Projective bundle formula for Heller' relative K_0

TL;DR

This paper proves a projective bundle formula for Heller's relative K_0 group in algebraic geometry, specifically for quasi-projective schemes over a ring, under flatness conditions, and describes the group explicitly in certain cases.

Contribution

It establishes the projective bundle formula for Heller's relative K_0, extending known results to this specific context and providing explicit descriptions under flatness assumptions.

Findings

Proves projective bundle formula for Heller's relative K_0

Provides explicit description of the relative K_0 group for affine, flat schemes

Extends algebraic K-theory results to new geometric settings

Abstract

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 $S.$