Uniform Symbolic Topologies via Multinomial Expansions

Abstract

When does a Noetherian commutative ring $R$ have uniform symbolic topologies on primes--read, when does there exist an integer $D>0$ such that the symbolic power $P^{(Dr)}\subseteq P^r$ for all prime ideals $P\subseteq R$ 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. It is natural to then sleuth for analogues 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 manuscript falls under the overlap of these research directions. Working with a prescribed type of prime ideal $Q$ inside of tensor products of domains of finite type over an algebraically closed field $\mathbb{F}$, we present binomial- and multinomial expansion criteria for containments of type $Q^{(Er)}\subseteq Q^r$, or even better, of type $Q^{(E(r-1)+1)}\subseteq Q^r$ for all $r>0$. The final section consolidates remarks on how often we can utilize these criteria, presenting an example.

10 pages, a follow-up to (arXiv:1608.02320). Rewritten both to address referee feedback and to reflect more recent developments in this research direction, as posted on arXiv in 2017. Abstract and Bibliography updated

Cited by in corpus (4)

• The Uniform Symbolic Topology Property for Diagonally $F$-regular Algebras
• Uniform Harbourne-Huneke Bounds via Flat Extensions
• Tensor-Multinomial Sums of Ideals: Primary Decompositions and Persistence of Associated Primes
• Uniform Symbolic Topologies in Normal Toric Rings