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The stable Adams operations on Hermitian K-theory

Jean Fasel, Olivier Haution

Abstract

We prove that exterior powers of (skew-)symmetric bundles induce a $λ$-ring structure on the ring $GW^0(X) \oplus GW^2(X)$, when $X$ is a scheme where $2$ is invertible. Using this structure, we define stable Adams operations on Hermitian $K$-theory. As a byproduct of our methods, we also compute the ternary laws associated to Hermitian $K$-theory.

The stable Adams operations on Hermitian K-theory

Abstract

We prove that exterior powers of (skew-)symmetric bundles induce a -ring structure on the ring , when is a scheme where is invertible. Using this structure, we define stable Adams operations on Hermitian -theory. As a byproduct of our methods, we also compute the ternary laws associated to Hermitian -theory.

Paper Structure

This paper contains 19 sections, 52 theorems, 215 equations.

Table of Contents

  1. Grothendieck--Witt groups and spectra
  2. Exterior powers and rank two symplectic bundles
  3. Grothendieck--Witt groups of representations
  4. The lambda-operations
  5. Exterior powers of metabolic forms
  6. The lambda-ring structure
  7. The Adams operations
  8. The unstable Adams operations
  9. Adams Operations on hyperbolic forms
  10. Adams operations on the universal rank two bundle
  11. Inverting (n)
  12. The stable Adams operations
  13. Ternary laws for Hermitian K-theory
  14. lambda-rings
  15. Graded rings
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $(E,\varepsilon)$ and $(F,\varphi)$ be vector bundles over a scheme $X$ equipped with bilinear forms, of respective ranks $e$ and $f$. Then we have an isometry

Theorems & Definitions (109)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 99 more