Pith. sign in

REVIEW 3 major objections 4 minor 3 cited by

{\delta}-rings, perfectoid towers, and lim Cohen-Macaulay sequences

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A complete mixed-characteristic local domain admitting a perfectoid tower has a lim Cohen–Macaulay sequence, linking tilting to the positivity conjecture.

desk verdict The core theorem connecting perfectoid towers to lim Cohen-Macaulay sequences is plausible and worth serious attention, but the proof has a gap: it never shows the tower layers are finitely generated over R, as Definition 3.16 requires. read the letter →

arxiv 2509.06527 v3 pith:BDKK76M6 submitted 2025-09-08 math.AC math.AGmath.NT

Shinnosuke Ishiro , Kazuma Shimomoto This is my paper · ORCID
classification math.ACmath.AGmath.NT MSC 13A3513D2213F5513F6514G45
keywords δ-ringFrobeniusliftperfectoidtowerlimCohen-Macaulaysequenceφ-stableidealmixedcharacteristiccommutativealgebrapurityintersectionmultiplicitypositivityconjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper attempts to connect two objects in commutative algebra: perfectoid towers and lim Cohen–Macaulay sequences. A perfectoid tower is a Noetherian approximation of a perfectoid ring arranged as a chain of ring maps whose reductions are purely inseparable; a lim Cohen–Macaulay sequence is a sequence of finitely generated modules whose local cohomology lengths grow slowly with the number of generators, a substitute for a Cohen–Macaulay module. The main theorem says that whenever a complete Noetherian local domain of mixed characteristic with perfect residue field admits a perfectoid tower, its layers form a lim Cohen–Macaulay sequence. The proof passes to the tilt, where the tower becomes a perfect tower of p-power roots, and perfect towers were already known to be lim Cohen–Macaulay. The paper then supplies construction methods for perfectoid towers from δ-rings and stable Frobenius lifts, with explicit examples from monomial, binomial, determinantal, and section rings; if the main theorem is right, the existence of such towers would give a route to the positivity conjecture on intersection multiplicities.

What carries the argument

The carrying mechanism is the tilting correspondence for perfectoid towers. From a tower satisfying the axioms, one takes the inverse limit of the layers modulo I_0 along the Frobenius projections; the result is a perfect tower, naturally isomorphic to adjoining successive p-power roots of the small tilt. The isomorphism R_i/f_0R_i ≅ R_i^♭/f_0^♭ R_i^♭ lets the lim Cohen–Macaulay property travel from the perfect tower back to the original one. On the construction side, the central object is a φ-stable ideal in a δ-ring: a ring A with a p-derivation δ whose associated Frobenius lift φ satisfies φ(I) ⊆ I. An ideal with this closure property lets the quotient R=A/I inherit a Frobenius lift, and

What would settle it

A perfectoid tower satisfying Definition 2.3 whose tilt is not F-finite, or with a transition map R_0 → R_1 that is not module-finite, would leave Theorem 3.19 unusable: Definition 3.16 requires each layer to be a finitely generated R-module. Concretely, one could look for a tower over R = W(k)[[x_2,...,x_d]]/I with squarefree monomial I in which the colimit layer R_1 is not finite as an R-module; if such a tower exists, the conclusion 'lim Cohen–Macaulay sequence' has no meaning under the paper's own definition.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 3.19: let (R,m,k) be a complete Noetherian local domain of mixed characteristic with perfect residue field of characteristic p. If a perfectoid tower ({R_i},{t_i}) arises from (R,I_0) for some ideal I_0, then the sequence {R_i} is a lim Cohen–Macaulay sequence of algebras. The proof fixes a generator f_0 of I_0 and uses the quoted facts that the I_0-torsion of each R_i vanishes and that the quotient by f_0 transfers the lim Cohen–Macaulay property. Because the reduced quotients R_i/f_0R_i are isomorphic to the corresponding quotients of the tilt R_i^♭/f_0^♭ R_i^♭, and because the tilt is a perfect tower isomorphic to R_0^♭ → (R_0^♭)^{1/p} → ..., already l

Load-bearing premise

The conclusion only makes sense if every layer R_i is finitely generated as an R-module, but the axioms of a perfectoid tower do not guarantee finite generation; the paper obtains it only under extra hypotheses such as an F-finite tilt and I_0-adic completeness.

