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

Brodmann, M.

Arch. Math. 79, No. 2, 87-92 (2002).

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 \(H^d_I(M)\).

Reviewer: Irena Swanson (Las Cruces)

MSC:

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