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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.

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
Full Text: DOI
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