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

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