-*- M2 -*-
Title: Quillen-Suslin
Description:
If M is a projective module over a polynomial ring k[x1,..,xn], the
Quillen-Suslin Theorem asserts that M is free. However, given a
presentation of M by generators and relations, it is not trivial to find a
set of free generators. There is a Maple implementation:
[http://wwwb.math.rwth-aachen.de/QuillenSuslin/], based on the first paper
below (thanks to Bernd Sturmfels for the link). Algorithms for doing this
are contained in the papers:
* Logar, Alessandro; Sturmfels, Bernd, Algorithms for the Quillen-Suslin
theorem. J. Algebra 145 (1992), no. 1, 231--239.
* Laubenbacher, Reinhard C.; Woodburn, Cynthia J. A new algorithm for the
Quillen-Suslin theorem. Beiträge Algebra Geom. 41 (2000), no. 1,
23--31.
and in the more general case of a module over a monomial ring in
* Laubenbacher, Reinhard C.; Woodburn, Cynthia J. An algorithm for the
Quillen-Suslin theorem for monoid rings. Algorithms for algebra
(Eindhoven, 1996). J. Pure Appl. Algebra 117/118 (1997), 395--429.
* Here is some Maple code the implements an algorithm, wiht some
added heuristics:
http://wwwb.math.rwth-aachen.de/QuillenSuslin/
Potential Application: If A is a 2-dimensional ring with Noether
Normalization k[x,y], then the integral closure B of A is a free module
over k[x,y]. Current algorithms to produce module generators of B may
produce sets of generators that are too large. An example is given by Doug
Leonard at his site[http://www.dms.auburn.edu/~leonada], under "example of
the qth-power algorithm with two free variables and a larger than expected
module generating set". This is a module of rank 9, given with 10
generators. The desired algorithm would produce a 9-generator presentation
(with no relations.)
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Proposed by: David Eisenbud
Potential Advisor:
Project assigned to:
Current status:
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Progress log: