String Cohomology of a Toroidal Singularity

We construct explicitly regular sequences in the semigroup ring $R=\CC[K]$ of lattice points of the graded cone $K$. We conjecture that the quotients of $R$ by these sequences describe locally string-theoretic cohomology of a toroidal singularity associated to $K$. As a byproduct, we give an elementary proof of the result of Hochster that semigroup rings of rational polyhedral cones are Cohen-Macaulay.