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The finiteness dimension of local cohomology modules and its dual notion. (English) Zbl 1156.13007

Let \(I\) be an ideal of a noetherian ring \(R\) and let \(M\) be a finitely generated \(R\)-module. In this paper the following invariants of \(M\) and \(I\) are studied: \[ f_I(M):=\inf \{ i| H^i_I(M)\text{ is not noetherian}\} \]
\[ q_I(M):=\sup \{ i| H^i_I(M)\text{ is not Artinian}\} \] It is known that \(f_I(M)\) is the smallest \(i\) such that no power of \(I\) annihilates \(H^i_I(M)\), i. e. \(f_I(M)=f^I_I(M)\) where \[ f^J_I(M):=\inf\{ i| J\not\subseteq \sqrt{\text{Ann}_R(H^i_I(M))}\} \] Here are the some major results from this paper:
If \((R,m)\) is local and \(f_I(M)<f^I_m(M)\) then \(H^{f_I(M)}_I(M)\) is not Artinian (the latter does not hold in general); in particular, \(f_I(M)\leq q_I(M)\).
If \(q_I(M)>0\) then \(H^{q_I(M)}_I(M)\) is not finitely generated; in particular, \(f_I(M)\leq q_I(M)\). \(H^{q_I(M)}_I(M)\otimes_R (R/I)\) is Artinian; this is in some sense dual to the previously known fact that \(\operatorname{Hom}_R(R/I,H^{f_I(M)}_I(M))\) is finitely generated. Nevertheless, in general, the Matlis dual of \(H^{q_I(M)}_I(M)\) has infinitely many associated prime ideals, (though \(\text{Ass} _R(H^{f_I(M)}_I(M))\) is known to be finite).

MSC:

13D45 Local cohomology and commutative rings
13E05 Commutative Noetherian rings and modules
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
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References:

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