Intersection theory of nef b-divisor classes

We prove that any nef Formula Presented-divisor class on a projective variety defined over an algebraically closed field of characteristic zero is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef Formula Presented-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint-free curve class is a power of a nef Formula Presented-divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of Formula Presented-divisors which were defined in our previous work.