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Uniform symbolic topologies via multinomial expansions


Author: Robert M. Walker
Journal: Proc. Amer. Math. Soc. 146 (2018), 3735-3746
MSC (2010): Primary 13H10; Secondary 14C20, 14M25.
DOI: https://doi.org/10.1090/proc/14073
Published electronically: June 13, 2018
MathSciNet review: 3825829
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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 search 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 $Q^{(E r)} \subseteq Q^r$, 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.


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Additional Information

Robert M. Walker
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan, 48109
MR Author ID: 1031098
Email: robmarsw@umich.edu

Keywords: Multinomial theorem, non-isolated singularities, symbolic powers, toric variety
Received by editor(s): May 11, 2017
Received by editor(s) in revised form: December 5, 2017
Published electronically: June 13, 2018
Communicated by: Irena Peeva
Article copyright: © Copyright 2018 American Mathematical Society