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

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.

arXiv commentAccepted version, 51 pages, to appear in Eur. J. Math