This material has been published in
"Algebra, Arithmetic and Geometry
with Applications," C. Christensen, G. Sundaram, A. Sathaye and C. Bajaj,
eds., Springer-Verlag, New York, 2004, pp. 337-356,
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The Hilbert series of Pfaffian rings
(20 pages)
Abstract.
We give three determinantal expressions for the Hilbert series
as well as the Hilbert function of a Pfaffian ring, and a closed form
product formula for its multiplicity.
An appendix outlining some basic facts about degeneracy
loci and applications to multiplicity formulae for Pfaffian rings is
also included.
The following versions are available:
Comment
In Footnote 7 of this paper, we remarked that most concepts and
results discussed in the appendix on Degeneracy Loci
would extend readily from the complex case to that of an
arbitrary ground field, at least in characteristic zero
case, if instead of cohomology rings, one works in the
Chow ring of algebraic cycles modulo rational
equivalence. Further, we stated that it is not clear to
us how the proof of the squaring principle in the paper
of Harris and Tu [Topology 23 (1984), 71-84] would go
through in the general case.
In this context, Professor Fulton has kindly informed us that the
`squaring principle' is, in fact, false in positive
characteristic. See his
e-mail for details.
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