2017/07/19 by Atazadeh, Ali, Sedghi, Monireh, Naghipour, Reza
#13D45 #13E05 #14B15 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1707.06536
Let (R, \frak m) be a local ring and M a finitely generated R-module. It is shown that if M is relative Cohen-Macaulay with respect to an ideal \frak a of R, then AnnR(H_\mathfraka^cd(\mathfraka, M)(M))=AnnRM/L=AnnRM and AssR(R/AnnRM)⊆ \\mathfrakp ∈ AssR M| \rm cd(\mathfraka, R/\mathfrakp)=cd(\mathfraka, M)\, where L is the largest submodule of M such that \rm cd(\mathfraka, L)< \rm cd(\mathfraka, M). We also show that if Hdim M_\mathfraka(M)=0, then AttR(Hdim M-1_\mathfraka(M))= \\mathfrakp ∈ Supp (M)| \rm cd(\mathfraka, R/\mathfrakp)=dim M-1\, and so the attached primes of Hdim M-1_\mathfraka(M) depends only on Supp (M). Finally, we prove that if M is an arbitrary module (not necessarily finitely generated) over a Noetherian ring R with \rm cd(\mathfraka, M)=\rm cd(\mathfraka, R/AnnRM), then AttR(H^\rm cd(\mathfraka, M)_\mathfraka(M))⊆\\mathfrakp ∈ V(AnnRM)| \rm cd(\mathfraka, R/\mathfrakp)=\rm cd(\mathfraka, M)\. As a consequence of this it is shown that if dim M=dim R, then AttR(Hdim M_\mathfraka(M))⊆\\mathfrakp ∈ AssR M| \rm cd(\mathfraka, R/\mathfrakp)=dim M\.