Finitely graded local cohomology and the depths of graded algebras
Author:
Thomas Marley
Journal:
Proc. Amer. Math. Soc. 123 (1995), 36013607
MSC:
Primary 13A30; Secondary 13C15, 13D45
MathSciNet review:
1283558
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The term "finitely graded" is introduced here to refer to graded modules which are nonzero in only finitely many graded pieces. We consider the question of when the local cohomology modules of a graded module are finitely graded. Using a theorem of Faltings concerning the annihilation of local cohomology, we obtain some partial answers to this question. These results are then used to compare the depths of the Rees algebra and the associated graded ring of an ideal in a local ring.
 [AHT]
Ian
M. Aberbach, Craig
Huneke, and Ngô
Vi\cfudot{e}t Trung, Reduction numbers, BriançonSkoda
theorems and the depth of Rees rings, Compositio Math.
97 (1995), no. 3, 403–434. MR 1353282
(96g:13002)
 [F]
Gerd
Faltings, Über die Annulatoren lokaler
Kohomologiegruppen, Arch. Math. (Basel) 30 (1978),
no. 5, 473–476 (German). MR 0506246
(58 #22058)
 [GH]
Shiro
Goto and Sam
Huckaba, On graded rings associated to analytic deviation one
ideals, Amer. J. Math. 116 (1994), no. 4,
905–919. MR 1287943
(95h:13003), http://dx.doi.org/10.2307/2375005
 [GS]
Shiro
Goto and Yasuhiro
Shimoda, On the Rees algebras of CohenMacaulay local rings,
Commutative algebra (Fairfax, Va., 1979) Lecture Notes in Pure and Appl.
Math., vol. 68, Dekker, New York, 1982, pp. 201–231. MR 655805
(84a:13021)
 [GW]
Shiro
Goto and Keiichi
Watanabe, On graded rings. I, J. Math. Soc. Japan
30 (1978), no. 2, 179–213. MR 494707
(81m:13021), http://dx.doi.org/10.2969/jmsj/03020179
 [Gr]
Robin
Hartshorne, Local cohomology, A seminar given by A.
Grothendieck, Harvard University, Fall, vol. 1961, SpringerVerlag,
BerlinNew York, 1967. MR 0224620
(37 #219)
 [HM1]
Sam
Huckaba and Thomas
Marley, Depth properties of Rees algebras and associated graded
rings, J. Algebra 156 (1993), no. 1,
259–271. MR 1213797
(94d:13006), http://dx.doi.org/10.1006/jabr.1993.1075
 [HM2]
Sam
Huckaba and Thomas
Marley, Depth formulas for certain graded rings associated to an
ideal, Nagoya Math. J. 133 (1994), 57–69. MR 1266362
(95d:13015)
 [JK]
Bernard
Johnston and Daniel
Katz, Castelnuovo regularity and graded
rings associated to an ideal, Proc. Amer. Math.
Soc. 123 (1995), no. 3, 727–734. MR 1231300
(95d:13005), http://dx.doi.org/10.1090/S00029939199512313001
 [M]
Hideyuki
Matsumura, Commutative ring theory, Cambridge Studies in
Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge,
1986. Translated from the Japanese by M. Reid. MR 879273
(88h:13001)
 [SUV]
Aron
Simis, Bernd
Ulrich, and Wolmer
V. Vasconcelos, CohenMacaulay Rees algebras and degrees of
polynomial relations, Math. Ann. 301 (1995),
no. 3, 421–444. MR 1324518
(96a:13005), http://dx.doi.org/10.1007/BF01446637
 [TI]
Ngô
Vi\cfudot{e}t Trung and Shin
Ikeda, When is the Rees algebra CohenMacaulay?, Comm. Algebra
17 (1989), no. 12, 2893–2922. MR 1030601
(91a:13009), http://dx.doi.org/10.1080/00927878908823885
 [V]
G.
Valla, Certain graded algebras are always CohenMacaulay, J.
Algebra 42 (1976), no. 2, 537–548. MR 0422249
(54 #10240)
 [ZS]
Oscar
Zariski and Pierre
Samuel, Commutative algebra. Vol. II, The University Series in
Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.
J.TorontoLondonNew York, 1960. MR 0120249
(22 #11006)
 [AHT]
 I. Aberbach, C. Huneke, and N.V. Trung, Reduction numbers, BriançonSkoda theorems and the depth of Rees rings, preprint. MR 1353282 (96g:13002)
 [F]
 G. Faltings, Über die Annulatoren lokaler Kohomologiegruppen, Arch. Math. 30 (1978), 473476. MR 0506246 (58:22058)
 [GH]
 S. Goto and S. Huckaba, On graded rings associated with analytic deviation one ideals, Amer. J. Math. 116 (1994), 905919. MR 1287943 (95h:13003)
 [GS]
 S. Goto and Y. Shimoda, On the Rees algebra of CohenMacaulay local rings, Commutative Algebra: Analytical Methods, Lecture Notes in Pure and Appl. Math., no. 68, Dekker, New York, 1982. MR 655805 (84a:13021)
 [GW]
 S. Goto and K. Watanabe, On graded rings I, J. Math. Soc. Japan 30 (1978), 179213. MR 494707 (81m:13021)
 [Gr]
 A. Grothendieck (notes by R. Hartshorne), Local cohomology, Lecture Notes in Math., vol. 41, SpringerVerlag, Berlin, 1967. MR 0224620 (37:219)
 [HM1]
 S. Huckaba and T. Marley, Depth properties of Rees algebras and associated graded rings, J. Algebra 156 (1993), 259271. MR 1213797 (94d:13006)
 [HM2]
 , Depth formulas for certain graded rings associated to an ideal, Nagoya Math. J. 133 (1994), 5769. MR 1266362 (95d:13015)
 [JK]
 B. Johnston and D. Katz, Castelnuovo regularity and graded rings associated to an ideal, Proc. Amer. Math. Soc. 123 (1995), 727734. MR 1231300 (95d:13005)
 [M]
 H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, 1980. MR 879273 (88h:13001)
 [SUV]
 A. Simis, B. Ulrich, and W. Vasconcelos, CohenMacaulay Rees algebras and degrees of polynomial relations, Math. Ann. (to appear). MR 1324518 (96a:13005)
 [TI]
 N.V. Trung and S. Ikeda, When is the Rees algebra CohenMacaulay?, Comm. Algebra 17 (1989), 28932922. MR 1030601 (91a:13009)
 [V]
 G. Valla, Certain graded algebras are always CohenMacaulay, J. Algebra 42 (1976), 537548. MR 0422249 (54:10240)
 [ZS]
 O. Zariski and P. Samuel, Commutative algebra, Vol. II, Van Nostrand, Princeton, NJ, 1960. MR 0120249 (22:11006)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
13A30,
13C15,
13D45
Retrieve articles in all journals
with MSC:
13A30,
13C15,
13D45
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512835580
PII:
S 00029939(1995)12835580
Keywords:
Local cohomology,
Rees algebra,
associated graded ring
Article copyright:
© Copyright 1995
American Mathematical Society
