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

Bavula, V.

doi:10.1090/S0002-9939-2012-11177-3

An analogue of Hilbert’s Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra
HTML articles powered by AMS MathViewer

by V. V. Bavula PDF

Proc. Amer. Math. Soc. 140 (2012), 3323-3331 Request permission

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 $A$: \[ \textrm {l.gldim}(\mathbb {S}_n(A))= \textrm {l.gldim}(A) +n.\] 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 $A$: \[ \textrm {w.dim} (\mathbb {S}_n(A))= \textrm {w.dim} (A) +n.\]

References

Maurice Auslander, On the dimension of modules and algebras. III. Global dimension, Nagoya Math. J. 9 (1955), 67–77. MR 74406
Reinhold Baer, Inverses and zero-divisors, Bull. Amer. Math. Soc. 48 (1942), 630–638. MR 6997, DOI 10.1090/S0002-9904-1942-07750-0
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Vladimir Bavula, Global dimension of generalized Weyl algebras, Representation theory of algebras (Cocoyoc, 1994) CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 81–107. MR 1388045
Vladimir Bavula, Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras, Bull. Sci. Math. 120 (1996), no. 3, 293–335. MR 1399845
V. V. Bavula, The algebra of one-sided inverses of a polynomial algebra, J. Pure Appl. Algebra 214 (2010), no. 10, 1874–1897. MR 2608115, DOI 10.1016/j.jpaa.2009.12.033
V. V. Bavula, The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra, arXiv:math.AG/0903.3049.
V. V. Bavula, $\textrm {K}_1(\mathbb {S}_1)$ and the group of automorphisms of the algebra $\mathbb {S}_2$ of one-sided inverses of a polynomial algebra in two variables, Journal of K-theory and its Appl. to Algebra, Geometry, Analysis and Topology, to appear, arXiv:0906.0600.
V. V. Bavula, The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra, II. arXiv:math.AG:0906.3733.
V. V. Bavula, The group $\textrm {K}_1(\mathbb {S}_n)$ of the algebra of one-sided inverses of a polynomial algebra, arXiv:math.KT:1005.3550.
Jan-Erik Björk, The global homological dimension of some algebras of differential operators, Invent. Math. 17 (1972), 67–78. MR 320078, DOI 10.1007/BF01390024
K. A. Brown, C. R. Hajarnavis, and A. B. MacEacharn, Noetherian rings of finite global dimension, Proc. London Math. Soc. (3) 44 (1982), no. 2, 349–371. MR 647437, DOI 10.1112/plms/s3-44.2.349
Samuel Eilenberg, Alex Rosenberg, and Daniel Zelinsky, On the dimension of modules and algebras. VIII. Dimension of tensor products, Nagoya Math. J. 12 (1957), 71–93. MR 98774
L. Gerritzen, Modules over the algebra of the noncommutative equation $yx=1$, Arch. Math. (Basel) 75 (2000), no. 2, 98–112. MR 1767169, DOI 10.1007/PL00000437
K. R. Goodearl, Global dimension of differential operator rings. II, Trans. Amer. Math. Soc. 209 (1975), 65–85. MR 382359, DOI 10.1090/S0002-9947-1975-0382359-7
N. Jacobson, Some remarks on one-sided inverses, Proc. Amer. Math. Soc. 1 (1950), 352–355. MR 36223, DOI 10.1090/S0002-9939-1950-0036223-1
Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
Irving Kaplansky, Fields and rings, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0269449
George S. Rinehart and Alex Rosenberg, The global dimensions of Ore extensions and Weyl algebras, Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), Academic Press, New York, 1976, pp. 169–180. MR 0409563
Alex Rosenberg and J. T. Stafford, Global dimension of Ore extensions, Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), Academic Press, New York, 1976, pp. 181–188. MR 0409564
Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953