Editorial extensions

If this is right

  • If Theorem 3.19 holds, constructing a perfectoid tower over any complete Noetherian local domain of mixed characteristic would produce the lim Cohen–Macaulay sequence needed for the positivity conjecture on intersection multiplicities, making perfectoid-tower existence the central question.
  • The φ-stable ideal construction in Theorem 4.6 shows that quotients of W(k)[[x]] by monomial or binomial ideals, and by ideals of 2×m-minors, carry perfectoid towers whenever they are p-torsion-free with reduced reduction, giving explicit lim Cohen–Macaulay sequences.
  • For section rings of smooth projective varieties with quasi-canonical liftings, the construction yields perfectoid towers whose layers are complete normal local domains; when the variety is an ordinary abelian variety of dimension at least 2, each layer is non-Cohen–Macaulay, so these are genuinely new lim Cohen–Macaulay examples.
  • In the p-torsion-free cases the tilt is explicitly the Frobenius-root tower over the residue field (Theorem 4.6(2), Corollaries 4.8 and 4.10), so the Frobenius structure of the tower is completely computable.
  • Lemma 2.10 and Corollary 4.11 turn perfectoid towers with pure transition maps into perfectoid purity of the base ring, yielding new perfectoid pure singularities such as lifts of wide Gorenstein ladder determinantal rings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: The finite-generation gap in Theorem 3.19 suggests that the natural strengthening replaces 'perfectoid tower exists' by 'perfectoid tower with F-finite tilt exists'; one could test whether the geometric examples of Section 4.3 satisfy this stronger hypothesis.
  • Editorial inference: The φ-stable ideal method is not tied to formal power series; any δ-ring quotient with a φ-stable ideal and reduced p-reduction should yield a perfectoid tower. A testable extension would be to determinantal ideals of k×m-minors for k>2 or to non-ladder binomial ideals, where φ-stability becomes a combinatorial condition on exponents.
  • Editorial inference: The p-torsion example in Section 4.4 is explicitly outside Theorem 3.19; a generalized definition of lim Cohen–Macaulay sequence allowing torsion or non-finite layers might recover a theorem for such towers and enlarge the supply of examples.
  • Editorial inference: Since the final step of the proof imports [3]'s theorem on perfect towers, any future strengthening of that theorem would automatically strengthen Theorem 3.19; conversely, a counterexample to that theorem would not necessarily invalidate the construction parts of the paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops connections among δ-rings, perfectoid towers, and lim Cohen-Macaulay sequences. Its central result, Theorem 3.19, asserts that any perfectoid tower over a complete Noetherian local domain of mixed characteristic with perfect residue field yields a lim Cohen-Macaulay sequence of algebras. The proof proceeds by reducing modulo a generator of the defining ideal, comparing with the tilt, and invoking results of Bhatt–Hochster–Ma on perfect towers. Sections 3 and 4 also study δ-rings and Frobenius lifts, introduce φ-stable ideals, and give constructions of perfectoid towers from monomial, binomial, and determinantal ideals, from section rings of smooth projective varieties, and from a new example with p-torsion elements.

Significance. If Theorem 3.19 is correct, it provides a broad new source of lim Cohen-Macaulay sequences and gives a concrete connection between perfectoid towers and Serre's positivity conjecture. The paper also contains useful explicit constructions, including the first example of a perfectoid tower with p-torsion elements, and a geometric construction via section rings. The main weakness is a missing finite-generation verification in the proof of the main theorem; this is a load-bearing gap, but it appears fixable by adding the missing hypotheses or proving finiteness from the cited definitions.

