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 and establish a weak comparison result between $\mathbb{A}^1$- and Suslin homology. As an example of relative $\mathbb{A}^1$-homology and as an application of the results above, we compute the relative $\mathbb{A}^1$-homology of the hyperplane embeddings $\mathbb{P}^{n-1} \hookrightarrow \mathbb{P}^{n}$.