Pure subrings of Du Bois singularities are Du Bois singularities

Charles Godfrey; Takumi Murayama

arxiv: 2208.14429 · v5 · pith:FRAXOGCAnew · submitted 2022-08-30 · 🧮 math.AG · math.AC

Pure subrings of Du Bois singularities are Du Bois singularities

Charles Godfrey , Takumi Murayama This is my paper

Pith reviewed 2026-05-24 11:11 UTC · model grok-4.3

classification 🧮 math.AG math.AC

keywords Du Bois singularitiescyclically pure mapsNoetherian Q-algebraslog canonical singularitiesGrothendieck topologiessingularity descent

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The pith

Cyclically pure maps of Noetherian Q-algebras descend Du Bois singularities from the larger ring to the smaller ring.

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

The paper proves that if R to S is a cyclically pure map of Noetherian algebras over the rationals and S has Du Bois singularities, then R has Du Bois singularities. The result holds even when the map is faithfully flat. The argument uses a characterization of the complex underline Omega zero X via sheafification in Grothendieck topologies, and it produces corollaries in positive and mixed characteristic. As an application, cyclically pure maps preserve log canonical singularities when the canonical divisor on the smaller ring is Cartier.

Core claim

Let R to S be a cyclically pure map of Noetherian Q-algebras. If S has Du Bois singularities, then R has Du Bois singularities. The same conclusion holds for the relevant notions in prime characteristic and mixed characteristic, and the method yields that log canonical singularities descend along such maps when K sub R is Cartier.

What carries the argument

Sheafification of the complex underline Omega zero X with respect to Grothendieck topologies, which characterizes Du Bois singularities on the schemes under study.

If this is right

Du Bois singularities descend along cyclically pure maps of Noetherian Q-algebras.
The descent statement extends to the corresponding notions of singularities in prime characteristic and mixed characteristic.
Log canonical singularities descend along cyclically pure maps of rings essentially of finite type over C when K sub R is Cartier.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The descent may allow reduction of singularity questions to simpler subrings in arithmetic settings.
Explicit examples such as hypersurface or quotient singularities could be checked to see the descent in action.
Similar descent statements might hold for other classes of singularities defined by vanishing of higher direct images.

Load-bearing premise

The Grothendieck-topology sheafification characterization of Du Bois singularities applies to the Noetherian schemes considered.

What would settle it

A concrete cyclically pure map R to S of Noetherian Q-algebras in which S has Du Bois singularities but R does not.

read the original abstract

Let $R \to S$ be a cyclically pure map of Noetherian $\mathbb{Q}$-algebras. In this paper, we show that if $S$ has Du Bois singularities, then $R$ has Du Bois singularities. Our result is new even when $R \to S$ is faithfully flat. Our proof also yields interesting results in prime characteristic and in mixed characteristic. As a consequence, we show that if $R \to S$ is a cyclically pure map of rings essentially of finite type over the complex numbers $\mathbb{C}$, $S$ has log canonical type singularities, and $K_R$ is Cartier, then $R$ has log canonical singularities. Along the way, we prove a version of the key injectivity theorem of Kov\'acs and Schwede for Noetherian schemes of equal characteristic zero that have isolated non-Du Bois points. Throughout the paper, we use the characterization of the complex $\underline{\Omega}^0_X$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Du Bois singularities descend under cyclically pure maps of Noetherian Q-algebras, but the argument rests on a Grothendieck topology characterization whose generality needs checking.

read the letter

The main thing to know is that the paper proves Du Bois singularities descend along cyclically pure maps of Noetherian Q-algebras. This holds even when the map is faithfully flat, and the same method produces statements in prime and mixed characteristic plus a version of the Kovács-Schwede injectivity theorem for isolated non-Du Bois points. As a side result they get descent for log canonical singularities when the rings are essentially finite type over C and the canonical divisor is Cartier. The approach relies on the characterization of the Du Bois complex via sheafification in the Grothendieck topology, which lets them move properties downward. That move is the technical core and appears to be applied cleanly to the descent question. The argument builds on existing characterizations without obvious circularity or self-referential fitting. The citation pattern is standard and appropriate for the area. The soft spot is the one flagged in the stress test. The sheafification characterization is typically established for schemes essentially of finite type over C. The paper states the result for arbitrary Noetherian Q-algebras, so if that characterization does not extend directly, the general claim has a gap at the foundation. The abstract does not add any finite type restriction, which makes the point worth verifying in the full text. If the extension holds or is justified, the rest looks solid. This paper is for people working on singularities in algebraic geometry and commutative algebra. A reader who cares about structural results on Du Bois or log canonical singularities would get direct value from the descent statement. It deserves a serious referee because the claim is precise, the method is technical, and the potential applications are immediate even if some clarification on scope is needed. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The paper claims that for a cyclically pure map R → S of Noetherian ℚ-algebras, Du Bois singularities descend from S to R. The argument relies throughout on a characterization of the Du Bois complex Ω⁰_X via sheafification in Grothendieck topologies, establishes an isolated-non-Du-Bois-points version of the Kovács–Schwede injectivity theorem in equal characteristic zero, and derives consequences for prime/mixed characteristic and for log canonical singularities when the rings are essentially of finite type over ℂ with K_R Cartier.

