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On the $K$-theory of the $p$-adic unit disk

Elden Elmanto, Noah Riggenbach

Abstract

In this note, we study the $p$-complete topological cyclic homology of the affine line relative to a ring $A$ which is smooth over a perfectoid ring $R$. Denoting by $NTC(A; \mathbb{Z}_p)$ the spectrum which measures the failure of $\mathbb{A}^1$-invariance on $A$, we observe a kind of Quillen-Lichtenbaum phenomena for $NTC(A; \mathbb{Z}_p)$ -- that it is isomorphic to its own $K(1)$-localization in a specified range of degrees which depends on the relative dimension of $A$. Somewhat surprisingly, this range is better than considerations following from a theorem of Bhatt-Mathew and étale-to-syntomic comparisons. Via the Dundas-Goodwillie-McCarthy theorem, we obtain a description of the algebraic $K$-theory of $p$-completed affine line over such rings.

On the $K$-theory of the $p$-adic unit disk

Abstract

In this note, we study the -complete topological cyclic homology of the affine line relative to a ring which is smooth over a perfectoid ring . Denoting by the spectrum which measures the failure of -invariance on , we observe a kind of Quillen-Lichtenbaum phenomena for -- that it is isomorphic to its own -localization in a specified range of degrees which depends on the relative dimension of . Somewhat surprisingly, this range is better than considerations following from a theorem of Bhatt-Mathew and étale-to-syntomic comparisons. Via the Dundas-Goodwillie-McCarthy theorem, we obtain a description of the algebraic -theory of -completed affine line over such rings.

Paper Structure

This paper contains 11 sections, 18 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Methods
  3. Notation
  4. Acknowledgements
  5. A priori bounds
  6. The Segal conjecture
  7. Coconnectivity of the Frobenius and the Segal conjecture
  8. Quillen-Lichtenbaum for NTC
  9. The case of polynomial algebras
  10. Bootstrap to smooth schemes
  11. Application: $p$-adic unit disks of curves over perfectoid bases

Key Result

Theorem 1.1

Let $R$ be a perfectoid ring. Let $A$ be the $p$-adic completion of smooth $R$-algebra of relative dimension $d$. Then the map is $(d-1)$-truncated and an isomorphism This is equivalent to saying that it is a $\tau_{\geq d}$-equivalence. in degree $d$. If $R$ is further assumed to be $p$-torsion free then this map is also $(d-2)$-truncated. In particular, the map $K(A\langle t\rangle, (t); \mathb

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 27 more