Koszul homology and extremal properties of Gin and Lex
Author:
Aldo Conca
Journal:
Trans. Amer. Math. Soc. 356 (2004), 2945-2961
MSC (2000):
Primary 13D02; Secondary 13P10, 13Fxx
DOI:
https://doi.org/10.1090/S0002-9947-03-03393-2
Published electronically:
November 18, 2003
MathSciNet review:
2052603
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: For every homogeneous ideal in a polynomial ring
and for every
we consider the Koszul homology
with respect to a sequence of
of generic linear forms. The Koszul-Betti number
is, by definition, the dimension of the degree
part of
. In characteristic
, we show that the Koszul-Betti numbers of any ideal
are bounded above by those of the gin-revlex
of
and also by those of the Lex-segment
of
. We show that
iff
is componentwise linear and that and
iff
is Gotzmann. We also investigate the set
of all the gin of
and show that the Koszul-Betti numbers of any ideal in
are bounded below by those of the gin-revlex of
. On the other hand, we present examples showing that in general there is no
is
such that the Koszul-Betti numbers of any ideal in
are bounded above by those of
.
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Additional Information
Aldo Conca
Affiliation:
Dipartimento di Matematica, Universita’ di Genova, Genova, I-16146 Italy
Email:
conca@dima.unige.it
DOI:
https://doi.org/10.1090/S0002-9947-03-03393-2
Keywords:
Koszul homology,
Betti numbers,
generic initial ideal,
Lex-segment
Received by editor(s):
December 3, 2002
Received by editor(s) in revised form:
April 30, 2003
Published electronically:
November 18, 2003
Article copyright:
© Copyright 2003
American Mathematical Society