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# 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

#### Author

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

#### Keywords

$\mathbb{A}^1$-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