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arxiv: 2607.01862 · v1 · pith:X422HKT6new · submitted 2026-07-02 · 🧮 math.AG · math.AT· math.KT

Motivic Hochschild homology of mod 2 motivic cohomology over algebraically closed fields

Tom Bachmann , Robert Burklund , Bj{\o}rn Ian Dundas , Paul Arne {\O}stv{\ae}r This is my paper

Pith reviewed 2026-07-03 06:11 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.KT
keywords motivic cohomologynormed motivic spectraBökstedt periodicitymultiplicative group schemetau-torsionextended powersmotivic homotopy
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The pith

The tensor of the multiplicative group scheme with the mod-2 motivic cohomology spectrum is a free algebra on a generator in bidegree (2,1).

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

The paper establishes a computation of the tensor product of the multiplicative group scheme with the mod-2 motivic cohomology spectrum within normed motivic spectra over the complex numbers. It concludes that this tensor forms a free algebra generated by a single element in bidegree (2,1). The proof relies on separating the tau-inverted case, which reduces to the classical topological Bökstedt periodicity through Betti realization, from the tau-reduced case handled by comparisons of algebra structures. This provides a motivic version of Bökstedt periodicity. The analysis of motivic extended powers reveals a rigid structure without the tau-torsion families seen in simplicial calculations.

Core claim

We compute the tensor of the multiplicative group scheme with the mod-2 motivic cohomology spectrum in normed motivic spectra over the complex numbers, and find that the resulting algebra is free on a generator in bidegree (2,1). This gives a motivic analog of Bökstedt periodicity. The proof proceeds by comparing the tau-inverted and tau-reduced forms of the tensor. After inverting tau, the calculation reduces to classical Bökstedt periodicity via Betti realization. The reduction modulo tau is governed by a comparison between normed algebra structures and derived algebra structures on cellular modules over motivic cohomology mod tau.

What carries the argument

The comparison between normed algebra structures and derived algebra structures on cellular modules over motivic cohomology mod tau that produces divided power operations and mixed Cartan and Adem relations.

If this is right

  • The result gives a motivic analog of Bökstedt periodicity.
  • The tau-torsion structure for the Gm-tensor is more rigid than in the simplicial-circle calculation.
  • Mixed Cartan and Adem relations intertwine normed and topological power operations.

Where Pith is reading between the lines

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

  • The rigidity of the algebraic structure may facilitate explicit computations of other invariants in motivic homotopy theory.
  • Methods based on extended powers and tau-torsion analysis could apply to tensors with other schemes.

Load-bearing premise

The comparison between normed algebra structures and derived algebra structures on cellular modules over motivic cohomology mod tau produces divided power operations and the mixed relations required to establish freeness.

What would settle it

An explicit computation revealing either a non-free relation in the algebra or a mismatch between the tau-inverted case and classical Bökstedt periodicity would disprove the claim.

read the original abstract

We compute the tensor of the multiplicative group scheme with the mod-$2$ motivic cohomology spectrum in normed motivic spectra over the complex numbers, and find that the resulting algebra is free on a generator in bidegree (2,1). This gives a motivic analog of B\"okstedt periodicity. The proof proceeds by comparing the tau-inverted and tau-reduced forms of the tensor. After inverting tau, the calculation reduces to classical B{\"o}kstedt periodicity via Betti realization. The reduction modulo tau is governed by a comparison between normed algebra structures and derived algebra structures on cellular modules over motivic cohomology mod tau. This comparison produces divided power operations and leads to mixed Cartan and Adem relations intertwining normed and topological power operations. A key input is a detailed analysis of motivic extended powers of spheres and their tau-torsion structure. In contrast with the corresponding simplicial-circle calculation due to Dundas-Hill-Ormsby-{\O}stv{\ae}r, the large families of tau-torsion classes disappear for the Gm-tensor, leaving a considerably more rigid algebraic structure.

