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On the vanishing of local cohomology modules. (English) Zbl 0717.13011
Let \(A\) be a commutative ring, \(I\) an ideal of \(A\). We call \[ \mathrm{cd}(A,I)=\min \{k\mid H^ q_ 1(M)=0, \text{ for all }A\text{-modules } M\text{ and all }q>k\} \] the cohomological dimension of \(A\). The subject of the paper is to give bounds for \(\mathrm{cd}(A,I)\). First, it is given a new proof of a theorem by A. Ogus [Ann. Math. (2) 98, 327–365 (1973; Zbl 0308.14003)] in the equicharacteristic zero case and by C. Peskine and L. Szpiro [Publ. Math., Inst. Hautes Étud. Sci. 42(1972), 47–119 (1973; Zbl 0268.13008)] in the equicharacteristic \(p>0\) case. This time the proof has nothing to do with the characteristic. Then, the authors give bounds which improve, with additional assumptions, the general results by G. Faltings [J. Reine Angew. Math. 313, 43–51 (1980; Zbl 0411.13010)]. Namely, if \((A,\mathfrak m)\) is a regular local ring of dimension \(d\) and \(I\) is an ideal of \(A\), let \(b=\max \{\mathrm{ht}(P)\mid P\) minimal over \(I\}\). It is shown that:
(a) If \(I\) is formally geometrically irreducible, then \[ \mathrm{cd}(A,I)\leq d-1-[(d-2)/b] \] and this is the best possible bound for all \(b\) and \(d\) such that \(0<b<d\).
(b) If \(A/I\) is normal, then \[ \mathrm{cd}(A,I)\leq d-[d/(b+1)]-[(d-1)/(b+1)]. \]
The authors also give a bound in the case where there is \(t\in\mathbb Z\) such that \(I_ P\) is generated up to the radical by \(t\) elements for all primes \(P\) with \(\dim(A/P)\geq 3\). Some special results are given in the case of regular local rings, extending further, in this case, the theorems of Ogus and Peskine-Szpiro cited above. Finally, these results are applied to give vanishing results for the relative singular homology groups with complex coefficients of subvarieties of projective space, using the fact that there is a relationship between the vanishing of local cohomology and the vanishing of relative singular homology. Examples are considered throughout the paper.

MSC:
13D05 Homological dimension and commutative rings
13D45 Local cohomology and commutative rings
13H05 Regular local rings
14B15 Local cohomology and algebraic geometry
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