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Finiteness properties of generalized local cohomology modules for minimax modules. (English) Zbl 1426.13009

Summary: Let \(R\) be a commutative Noetherian ring, \(I\) an ideal of \(R, M\) be a finitely generated \(R\)-module and \(t\) be a non-negative integer. In this paper, we introduce the concept of \(I, M\)-minimax \(R\)-modules. We show that \(\mathrm{Hom}_R(R/I,H_I^t(M,N)/K)\) is \(I,M\)-minimax, for all \(I,M\)-minimax submodules \(K\) of \(H_I^t(M,N)\), whenever \(N\) and \(H_I^0(M)\), \(H_I^1(M),\dots,H_I^{t-1}(M)\) are \(I, M\)-minimax \(R\)-modules. As consequence, it is shown that \(\mathrm{Ass}_RH_I^t(M,N)/K\) is a finite set.

MSC:

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