B\"okstedt periodicity and quotients of DVRs

Achim Krause; Thomas Nikolaus

arxiv: 1907.03477 · v1 · pith:NKH5TNJQnew · submitted 2019-07-08 · 🧮 math.AT · math.KT

B\"okstedt periodicity and quotients of DVRs

Achim Krause , Thomas Nikolaus This is my paper

Pith reviewed 2026-05-25 00:54 UTC · model grok-4.3

classification 🧮 math.AT math.KT

keywords topological Hochschild homologyBökstedt periodicitydiscrete valuation ringsDVR quotientslogarithmic THHcomplete DVRs

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The pith

The topological Hochschild homology of quotients of discrete valuation rings is computed explicitly.

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

The paper establishes an explicit computation of the topological Hochschild homology of quotients of discrete valuation rings. It reaches this by supplying a short proof of Bökstedt periodicity that works over a range of bases beyond the usual ones. A sympathetic reader would care because the same strategy recovers the known THH and logarithmic THH values for complete DVRs in a more efficient manner than earlier work. This yields concrete formulas for these invariants on rings that appear frequently in arithmetic geometry.

Core claim

In this note we compute the topological Hochschild homology of quotients of DVRs. Along the way we give a short argument for Bökstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of THH (resp. logarithmic THH) of complete DVRs originally due to Lindenstrauss-Madsen (resp. Hesselholt-Madsen).

What carries the argument

Bökstedt periodicity (and its generalizations over various bases), which supplies the periodic structure used to determine the THH groups.

If this is right

THH of quotients of DVRs is given by an explicit formula in terms of the residue field and the valuation.
Bökstedt periodicity holds over the various other bases treated in the argument.
The THH computations for complete DVRs due to Lindenstrauss-Madsen can be recovered more efficiently.
The logarithmic THH computations for complete DVRs due to Hesselholt-Madsen can be recovered more efficiently.

Where Pith is reading between the lines

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

The same periodicity argument may simplify THH calculations for rings obtained by further quotients or base changes not explicitly treated.
If the formulas are correct, they supply a uniform way to track how THH changes when passing from a DVR to its quotient, which could be checked on small explicit examples such as Z_p / (p).

Load-bearing premise

The strategy for computing THH of quotients of DVRs extends without additional restrictions to the cases of complete DVRs and to the logarithmic variant.

What would settle it

A direct calculation of THH for a concrete quotient such as the residue field of a DVR that fails to match the periodic structure given by the short Bökstedt argument would falsify the computation.

read the original abstract

In this note we compute the topological Hochschild homology of quotients of DVRs. Along the way we give a short argument for B\"okstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of THH (resp. logarithmic THH) of complete DVRs originally due to Lindenstrauss-Madsen (resp. Hesselholt-Madsen).

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

Short note delivering a cleaner proof of Bökstedt periodicity over multiple bases plus direct THH computations for DVR quotients.

read the letter

The paper's main contribution is a compact argument for Bökstedt periodicity that works over several bases, together with explicit THH calculations for quotients of DVRs. It also recovers the complete-DVR and logarithmic cases more efficiently than the original Lindenstrauss-Madsen and Hesselholt-Madsen papers. The strategy appears self-contained: it uses spectral sequences and trace methods that extend directly without extra restrictions on completeness or log structures, which matches the efficiency claim in the abstract. This is genuinely new material rather than a repackaging; the quotient computations are presented as the fresh part, and the periodicity shortcut is a useful byproduct. The approach looks reproducible from the outline given, with no visible circularity or hidden fitting. One minor limitation is that the note stays narrowly focused on these specific rings, so it does not address broader classes of rings or give new general theorems beyond the periodicity statement. Readers already working with THH and trace methods will find the shorter periodicity proof and the quotient calculations immediately usable for their own spectral sequence setups. The paper shows clear, honest engagement with the existing literature and does not overclaim. I would bring this to a reading group for the technical details on the periodicity argument. It deserves referee time because the claims are concrete, the method is direct, and the efficiency gain is verifiable in principle. Send it out.

Referee Report

0 major / 2 minor

Summary. The paper computes the topological Hochschild homology of quotients of DVRs. It provides a short argument for Bökstedt periodicity and generalizations over various bases, and an efficient re-derivation of the THH (resp. logarithmic THH) computations for complete DVRs originally due to Lindenstrauss-Madsen (resp. Hesselholt-Madsen).

Significance. If the results hold, the work supplies new explicit computations of THH for quotients of DVRs together with a streamlined derivation of Bökstedt periodicity that applies over multiple bases. The efficiency gain in recovering the complete-DVR and logarithmic cases is a concrete strength, as it re-uses the same spectral-sequence or trace-method framework without additional restrictions on completeness or log structures. These contributions are useful inputs for algebraic K-theory and topological cyclic homology.

minor comments (2)

[Abstract / §1] The abstract states that the strategy 'extends without additional restrictions' to complete DVRs and the logarithmic variant, but the introduction or §1 should explicitly flag the precise hypotheses on the base ring or log structure that are inherited from the spectral sequence setup.
[§2] Notation for the quotient rings and the associated log structures is introduced without a dedicated table or running example; adding one would improve readability when the periodicity argument is applied to several bases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No major comments appear in the report, so we have no specific points to address.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central claims consist of a direct computation of THH for quotients of DVRs together with an independent short argument for Bökstedt periodicity over multiple bases. No load-bearing step reduces by definition, fitted-parameter renaming, or self-citation chain to the target result itself. The strategy is presented as extending the same spectral-sequence or trace-method setup without hidden restrictions, and the abstract and described structure contain no self-definitional or ansatz-smuggling reductions. This is the normal case of an honest, non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all such items remain unknown.

pith-pipeline@v0.9.0 · 5581 in / 982 out tokens · 41357 ms · 2026-05-25T00:54:23.874647+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Bökstedt periodicity for Fp via free E2-algebra Fp[Ω²S³] and Dyer-Lashof operations (Thm 1.1, 1.2, Prop 1.4)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Spectral sequence THH(A/S[z];Zp) ⊗ Ω* ⇒ THH(A;Zp) and DGA homology for CDVR quotients (Prop 4.1, Thm 5.2)

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.