Significance. If correct, the result is new even for faithfully flat maps and supplies a downward-transfer statement for Du Bois singularities that is unavailable from existing literature; the isolated-points injectivity theorem and the log-canonical consequence are additional contributions of independent interest.

major comments (1)

[Abstract] Abstract (final paragraph) and the statement of the main theorem: the characterization of Ω⁰_X and of Du Bois singularities via Grothendieck-topology sheafification is invoked for arbitrary Noetherian ℚ-schemes. Standard references for this characterization (e.g., via the h-topology) are proved only under the hypothesis that X is essentially of finite type over ℂ. No additional argument or reference is supplied to remove this hypothesis, so the claimed generality for arbitrary Noetherian ℚ-algebras (including formal power series rings) is not justified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point about the scope of the cited characterization. We address the comment below.

read point-by-point responses

Referee: [Abstract] Abstract (final paragraph) and the statement of the main theorem: the characterization of Ω⁰_X and of Du Bois singularities via Grothendieck-topology sheafification is invoked for arbitrary Noetherian ℚ-schemes. Standard references for this characterization (e.g., via the h-topology) are proved only under the hypothesis that X is essentially of finite type over ℂ. No additional argument or reference is supplied to remove this hypothesis, so the claimed generality for arbitrary Noetherian ℚ-algebras (including formal power series rings) is not justified.

Authors: We agree that the standard references establishing the equivalence between the Du Bois complex and its sheafification in the h-topology (or other Grothendieck topologies) are stated for schemes essentially of finite type over ℂ. The manuscript invokes this characterization for arbitrary Noetherian ℚ-schemes without supplying an additional reference or argument to justify the extension. In the revised version we will either (i) add a self-contained argument or citation showing that the relevant sheafification property holds for Noetherian ℚ-schemes in general, or (ii) restrict the main statements to the essentially finite-type-over-ℚ case while noting that the algebraic applications (including the faithfully flat case) remain new. We believe the core descent result continues to hold in the stated generality once the foundational point is clarified. revision: yes

Circularity Check

0 steps flagged

No circularity: theorem proved from independent prior results

full rationale

The paper establishes that cyclically pure maps preserve Du Bois singularities for Noetherian Q-algebras by invoking the Grothendieck-topology sheafification characterization of Ω^0_X (standard in the literature) and proving a version of the Kovács-Schwede injectivity theorem for isolated non-Du Bois points. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claim has independent mathematical content beyond its inputs. The characterization is treated as an external tool rather than derived within the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the Grothendieck-topology characterization of the Du Bois complex (final sentence of abstract) and on a version of the Kovács-Schwede injectivity theorem proved for isolated non-Du Bois points.

axioms (1)

domain assumption The characterization of Ω^0_X and Du Bois singularities via sheafification in Grothendieck topologies holds for the Noetherian schemes considered.
Invoked throughout the paper per the abstract.

pith-pipeline@v0.9.0 · 5715 in / 1208 out tokens · 26063 ms · 2026-05-24T11:11:42.408438+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Equidimensional morphisms onto splinters are pure
math.AG 2025-12 unverdicted novelty 8.0

A Noetherian ring R is a splinter if and only if every equidimensional surjective morphism Spec(S) to Spec(R) makes R to S pure.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · cited by 1 Pith paper

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[1] [1]

Braun, D

L. Braun, D. Greb, K. Langlois, J. Moraga. ``Reductive quotients of klt singularities.'' Dec. 22, 2021. https://arxiv.org/abs/2111.02812v2 arXiv:2111.02812v2 https://arxiv.org/abs/2111.02812v2 [math.AG]

work page arXiv 2021

[2] [2]

J.-F. Boutot. ``Singularit\'es rationnelles et quotients par les groupes r\'eductifs.'' Invent. Math. 88.1 (1987), pp. 65--68. doi : https://doi.org/10.1007/BF01405091 10.1007/BF01405091 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=877006 877006

work page doi:10.1007/bf01405091 1987

[3] [3]

