ON THE MINIMAXNESS AND ARTINIANNESS DIMENSIONS

Document Type : Original Manuscript

Authors

1 Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran.

2 Department of Mathematics, University of Tabriz, Tabriz, Iran; and School of Mathematics, Institute for research in fundamental sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

Abstract

Let R be a commutative Noetherian ring, I, J be ideals of R such that
J ⊆ I, and M be a finitely generated R-module. In this paper, we prove that the
invariants AJI
(M) := inf{i ∈ N0 | JtHi
I (M) is not Artinian for all t ∈ N0} and inf{i ∈
N0 | JtHi
I (M) is not minimax for all t ∈ N0} are equal. In particular, we show that the
invariants AII
(M) and inf{i ∈ N0 | Hi
I (M) is not minimax} are equal. We also establish
the local-global principle, AJI
(M) = inf{AJRp
IRp
(Mp)|p ∈ Spec (R)}, in some cases.

Keywords


 [1] M. Aghapour and L. Melkersson, Finiteness properties of minimax and coatomic local cohomology modules, Arch. Math., 94 (2010), 519–528.
 
[2] D. Asadollahi and R. Naghipour, A new proof of Faltings’ local-global principle for the finiteness of local cohomology modules, Arch. Math., 103 (2014), 451–459.
 
[3] J. A’zami, R. Naghipour and B. Vakili, Finiteness properties of local cohomology modules for a-minimax modules, Proc. Amer. Math. Soc., 137(2) (2008), 439–448.
 
[4] K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc., 136 (2008), 2359–2363.
 
[5] M. P. Brodmann and R. Y. Sharp, Local cohomology; an algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998.
 
[6] M. R. Doustimehr, Faltings’ local-global principle and annihilator theorem for the finiteness dimensions, Comm. Algebra., 47(5) (2019), 1853–1861.
 
[7] M. R. Doustimehr and R. Naghipour, Faltings’ local-global principle for minimaxness of local cohomology modules, Comm. Algebra, 43 (2015), 400–411.
 
[8] K. Khashyarmanesh and Sh. Salarian, Faltings; theorem for the annihilation of local cohomology modules over a Gorenstein ring, Proc. Amer. Math. Soc., 132 (2004), 2215–2220.
 
[9] H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, UK, 1986.
 
[10] L. Melkersson, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Cambridge Philos. Soc., 107 (1990), 267–271.
 
[11] H. Zo¨schinger, Minimax modules, J. Algebra, 102 (1986), 1–32.
 
[12] H. Zo¨schinger, U¨ber die maximalbedingung fu¨r radikalvolle untermoduln, Hokkaido Math. J., 17 (1988), 101–116.