pe]x^{p^e}\in I^{[p^e]} for all sufficiently large ee.

Tight closure is the closure operation at the center of this story: for an ideal I⊆RI\subseteq R, an element xx lies in the tight closure I∗I^{*} of II if there is an element cc outside every minimal prime of RR such that cxpe∈I[pe]cx^{p^e}\in I^{[p^e]} for all e≥1e\geq 1, where I[pe]I^{[p^e]} denotes the ideal generated by the pep^e-th powers of elements of II. Whether tight closure commutes with localization — whether S−1(I∗)=(S−1I)∗S^{-1}(I^{*})=(S^{-1}I)^{*} for every multiplicative set — is a fundamental problem of the theory: Brenner and Monsky constructed a counterexample in general (arXiv:0710.2913), and the refined question of whether tight closure localizes at single elements remains open. Epstein (2024) recast the single-element case geometrically: tight closure localizes at single elements if and only if the sheafified tight closure operation on ideals of coherent sheaves preserves quasi-coherence (a sheaf of modules is quasi-coherent when it is locally presentable by generators and relations over the structure sheaf), which makes coherence-type conditions the natural algebraic hypothesis. F-coherence is exactly such a condition: in a reduced F-coherent ring, tight closure equals Frobenius closure and therefore commutes with localization (Shimomoto, Corollary 3.12 of the preprint version).

What the theory lacks is ascent. Descent along faithful flatness is known, but no technique produces F-coherence upstairs, and as of 2024 the only known examples of F-coherent rings are purely inseparable extensions or inclusions of regular rings (Simpson), so not even a test case for ascent is available. Asgharzadeh and Shimomoto (2012) proved the one positive ascent result: a one-dimensional (local) F-coherent ring has F-coherent polynomial ring. Their route uses the bound that the global dimension of R∞R^{\infty} is at most dim⁡R+1\dim R+1, which is at most 2 precisely in dimension one, together with the classical coherence criterion for polynomial algebras cited above; the same bound blocks the method in dimension 2 and higher. Shimomoto posed the ascent problem as Question 1 of the preprint version — asking what conditions on the fibers of a faithfully flat map R→SR\to S with RR F-coherent force SS to be F-coherent, noting that even the smooth case and even the polynomial case R[x]R[x] were unknown — together with the companion lifting question (Question 2 of the preprint): if t∈Rt\in R is a non-zero-divisor and R/tRR/tR is F-coherent, must RR be F-coherent? Coherence does not in general pass from a quotient by a non-zero-divisor back to the ring, so this question is not formal.

Problem Statement

Let RR be a Noetherian ring of characteristic p>0p>0 and R∞R^{\infty} its perfect closure; RR is F-coherent when R∞R^{\infty} is a coherent ring. Determine the ascent behavior of F-coherence under the basic constructions of commutative algebra, as posed by Shimomoto:

  1. (Flat ascent with fiber control.) For a faithfully flat map R→SR\to S of Noetherian rings with RR F-coherent, identify conditions on the fibers of R→SR\to S under which SS is F-coherent.

  2. (Smooth and polynomial ascent.) The base case of 1: for every F-coherent Noetherian ring RR and every n≥1n\geq 1, is the polynomial ring R[x1,…,xn]R[x_1,\dots,x_n] F-coherent? Equivalently — using the known stability of F-coherence under localization and under étale maps, together with the fact that a smooth map is Zariski-locally standard étale over a polynomial ring — does F-coherence ascend along smooth ring maps?

  3. (Hypersurface lifting.) If t∈Rt\in R is a non-zero-divisor and R/tRR/tR is F-coherent, must RR be F-coherent?

These three questions share one scientific objective: establishing F-coherence as a base-change-stable singularity condition, the algebraic counterpart of quasi-coherence for sheafified tight closure. They are the flat-ascent, polynomial/smooth-ascent, and hypersurface-lifting faces of one permanence problem: question (1) asks which fiber conditions force SS to be F-coherent, and questions (2) and (3) ask for a proof or a counterexample. The polynomial case (2) is the litmus test of the fiber-condition question (1), and the lifting question (3) is the converse-direction base-change face of the same permanence problem. Questions (1)-(3) are posed for arbitrary Noetherian rings of characteristic p>0p>0; no excellence (a standard descent condition on completions and formal fibers) or F-finiteness (Frobenius presenting RR as a finite module over its subring of pp-th powers) hypothesis is assumed in (2) and (3) unless it is part of an explicitly stated fiber condition in (1). Two partial results are known: polynomial ascent holds for one-dimensional local F-coherent rings, and RR is at least a domain when R/tRR/tR is reduced and F-coherent.

References

  1. Kazuma Shimomoto, "F-coherent rings with applications to tight closure theory", Journal of Algebra 338 (2011) 24-34, DOI 10.1016/j.jalgebra.2011.05.006, arXiv:0908.0630, https://arxiv.org/abs/0908.0630
  2. Mohsen Asgharzadeh and Kazuma Shimomoto, "Almost Cohen-Macaulay and almost regular algebras via almost flat extensions", Journal of Commutative Algebra 4 (2012), Issue 4, DOI 10.1216/jca-2012-4-4-445, arXiv:1003.0265, https://arxiv.org/abs/1003.0265
  3. Shravan Patankar, "Coherence of absolute integral closures", Proceedings of the American Mathematical Society, Series B 9 (2022) 75-89, DOI 10.1090/bproc/121, arXiv:2107.14743, https://arxiv.org/abs/2107.14743
  4. Austyn Simpson, "The perfection can be a noncoherent GCD domain", Journal of Commutative Algebra 16 (2024), Issue 3, 363-367, DOI 10.1216/jca.2024.16.363, arXiv:2401.00803, https://arxiv.org/abs/2401.00803
  5. Neil Epstein, "Tight Closure, Coherence, and Localization at Single Elements", Acta Mathematica Vietnamica 49 (2024), Issue 2, 201-215, DOI 10.1007/s40306-024-00533-9, https://doi.org/10.1007/s40306-024-00533-9
  6. Holger Brenner and Paul Monsky, "Tight closure does not commute with localization", arXiv:0710.2913 (2007, final version 2009), https://arxiv.org/abs/0710.2913
  7. Farzad Shaveisi and Mostafa Amini, "On the n-coherence of the perfect closure of some ring extensions", Asian-European Journal of Mathematics 11 (2018) 1850077, DOI 10.1142/s1793557118500778, https://doi.org/10.1142/s1793557118500778
  8. Mohsen Asgharzadeh, "Perfect closure detects injective dimension", arXiv:2606.30416 (2026), https://arxiv.org/abs/2606.30416
  9. Mohsen Asgharzadeh and Shravan Patankar, "Remarks on some Homological Problems regarding Infinite Integral Extensions", arXiv:2608.09473 (2026), https://arxiv.org/abs/2608.09473