pith. sign in

arxiv: 2607.01178 · v1 · pith:RUJ6MGB4new · submitted 2026-07-01 · 🧮 math.AG · math.KT· math.NT

Sheared Witt Vectors

Bhargav Bhatt , Akhil Mathew , Vadim Vologodsky This is my paper

Pith reviewed 2026-07-02 05:46 UTC · model grok-4.3

classification 🧮 math.AG math.KTmath.NT
keywords sheared Witt vectorsWitt vectorsDrinfeld-Lau constructionperfect F_p-algebrap-typical Witt vectorsalgebraic geometryp-adic rings
0
0 comments X

The pith

Sheared Witt vectors are a decompletion of p-typical Witt vectors for rings with perfect reduction modulo p.

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

This paper offers an exposition of the sheared Witt vectors, a construction introduced by Drinfeld and Lau as a decompletion of the p-typical Witt vectors following Zink. It works on the category of rings R such that R modulo p is a perfect algebra over the finite field with p elements. A sympathetic reader would care because the presentation makes this technical tool available without requiring the full background of the original sources.

Core claim

V. Drinfeld and E. Lau introduced a decompletion of the ring of p-typical Witt vectors, following earlier work of T. Zink. The paper offers an exposition of this construction, which we call the sheared Witt vectors, on the category of rings R whose reduction is a perfect F_p-algebra.

What carries the argument

The sheared Witt vectors construction, a decompletion of p-typical Witt vectors defined on rings whose reduction modulo p is perfect over F_p.

If this is right

  • The construction applies directly to rings R with perfect F_p reduction.
  • It provides a decompleted version of the p-typical Witt vectors.
  • It builds explicitly on Zink's prior work on Witt vectors.
  • The exposition targets readers working in the relevant category of rings in algebraic geometry.

Where Pith is reading between the lines

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

  • Researchers may now apply the construction in their own work without first mastering the original technical references.
  • Similar decompletion techniques could be developed for other variants of Witt vectors or related p-adic structures.
  • The approach may simplify calculations in contexts like p-adic cohomology where full Witt vectors are too complete.

Load-bearing premise

The Drinfeld-Lau construction can be accurately and usefully re-presented in expository form without introducing errors or requiring extra technical background.

What would settle it

A direct comparison showing that the re-presented construction fails to match the properties or definitions in the original Drinfeld-Lau paper would demonstrate the exposition is inaccurate.

read the original abstract

V. Drinfeld and E. Lau introduced a ``decompletion'' of the ring of $p$-typical Witt vectors, following earlier work of T. Zink. The goal of this paper is to offer an exposition of this construction, which we call the sheared Witt vectors, on the category of rings $R$ whose reduction is a perfect $\mathbb{F}_p$-algebra.

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.

Referee Report

0 major / 0 minor

Summary. The paper provides an exposition of the sheared Witt vectors construction introduced by V. Drinfeld and E. Lau (following T. Zink) on the category of rings R whose reduction is a perfect F_p-algebra.

Significance. A clear and accurate re-presentation of the Drinfeld-Lau construction would be useful for researchers in p-adic algebraic geometry and Witt vector theory, as it makes an existing advanced tool more accessible without asserting new theorems or derivations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, clear summary of the manuscript's purpose as an exposition of the Drinfeld-Lau sheared Witt vectors construction, and recommendation to accept. We are pleased that the referee finds the re-presentation useful for making this tool more accessible in p-adic algebraic geometry and Witt vector theory.

Circularity Check

0 steps flagged

Expository paper presents no original derivations or predictions

full rationale

The paper is framed explicitly as an exposition of the existing Drinfeld-Lau sheared Witt vector construction (following Zink) on rings with perfect F_p reduction. No original theorems, parameter fits, predictions, or derivation chains are asserted; the central claim is accurate re-presentation of prior external results by different authors. Absent any self-referential steps, fitted inputs called predictions, or load-bearing self-citations, the work is self-contained against external benchmarks with no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is an exposition of prior work; the abstract introduces no new free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5583 in / 890 out tokens · 22650 ms · 2026-07-02T05:46:06.097148+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 5 canonical work pages · 2 internal anchors

  1. [1]

    Aberbach and Melvin Hochster

    Ian M. Aberbach and Melvin Hochster. Finite T or dimension and failure of coherence in absolute integral closures. J. Pure Appl. Algebra , 122(3):171--184, 1997

