The Steinberg group of a monoid ring, nilpotence, and algorithms

For a regular ring R and an affine monoid M the homotheties of M act nilpotently on the Milnor unstable groups of R[M]. This strengthens the K2 part of the main result of [J. Gubeladze, The nilpotence conjecture in K-theory of toric varieties, Invent. Math. 160 (2005) 173–216] in two ways: the coefficient field of characteristic 0 is extended to any regular ring and the stable K2-group is substituted by the unstable ones. The proof is based on a polyhedral/combinatorial technique, computations in Steinberg groups, and a substantially corrected version of an old result on elementary matrices by Mushkudiani [Z. Mushkudiani, K2-groups of monoid algebras over regular rings, Proc. A. Razmadze Math. Inst. 113 (1995) 120–137]. A similar stronger nilpotence result for K1 and algorithmic consequences for factorization of high Frobenius powers of invertible matrices are also derived.