Uniform Symbolic Topologies in Non-Regular Rings

Abstract

When does a Noetherian commutative ring R have uniform symbolic topologies (USTP) on primes -- read, when does there exist an integer D>0 such that the symbolic power P^{(Dr)} lies in P^r for all prime ideals P in R and all r >0? Groundbreaking work of Ein -- Lazarsfeld -- Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. Their work shows that there exists a D depending only on the Krull dimension: in other words, the exact same D works for all regular rings as stated of a fixed dimension.

Referring to this last observation, we say in the thesis that the class of excellent regular rings enjoys class solidarity relative to the uniform symbolic topology property (USTP class solidarity), a strong form of uniformity. In contrast, this thesis shows that for certain classes of non-regular rings including rational surface singularities and select normal toric rings, a uniform bound D does exist but depends on the ring, not just its dimension. In particular, for rational double point surface singularities over the field C of complex numbers, we show that USTP solidarity is plainly impossible.

It is natural to sleuth for analogues of the Improved Ein -- Lazarsfeld -- Smith Theorem where the ring R is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This thesis lies in the overlap of these research directions, working with Noetherian domains.