-*- 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: