Hilbert-Kirby Polynomials in Generalized Local Cohomology Modules

Let \(R = \oplus _{n\in \mathbb {N}_{0}}R_{n}\) be a Noetherian homogeneous ring with irrelevant ideal \(R_{+} = \oplus _{n\in \mathbb {N}} R_{n}\) and with local base ring \((R_{0},\mathfrak {m}_{0})\). Let M, N be two finitely generated \(\mathbb {Z}\)-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 \(H^{i}_{R_{+}}(M, N)\) are anti-polynomial. Under some mild assumptions, the Artinianness of \(H^{i}_{R_{+}}(M, N)\) and the asymptotic behavior of the R₀-modules \(H^{i}_{R_{+}}(M, N)_{n}\) for \(n\rightarrow -\infty \) in the range \(i\leq \inf \{i\in \mathbb {N}_{0} \vert \sharp \{n\vert \ell _{R_{0}}\) \((H^{i}_{ R_{+}}(M , N)_{n}) = \infty \}=\infty \}\) will be studied. Moreover, it has been proved that, if u is the least integer i for which \(H^{i}_{R_{+}}(M,N)\) is not Artinian and \(\mathfrak {q}_{0}\) is an \(\mathfrak {m}_{0}\)-primary ideal of R₀, then \(H^{u}_{R_{+}}(M,N)/\mathfrak q_{0}H^{u}_{R_{+}}(M,\) N) is Artinian with Hilbert-Kirby polynomial of degree less than u. In particular, with M = R, we deduce the correspondent result for ordinary local cohomology module \(H^{i}_{R_{+}}(N)\).