Analytic spread of filtrations on two dimensional normal local rings
Abstract
In this paper we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two dimensional normal excellent local ring $R$, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on $R$ has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of $R$. The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.05935
- arXiv:
- arXiv:2203.05935
- Bibcode:
- 2022arXiv220305935C
- Keywords:
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- Mathematics - Commutative Algebra;
- Mathematics - Algebraic Geometry;
- 13C05;
- 13H15;
- 14C17;
- 14C20
- E-Print:
- 28 pages