Published online by Cambridge University Press: 20 November 2018
In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety $X$ over
$\mathbf{k}\,=\,\mathbb{C}$. In this short note, we first extend this intersection theory to an arbitrary algebraically closed ground field
$\mathbf{k}$. Secondly we give an isomorphism between the group of Cartier
$b$-divisors on the birational class of
$X$ and the Grothendieck group of the semigroup of subspaces of rational functions on
$X$. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative point of view on Cartier
$b$-divisors and their intersection theory.