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

## TL;DR

This paper develops a relative $ ext{A}^1$-homology theory, proving key theorems like Whitehead and excision, and applies these to compute homology in projective space embeddings.

## Contribution

It introduces a general theory of relative $ ext{A}^1$-homology and homotopy sheaves, extending foundational results in $ ext{A}^1$-homotopy theory.

## Key findings

- Proved an $ ext{A}^1$-homology version of the Whitehead theorem.
- Established an excision theorem for $ ext{A}^1$-homology and related sheaves.
- Computed the relative $ ext{A}^1$-homology of a hyperplane embedding in projective space.

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