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Cdh Descent for Homotopy Hermitian $K$-Theory of Rings with Involution

Daniel Carmody

Abstract

We provide a geometric model for the classifying space of automorphism groups of Hermitian vector bundles over a ring with involution $R$ such that $\frac{1}{2} \in R$; this generalizes a result of Schlichting-Tripathi \cite{SchTri}. We then prove a periodicity theorem for Hermitian $K$-theory and use it to construct an $E_\infty$ motivic ring spectrum $\mathbf{KR}^{\mathrm{alg}}$ representing homotopy Hermitian $K$-theory. From these results, we show that $\mathbf{KR}^{\mathrm{alg}}$ is stable under base change, and cdh descent for homotopy Hermitian $K$-theory of rings with involution is a formal consequence.

Cdh Descent for Homotopy Hermitian $K$-Theory of Rings with Involution

Abstract

We provide a geometric model for the classifying space of automorphism groups of Hermitian vector bundles over a ring with involution such that ; this generalizes a result of Schlichting-Tripathi \cite{SchTri}. We then prove a periodicity theorem for Hermitian -theory and use it to construct an motivic ring spectrum representing homotopy Hermitian -theory. From these results, we show that is stable under base change, and cdh descent for homotopy Hermitian -theory of rings with involution is a formal consequence.

Paper Structure

This paper contains 21 sections, 37 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Equivariant Topologies and the Equivariant Motivic Homotopy Category
  4. The Equivariant and Isovariant Étale Topologies
  5. The Equivariant Nisnevich Topology
  6. The Equivariant cdh Topology
  7. The Equivariant Motivic Homotopy Category
  8. Computations with Equivariant Spheres
  9. Hermitian Forms on Schemes
  10. Definitions
  11. Properties
  12. Hermitian Forms on Semilocal Rings
  13. Higher Grothendieck-Witt Groups
  14. Representability of Automorphism Groups of Hermitian Forms
  15. The definition of the Hermitian Grassmannian $\mathbb R\mathrm{Gr}$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $S$ be a Noetherian scheme of finite Krull dimension with an ample family of line bundles and $\frac{1}{2} \in S$. There is an equivalence of motivic spaces on $\mathbf{Sm}_{S,qp}^{C_2}$

Theorems & Definitions (119)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Remark 2.7
  • Definition 2.9
  • ...and 109 more