F-thresholds, tight closure, integral closure, and multiplicity bounds
Abstract
The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect the containment in the integral closure or the tight closure of a parameter ideal using F-thresholds. We formulate a conjecture bounding $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals, and $J$ is generated by a system of parameters. We prove the conjecture when $J$ is a monomial ideal in a polynomial ring, and also when $\a$ and $J$ are generated by homogeneous systems of parameters in a Cohen-Macaulay graded $k$-algebra.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2007
- DOI:
- 10.48550/arXiv.0708.2394
- arXiv:
- arXiv:0708.2394
- Bibcode:
- 2007arXiv0708.2394H
- Keywords:
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- Mathematics - Commutative Algebra;
- Mathematics - Algebraic Geometry;
- 13A35 (Primary);
- 13B22;
- 13H15;
- 14B05 (Secondary)
- E-Print:
- 22 pages