$\mathbb{A}^1$-homotopy theory of log schemes

TL;DR

This paper develops an $A^1$-homotopy theory framework for fs log schemes, extending motivic cohomology, K-theory, and algebraic cobordism to this setting, and establishing foundational properties like localization and six functors formalism.

Contribution

It constructs the $A^1$-local stable motivic homotopy category for fs log schemes, extending classical theories and proving key properties like localization.

Findings

Established the $A^1$-local stable motivic homotopy category for fs log schemes.

Extended motivic cohomology, K-theory, and cobordism to fs log schemes.

Proved the localization property and six functors formalism for strict morphisms.

Abstract

We construct the $\mathbb{A}^1$-local stable motivic homotopy categories of fs log schemes. For schemes with the trivial log structure, our construction is equivalent to the original construction of Morel-Voevodsky. We prove the localization property. As a consequence, we obtain the Grothendieck six functors formalism for strict morphisms of fs log schemes. We extend $\mathbb{A}^1$-invariant cohomology theories of schemes to fs log schemes. In particular, we define motivic cohomology, homotopy $K$-theory, and algebraic cobordism of fs log schemes. For any fs log scheme log smooth over a scheme, we express cohomology of its boundary in terms of cohomology of schemes.