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Gersten conjecture for K-theory on Henselian schemes and $φ$-motivic localisation

Andrei E Druzhinin

TL;DR

The work establishes a Gersten-type acyclicity for K-theory on essentially smooth local Henselian bases by introducing φ-motivic localization, which enforces φ-equivalences and governs A^1-homotopies within an ambient ∞-categorical setup. Central to the approach are the constructions of the A_B, vecA_B, and O_B frameworks, endofunctors tied to blow-ups and cubical directions, and the demonstration that Cousin complexes and support-extension maps become trivial in the φ-motivic context. A key outcome is the Injectivity Theorem and the acyclicity of the relevant Cousin complexes, which together yield the Gersten conjecture for K-theory in this base-change setting and generalize to Cousin-type complexes for φ- and □-homotopies. The paper also lays out a comprehensive localization machinery and outlines a comparison with traditional motivic homotopy theory, highlighting potential extensions via blow-up techniques and invariant objects, thereby broadening the scope of Gersten-type results in motivic homotopy theory over bases other than fields.

Abstract

A key triviality result for support extension maps for motivic $\mathbb{A}^1$-homotopies of cellular motivic spaces $S$ over a DVR spectrum $B$ is proven. Combining with earlier known results on Gersten complex and the K-theory motivic spectrum we achieve a proof of the Gersten Conjecture for K-theory on essentially smooth local Henselian $B$-schemes. Additionally, we outline generalisations for Cousin complexes associated to motivic $\mathbb{A}^1$- and $\square$-homotopies of cellular $B$-spectra. The proof is based on two ingredients: (1) A new ``motivic localisation'' over $B$, called \emph{$φ$-motivic}, % localisation giving rise to the $φ$-motivic homotopy category such that the triviality of the support extension maps and the acyclicity of Cousin complexes hold for all objects $S$, not necessarily cellular. (2) An interpretation of some classes in the motivic $\mathbb{A}^1$-homotopies with support defined with respect to the Morel-Voevodsky motivic homotopy category of smooth $B$-schemes in terms of the construction of $φ$-motivic homotopy category mentioned in Point (1).

Gersten conjecture for K-theory on Henselian schemes and $φ$-motivic localisation

TL;DR

The work establishes a Gersten-type acyclicity for K-theory on essentially smooth local Henselian bases by introducing φ-motivic localization, which enforces φ-equivalences and governs A^1-homotopies within an ambient ∞-categorical setup. Central to the approach are the constructions of the A_B, vecA_B, and O_B frameworks, endofunctors tied to blow-ups and cubical directions, and the demonstration that Cousin complexes and support-extension maps become trivial in the φ-motivic context. A key outcome is the Injectivity Theorem and the acyclicity of the relevant Cousin complexes, which together yield the Gersten conjecture for K-theory in this base-change setting and generalize to Cousin-type complexes for φ- and □-homotopies. The paper also lays out a comprehensive localization machinery and outlines a comparison with traditional motivic homotopy theory, highlighting potential extensions via blow-up techniques and invariant objects, thereby broadening the scope of Gersten-type results in motivic homotopy theory over bases other than fields.

Abstract

A key triviality result for support extension maps for motivic -homotopies of cellular motivic spaces over a DVR spectrum is proven. Combining with earlier known results on Gersten complex and the K-theory motivic spectrum we achieve a proof of the Gersten Conjecture for K-theory on essentially smooth local Henselian -schemes. Additionally, we outline generalisations for Cousin complexes associated to motivic - and -homotopies of cellular -spectra. The proof is based on two ingredients: (1) A new ``motivic localisation'' over , called \emph{-motivic}, % localisation giving rise to the -motivic homotopy category such that the triviality of the support extension maps and the acyclicity of Cousin complexes hold for all objects , not necessarily cellular. (2) An interpretation of some classes in the motivic -homotopies with support defined with respect to the Morel-Voevodsky motivic homotopy category of smooth -schemes in terms of the construction of -motivic homotopy category mentioned in Point (1).

Paper Structure

This paper contains 56 sections, 38 theorems, 152 equations.

Table of Contents

  1. Introdction
  2. Gersten Conjecture for K-theory
  3. Claim and Results
  4. Generalisations over $k$ and obstructions over $B$
  5. Results for "parts or modifications" of K-theory
  6. Reduction to one-dimensional schemes
  7. Reduction to the injectivity claim
  8. Proof for essentially smooth local Henselian schemes
  9. $\phi$-motivic localisation and homotopy category
  10. Interpretation of Morel-Voevodsky $\mathbb{A}^1$-homotopies in the $\infty$-category $O_B$
  11. Awknowlagement
  12. Notation
  13. $\phi$-motivic localisation
  14. Schemes category preparations
  15. Line bundles and sections
  16. ...and 41 more sections

Key Result

Theorem 1

Let $B\in \mathrm{Sch}$ be regular of Krull dimension one. Let $U$ be an essentially smooth local henselian scheme over $B$. Let $\rho$ denote a regular function on $B$ that vanishing locus equals a closed point in $B$ as well as its inverse image on $U$. For any non-negative integer $n$, the morphi is trivial for any $S\in \mathrm{H}^\bullet_{\mathbb{A}^1,\tau}(B)$ generated by $\mathbb{P}^{\wedg

Theorems & Definitions (152)

  • Remark 1
  • Remark 2: Stable Connectivity over $B$
  • Remark 3: Counter-example over $B$
  • Remark 4: Surjectivity Property
  • Theorem : Theorems \ref{['th:RLP:HsupK']}, \ref{['th:Injectivity']}
  • Remark 5
  • Remark 6: Illuminating $(\shortdownarrow,r,\varsigma)$-equivalences and the data $\rho$
  • Remark 7: Continuous presheaves without $\phi$-invariance condition
  • Definition 2.1
  • Remark 8
  • ...and 142 more