Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Grothendieck ring of varieties and algebraic K-theory of spaces

Oliver Röndigs

Abstract

Waldhausen's algebraic K-theory machinery is applied to motivic homotopy theory, producing an interesting motivic homotopy type. Over a field F of characteristic zero, its path components receive a surjective ring homomorphism from the Grothendieck ring of varieties over F.

The Grothendieck ring of varieties and algebraic K-theory of spaces

Abstract

Waldhausen's algebraic K-theory machinery is applied to motivic homotopy theory, producing an interesting motivic homotopy type. Over a field F of characteristic zero, its path components receive a surjective ring homomorphism from the Grothendieck ring of varieties over F.

Paper Structure

This paper contains 6 sections, 37 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. $K$-theory of model categories
  3. $\mathbb{A}^1$-homotopy theory
  4. Algebraic $K$-theory of $\mathbb{A}^1$-homotopy theory
  5. Grothendieck rings
  6. A trace map

Key Result

Theorem 1

Let $F$ be a field of characteristic zero. Sending a smooth projective variety to its natural class in $\pi_0\mathrm{A}(F)$ defines a surjective ring homomorphism from the Grothendieck ring of varieties over $F$.

Theorems & Definitions (84)

  • Theorem 1
  • Theorem 2: Waldhausen
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • ...and 74 more