Relative ${\mathbb{A}}^1$-homology and its applications

Abstract

In this paper, we prove an ${\mathbb{A}}^1$-homology version of the Whitehead theorem with dimension bound. We also prove an excision theorem for ${\mathbb{A}}^1$-homology, Suslin homology and ${\mathbb{A}}^1$-homotopy sheaves. In order to prove these results, we develop a general theory of relative ${\mathbb{A}}^1$-homology and ${\mathbb{A}}^1$-homotopy sheaves. As an application, we compute the relative ${\mathbb{A}}^1$-homology of a hyperplane embedding ${\mathbb{P}}^{n-1}\hookrightarrow {\mathbb{P}}^n$.

To appear in Homology, Homotopy, and Applications