major comments (3)
  1. [Theorem 3.19, Definition 3.16, Lemma 2.7] Definition 3.16 defines a lim Cohen-Macaulay sequence only for finitely generated R-modules. The proof of Theorem 3.19 applies Proposition 3.17 to the sequence {R_i} without proving that each R_i is finitely generated over R. Definition 2.3 does not include this property: axiom (e) only says R_i is I0-adically Zariskian. Lemma 2.7 gives module-finiteness of the transition maps only under extra hypotheses (F-finite tilt and I0-adic completeness/separatedness of each R_i), neither of which is assumed in Theorem 3.19. Thus on the stated hypotheses the object named in the conclusion may not even satisfy Definition 3.16. The same gap affects Corollary 3.18, where finiteness of R_i/pR_i and equality of Krull dimensions do not imply R_i is finite over R without additional completeness or finite-generation information. The theorem should be repaired by adding an explicit finite-generation/comple
  2. [Theorem 3.19, tilt comparison] After reducing modulo f0, the proof asserts that {R_i/f0R_i} is lim Cohen-Macaulay if and only if {R_i^♭/f0^♭R_i^♭} is, based on the ring isomorphisms R_i^♭/f0^♭R_i^♭ ≅ R_i/f0R_i. However Proposition 3.17 is stated for sequences of finitely generated modules over a fixed base local ring. The proof does not identify the base rings R/(f0) and R^♭_0/(f0^♭), nor does it check that either sequence satisfies the finite-generation hypothesis required by Proposition 3.17. The first point is likely repairable because R^♭_0/(f0^♭) ≅ R/(f0) by construction of the small tilt, but the second is the same finiteness gap as in the previous comment.
  3. [Definition 3.2, Definition 4.1, Discussion 4.7] The definition of A_i (and R_i) as a 'finite colimit' of a diagram A --φ--> A --φ--> ... --φ--> A is literally the last copy of A, so with the written definition A_i = A for all i. This is inconsistent with later claims such as A_i ≅ W(k)[[x_1^{1/p^i}, ..., x_d^{1/p^i}]] in Discussion 4.7 and the computation of R_i in Corollary 4.8. The authors presumably intend the i-th stage of the direct limit, i.e. the subring of A^{1/p^∞} generated by the φ^i-th roots of elements of A, or an equivalent formulation. This is not merely a typo, because Corollary 3.18 and the constructions in Section 4 rely on the explicit form of R_i as a p^{1/p^i}-root extension.
minor comments (4)
  1. [Throughout] Several cross-references use 'Theorem' where 'Lemma' or 'Definition' is meant, e.g. 'Theorem 3.4' in the proof of Lemma 3.4, 'Theorem 2.5 (3)' for Lemma 2.5, 'Theorem 3.13 (2)' for Lemma 3.13, and 'Theorem 4.17' for Lemma 4.17. Please correct.
  2. [Section 4.2 heading] Typo in heading: 'determiantal' should be 'determinantal'.
  3. [Theorem 4.18(1)] The proof says the tower satisfies 'conditions (i), (ii), (iii) in Theorem 4.6', but Theorem 4.6 states only conditions (i) and (ii). Clarify the reference.
  4. [Definition 3.16] In the definition, ℓ_R(H_i(x;M_n)) uses the length over R, but the condition is written with a system of parameters of R. It would be helpful to state explicitly that the same system of parameters is used for all n and that the length is taken after viewing M_n as an R-module via the structure map.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main theorem's proof chain ends in external results [3]; the self-citation to [18] supplies the definition and tilt properties but does not smuggle in the conclusion; the unverified finite-generation step is a correctness gap, not a circular reduction.

full rationale

The central derivation in Theorem 3.19 is not circular. The proof reduces the claim that a perfectoid tower is a lim Cohen-Macaulay sequence to three ingredients: (1) the tilt of a perfectoid tower is a perfect tower (Lemma 2.5(1), quoted from [18, Prop. 3.10(2)]); (2) reduction modulo a nonzero divisor preserves lim Cohen-Macaulayness (Proposition 3.17, quoted from [3, Prop. 4.12]); and (3) a perfect tower R0^flat -> (R0^flat)^{1/p} -> ... is a lim Cohen-Macaulay sequence by [3, Thm. 5.4 & Rem. 5.5]. The final step is an external theorem, and the input 'there exists a perfectoid tower' does not by construction contain the conclusion. The self-citation to [18] is load-bearing in the sense that it supplies the definition of the paper's central object and its tilt properties, but those are prior results with stated assumptions that do not include Theorem 3.19; under the stated rules, such a citation is genuine evidence and does not by itself raise the circularity score. The real concern is a missing finiteness check, not a circular one: Definition 3.16 requires each member of a lim Cohen-Macaulay sequence to be a finitely generated R-module, whereas Definition 2.3 of a perfectoid tower does not require R_i to be module-finite over R. Lemma 2.7 obtains module-finiteness only under extra completeness/F-finite hypotheses. The proof of Theorem 3.19 applies Proposition 3.17 without verifying finite generation ('because we can use Theorem 2.6, Theorem 3.17, and f0 is a nonzero divisor'), and Corollary 3.18 similarly asserts a 'short exact sequence of finitely generated R-modules' without proof. This is a correctness gap that is fixable by adding a module-finiteness/completeness hypothesis; it is not a reduction of the conclusion to the input. The paper honestly notes in Example 4.27 that Theorem 3.19 cannot be applied to its p-torsion tower, so it does not overclaim in that direction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new primitive entities. Its central claims rest on the existence and properties of perfectoid towers (from [18], partly self-cited), on the known lim Cohen-Macaulay behavior of finite perfections ([3]), and on the domain hypotheses for phi-stable ideals and Frobenius splittings.

