Relative Cartier divisors and K-theory

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 $N{K}_n(f)$ are continuous $W(A)$-modules.