Growth of graded noetherian rings
Authors:
Darin R. Stephenson and James J. Zhang
Journal:
Proc. Amer. Math. Soc. 125 (1997), 15931605
MSC (1991):
Primary 16P90, 16W50, 16E10
MathSciNet review:
1371143
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We show that every graded locally finite right noetherian algebra has subexponential 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 GKdimension. Using this, we can classify all connected graded noetherian algebras of global dimension two.
 [AS]
Michael
Artin and William
F. Schelter, Graded algebras of global dimension 3, Adv. in
Math. 66 (1987), no. 2, 171–216. MR 917738
(88k:16003), http://dx.doi.org/10.1016/00018708(87)90034X
 [ATV]
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
(93e:16055), http://dx.doi.org/10.1007/BF01243916
 [CF]
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,
1967. MR
0215665 (35 #6500)
 [Di]
Warren
Dicks, On the cohomology of onerelator associative algebras,
J. Algebra 97 (1985), no. 1, 79–100. MR 812171
(87h:16041), http://dx.doi.org/10.1016/00218693(85)900754
 [KL]
G.
R. Krause and T.
H. Lenagan, Growth of algebras and Gel′fandKirillov
dimension, Research Notes in Mathematics, vol. 116, Pitman
(Advanced Publishing Program), Boston, MA, 1985. MR 781129
(86g:16001)
 [MR]
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 WileyInterscience Publication. MR 934572
(89j:16023)
 [MS]
J.
C. McConnell and J.
T. Stafford, Gel′fandKirillov dimension and associated
graded modules, J. Algebra 125 (1989), no. 1,
197–214. MR 1012671
(90i:16002), http://dx.doi.org/10.1016/00218693(89)903013
 [NV]
C.
Năstăsescu and F.
van Oystaeyen, Graded ring theory, NorthHolland Mathematical
Library, vol. 28, NorthHolland Publishing Co., Amsterdam, 1982. MR 676974
(84i:16002)
 [RS]
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
(94m:16029), http://dx.doi.org/10.1112/blms/25.6.549
 [S1]
Martha
K. Smith, Growth of twisted Laurent extensions, Duke Math. J.
49 (1982), no. 1, 79–85. MR 650370
(83j:16034)
 [S2]
Martha
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/S00029939197604195345
 [SmZ]
S. P. Smith and J. J. Zhang, Some nonnoetherian regular graded rings, in preparation.
 [SZ]
J.
T. Stafford and J.
J. Zhang, Homological properties of (graded) Noetherian
𝑃𝐼 rings, J. Algebra 168 (1994),
no. 3, 988–1026. MR 1293638
(95h:16030), http://dx.doi.org/10.1006/jabr.1994.1267
 [St]
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
(81i:05015)
 [Ste]
D. R. Stephenson, ArtinSchelter regular algebras of global dimension three, Ph.D. thesis, University of Michigan, UMI, 1994.
 [W]
C.
T. C. Wall, Generators and relations for the Steenrod algebra,
Ann. of Math. (2) 72 (1960), 429–444. MR 0116326
(22 #7117)
 [Zh]
J. J. Zhang, Connected graded Gorenstein algebras with enough normal elements, J. Algebra, to appear.
 [AS]
 M. Artin and W. Schelter, Graded Algebras of Dimension 3, Adv. Math. 66 (1987), 172216. MR 88k:16003
 [ATV]
 M. Artin, J. Tate and M. Van Den Bergh, Modules over Regular Algebras of Dimension 3, Inventions Math., vol. 106, 1991, pp. (335389). MR 93e:16055
 [CF]
 J. W. S. Cassels and A. Fröhlich, Eds., Algebraic Number Theory, Proceedings of an Instructional Conference, Academic Press, 1967. MR 35:6500
 [Di]
 W. Dicks, On the cohomology of onerelator associative algebras, J. Alg. 97 (1985), 79100. MR 87h:16041
 [KL]
 G. Krause and T. H. Lenagan, Growth of algebras and GelfandKirillov dimension, Research Notes in Mathematics, Pitman Adv. Publ. Program, vol. 116 (1985). MR 86g:16001
 [MR]
 J. C. McConnell and J. C. Robson, NonCommutative Noetherian Rings, WileyInterscience, Chichester, 1987. MR 89j:16023
 [MS]
 J. C. McConnell and J. T. Stafford, GelfandKirillov dimension and associated graded rings, J. Algebra 125 (1989), 197214. MR 90i:16002
 [NV]
 C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, North Holland, Amsterdam, 1982. MR 84i:16002
 [RS]
 R. Resco and L. Small, Affine noetherian algebras and extensions of the base field, Bull. London Math. Soc. 25 (1993), 549552. MR 94m:16029
 [S1]
 M. K. Smith, Growth of twisted Laurent extensions, Duke Math. J. 49 (1982), 7985. MR 83j:16034
 [S2]
 , Universal enveloping algebras with subexponential but not polynomially bounded growth, Proc. Amer. Math. Soc. 60 (1976), 2224. MR 54:7555
 [SmZ]
 S. P. Smith and J. J. Zhang, Some nonnoetherian regular graded rings, in preparation.
 [SZ]
 J. T. Stafford and J. J. Zhang, Homological properties of (graded) noetherian PI rings, J. Algebra 168 (1994), 9881026. MR 95h:16030
 [St]
 R. P. Stanley, Generating Functions (G.C. Rota, ed.), Studies in Combinatorics, Vol. 17, MAA Studies in Math., Washington: MAA, Inc., 1978, pp. (100141). MR 81i:05015
 [Ste]
 D. R. Stephenson, ArtinSchelter regular algebras of global dimension three, Ph.D. thesis, University of Michigan, UMI, 1994.
 [W]
 C. T. C. Wall, Generators and relation for the Steenrod algebra, Ann. Math. 72 (1960), 429444. MR 22:7117
 [Zh]
 J. J. Zhang, Connected graded Gorenstein algebras with enough normal elements, J. Algebra, to appear.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
16P90,
16W50,
16E10
Retrieve articles in all journals
with MSC (1991):
16P90,
16W50,
16E10
Additional Information
Darin R. Stephenson
Affiliation:
Department of Mathematics0112, University of California at San Diego, La Jolla, California 920930112
Email:
dstephen@math.ucsd.edu
James J. Zhang
Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email:
zhang@math.washington.edu
DOI:
http://dx.doi.org/10.1090/S0002993997037520
PII:
S 00029939(97)037520
Keywords:
Subexponential growth,
GKdimension,
graded ring,
global dimension
Received by editor(s):
December 5, 1995
Additional Notes:
The second author was supported by the NSF
Communicated by:
Lance W. Small
Article copyright:
© Copyright 1997 American Mathematical Society