    1997

  2. [2]

    K -theory of endomorphisms

    Gert Almkvist. K -theory of endomorphisms. J. Algebra , 55(2):308--340, 1978

    1978

  3. [3]

    Norm functors and effective zero cycles

    Vladimir Baranovsky. Norm functors and effective zero cycles. Proc. Edinb. Math. Soc. (2) , 53(3):557--574, 2010

    2010

  4. [4]

    Acyclic models and triples

    Michael Barr and Jon Beck. Acyclic models and triples. In Proc. C onf. C ategorical A lgebra ( L a J olla, C alif., 1965) , pages 336--343. Springer-Verlag New York, Inc., New York, 1966

    1965

  5. [5]

    Triples, algebras and cohomology

    Jonathan Mock Beck. Triples, algebras and cohomology. Repr. Theory Appl. Categ. , (2):1--59, 2003

    2003

  6. [6]

    Torsion completions are bounded

    Bhargav Bhatt. Torsion completions are bounded. J. Pure Appl. Algebra , 223(5):1940--1945, 2019

    1940

  7. [7]

    Sheared prismatization

    Bhargav Bhatt, Artem Kanaev, Akhil Mathew, Vadim Vologodsky, and Mingjia Zhang. Sheared prismatization . In preparation

  8. [8]

    A R iemann- H ilbert correspondence in positive characteristic

    Bhargav Bhatt and Jacob Lurie. A R iemann- H ilbert correspondence in positive characteristic. Camb. J. Math. , 7(1-2):71--217, 2019

    2019

  9. [9]

    Absolute prismatic cohomology

    Bhargav Bhatt and Jacob Lurie. Absolute prismatic cohomology. https://arxiv.org/abs/2201.06120, 2022

    work page arXiv 2022

  10. [10]

    Integral p -adic H odge theory

    Bhargav Bhatt, Matthew Morrow, and Peter Scholze. Integral p -adic H odge theory. Publ. Math. Inst. Hautes \'Etudes Sci. , 128:219--397, 2018

    2018

  11. [11]

    The pro-\' e tale topology for schemes

    Bhargav Bhatt and Peter Scholze. The pro-\' e tale topology for schemes. Ast\' e risque , (369):99--201, 2015

    2015

  12. [12]

    Prisms and prismatic cohomology

    Bhargav Bhatt and Peter Scholze. Prisms and prismatic cohomology. Ann. of Math. (2) , 196(3):1135--1275, 2022

    2022

  13. [13]

    Groupes formels associ\'es aux anneaux de W itt g\'en\'eralis\'es

    Pierre Cartier. Groupes formels associ\'es aux anneaux de W itt g\'en\'eralis\'es. C. R. Acad. Sci. Paris S\'er. A-B , 265:A49--A52, 1967

    1967

  14. [14]

    Groupes alg\' e briques

    Michel Demazure and Pierre Gabriel. Groupes alg\' e briques. T ome I : G \' e om\' e trie alg\' e brique, g\' e n\' e ralit\' e s, groupes commutatifs . Masson & Cie, \' E diteurs, Paris; North-Holland Publishing Co., Amsterdam, 1970. Avec un appendice Corps de classes local par Michiel Hazewinkel

    1970

  15. [15]

    Christopher Davis and Kiran S. Kedlaya. On the W itt vector F robenius. Proc. Amer. Math. Soc. , 142(7):2211--2226, 2014

    2014

  16. [16]

    Prismatization

    Vladimir Drinfeld. Prismatization. Selecta Math. (N.S.) , 30(3):Paper No. 49, 150, 2024

    2024

  17. [17]

    Ring stacks conjecturally related to the stacks BT_n^ G,

    Vladimir Drinfeld. Ring stacks conjecturally related to the stacks BT_n^ G, . arXiv preprint arXiv:2510.04958 , Oct 2025

    work page arXiv 2025

  18. [18]

    On the naturality of Pic ,\;SK_0 and SK_1

    Barry Dayton and Charles Weibel. On the naturality of Pic ,\;SK_0 and SK_1 . In Algebraic K -theory: connections with geometry and topology ( L ake L ouise, AB , 1987) , volume 279 of NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. , pages 1--28. Kluwer Acad. Publ., Dordrecht, 1989

