UNIFORM SYMBOLIC TOPOLOGIES VIA MULTINOMIAL EXPANSIONS

作者：Walker Robert M^*

来源：PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2018, 146(9): 3735-3746.

DOI：10.1090/proc/14073

• Multinomial theorem
• non-isolated singularities
• symbolic powers
• toric variety

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) subset of P-r for all prime ideals P subset of 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 F, we present binomial and multinomial expansion criteria for containments of type Q((Er)) subset of Q(r), or even better, of type Q((E(r-1)=1) subset of Q(r) for all r > 0. The final section consolidates remarks on how often we can utilize these criteria, presenting an example.

• 出版日期2018-7-15