Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 2984055
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. M. Aschenbrenner, A. Leykin, Degree Bounds for Gröbner Bases in Algebras of Solvable Type, J. of Pure and Applied Alg. 213 (2009), 1578-1605. MR 2517995 (2010g:13042)
  • 2. M. Assi, F. J. Castro-Jiménez, M. Granger, The Gröbner Fan of an $ A_n$-module, J. of Pure and Applied Alg. 150 (2000), 27-39. MR 1762918 (2001j:16036)
  • 3. J. P. Bell, Affine Rings of Low GK Dimension, Dissertation, UCSD, 2002. MR 2703199
  • 4. J. N. Bernstein, Modules over a Ring of Differential Operators. Study of the Fundamental Solutions of Equations with Constant Coefficients, translated from the Russian original, Funktsional'nyi Analiz i Ego Prilozheniya 5(2) (1971), 1-16. MR 0290097 (44:7282)
  • 5. R. Boldini, Finiteness of Leading Monomial Ideals and Critical Cones of Characteristic Varieties, Ph.D. Dissertation, UZH, 2012.
  • 6. A. Borel et al., Algebraic $ D$-Modules, Perspectives in Mathematics 2, Academic Press, 1987. MR 882000 (89g:32014)
  • 7. M. Brodmann, Algebren von Differentialoperatoren, Lecture Notes, UZH, 1986.
  • 8. S. C. Coutinho, A Primer of Algebraic $ D$-Modules, LMS Student Texts 33, Cambridge University Press, 1995. MR 1356713 (96j:32011)
  • 9. D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, Second Edition, Springer, 1997. MR 1417938 (97h:13024)
  • 10. A. Kandry-Rody, V. Weispfenning, Non-commutative Gröbner Bases in Algebras of Solvable Type, J. of Symb. Comp. 9 (1990), 1-26. MR 1044911 (91e:13025)
  • 11. G. R. Krause, T. H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, Graduate Studies in Mathematics 22, AMS, 2000. MR 1721834 (2000j:16035)
  • 12. H. Li, Noncommutative Gröbner Bases and Filtered-Graded Transfer, Lecture Notes in Mathematics 1795, Springer, 2002. MR 1947291 (2003i:16065)
  • 13. H. Li, F. van Oystaeyen, Zariskian Filtrations, $ K$-Monographs in Mathematics 2, Kluwer Academic Publishers, 1996. MR 1420862 (97m:16083)
  • 14. J. C. McConnell, J. C. Robson, Noncommutative Noetherian Rings, Wiley, 1987. MR 934572 (89j:16023)
  • 15. M. Saito, B. Sturmfels, N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics 6, Springer, 2000. MR 1734566 (2001i:13036)
  • 16. B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series 8, AMS, 1996. MR 1363949 (97b:13034)
  • 17. V. Weispfenning, Constructing Universal Gröbner Bases, Lecture Notes in Computer Sciences 356, 408-417, Springer, 1989. MR 1008554 (91e:13029)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 13C15, 13N10, 13P10, 16P90, 16W70

Retrieve articles in all journals with MSC (2010): 13C15, 13N10, 13P10, 16P90, 16W70

Additional Information

Roberto Boldini
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland

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.

American Mathematical Society