Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

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

Pages: 129 – 141

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a7


Yuri Shimizu (Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo, Japan)


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 $\mathbb{P}^{n-1} \to \mathbb{P}^n$.


$\mathbb{A}^1$-homology, $\mathbb{A}^1$-homotopy, algebraic geometry

2010 Mathematics Subject Classification

14F42, 19E15

This work was supported by JSPS KAKENHI Grant Number JP19J21433.

Received 3 May 2020

Received revised 22 January 2021

Accepted 5 February 2021

Published 6 April 2022