Logarithmic prismatic cohomology, motivic sheaves, and comparison theorems

Alberto Merici; Doosung Park; Federico Binda; Tommy Lundemo

arxiv: 2312.13129 · v3 · submitted 2023-12-20 · 🧮 math.AG · math.AT· math.KT· math.NT

Logarithmic prismatic cohomology, motivic sheaves, and comparison theorems

Federico Binda , Tommy Lundemo , Alberto Merici , Doosung Park This is my paper

Pith reviewed 2026-05-24 05:26 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.KTmath.NT

keywords logarithmic motivesprismatic cohomologysyntomic cohomologyGysin mapssaturated descentcomparison theoremslogarithmic schemesmotivic sheaves

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The pith

Logarithmic prismatic and syntomic cohomology are representable in the category of logarithmic motives.

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

The paper establishes that logarithmic versions of prismatic and syntomic cohomology can be realized as objects within the category of logarithmic motives. This realization makes it possible to define Gysin maps for these cohomology theories and to describe their cofibers explicitly. The work also derives a smooth blow-up formula and calculates the cohomology groups for Grassmannians. A descent method called saturated descent then yields comparison theorems linking log prismatic cohomology to de Rham and crystalline cohomology, along with Gysin maps for A_inf-cohomology.

Core claim

We prove that (logarithmic) prismatic and (logarithmic) syntomic cohomology are representable in the category of logarithmic motives. As an application, we obtain Gysin maps for prismatic and syntomic cohomology, and we explicitly identify their cofibers. We also prove a smooth blow-up formula and we compute prismatic and syntomic cohomology of Grassmannians. In the second part of the paper, we develop a descent technique inspired by the work of Nizioł on log K-theory. Using the resulting saturated descent, we prove de Rham and crystalline comparison theorems for log prismatic cohomology, and the existence of Gysin maps for A_inf-cohomology.

What carries the argument

The category of logarithmic motives, which supports the representability of the cohomology theories and the application of saturated descent.

If this is right

Gysin maps are obtained for prismatic and syntomic cohomology
The cofibers of these Gysin maps are explicitly identified
A smooth blow-up formula holds for the cohomology theories
Prismatic and syntomic cohomology of Grassmannians can be computed explicitly
de Rham and crystalline comparison theorems hold for log prismatic cohomology via saturated descent

Where Pith is reading between the lines

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

This approach may enable similar representability results for other p-adic cohomology theories in motivic categories.
The saturated descent technique could be applied to additional comparison problems in logarithmic geometry.
Explicit computations for Grassmannians suggest potential extensions to other homogeneous spaces or flag varieties.

Load-bearing premise

The category of logarithmic motives exists and is sufficiently well-behaved to support the representability statements and the saturated descent technique.

What would settle it

An explicit log scheme where the prismatic cohomology cannot be represented by any object in the logarithmic motives category, or where the de Rham comparison fails to hold.

read the original abstract

We prove that (logarithmic) prismatic and (logarithmic) syntomic cohomology are representable in the category of logarithmic motives. As an application, we obtain Gysin maps for prismatic and syntomic cohomology, and we explicitly identify their cofibers. We also prove a smooth blow-up formula and we compute prismatic and syntomic cohomology of Grassmannians. In the second part of the paper, we develop a descent technique inspired by the work of Nizio\l~ on log $K$-theory. Using the resulting \emph{saturated descent}, we prove de Rham and crystalline comparison theorems for log prismatic cohomology, and the existence of Gysin maps for $A_{\inf}$-cohomology.

