An algorithm for the Quillen-Suslin theorem for monoid rings (Q1358921)

Let \(M\) be a commutative, finitely generated, torsion-free, cancellative seminormal monoid without nontrivial units (the authors call it toric monoid); \(kM\) be its monoid ring over a field \(k\) (it can be realized as a subalgebra of a polynomial ring \(k[x_1,x_2,\dots, x_n]\), generated by monomials); \(P\) be a finitely generated module over \(kM\), given as the cokernel of a matrix with entries in \(kM\). \textit{I. Dzh. Gubeladze} has proved that if \(P\) is projective, it must be free (over \(kM\)) [Math. USSR, Sb. 63, No. 1, 165-180 (1989); translation from Mat. Sb., Nov. Ser. 135(177), No. 2, 169-185 (1988; Zbl 0654.13013)]. Following the work of \textit{R. G. Swan} [in: Azumaya algebras, actions, and modules, Proc. Conf. Honor Azumaya's 70th birthday, Bloomington 1990, Contemp. Math. 124, 215-250 (1992; Zbl 0742.13005)], algebraizing the Gubeladze proof, the authors present an algorithm for computing a free basis of \(P\). In passing, a new constructive proof of the ``ordinary'' Quillen-Suslin theorem on projective modules over polynomial rings is obtained, using a modification of the algorithm by \textit{A. Logar} and \textit{B. Sturmfels} [J. Algebra 145, No. 1, 231-239 (1992; Zbl 0747.13020)] in combination with the algorithm given by \textit{H. Park} and \textit{C. Woodburn} [J. Algebra 178, No. 1, 277-298 (1995; Zbl 0841.19001)]. The corresponding results related to a Laurent polynomial ring \(k[x^{\pm 1}_1, x^{\pm 1}_2,\dots, x^{\pm 1}_n ]\) are indicated, as well.