36 research outputs found
Parameter test ideals of Cohen Macaulay rings
We describe an algorithm for computing parameter-test-ideals in certain local
Cohen-Macaulay rings. The algorithm is based on the study of a Frobenius map on
the injective hull of the residue field of the ring and on the application of
Rodney Sharp's notion of ``special ideals''.
Our techniques also provide an algorithm for computing indices of nilpotency
of Frobenius actions on top local cohomology modules of the ring and on the
injective hull of its residue field. The study of nilpotent elements on
injective hulls of residue fields also yields a great simplification of the
proof of the fact that for a power series ring of prime characteristic, for
all nonzero , generates as a -module.Comment: 16 pages To appear in Compositio Mathematic
An example of an infinite set of associated primes of a local cohomology module
Let be a local Noetherian ring, let be any ideal and let be a finitely generated -module. In 1990 Craig Huneke conjectured that the local cohomology modules have finitely many associated primes for all . In this paper I settle this conjecture by constructing a local cohomology module of a local -algebra with an infinite set of associated primes, and I do this for any field
An algorithm for computing compatibly Frobenius split subvarieties
Let be a ring of prime characteristic , and let denote
viewed as an -module via the th iterated Frobenius map. Given a
surjective map (for example a Frobenius splitting), we
exhibit an algorithm which produces all the -compatible ideals.
We also explore a variant of this algorithm under the hypothesis that
is not necessarily a Frobenius splitting (or even surjective). This algorithm,
and the original, have been implemented in Macaulay2.Comment: 15 pages, many statements clarified and numerous other substantial
improvements to the exposition (thanks to the referees). To appear in the
Journal of Symbolic Computatio
Characteristic-independence of Betti numbers of graph ideals
In this paper, we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first 6 Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n = 11, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertices whose homology has torsion.
In each of the 4 examples mentioned above the 8th Betti numbers depend on the ground field and so we conclude that the highest Betti number which is always independent of the ground field is either 6 or 7; if the former is true then we show that there must exist a graph with 12 vertices whose 7th Betti number depends on the ground field. (c) 2005 Elsevier Inc. All rights reserved