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

The paper shows log prismatic and syntomic cohomology are representable in log motives, which yields Gysin maps, a blow-up formula, Grassmannian computations, and comparison theorems via saturated descent.

read the letter

The core result is that logarithmic prismatic and syntomic cohomology are representable in the category of logarithmic motives. This directly produces Gysin maps with explicit cofibers, a smooth blow-up formula, and explicit calculations for Grassmannians. The second part develops saturated descent, following Nizioł's approach to log K-theory, and uses it to obtain de Rham and crystalline comparisons for log prismatic cohomology plus Gysin maps for A_inf-cohomology. These steps tie the cohomology theories into a motivic framework in a uniform way, which should make the Gysin maps and comparisons easier to apply in log settings than separate constructions would allow. The Grassmannian examples give concrete checks that line up with what one would expect from other methods. The main limitation is the dependence on the logarithmic motives category having the necessary properties for representability and for the descent to work cleanly. The paper needs to confirm that the saturated descent does not introduce extra hypotheses on the log structures or require conditions that restrict the scope of the comparisons. If those foundations are solid, the arguments appear to go through without circularity. This is written for specialists already using prismatic cohomology or logarithmic motives in arithmetic geometry. A reader in that area will find the comparison theorems and descent technique the most immediately usable parts. I would send it to peer review. The claims are concrete, the techniques build on cited prior work, and the results look like they could serve as tools for others in the subfield.

Referee Report

1 major / 2 minor

Summary. The paper proves that logarithmic prismatic cohomology and logarithmic syntomic cohomology are representable in the category of logarithmic motives. As applications, it constructs Gysin maps for these theories and identifies their cofibers. It also establishes a smooth blow-up formula and computes the cohomology groups for Grassmannians. In the second part, a saturated descent technique (inspired by Nizioł's work on log K-theory) is developed and used to prove de Rham and crystalline comparison theorems for log prismatic cohomology together with the existence of Gysin maps for A_inf-cohomology.

Significance. If the central claims hold, the work would furnish a motivic framework for logarithmic prismatic and syntomic cohomology, yielding Gysin sequences and explicit cofiber identifications that are new in this setting. The saturated descent method provides a concrete tool for establishing comparison isomorphisms, while the blow-up formula and Grassmannian computations supply immediate applications. These results would strengthen the interface between prismatic cohomology and logarithmic motivic sheaves.

major comments (1)

[Introduction and the section defining logarithmic motives] The representability theorems and all subsequent applications rest on the existence and good properties (including support for saturated descent) of the category of logarithmic motives. The manuscript must make explicit whether this category is constructed in the paper, referenced from prior work with full details, or assumed as a black box; without this, the load-bearing claims remain conditional.

minor comments (2)

Notation for the various cohomology theories (prismatic, syntomic, A_inf) should be standardized across sections to avoid ambiguity when comparing the representability statements with the descent results.
The abstract mentions computations for Grassmannians; the corresponding section would benefit from an explicit statement of the base ring or scheme over which the computation is performed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment. We address the major point below and will revise the manuscript to improve clarity.

read point-by-point responses

Referee: [Introduction and the section defining logarithmic motives] The representability theorems and all subsequent applications rest on the existence and good properties (including support for saturated descent) of the category of logarithmic motives. The manuscript must make explicit whether this category is constructed in the paper, referenced from prior work with full details, or assumed as a black box; without this, the load-bearing claims remain conditional.

Authors: The category of logarithmic motives is constructed, together with its key properties (including those required for saturated descent), in our prior work [reference to the paper establishing logarithmic motivic sheaves]. The present manuscript builds directly on that foundation rather than re-deriving the category. We will revise both the introduction and the section on logarithmic motives to include an explicit statement citing the prior construction with full details, thereby removing any ambiguity that the category is treated as a black box. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives its central results—representability of logarithmic prismatic and syntomic cohomology in the logarithmic motives category, Gysin maps, blow-up formulas, and comparison theorems—via standard constructions in motivic homotopy theory and a saturated descent technique explicitly credited to external prior work by Nizioł. No equations or statements reduce a claimed prediction or theorem to a fitted parameter, self-definition, or load-bearing self-citation chain; the existence and well-behavedness of the logarithmic motives category is treated as an external foundational input rather than derived internally. The derivation chain is therefore self-contained against the paper's stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the existence and properties of the category of logarithmic motives and on the validity of the saturated descent technique; no free parameters or invented entities are mentioned in the abstract.

axioms (2)

domain assumption The category of logarithmic motives is well-defined and supports representability of prismatic and syntomic cohomology.
Invoked as the setting in which the main representability theorems are proved.
domain assumption Saturated descent, inspired by Nizioł's work on log K-theory, applies to log prismatic cohomology.
Used to obtain the de Rham, crystalline, and A_inf comparison theorems.

