ULRICH ELEMENTS IN NORMAL SIMPLICIAL AFFINE SEMIGROUPS

Let H subset of N-d be a normal affine semigroup, R = K[H] its semigroup ring over the field K and omega(R) its canonical module. The Ulrich elements for H are those h in H such that for the multiplication map by x(h) from R into omega(R), the cokernel is an Ulrich module. We say that the ring R is almost Gorenstein if Ulrich elements exist in H. For the class of slim semigroups that we introduce, we provide an algebraic criterion for testing the Ulrich property. When d = 2, all normal affine semigroups are slim. Here we have a simpler combinatorial description of the Ulrich property. We improve this result for testing the elements in H which are closest to zero. In particular, we give a simple arithmetic criterion for when is. (1, 1) an Ulrich element in H.