Note on the Grothendieck group of subspaces of rational functions and Shokurov's b-divisors

In a previous paper the authors develop an intersection theory for subspaces of rational functions on an algebraic variety X over complex numbers. In this note, we first extend this intersection theory to an arbitrary algebraically closed ground field. Secondly we give an isomorphism between the group of 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 approach to b-divisors and their intersection theory.