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Associated primes of local cohomology modules. (English) Zbl 1103.13010
For an ideal \(a\) of a noetherian ring \(R\) and \(R\)-modules \(M,N\) generalized local cohomology is defined as \(H^i_a(M,N)=\varinjlim _n \text{Ext}^i_R(M/a^nM,N)\); for \(M=R\) one gets (usual) local local cohomology. Results about the associated primes of generalized local cohomology are proven, partially extensions of results known for the case of usual local cohomology. It is shown that for any natural number \(t\) one has \[ \text{Ass}_R(H^t_a(M,N))\subseteq \bigcup_{i=0}^t \text{Ass}_R (\text{Ext}^i_R(M,H^{t-i}_a(N))); \] in addition, if both \(d=\text{pd}(M)\) and \(n=\dim(N)\) are finite, the set \(\text{Ass}_R(H^{n+d}_a(M,N))\) is finite.

MSC:
13D45 Local cohomology and commutative rings
13E99 Chain conditions, finiteness conditions in commutative ring theory
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