## A rigidity result for highest order local cohomology modules.(English)Zbl 1090.13011

The author explicitly describes the set $$\text{Supp}(H^d_I(M))$$, where $$R$$ is a noetherian ring of dimension $$d$$, $$I$$ is an ideal, and $$M$$ is a finitely generated faithful $$R$$-module, as follows: \begin{multline*} \text{Supp}(H^d_I(M)) = \{ P \in \text{Spec}(R) \mid \text{height}(P) = d, I \subseteq P \\ \exists Q \in \text{Spec}(\widehat{R_P}) \text{ such that } \dim(\widehat{R_P}/Q) = d, \dim(\widehat{R_P}/(Q + I\widehat{R_P}) = 0\}. \end{multline*} The author calls the ideal $$I$$ formally isolated at a prime ideal $$P$$ if the condition onn the right side above holds. The author also defines “normally isolated”, analyzes connections between formally isolated and normally isolated, gives other descriptions of $$\text{Supp}(H^d_I(M))$$, and gives some criteria for finite generation of the module $$H^d_I(M)$$.

### MSC:

 13D45 Local cohomology and commutative rings 13E05 Commutative Noetherian rings and modules 14B15 Local cohomology and algebraic geometry
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