On the finiteness of Bass numbers of local cohomology modules

Kawasaki, Ken-ichiroh

doi:10.1090/S0002-9939-96-03399-0

On the finiteness of Bass numbers of local cohomology modules
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Proc. Amer. Math. Soc. 124 (1996), 3275-3279 Request permission

Abstract:

In this note we show that, if $I$ is an ideal of dimension 1 of an analytically irreducible local ring, then the Bass numbers of local cohomology modules with support in $V(I)$ are finite.

References

Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
N. Bourbaki, Éléments de mathématique. Fascicule XXVII. Algèbre commutative. Chapitre 1: Modules plats. Chapitre 2: Localisation, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1290, Hermann, Paris, 1961 (French). MR 0217051
Donatella Delfino, On the cofiniteness of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 79–84. MR 1253283, DOI 10.1017/S0305004100071929
Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620, DOI 10.1007/BFb0073971
Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. MR 0476737
Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/70), 145–164. MR 257096, DOI 10.1007/BF01404554
Robin Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403–450. MR 232780, DOI 10.2307/1970720
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Craig Huneke and Jee Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 3, 421–429. MR 1120477, DOI 10.1017/S0305004100070493
Craig L. Huneke and Rodney Y. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), no. 2, 765–779. MR 1124167, DOI 10.1090/S0002-9947-1993-1124167-6
Craig Huneke, Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990) Res. Notes Math., vol. 2, Jones and Bartlett, Boston, MA, 1992, pp. 93–108. MR 1165320
Gennady Lyubeznik, Finiteness properties of local cohomology modules (an application of $D$-modules to commutative algebra), Invent. Math. 113 (1993), no. 1, 41–55. MR 1223223, DOI 10.1007/BF01244301
I. G. MacDonald and Rodney Y. Sharp, An elementary proof of the non-vanishing of certain local cohomology modules, Quart. J. Math. Oxford Ser. (2) 23 (1972), 197–204. MR 299598, DOI 10.1093/qmath/23.2.197
Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360, DOI 10.2140/pjm.1958.8.511
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
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
Robert Speiser, Cohomological dimension and Abelian varieties, Amer. J. Math. 95 (1973), 1–34. MR 325617, DOI 10.2307/2373641
Jan R. Strooker, Homological questions in local algebra, London Mathematical Society Lecture Note Series, vol. 145, Cambridge University Press, Cambridge, 1990. MR 1074178, DOI 10.1017/CBO9780511629242
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition. MR 0389876