Unimodular rows over monoid rings. (English) Zbl 1401.19003
This work finishes a project that the author started with J. Gubeladze [J. Algebra 148, No. 1, 135–161 (1992; Zbl 0795.20056)]. Let \(R\) be a commutative Noetherian ring of dimension \(d\) and let \(M\) be a commutative cancellative monoid. So \(M\) is a submonoid of an abelian group. It is shown that the elementary action on unimodular \(n\)-rows over the monoid ring \(R[M]\) is transitive for \(n \geq \max(d + 2, 3)\). The starting point is the case \(R[t_1,\cdots,t_k,s_1^\pm,\cdots,s_l^\pm]\) considered by A. A. Suslin [Math. USSR, Izv. 11, 221–238 (1977; Zbl 0378.13002)]. One main ingredient of the proof is the author’s pyramidal descent method for monoid rings. But he now also needs to cover his monoid rings with polynomial rings where suitable non-monomial endomorphisms are used to produce monic polynomials.
Reviewer: Wilberd van der Kallen (Utrecht)
Keywords:
unimodular row; elementary action; Milnor patching; monoid ring; cancellative monoid; normal monoidReferences:
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