id: 01570908
dt: j
an: 01570908
au: Laubenbacher, Reinhard; Schlauch, Karen
ti: An algorithm for the Quillen-Suslin theorem for quotients of polnomial
    rings by monomial ideals.
so: J. Symb. Comput. 30, No.5, 555-571 (2000).
py: 2000
pu: Elsevier Science (Academic Press), London
la: EN
cc: 
ut: monomial ideal; free module; Gröbner basis; square-free; Quillen-Suslin
    theorem; Stanley-Reisner ring
ci: 
li: doi:10.1006/jsco.2000.0367
ab: The Quillen-Suslin theorem states the following. Consider a $m\times
    m$-matrix $M$ with polynomial entries from $k[x].$ The columns generate
    a submodule $P$ of $k[x]^m.$ If $P$ is projective then the submodule
    $\ker(M)$ has a free basis. The authors generalize this result to the
    case that $k[x]$ is replaced by the quotient ring $k[x]/I$ where I is a
    monomial ideal. An algorithm for the construction of a free basis of
    ker(M) is presented. The first step is the redution to the square-free
    case. Ideas of Vorst are used to represent the square-free quotient
    ring as a pullback of two quotient rings. The construction uses a
    Stanley-Reisner ring of a simplicial complex. This idea of two rings is
    repeated until polynomial rings are found. Here the original
    Quillen-Suslin theorem can be used and the algorithm works backward to
    reconstruct the free basis of the original $\ker(M)$. The lifting
    algorithm involves Gröbner bases. The last section illustrates the
    algorithm by an example. This paper will be interesting to people
    working in commutative algebra and computational algebra.
rv: Karin Gatermann (Berlin)