assumptions (5)
  • domain assumption Axioms defining a perfectoid tower (Definition 2.3, from [18]) are satisfied; in particular the tilt is a perfect tower with the stated properties.
    Theorems 3.19 and 4.6 take such a tower as input; correctness of the tilt properties is quoted from [18].
  • standard math The perfect tower of finite perfections of a complete local F-finite domain is a lim Cohen-Macaulay sequence ([3, Theorem 5.4]).
    Used in the proof of Theorem 3.19 to conclude the tilt of a perfectoid tower is lim Cohen-Macaulay.
  • domain assumption Existence of a delta-ring structure with Frobenius lift phi and a phi-stable ideal I in Main Theorem 2 (Definition 3.9).
    Hypotheses of Main Theorem 2; the construction uses the colimit of the quotients by phi-stable ideals.
  • domain assumption Frobenius splitting and Gorenstein assumptions in Lemma 3.5 and Corollary 4.8(3), namely A/pA is F-finite, F-split, and Gorenstein.
    Needed to prove the Frobenius lift splits and to obtain purity of the tower maps.
  • standard math Local duality and the Rees lemma (Bruns-Herzog [9]) used in Lemma 3.5.
    Standard homological algebra input for the splitting claim.

how reviews work

0 comments
Cite this review

Pith. "Pith review of {\delta}-rings, perfectoid towers, and lim Cohen-Macaulay sequences." pith.science (2026). https://pith.science/paper/BDKK76M6

@misc{pith2026250906527,
  author       = {Pith},
  title        = {Pith review of: \delta-rings, perfectoid towers, and lim Cohen-Macaulay sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BDKK76M6}},
  note         = {Machine review of arXiv:2509.06527}
}
abstract

The aim of this article is to study basic structures and interrelations of $\delta$-rings, perfectoid towers, and lim Cohen--Macaulay sequences over Noetherian rings in positive or mixed characteristic. We also discuss the deformation of perfectoid purity via perfectoid towers. In the latter part of this paper, we discuss some methods for constructing perfectoid towers, dealing with $p$-torsion-free and $p$-torsion cases, respectively. Some interesting examples arise as quotients by monomial or binomial ideals or determinantal rings. We also explain a geometric method with a view toward constructing rings with certain singularities.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Regular rings and perfectoid towers

    math.AC 2026-05 unverdicted novelty 6.0 of 10

    A Noetherian local ring of residue characteristic p is regular iff it admits a flat map to a Noetherian ring that extends to a perfectoid tower.

  2. Structural properties and tilting correspondences of perfectoid towers

    math.AC 2026-05 unverdicted novelty 5.0 of 10

    Proves fiber-product decomposition of perfectoid towers plus tilting invariance of étale cohomology and Koszul homology, implying preservation of several Noetherian local ring properties under tilting.

  3. A characterization of perfectoid towers in terms of conormal cones

    math.AC 2026-05 unverdicted novelty 5.0 of 10

    Characterizes perfectoid towers via conormal cones by refining the relationship between torsion for a principal ideal and the associated conormal cone, extending Gabber-Ramero.

