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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Critical cones of characteristic varieties


Author: Roberto Boldini
Journal: Trans. Amer. Math. Soc. 365 (2013), 143-160
MSC (2010): Primary 13C15, 13N10, 13P10, 16P90, 16W70
Published electronically: July 25, 2012
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Abstract: Let $ M$ be a left module over a Weyl algebra in characteristic zero. Given natural weight vectors $ \nu $ and $ \omega $, we show that the characteristic varieties arising from filtrations with weight vector $ \nu +s\omega $ stabilize to a certain variety determined by $ M$, $ \nu $, $ \omega $ as soon as the natural number $ s$ grows beyond a bound which depends only on $ M$ and $ \nu $ but not on $ \omega $.

As a consequence, in the notable case when $ \nu $ is the standard weight vector, these characteristic varieties deform to the critical cone of the $ \omega $-characteristic variety of $ M$ as soon as $ s$ grows beyond an invariant of $ M$.

As an application, we give a new, easy, non-homological proof of a classical result, namely, that the $ \omega $-characteristic varieties of $ M$ all have the same Krull dimension.

The set of all $ \omega $-characteristic varieties of $ M$ is finite. We provide an upper bound for its cardinality in terms of supports of universal Gröbner bases in the case when $ M$ is cyclic. By the above stability result, we conjecture a second upper bound in terms of total degrees of universal Gröbner bases and of Fibonacci numbers in the case when $ M$ is cyclic over the first Weyl algebra.


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Additional Information

Roberto Boldini
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland
Email: roberto.boldini@math.uzh.ch

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05531-0
PII: S 0002-9947(2012)05531-0
Received by editor(s): July 21, 2010
Received by editor(s) in revised form: December 4, 2010, and December 24, 2010
Published electronically: July 25, 2012
Additional Notes: The author thanks Professor Markus Brodmann and Professor Joseph Ayoub, University of Zurich
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.