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Local Cohomology of Multi-Rees Algebras with Applications to joint Reductions and Complete Ideals
Shreedevi K. Masuti, Tony J. Puthenpurakal, J. K. Verma

 

Abstract

Let (R,m) be a Cohen-Macaulay local ring of dimension d and I=(I 1,…,I d ) be mprimary ideals in R. We prove that λR([Hd(xiiti:1id)(R(F)]n) <, for all nNd, where F={F(n):nZd} is an I−admissible filtration and (x i j ) is a strict complete reduction of F and R(F) is the extended multi-Rees algebra of F. As a consequence, we prove that the normal joint reduction number of I,J,K is zero in an analytically unramified Cohen-Macaulay local ring of dimension 3 if and only if e¯¯¯3(IJK)[e¯¯¯3(IJ)+e¯¯¯3(IK) +e¯¯¯3(JK)]+e¯¯¯3(I)+e¯¯¯3(J)+e¯¯¯3(K)=0. This generalizes a theorem of Rees on joint reduction number zero in dimension 2. We apply this theorem to generalize a theorem of M. A. Vitulli in dimension 3.

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