Growth of graded noetherian rings
Darin R. Stephenson and James J. Zhang
Proc. Amer. Math. Soc. 125 (1997), 1593-1605
Primary 16P90, 16W50, 16E10
Full-text PDF Free Access
Similar Articles |
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.
Artin and William
F. Schelter, Graded algebras of global dimension 3, Adv. in
Math. 66 (1987), no. 2, 171–216. MR 917738
Tate, and M.
Van den Bergh, Modules over regular algebras of dimension 3,
Invent. Math. 106 (1991), no. 2, 335–388. MR 1128218
Algebraic number theory, Proceedings of an instructional
conference organized by the London Mathematical Society (a NATO Advanced
Study Institute) with the support of the Inter national Mathematical Union.
Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London,
0215665 (35 #6500)
Dicks, On the cohomology of one-relator associative algebras,
J. Algebra 97 (1985), no. 1, 79–100. MR 812171
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
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
C. McConnell and J.
T. Stafford, Gel′fand-Kirillov dimension and associated
graded modules, J. Algebra 125 (1989), no. 1,
197–214. MR 1012671
Năstăsescu and F.
van Oystaeyen, Graded ring theory, North-Holland Mathematical
Library, vol. 28, North-Holland Publishing Co., Amsterdam, 1982. MR 676974
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
K. Smith, Growth of twisted Laurent extensions, Duke Math. J.
49 (1982), no. 1, 79–85. MR 650370
K. Smith, Universal enveloping algebras with
subexponential but not polynomially bounded growth, Proc. Amer. Math. Soc. 60 (1976), 22–24 (1977). MR 0419534
(54 #7555), http://dx.doi.org/10.1090/S0002-9939-1976-0419534-5
S. P. Smith and J. J. Zhang, Some non-noetherian regular graded rings, in preparation.
T. Stafford and J.
J. Zhang, Homological properties of (graded) Noetherian
𝑃𝐼 rings, J. Algebra 168 (1994),
no. 3, 988–1026. MR 1293638
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.
T. C. Wall, Generators and relations for the Steenrod algebra,
Ann. of Math. (2) 72 (1960), 429–444. MR 0116326
J. J. Zhang, Connected graded Gorenstein algebras with enough normal elements, J. Algebra, to appear.
- M. Artin and W. Schelter, Graded Algebras of Dimension 3, Adv. Math. 66 (1987), 172-216. MR 88k:16003
- M. Artin, J. Tate and M. Van Den Bergh, Modules over Regular Algebras of Dimension 3, Inventions Math., vol. 106, 1991, pp. (335-389). MR 93e:16055
- J. W. S. Cassels and A. Fröhlich, Eds., Algebraic Number Theory, Proceedings of an Instructional Conference, Academic Press, 1967. MR 35:6500
- W. Dicks, On the cohomology of one-relator associative algebras, J. Alg. 97 (1985), 79-100. MR 87h:16041
- G. Krause and T. H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics, Pitman Adv. Publ. Program, vol. 116 (1985). MR 86g:16001
- J. C. McConnell and J. C. Robson, Non-Commutative Noetherian Rings, Wiley-Interscience, Chichester, 1987. MR 89j:16023
- J. C. McConnell and J. T. Stafford, Gelfand-Kirillov dimension and associated graded rings, J. Algebra 125 (1989), 197-214. MR 90i:16002
- C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, North Holland, Amsterdam, 1982. MR 84i:16002
- R. Resco and L. Small, Affine noetherian algebras and extensions of the base field, Bull. London Math. Soc. 25 (1993), 549-552. MR 94m:16029
- M. K. Smith, Growth of twisted Laurent extensions, Duke Math. J. 49 (1982), 79-85. MR 83j:16034
- -, Universal enveloping algebras with sub-exponential but not polynomially bounded growth, Proc. Amer. Math. Soc. 60 (1976), 22-24. MR 54:7555
- 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 PI rings, J. Algebra 168 (1994), 988-1026. MR 95h:16030
- R. P. Stanley, Generating Functions (G.-C. Rota, ed.), Studies in Combinatorics, Vol. 17, MAA Studies in Math., Washington: MAA, Inc., 1978, pp. (100-141). MR 81i:05015
- 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 relation for the Steenrod algebra, Ann. Math. 72 (1960), 429-444. MR 22:7117
- J. J. Zhang, Connected graded Gorenstein algebras with enough normal elements, J. Algebra, to appear.
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
Retrieve articles in all journals
with MSC (1991):
Darin R. Stephenson
Department of Mathematics-0112, University of California at San Diego, La Jolla, California 92093-0112
James J. Zhang
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Received by editor(s):
December 5, 1995
The second author was supported by the NSF
Lance W. Small
© Copyright 1997 American Mathematical Society