Bhatt, K

B. Bhatt, K. Schwede, and S. Takagi. ``The weak ordinarity conjecture and \(F\)-singularities.'' Higher dimensional algebraic geometry---in honour of Professor Yujiro Kawamata's sixtieth birthday. Adv. Stud. Pure Math., Vol. 74. Tokyo: Math. Soc. Japan, 2017, pp. 11--39. doi : https://doi.org/10.2969/aspm/07410011 10.2969/aspm/07410011 . mr : https://math...

work page doi:10.2969/aspm/07410011 2017

[4] [4]

Chakraborty, R

S. Chakraborty, R. V. Gurjar, and M. Miyanishi. ``Pure subrings of commutative rings.'' Nagoya Math. J. 221.1 (2016), pp. 33--68. doi : https://doi.org/10.1017/nmj.2016.2 10.1017/nmj.2016.2 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=3508743 3508743

work page doi:10.1017/nmj.2016.2 2016

[5] [5]

Datta and T

R. Datta and T. Murayama. ``Permanence properties of \(F\)-injectivity.'' To appear in Math. Res. Lett. August 7, 2023. https://arxiv.org/abs/1906.11399v4 arXiv:1906.11399v4 https://arxiv.org/abs/1906.11399v4 [math.AC]

work page arXiv 2023

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Hochster and C

M. Hochster and C. Huneke. ``Applications of the existence of big Cohen-Macaulay algebras.'' Adv. Math. 113.1 (1995), pp. 45--117. doi : https://doi.org/10.1006/aima.1995.1035 10.1006/aima.1995.1035 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=1332808 1332808

work page doi:10.1006/aima.1995.1035 1995

[7] [7]

Heitmann and L

R. Heitmann and L. Ma. ``Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic.'' Algebra Number Theory 12.7 (2018), pp. 1659--1674. doi : https://doi.org/10.2140/ant.2018.12.1659 10.2140/ant.2018.12.1659 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=3871506 3871506

work page doi:10.2140/ant.2018.12.1659 2018

[8] [8]

Hochster

M. Hochster. ``Cyclic purity versus purity in excellent Noetherian rings.'' Trans. Amer. Math. Soc. 231.2 (1977), pp. 463--488. doi : https://doi.org/10.2307/1997914 10.2307/1997914 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=463152 463152

work page doi:10.2307/1997914 1977

[9] [9]

Hochster and J

M. Hochster and J. L. Roberts. ``Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay.'' Advances in Math. 13 (1974), pp. 115--175. doi : https://doi.org/10.1016/0001-8708(74)90067-X 10.1016/0001-8708(74)90067-X . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=347810 347810

work page doi:10.1016/0001-8708(74)90067-x 1974

[10] [10]

Hochster and J

M. Hochster and J. L. Roberts. ``The purity of the Frobenius and local cohomology.'' Advances in Math. 21.2 (1976), pp. 117--172. doi : https://doi.org/10.1016/0001-8708(76)90073-6 10.1016/0001-8708(76)90073-6 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=417172 417172

work page doi:10.1016/0001-8708(76)90073-6 1976

[11] [11]

Kawamata

Y. Kawamata. ``The cone of curves of algebraic varieties.'' Ann. of Math. (2) 119.3 (1984), pp. 603--633. doi : https://doi.org/10.2307/2007087 10.2307/2007087 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=744865 744865

work page doi:10.2307/2007087 1984

[12] [12]

G. Kempf. ``The Hochster-Roberts theorem of invariant theory.'' Michigan Math. J. 26.1 (1979), pp. 19--32. doi : https://doi.org/10.1307/mmj/1029002160 10.1307/mmj/1029002160 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=514958 514958

work page doi:10.1307/mmj/1029002160 1979

[13] [13]

Koll\'ar and S

J. Koll\'ar and S. J. Kov\'acs. ``Log canonical singularities are Du Bois.'' J. Amer. Math. Soc. 23.3 (2010), pp. 791--813. doi : https://doi.org/10.1090/S0894-0347-10-00663-6 10.1090/S0894-0347-10-00663-6 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=2629988 2629988

work page doi:10.1090/s0894-0347-10-00663-6 2010

[14] [14]

1998 , PAGES =

J. Koll\'ar and S. Mori. Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Math., Vol. 134. Cambridge: Cambridge Univ. Press, 1998. doi : https://doi.org/10.1017/CBO9780511662560 10.1017/CBO9780511662560 . mr : https://mathscinet.ams.org/mathsci...

work page doi:10.1017/cbo9780511662560 1998

[15] [15]

S. J. Kov\'acs. ``Rational, log canonical, Du Bois singularities: on the conjectures of Koll\'ar and Steenbrink.'' Compositio Math. 118.2 (1999), pp. 123--133. doi : https://doi.org/10.1023/A:1001120909269 10.1023/A:1001120909269 . mr : https://mathscinet.ams.org/mathscinet-getitem?mr=1713307 1713307

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