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Bass numbers of local cohomology modules. (English) Zbl 0785.13005
Let \(A\) be a regular ring of characteristic \(p>0\), \(b\subset A\) an ideal and \(H^ j_ b(A)\), \(j\geq 0\) the local cohomology modules of \(A\) with respect to \(b\). Then the Bass numbers of \(H^ j_ b(A)\) and \(\text{Ass}_ A(H^ j_ b(A))\) are finite, in fact \(\mu^ i(q,H^ j_ b(A))\leq\mu^ i(q,\text{Ext} c^ j_ A(A/b,A))\) and \(\text{Ass}_ A(H^ j_ b(A)\subset\text{Ass}_ A(\text{Ext}^ j_ A(A/b,A))\) for all \(i,j\geq 0\), \(q\in\text{Spec} A\). Moreover if \((A,m)\) is local the injective dimension of \(H^ j_ b(A)\) is bounded by \(\dim A\) and \(H^ i_ m(H^ j_ b(A))\) is injective for all \(i,j\geq 0\). The proofs use Frobenius in a nice way. Also there exist some small results of this type in characteristic zero but here there are many open questions.

MSC:
13D45 Local cohomology and commutative rings
13H05 Regular local rings
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14B15 Local cohomology and algebraic geometry
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[1] Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8 – 28. · Zbl 0112.26604
[2] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin-New York, 1967. · Zbl 0237.14008
[3] -, Cohomologie locale des faisceaux coherents at théorèmes de Lefschetz locaux et globaux (\( SGA\;2\)), North-Holland, Amsterdam, 1968.
[4] Robin Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403 – 450. · Zbl 0169.23302
[5] Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145 – 164. · Zbl 0196.24301
[6] Robin Hartshorne and Robert Speiser, Local cohomological dimension in characteristic \?, Ann. of Math. (2) 105 (1977), no. 1, 45 – 79. · Zbl 0362.14002
[7] Craig Huneke and Jee Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 3, 421 – 429. · Zbl 0749.13007
[8] C. Huneke and G. Lyubeznik, On the vanishing of local cohomology modules, Invent. Math. 102 (1990), no. 1, 73 – 93. · Zbl 0717.13011
[9] Ernst Kunz, Characterizations of regular local rings for characteristic \?, Amer. J. Math. 91 (1969), 772 – 784. · Zbl 0188.33702
[10] Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511 – 528. · Zbl 0084.26601
[11] Eben Matlis, Modules with descending chain condition, Trans. Amer. Math. Soc. 97 (1960), 495 – 508. · Zbl 0094.25203
[12] 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). · Zbl 0268.13008
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