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Title: Frobenius Multiplicities
Description: For rings of characteristic p > 0, the Frobenius morphism x |-> x^p provides an
honest ring endomorphism with surprising utility. One application is the study
of Hilbert-Kunz and related multiplicities.
Implement the Frobenius and its action on complexes of free modules as a way to
quickly compute approximations of Hilbert-Kunz and higher derived multiplicities of a ring.
See:
1. Huneke, Craig
Tight closure and its applications.
With an appendix by Melvin Hochster.
CBMS Regional Conference Series in Mathematics, 88.
Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the
American Mathematical Society, Providence, RI, 1996.
2. Watanabe, Kei-ichi; Yoshida, Ken-ichi
Hilbert-Kunz multiplicity and an inequality between multiplicity and colength.
J. Algebra 230 (2000), no. 1, 295--317.
3. Monsky, P.
The Hilbert\mhy Kunz function.
Math. Ann. 263 (1983), no. 1, 43--49.
4. Li, Jinjia
Characterizations of regular local rings in positive characteristics.
Proc. Amer. Math. Soc. 136 (2008), no. 5, 1553--1558.
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Proposed by: Jason McCullough
Potential Advisor: Dan Grayson
Project assigned to: Jason McCullough
Current status: A working project file is completed. It needs documentation and
error checking. Extensions to computing Hilbert-Kunz multiplicities of m-primary
ideals are possible.
As of now the ring in question much be a quotient of a polynomial ring by a
homogeneous ideal.
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Progress log: 6/4/09 - Package is fully documented and will be uploaded shortly.