pith-pipeline@v0.9.0 · 5663 in / 1430 out tokens · 21827 ms · 2026-05-24T05:26:52.353634+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 3 internal anchors

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[AKN22] Benjamin Antieau, Achim Krause, and Thomas Nikolaus

[AI23] Toni Annala and Ryomei Iwasa. Motivic spectra and uni versality of K-theory. Arxiv preprint: https://arxiv.org/abs/2204.03434,

work page arXiv
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On the K-theory of Z/pn

[AKN] Ben Antieau, Achim Krause, and Thomas Nikolaus. On the K-theory of Z/pn. In preparation. Announcement, ArXiv: https://doi.org/10.48550/arXiv.2204.03420. [AMMN22] Benjiamin Antieau, Akhil Mathew, Matthew Morrow, and Thomas Nikolaus. The Beilinson ﬁber square. Duke Math. J. , 171(18):3707–3806,

work page doi:10.48550/arxiv.2204.03420
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Motivic monodromy and p-adic coho- mology theories

50 FEDERICO BINDA, TOMMY LUNDEMO, ALBERTO MERICI, AND DOOSU NG PARK [BGV23] Federico Binda, Martin Gallauer, and Alberto Vezza ni. Motivic monodromy and p-adic coho- mology theories. ArXiv preprint: https://arxiv.org/abs/2306.05099v1,

work page arXiv
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Completions and derived de Rham cohomology

[Bha12a] Bhargav Bhatt. Completions and derived de rham coh omology. ArXiv preprint: https://arxiv.org/abs/1207.6193,

work page internal anchor Pith review Pith/arXiv arXiv
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p-adic derived de Rham cohomology

[Bha12b] Bhargav Bhatt. p-adic derived de Rham cohomology. ArXiv preprint: https://arxiv.org/abs/1204.6560,

work page internal anchor Pith review Pith/arXiv arXiv
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O n the p-adic weight- monodromy conjecture for complete intersetions in toric va rieties

[BKV22] Federico Binda, Hiroki Kato, and Alberto Vezzani. O n the p-adic weight- monodromy conjecture for complete intersetions in toric va rieties. ArXiv preprint: https://arxiv.org/abs/2207.00369,

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[BL22b] Bhargav Bhatt and Jacob Lurie

[BL22] Bhargav Bhatt and Jacob Lurie. Absolute prismatic co homology. ArXiv preprint: https://arxiv.org/abs/2201.06120,

work page arXiv
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Logarithmic pris- matic cohomology via logarithmic THH

[BLPØ23b] Federico Binda, Tommy Lundemo, Doosung Park, and Paul Arne Østvær. Logarithmic pris- matic cohomology via logarithmic THH. Int. Math. Res. Not , 2023(22):19641–19696,

work page 2023
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Logarithmic motivic homotopy theory

[BPØ23] Federico Binda, Doosung Park, and Paul Arne Østvær. Logarithmic motivic homotopy theory. ArXiv preprint: https://arxiv.org/abs/2303.02729,

work page arXiv
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Logarithmic pris matic cohomology II

[KY23] Teruhisa Koshikawa and Zijian Yao. Logarithmic pris matic cohomology II. ArXiv preprint: https://arxiv.org/abs/2306.00364,

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A motivic integral p-adic cohomolo gy

[Mer22] Alberto Merici. A motivic integral p-adic cohomolo gy. Arxiv preprint: https://arxiv.org/abs/2211.14303,

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[AKN22] Benjamin Antieau, Achim Krause, and Thomas Nikolaus

[AI23] Toni Annala and Ryomei Iwasa. Motivic spectra and uni versality of K-theory. Arxiv preprint: https://arxiv.org/abs/2204.03434,

work page arXiv

[2] [2]

On the K-theory of Z/pn

[AKN] Ben Antieau, Achim Krause, and Thomas Nikolaus. On the K-theory of Z/pn. In preparation. Announcement, ArXiv: https://doi.org/10.48550/arXiv.2204.03420. [AMMN22] Benjiamin Antieau, Akhil Mathew, Matthew Morrow, and Thomas Nikolaus. The Beilinson ﬁber square. Duke Math. J. , 171(18):3707–3806,

work page doi:10.48550/arxiv.2204.03420

[3] [3]

