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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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. 140 (2012), 3323-3331
MSC (2010): Primary 16E10, 16S85, 16S99
Published electronically: 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 $ A$:

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

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 $ A$:

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


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Additional Information

V. V. Bavula
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
Email: v.bavula@sheffield.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11177-3
PII: S 0002-9939(2012)11177-3
Keywords: The algebras of one-sided inverses, the algebra of one-sided inverses of a polynomial algebra, the global dimension, Hilbert’s Syzygy Theorem, the projective dimension, the weak dimension.
Received by editor(s): June 2, 2010
Received by editor(s) in revised form: April 1, 2011
Published electronically: February 2, 2012
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.