Test ideals in diagonal hypersurface rings, II

Abstract

Let R = k[x 1 , . . . , x n ]/(x d 1 + · · · + x d n ), where k is a field of characteristic p, p does not divide d and n ≥ 3. We describe a method for computing the test ideal for these diagonal hypersurface rings. This method involves using a characterization of test ideals in Gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm. In addition, we compute examples of test ideals in diagonal hypersurface rings of small characteristic (relative to d) including several that are not integrally closed. These examples provide a negative answer to Smith's question of whether the test ideal in general is always integrally closed [Sm2].