Growth of graded noetherian rings
HTML articles powered by AMS MathViewer
- by Darin R. Stephenson and James J. Zhang
- Proc. Amer. Math. Soc. 125 (1997), 1593-1605
- DOI: https://doi.org/10.1090/S0002-9939-97-03752-0
- PDF | Request permission
Abstract:
We show that every graded locally finite right noetherian algebra has sub-exponential growth. As a consequence, every noetherian algebra with exponential growth has no finite dimensional filtration which leads to a right (or left) noetherian associated graded algebra. We also prove that every connected graded right noetherian algebra with finite global dimension has finite GK-dimension. Using this, we can classify all connected graded noetherian algebras of global dimension two.References
- Michael Artin and William F. Schelter, Graded algebras of global dimension $3$, Adv. in Math. 66 (1987), no. 2, 171–216. MR 917738, DOI 10.1016/0001-8708(87)90034-X
- M. Artin, J. Tate, and M. Van den Bergh, Modules over regular algebras of dimension $3$, Invent. Math. 106 (1991), no. 2, 335–388. MR 1128218, DOI 10.1007/BF01243916
- J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
- Warren Dicks, On the cohomology of one-relator associative algebras, J. Algebra 97 (1985), no. 1, 79–100. MR 812171, DOI 10.1016/0021-8693(85)90075-4
- G. R. Krause and T. H. Lenagan, Growth of algebras and Gel′fand-Kirillov dimension, Research Notes in Mathematics, vol. 116, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 781129
- J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
- J. C. McConnell and J. T. Stafford, Gel′fand-Kirillov dimension and associated graded modules, J. Algebra 125 (1989), no. 1, 197–214. MR 1012671, DOI 10.1016/0021-8693(89)90301-3
- C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 676974
- Richard Resco and L. W. Small, Affine Noetherian algebras and extensions of the base field, Bull. London Math. Soc. 25 (1993), no. 6, 549–552. MR 1245080, DOI 10.1112/blms/25.6.549
- Martha K. Smith, Growth of twisted Laurent extensions, Duke Math. J. 49 (1982), no. 1, 79–85. MR 650370
- Martha K. Smith, Universal enveloping algebras with subexponential but not polynomially bounded growth, Proc. Amer. Math. Soc. 60 (1976), 22–24 (1977). MR 419534, DOI 10.1090/S0002-9939-1976-0419534-5
- S. P. Smith and J. J. Zhang, Some non-noetherian regular graded rings, in preparation.
- J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian $\textrm {PI}$ rings, J. Algebra 168 (1994), no. 3, 988–1026. MR 1293638, DOI 10.1006/jabr.1994.1267
- Richard P. Stanley, Generating functions, Studies in combinatorics, MAA Stud. Math., vol. 17, Math. Assoc. America, Washington, D.C., 1978, pp. 100–141. MR 513004
- D. R. Stephenson, Artin-Schelter regular algebras of global dimension three, Ph.D. thesis, University of Michigan, UMI, 1994 .
- C. T. C. Wall, Generators and relations for the Steenrod algebra, Ann. of Math. (2) 72 (1960), 429–444. MR 116326, DOI 10.2307/1970225
- J. J. Zhang, Connected graded Gorenstein algebras with enough normal elements, J. Algebra, to appear.
Bibliographic Information
- Darin R. Stephenson
- Affiliation: Department of Mathematics-0112, University of California at San Diego, La Jolla, California 92093-0112
- Email: dstephen@math.ucsd.edu
- James J. Zhang
- Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
- MR Author ID: 314509
- Email: zhang@math.washington.edu
- Received by editor(s): December 5, 1995
- Additional Notes: The second author was supported by the NSF
- Communicated by: Lance W. Small
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1593-1605
- MSC (1991): Primary 16P90, 16W50, 16E10
- DOI: https://doi.org/10.1090/S0002-9939-97-03752-0
- MathSciNet review: 1371143