In this article, we study the Heller relative $K_{ 0}$ group of the map $\mathbb{ P}_{ X}^{ r} \to \mathbb{ P}_{ S}^{ r},$ 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 $K_{ 0}(\mathbb{ P}_{ X}^{ r} \to \mathbb{ P}_{ S}^{ r})$ in terms of generators and relations, provided $X$ is affine and flat over $S.$
Categories: math.AG