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An analogue of Hilbert's Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra

Author: V. V. Bavula
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 16E10, 16S85, 16S99
Posted: February 2, 2012
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Abstract: An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mathbb{S}_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, \ldots , x_n]$ over an arbitrary ring :

$\displaystyle \textrm {l.gldim}(\mathbb{S}_n(A))= \textrm {l.gldim}(A) +n.$

The algebra $\mathbb{S}_n(A)$ is noncommutative, neither left nor right Noetherian and not a domain. The proof is based on a generalization of the Theorem of Kaplansky (on the projective dimension) obtained in the paper. As a consequence it is proved that for a left or right Noetherian algebra

$\displaystyle \textrm {w.dim} (\mathbb{S}_n(A))= \textrm {w.dim} (A) +n.$

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