Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Motivic six-functor formalism for log schemes

Doosung Park

Abstract

We establish the motivic six-functor formalism for fs log schemes. In particular, we prove the exact base change property, projection formula, and Poincaré duality. We also define Borel-Moore motivic homology, G-theory, and Chow homology of fs log schemes and the category of Chow motives over fs log schemes.

Motivic six-functor formalism for log schemes

Abstract

We establish the motivic six-functor formalism for fs log schemes. In particular, we prove the exact base change property, projection formula, and Poincaré duality. We also define Borel-Moore motivic homology, G-theory, and Chow homology of fs log schemes and the category of Chow motives over fs log schemes.
Paper Structure (21 sections, 59 theorems, 174 equations, 2 figures)

This paper contains 21 sections, 59 theorems, 174 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cohomology and Borel-Moore homology of log schemes
  3. Summary of the previous paper
  4. The result in this paper
  5. Chow motives over the standard log point
  6. Purity
  7. Thom transformations
  8. Exchange transformations
  9. Composition transformations
  10. Purity transformations
  11. Naturality of the purity transformations
  12. Poincaré duality
  13. Affine toric log schemes
  14. Motivic invariance under virtual isomorphisms
  15. Base change property
  16. ...and 6 more sections

Key Result

Theorem 1.2.1

The above functor $\mathscr{T}^\mathrm{ex}$ satisfies the following properties.

Figures (2)

  • Figure 1: The fiber of $E_{\log}\to S^1$
  • Figure 2: The illustrations of $C_i^\vee$ and $D_i^\vee$

Theorems & Definitions (138)

  • Theorem 1.2.1
  • proof
  • Theorem 1.3.1
  • proof
  • Proposition 1.4.1: Proposition \ref{['logChow.15']}
  • Definition 2.1.1
  • Definition 2.1.2
  • Proposition 2.1.3
  • proof
  • Proposition 2.1.4
  • ...and 128 more