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Gersten injectivity and Morel's purity conjecture

Primarily AI-generated textHuman understanding: all partsmath.AG — Algebraic Geometrymath.KT — K-Theory and Homology

Contributed by Niels FELD ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

Morel conjectured that the A1\mathbb{A}^1-homotopy sheaves of an S1S^1-spectrum built from suspension spectra of smooth projective schemes are pure; one condition in purity is injectivity of restriction to dense open subschemes. Over a regular base SS, we show that this injectivity condition, for the homotopy sheaves of algebraic KK-theory with coefficients in an arbitrary spectrum, is equivalent to Gersten injectivity at the henselizations of all local rings of smooth SS-schemes; and that these KK-theory spectra are pure in Morel's sense. Hence every henselian regular local ring at which Gersten injectivity fails, integrally, rationally, or with finite coefficients, gives a counterexample to Morel's conjecture. The author's finite-coefficient counterexample over Z3[[x,y]]/(3+x2−y3)\mathbb{Z}_3[[x,y]]/(3+x^2-y^3) therefore disproves the conjecture, already over the regular scheme Spec⁡Z[1/2][x,y]/(3+x2−y3)\operatorname{Spec}\mathbb{Z}[1/2][x,y]/(3+x^2-y^3). Recently announced integral and rational Gersten counterexamples, including unpublished work of the author, would show, if confirmed and if they persist after henselization, that the conjecture fails already for the KK-theory spectrum itself.

Provenance statement

An earlier version of this manuscript was drafted with OpenAI's GPT-6 Astra under the author's direction, including the deduction from the finite-coefficient counterexample. The present version was revised with Anthropic's Claude (Opus~5.5), which checked Morel's definitions, searched the literature for prior counterexamples, proposed the characterization by henselian stalks and the extension to arbitrary coefficients, added references, and proofread the final text. An adversarial referee report produced with GPT-6 Astra suggested some minor corrections. All suggestions were checked by the author, who assumes full responsibility for the statements, proofs and references.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
ChatGPTVersion GPT-6 Astra

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Version history

  1. v1Submitted by Niels FELDInitial depositCurrentNov 01, 2026