Multiplicities of graded components of local cohomology modules
Let \(M\) be a finitely generated graded module over a Noetherian homogeneous ring \(R\) with local base ring \((R_0,m_0).\) Let \(H^i_{R_+}(M)_n\) be the \(i\)-th local cohomology module of \(M\) with respect to the irrelevant ideal \(R_+\) of \(R.\) The authors study the multiplicity of \(H^i_{R_+}(M)_n\) relative to an \(m_0\)-primary ideal of \(R_0.\) They prove that it is antipolynomial in \(n\) of degree less than \(i\) when \(i\) is not greater than the cohomological finite length dimension of \(M\) or \(\dim (R_0) = 2.\) A similar property of the lengths of the graded components of various graded submodules of \(H^i_{R_+}(M)\) is also described. Among other things the cases \(\dim (R_0) = 0\) and \(\dim (R_0) = 1\) are analyzed following \textit{M. P. Brodmann} and \textit{R. Y. Sharp} [``Local cohomology. An algebraic introduction with geometric applications.'' Cambridge: Cambridge University Press (1998; Zbl 0903.13006)] and \textit{M. Brodmann, S. Fumasoli} and \textit{R. Tajarod} [Proc. Am. Math. Soc. 131, No. 10, 2977--2985 (2003; Zbl 1041.13012)], respectively.
scientific article; zbMATH DE number 2157528