    1987

  19. [19]

    Formal groups and applications

    Michiel Hazewinkel. Formal groups and applications . AMS Chelsea Publishing, Providence, RI, 2012. Corrected reprint of the 1978 original

    2012

  20. [20]

    Sheared displays and $p$-divisible groups

    Manuel Hoff and Eike Lau. Sheared displays and p -divisible groups . arXiv preprint arXiv:2601.12565 , Jan 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  21. [21]

    -anneaux et vecteurs de W itt

    Andr\' e Joyal. -anneaux et vecteurs de W itt. C. R. Math. Rep. Acad. Sci. Canada , 7(3):177--182, 1985

    1985

  22. [22]

    Kiran S. Kedlaya. Notes on prismatic cohomology, 2024. Version dated January 18, 2024. Available at https://kskedlaya.org/prismatic/

    2024

  23. [23]

    Smoothness of the truncated display functor

    Eike Lau. Smoothness of the truncated display functor. J. Amer. Math. Soc. , 26(1):129--165, 2013

    2013

  24. [24]

    Relations between D ieudonn\' e displays and crystalline D ieudonn\' e theory

    Eike Lau. Relations between D ieudonn\' e displays and crystalline D ieudonn\' e theory. Algebra Number Theory , 8(9):2201--2262, 2014

    2014

  25. [25]

    The Shimurian BT stack is a gerbe over truncated displays

    Eike Lau. The Shimurian BT stack is a gerbe over truncated displays . arXiv preprint arXiv:2510.23207 , Oct 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  26. [26]

    Witt vectors and ‑Cartier rings

    Kirill Magidson. Witt vectors and ‑Cartier rings . arXiv preprint arXiv:2409.03877v2 , September 2024

    work page arXiv 2024

  27. [27]

    The mod- p R iemann- H ilbert correspondence and the perfect site

    Akhil Mathew. The mod- p R iemann- H ilbert correspondence and the perfect site. Tunis. J. Math. , 5(2):369--403, 2023

    2023

  28. [28]

    Introduction to algebraic K -theory , volume No

    John Milnor. Introduction to algebraic K -theory , volume No. 72 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1971

    1971

  29. [29]

    Daniel G. Quillen. Homotopical algebra . Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin-New York, 1967

    1967

  30. [30]

    On the (co-) homology of commutative rings

    Daniel Quillen. On the (co-) homology of commutative rings. In Applications of C ategorical A lgebra ( P roc. S ympos. P ure M ath., V ol. XVII , N ew Y ork, 1968) , Proc. Sympos. Pure Math., XVII, pages 65--87. Amer. Math. Soc., Providence, RI, 1970

    1968

  31. [31]

    Tate duality in positive dimension over function fields

    Zev Rosengarten. Tate duality in positive dimension over function fields. arXiv preprint , 2018

    2018

  32. [32]

    Lectures on condensed mathematics, 2026

    Peter Scholze. Lectures on condensed mathematics, 2026

    2026

  33. [33]

    A note on extensions of algebraic and formal groups

    Tsutomu Sekiguchi and Noriyuki Suwa. A note on extensions of algebraic and formal groups. III . Tohoku Math. J. (2) , 49(2):241--257, 1997

    1997

  34. [34]

    The stacks project

    The Stacks project authors . The stacks project. https://stacks.math.columbia.edu, 2023

    2023

  35. [35]

    Richard G. Swan. On seminormality. J. Algebra , 67(1):210--229, 1980

    1980

  36. [36]

    Purity for flat cohomology

    K e stutis C esnavi c ius and Peter Scholze. Purity for flat cohomology. Ann. of Math. (2) , 199(1):51--180, 2024

    2024

  37. [37]

    Charles A. Weibel. The K -book , volume 145 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2013

    2013

  38. [38]

    A D ieudonn\' e theory for p -divisible groups

    Thomas Zink. A D ieudonn\' e theory for p -divisible groups. In Class field theory---its centenary and prospect ( T okyo, 1998) , volume 30 of Adv. Stud. Pure Math. , pages 139--160. Math. Soc. Japan, Tokyo, 2001

    1998

  39. [39]

    The display of a formal p -divisible group

    Thomas Zink. The display of a formal p -divisible group. Ast\' e risque , (278):127--248, 2002. Cohomologies p -adiques et applications arithm\' e tiques, I

    2002