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Hermitian K-theory of quadric hypersurfaces

Heng Xie

Abstract

Let k be a commutative ring in which 2 is invertible. We prove that the Hermitian K-theory of quadric hypersurfaces over k admits fibration sequences relating it to the base ring and to Clifford algebras equipped with various duality coefficients. All shifts and twists are taken into account.

Hermitian K-theory of quadric hypersurfaces

Abstract

Let k be a commutative ring in which 2 is invertible. We prove that the Hermitian K-theory of quadric hypersurfaces over k admits fibration sequences relating it to the base ring and to Clifford algebras equipped with various duality coefficients. All shifts and twists are taken into account.

Paper Structure

This paper contains 18 sections, 13 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Dg $k$-modules
  4. Clifford algebras and Tate resolution
  5. Quadric hypersurfaces
  6. Clifford sequences
  7. Tate resolution
  8. Modification
  9. Construction of symmetric forms
  10. Symmetric forms on Tate resolution
  11. Trivial duality
  12. Twisted duality
  13. Proof of Theorem \ref{['theo:main-theorem']}
  14. Proof of the main theorem
  15. Semi-orthogonal decomposition and duality
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathscr{L}$ be a line bundle on $Q_d$. The following statements hold in the stable homotopy category of spectra $\mathcal{SH}$. Moreover, the spectrum $\mathrm{GW}^{[i]}(\mathfrak{A}_{\mathscr{L}})$ has the following properties:

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 3.1: Swan bundle
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • ...and 31 more