2010/05/19 by V. V. Bavula, Bavula, V. V.
Mathematics · #14H37 #16W20 #19B99 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA) #math.AG #math.KT #math.RA #msc:14H37 #msc:16W20 #msc:19B99
paper · pdf · doi:10.48550/arxiv.1005.3550
23 pages. arXiv admin note: text overlap with arXiv:0906.3733
arxiv created 2013/05/03 · arxiv updated 2013/05/06
The algebra \mSn of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting, \em left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra \mSn is a noncommutative, non-Noetherian algebra of classical Krull dimension 2n and of global dimension n which is not a domain. If the ground field K has characteristic zero then the algebra \mSn is canonically isomorphic to the algebra K< (\der)/(\der x1), ..., frac\der\der xn, ∫1, ..., ∫n> of scalar integro-differential operators. %Ignoring non-Noetherian % property, the algebra \mSn belongs to a family of algebras % like the nth Weyl algebra An and the polynomial algebra %P2n. It is proved that \rm K1(\mSn)≃ K^*. The main idea is to show that the group \GL_∞ (\mSn) is generated by K^*, the group of elementary matrices E_∞ (\mSn) and (n-2)2n-1+1 explicit (tricky) matrices and then to prove that all the matrices are elementary. For each nonzero idempotent prime ideal \gp of height m of the algebra \mSn, it is proved that \rm K1(\mSn, \gp)≃ K^*, ifm=1, \Z(m(m-1))/(2)× K*m ifm> 1.