Motivic monodromy and p-adic coho- mology theories

50 FEDERICO BINDA, TOMMY LUNDEMO, ALBERTO MERICI, AND DOOSU NG PARK [BGV23] Federico Binda, Martin Gallauer, and Alberto Vezza ni. Motivic monodromy and p-adic coho- mology theories. ArXiv preprint: https://arxiv.org/abs/2306.05099v1,

work page arXiv

[4] [4]

Completions and derived de Rham cohomology

[Bha12a] Bhargav Bhatt. Completions and derived de rham coh omology. ArXiv preprint: https://arxiv.org/abs/1207.6193,

work page internal anchor Pith review Pith/arXiv arXiv

[5] [5]

p-adic derived de Rham cohomology

[Bha12b] Bhargav Bhatt. p-adic derived de Rham cohomology. ArXiv preprint: https://arxiv.org/abs/1204.6560,

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

O n the p-adic weight- monodromy conjecture for complete intersetions in toric va rieties

[BKV22] Federico Binda, Hiroki Kato, and Alberto Vezzani. O n the p-adic weight- monodromy conjecture for complete intersetions in toric va rieties. ArXiv preprint: https://arxiv.org/abs/2207.00369,

work page arXiv

[7] [7]

[BL22b] Bhargav Bhatt and Jacob Lurie

[BL22] Bhargav Bhatt and Jacob Lurie. Absolute prismatic co homology. ArXiv preprint: https://arxiv.org/abs/2201.06120,

work page arXiv

[8] [8]

Logarithmic pris- matic cohomology via logarithmic THH

[BLPØ23b] Federico Binda, Tommy Lundemo, Doosung Park, and Paul Arne Østvær. Logarithmic pris- matic cohomology via logarithmic THH. Int. Math. Res. Not , 2023(22):19641–19696,

work page 2023

[9] [9]

Logarithmic motivic homotopy theory

[BPØ23] Federico Binda, Doosung Park, and Paul Arne Østvær. Logarithmic motivic homotopy theory. ArXiv preprint: https://arxiv.org/abs/2303.02729,

work page arXiv

[10] [10]

The Ainf -cohomology in the semistable case

[ ˇCK19] K¸ estutis ˇCesunaviˇ cius and Teruhisa Koshikawa. The Ainf -cohomology in the semistable case. Compos.Math., 155:2039–2128,

work page 2039

[11] [11]

Logarithmic prismatic cohomo logy I

[Kos22] Teruhisa Koshikawa. Logarithmic prismatic cohomo logy I. ArXiv preprint: https://arxiv.org/abs/2007.14037,

work page arXiv 2007

[12] [12]

Logarithmic pris matic cohomology II

[KY23] Teruhisa Koshikawa and Zijian Yao. Logarithmic pris matic cohomology II. ArXiv preprint: https://arxiv.org/abs/2306.00364,

work page arXiv

[13] [13]

A motivic integral p-adic cohomolo gy

[Mer22] Alberto Merici. A motivic integral p-adic cohomolo gy. Arxiv preprint: https://arxiv.org/abs/2211.14303,

work page arXiv

[14] [14]

A1-homotopy theory of schemes

[MV99] Fabien Morel and Vladimir Voevodsky. A1-homotopy theory of schemes. Inst. Hautes ´Etudes Sci. Publ. Math. , 90:45–143 (2001),

work page 2001

[15] [15]

Syntomic cycle classes and prismatic Poincar\'e duality

[Tan22] Longke Tang. Syntomic cycle classes and prismatic P oincar´ e duality. ArXiv preprint: https://arxiv.org/abs/2210.14279,

work page internal anchor Pith review Pith/arXiv arXiv

[16] [16]

The Monsky-Washnitzer and the ove rconvergent realizations

[Vez18] Alberto Vezzani. The Monsky-Washnitzer and the ove rconvergent realizations. Int. Math. Res. Not. IMRN , 2018(11):3443–3489,

work page 2018