Bass numbers of local cohomology modules

Huneke, Craig L.; Sharp, Rodney Y.

doi:10.1090/S0002-9947-1993-1124167-6

Bass numbers of local cohomology modules
HTML articles powered by AMS MathViewer

by Craig L. Huneke and Rodney Y. Sharp

Trans. Amer. Math. Soc. 339 (1993), 765-779

DOI: https://doi.org/10.1090/S0002-9947-1993-1124167-6

PDF | Request permission

Abstract:

Let $A$ be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of $A$ itself, but with respect to an arbitrary ideal of $A$. The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that $A$ is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic $0$ are true.

References

Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
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

Cohomologie locale des faisceaux coherents at théorèmes de Lefschetz locaux et globaux

Robin Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403–450. MR 232780, DOI 10.2307/1970720
Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/70), 145–164. MR 257096, DOI 10.1007/BF01404554
Robin Hartshorne and Robert Speiser, Local cohomological dimension in characteristic $p$, Ann. of Math. (2) 105 (1977), no. 1, 45–79. MR 441962, DOI 10.2307/1971025
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
C. Huneke and G. Lyubeznik, On the vanishing of local cohomology modules, Invent. Math. 102 (1990), no. 1, 73–93. MR 1069240, DOI 10.1007/BF01233420
Ernst Kunz, Characterizations of regular local rings of characteristic $p$, Amer. J. Math. 91 (1969), 772–784. MR 252389, DOI 10.2307/2373351
Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360
Eben Matlis, Modules with descending chain condition, Trans. Amer. Math. Soc. 97 (1960), 495–508. MR 169879, DOI 10.1090/S0002-9947-1960-0169879-4
C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119 (French). MR 374130