Relative Cartier divisors and K-theory

TL;DR

This paper investigates the structure of relative Picard groups, Cartier divisors, and K-theory for scheme maps, revealing their properties as continuous modules and their relationships via filtrations.

Contribution

It establishes the connection between relative Picard groups, Cartier divisors, and K-theory, and demonstrates the continuity of certain nil groups as modules over the Witt ring.

Findings

Pic(f) is the top quotient of the $ extgamma$-filtration on $K_0(f)$

When induced by ring homomorphisms, $NPic(f)$ and $NK_n(f)$ are continuous $W(A)$-modules

The relative Picard group $Pic(f)$ equals the group of relative Cartier divisors $I(f)$ when $f$ is faithful affine

Abstract

We study the relative Picard group $Pic(f)$ of a map $f:X\to S$ of schemes. If $f$ is faithful affine, it is the relative Cartier divisor group $I(f)$. The relative group $K_0(f)$ has a $\gamma$-filtration, and $Pic(f)$ is the top quotient for the $\gamma$-filtration. When $f$ is induced by a ring homomorphism $A\to B$, we show that the relative "nil" groups $NPic(f)$ and $NK_n(f)$ are continuous $W(A)$-modules.