Relative Cartier Divisors and Laurent Polynomial Extensions

Abstract

If $i:A\subset B$ is a commutative ring extension, we show that the group $\mathcal{I}(A,B)$ of invertible $A$-submodules of $B$ is contracted in the sense of Bass, with $L\mathcal{I}(A,B)=H_{et}^0(A,i_{\ast }\mathbb{Z}/\mathbb{Z})$. This gives a canonical decomposition for $\mathcal{I}(A[t,\frac{1}{t}],B[t,\frac{1}{t}])$.

Section numbers changed in v3 to agree with published version. Note that versions v1 and v2 are identical