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Acta Mathematica Vietnamica

Hilbert-Kirby Polynomials in Generalized Local Cohomology Modules

M. Shafiei , icon-email A. Khojali , A. Azari , N. Zamani

Abstract

Let R=nN0Rn be a Noetherian homogeneous ring with irrelevant ideal R+=nNRn and with local base ring (R0,m0). Let M, N be two finitely generated Z-graded R-modules. We show that the lengths of the graded components of various graded submodules and quotients of the i-th generalized local cohomology HR+i(M,N) are anti-polynomial. Under some mild assumptions, the Artinianness of HR+i(M,N) and the asymptotic behaviour of the R0-modules HR+i(M,N)n for n in the range iinf{iN0|{n|R0(HR+i(M,N)n) =}=} will be studied. Moreover, it has been proved that, if u is the least integer i for which HR+i(M,N) is not Artinian and q0 is an m0-primary ideal of R0, then HR+u(M,N)/q0HR+u(M, N) is Artinian with Hilbert-Kirby polynomial of degree less than u. In particular, with M=R, we deduce the correspondence result for ordinary local cohomology module HR+i(N).