Reference graph

Works this paper leans on

38 extracted references · 36 canonical work pages · cited by 3 Pith papers

  1. [18]

    Ishiro, K

    S. Ishiro, K. Nakazato, and K. Shimomoto,Perfectoid towers and their tilts : with an application to the étale cohomology groups of local log-regular rings, to appear in Algebra and Number Theory

    work page

  2. [19]

    Ishizuka,Perfectoid towers generated from prisms,https://arxiv.org/abs/2409.15785v2

    R. Ishizuka,Perfectoid towers generated from prisms,https://arxiv.org/abs/2409.15785v2

    work page arXiv

  3. [21]

    Ishizuka and K

    R. Ishizuka and K. Shimomoto,Quasi-canonical lifting of projective varieties in positive characteristic, in prepa- ration

    work page

  4. [1]

    André,Le lemme d’Abhyankar perfectoide, Publ

    Y. André,Le lemme d’Abhyankar perfectoide, Publ. Math. I.H.E.S.127(2018), 1–70

    work page 2018

  5. [2]

    André,La conjecture du facteur direct, Publ

    Y. André,La conjecture du facteur direct, Publ. Math. I.H.E.S.127(2018), 71–93

    work page 2018

  6. [3]

    Bhatt, M

    B. Bhatt, M. Hochster and L. Ma,Lim Cohen-Macaulay sequences of modules,https://arxiv.org/abs/2410. 18372

    work page

  7. [4]

    Bhatt, M

    B. Bhatt, M. Morrow, and P. Scholze,Integralp-adic Hodge theory, Publ. Math. I.H.É.S.128(2018), 219–397

    work page 2018

  8. [5]

    Bhatt and P

    B. Bhatt and P. Scholze,Prisms and prismatic cohomology, Ann. of Math.196(2022), 1135–1275

    work page 2022

Show all 38 references
  1. [6]

    Bhatt, L

    B. Bhatt, L. Ma, Z. Patakfalvi, K. Schwede, K. Tucker, J. Waldron and J. Witaszek,Globally+++-regular varieties and the minimal model program for threefolds in mixed characteristic, Publications Math. I.H.E.S.138(2023), 69–227 (2023)

    2023

  2. [7]

    Bhatt, L

    B. Bhatt, L. Ma, Z. Patakfalvi, K. Schwede, K. Tucker, J. Waldron, J. Witaszek,Perfectoid pure singularities, https://arxiv.org/abs/2409.17965

    arXiv

  3. [8]

    Bruns and A

    W. Bruns and A. Conca,Gröbner Bases and Determinantal Ideals, In: Herzog, J., Vuletescu, V. (eds) Com- mutative Algebra, Singularities and Computer Algebra. NATO Science Series, vol 115. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1092-4_2, (2003)

    2003 doi

  4. [9]

    Bruns and J

    W. Bruns and J. Herzog,Cohen-Macaulay rings, Cambridge University Press39, revised edition 1998

    1998

  5. [10]

    Conca and J

    A. Conca and J. Herzog,Ladder determinantal rings have rational singularities, Advances in Mathematics132(1) (1997), 120–147

    1997

  6. [11]

    Chiriacescu,On the theory of Japanese rings, Rev

    G. Chiriacescu,On the theory of Japanese rings, Rev. Roumaine Math. Pures Appl.27(1982), 945–948

    1982

  7. [12]

    Dietz,Big Cohen-Macaulay algebras and seeds, Trans

    G. Dietz,Big Cohen-Macaulay algebras and seeds, Trans. Amer. Math. Soc.359(2007), 5959–5989

    2007

  8. [13]

    Gabber and L

    O. Gabber and L. Ramero,Almost ring theory, Lecture Notes in Mathematics1800, Springer

  9. [14]

    Gabber and L

    O. Gabber and L. Ramero,Almost ring theory and perfectoid spaces,https://pro.univ-lille.fr/ lorenzo-ramero/publications/

  10. [15]

    Grothendieck,Élements de Géométrie Algébrique III, Publications Math

    A. Grothendieck,Élements de Géométrie Algébrique III, Publications Math. I.H.E.S.11(1961),17(1963)

    1961

  11. [16]

    M. Hochster,Homological conjectures and lim Cohen-Macaulay sequences, Homological and computational meth- ods incommutative algebra, Springer INdAM Ser.,20, Springer, Cham, (2017), 173–197

    2017

  12. [17]

    Illusie,Grothendieck’s existence theorem in formal geometry, Fundamental Algebraic Geometry, Mathematical Surveys and Monographs123AMS 2005