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

1 major / 1 minor

Summary. The manuscript computes the tensor product of the multiplicative group scheme Gm with the mod-2 motivic cohomology spectrum in the category of normed motivic spectra over the complex numbers. It concludes that the resulting algebra is free on a single generator in bidegree (2,1), yielding a motivic analog of Bökstedt periodicity. The argument proceeds by comparing the tau-inverted and tau-reduced forms of the tensor: tau-inversion reduces to classical Bökstedt periodicity via Betti realization, while the mod-tau case is handled by comparing normed algebra structures with derived algebra structures on cellular modules over motivic cohomology mod tau. This comparison is said to produce divided power operations together with mixed Cartan and Adem relations that intertwine normed and topological power operations. A detailed analysis of motivic extended powers of spheres and their tau-torsion is used to show that, unlike the simplicial-circle case of Dundas-Hill-Ormsby-Østvær, large families of tau-torsion classes disappear, leaving a rigid algebraic structure.

Significance. If the central claim holds, the result would be a notable contribution to motivic homotopy theory by furnishing a Gm-analog of Bökstedt periodicity and exhibiting new rigidity phenomena for the Gm-tensor that are absent in the simplicial setting. The explicit comparison of normed versus derived structures and the analysis of extended powers constitute concrete technical advances that could be useful for further computations in normed motivic spectra.

major comments (1)
  1. [Abstract] Abstract: the claim that the mixed Cartan and Adem relations arising from the normed-versus-derived comparison are exhaustive (and therefore kill every potential extra generator or surviving tau-torsion class) is load-bearing for the freeness conclusion. The abstract supplies no explicit equations for these relations, no verification that the divided-power operations are identified completely with the normed structure, and no check that the resulting ideal is precisely the one needed to leave only the (2,1) generator. Without such verification the reduction from the Gm-tensor to a free algebra remains formally incomplete.
minor comments (1)
  1. The title states the result holds over algebraically closed fields while the abstract restricts the computation to the complex numbers; the precise scope should be clarified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the load-bearing aspects of the argument. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the mixed Cartan and Adem relations arising from the normed-versus-derived comparison are exhaustive (and therefore kill every potential extra generator or surviving tau-torsion class) is load-bearing for the freeness conclusion. The abstract supplies no explicit equations for these relations, no verification that the divided-power operations are identified completely with the normed structure, and no check that the resulting ideal is precisely the one needed to leave only the (2,1) generator. Without such verification the reduction from the Gm-tensor to a free algebra remains formally incomplete.

    Authors: The abstract is a high-level summary and therefore omits the explicit equations and verifications, which appear in the body. The mixed Cartan and Adem relations are derived explicitly from the comparison of normed algebra structures with derived algebra structures on cellular modules over motivic cohomology mod tau; the divided-power operations are identified with the normed structure via the detailed analysis of motivic extended powers of spheres and their tau-torsion (which shows that the large families of tau-torsion classes present in the simplicial-circle case of Dundas-Hill-Ormsby-Østvær disappear for the Gm-tensor). This analysis establishes that the resulting ideal is precisely the one that eliminates all extra generators and surviving tau-torsion, leaving only the generator in bidegree (2,1). The full argument is therefore complete in the manuscript. We agree that the abstract could be strengthened by a brief additional sentence referencing these relations and the key theorem on freeness. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation reduces to external classical input and new comparison

full rationale

The abstract states that after inverting tau the calculation reduces to classical Bökstedt periodicity via Betti realization (external). The mod-tau case is handled by a comparison of normed and derived algebra structures on cellular modules that produces divided power operations and mixed Cartan/Adem relations; this is presented as a new analysis of motivic extended powers, explicitly contrasted to the simplicial case in prior work by overlapping authors. No equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the freeness claim to its own inputs are visible. The derivation is therefore self-contained against external benchmarks.

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 background structures (normed motivic spectra, tau, Betti realization) are treated as standard inputs from prior literature.

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