The Cd-index of Fans and Lattices

where hk = dim(A(∆)⊗AR) k. When the fan ∆ is rational, it corresponds to a toric variety X(∆) and the numbers hk are the even betti numbers of the cohomology of X(∆). In any case, they satisfy the Poincaré duality hk = hn−k. When the fan ∆ is complete but not necessarily simplicial, one can apply the same construction to the first barycentric subdivision B(∆) of ∆. We can label a 1-dimensional cone of B(∆) according to the dimension of the cone in ∆ that it is a barycenter of. This gives a multi-grading on A(B(∆)) by Nn. It is also possible to adjust the A-module structure of A(B(∆)) so that it preserves this multigrading. The corresponding Poincaré polynomial is