    L. Illusie,Grothendieck’s existence theorem in formal geometry, Fundamental Algebraic Geometry, Mathematical Surveys and Monographs123AMS 2005. 30 S. ISHIRO AND K. SHIMOMOTO

    2005

  13. [20]

    Ishizuka and K

    R. Ishizuka and K. Nakazato,Prismatic Kunz’s theorem,https://arxiv.org/abs/2402.06207

  14. [22]

    Kawakami and T

    T. Kawakami and T. Takamatsu,On Frobenius liftability of surface singularities,https://arxiv.org/abs/2402. 08152

  15. [23]

    Kinouchi and K

    R. Kinouchi and K. Shimomoto,A study of perfectoid rings via Galois cohomology, arXiv:2506.01345

    arXiv

  16. [24]

    Kollár,Singularities of the minimal model program, With a collaboration of Sándor Kovács, Cambridge Uni- versity Press, Cambridge (2013)

    J. Kollár,Singularities of the minimal model program, With a collaboration of Sándor Kovács, Cambridge Uni- versity Press, Cambridge (2013)

    2013

  17. [25]

    Kollár and S

    J. Kollár and S. Mori,Birational geometry of algebraic varieties, Cambridge University Press, Cambridge (1998)

    1998

  18. [26]

    Ma,Lim Ulrich sequences and Lech’s conjecture, Invent

    L. Ma,Lim Ulrich sequences and Lech’s conjecture, Invent. Math.231(2023), 407–429

    2023

  19. [27]

    Matsumura,Commutative ring theory, Cambridge University Press.8(1986)

    H. Matsumura,Commutative ring theory, Cambridge University Press.8(1986)

    1986

  20. [28]

    Mumford,Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Oxford University Press, London, 1970

    D. Mumford,Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Oxford University Press, London, 1970

    1970

  21. [29]

    Murayama,A uniform treatment of Grothendieck’s localization problem, Compositio Math.158(2022), 57–88

    T. Murayama,A uniform treatment of Grothendieck’s localization problem, Compositio Math.158(2022), 57–88

    2022

  22. [30]

    Narasimhan,The irreducibility of ladder determinantal varieties, Journal of Algebra102(1986), 162–185

    H. Narasimhan,The irreducibility of ladder determinantal varieties, Journal of Algebra102(1986), 162–185

    1986

  23. [31]

    Rezk,Étale extensions ofλ-rings,https://rezk.web.illinois.edu/etale-lambda.pdf

    C. Rezk,Étale extensions ofλ-rings,https://rezk.web.illinois.edu/etale-lambda.pdf

  24. [32]

    Shimomoto,On the Witt vectors of perfect rings in positive characteristic, Communications in Algebra43 (2015), 5328–5342

    K. Shimomoto,On the Witt vectors of perfect rings in positive characteristic, Communications in Algebra43 (2015), 5328–5342

    2015

  25. [33]

    Shimomoto,On the semicontinuity problem of fibers and globalF-regularity, Communications in Algebra45 (2017), 1057–1075

    K. Shimomoto,On the semicontinuity problem of fibers and globalF-regularity, Communications in Algebra45 (2017), 1057–1075

    2017

  26. [34]

    Shimomoto and E

    K. Shimomoto and E. Tavanfar,On local rings without small Cohen-Macaulay algebras in mixed characteristic, to appear in Mathematical research letters

  27. [35]

    A. K. Singh,Failure ofF-purity andF-regularity in certain rings of invariants, Illinois Journal of Mathematics 42(1998), 441–448

    1998

  28. [36]

    The Stacks Project Authors,http://stacks.math.columbia.edu/browse

  29. [37]

    Strahan,Content and Q-sequences in mixed characteristic local rings, Journal of Algebra666(2025), 633–656

    O. Strahan,Content and Q-sequences in mixed characteristic local rings, Journal of Algebra666(2025), 633–656

    2025

  30. [38]

    Zdanowicz,Liftability of singularities and their Frobenius morphism modulop2, International Mathematics Research Notices2018(2017), 4513–4577

    M. Zdanowicz,Liftability of singularities and their Frobenius morphism modulop2, International Mathematics Research Notices2018(2017), 4513–4577. National Institute of Technology, Gunma College, 580 Toriba-machi, Maebashi-shi, Gunma 371- 8530, Japan Email address:shinnosukeish...

    2017

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.