Finiteness dimensions and cofiniteness of generalized local cohomology modules

2018/10/24 by Alireza Vahidi, Vahidi, Alireza, Moharram Aghapournahr +3
Mathematics · #Commutative Algebra and Its Applications #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1810.10223

Abstract

Let R be a commutative Noetherian ring with non-zero identity, \mathfraka and ideal of R, M a finite R--module, and n a non-negative integer. In this paper, for an arbitrary R--module X which is not necessarily finite, we study the finiteness dimension f_\mathfraka(M,X) and the n-th finiteness dimension fⁿ_\mathfraka(M,X) of M and X with respect to \mathfraka. Assume that Extⁱ_R(R/\mathfraka,X) is finite for all i≤ f²_\mathfraka(M,X) (resp. i< f¹_\mathfraka(M,X)). We show that Hⁱ_\mathfraka(M,X) is \mathfraka--cofinite for all i< f²_\mathfraka(M,X) (resp. i< f¹_\mathfraka(M,X)) and Ass_R(H^f²_\mathfraka(M,X)_\mathfraka(M,X)) (resp. if Ext^f¹_\mathfraka(M,X)_R(R/\mathfraka,X) is finite, then Ass_R(H^f¹_\mathfraka(M,X)_\mathfraka(M,X))) is finite.

Related

• The finiteness dimension of local cohomology modules and its dual notion
• Faltings' local-global principle for the finiteness of local cohomology modules over Noetherian rings
• Upper bounds for finiteness of generalized local cohomology modules
• Cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules
• Cofiniteness of generalized local cohomology modules for ideals of small dimension