Critical cones of characteristic varieties
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- by Roberto Boldini PDF
- Trans. Amer. Math. Soc. 365 (2013), 143-160 Request permission
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
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 143-160
- MSC (2010): Primary 13C15, 13N10, 13P10, 16P90, 16W70
- DOI: https://doi.org/10.1090/S0002-9947-2012-05531-0
- MathSciNet review: 2984055