Projective bundle formula for Heller's relative $K_{0}$

Abstract

In this article, we study the Heller relative $K_{0}$ group of the map $\mathbf{P}_{X}^{r} \to \mathbf{P}_{S}^{r}$, where $X$ and $S$ are quasi-projective schemes over a commutative ring. More precisely, we prove that the projective bundle formula holds for Heller's relative $K_{0}$, provided $X$ is flat over $S$. As a corollary, we get a description of the relative group $K_{0}(\mathbf{P}_{X}^{r} \to \mathbf{P}_{S}^{r})$ in terms of generators and relations, provided $X$ is affine and flat over $S$.