on the lengths and multiplicities of some graded generalized local cohomology modules

Let r = ⊕n∈n0 rn be a noetherian homogeneous ring with a local base ring (r0, m0) of krull dimension 2. let r+ = ⊕n∈nrn be the irrelevant ideal of r. let m and n be two finitely generated z-graded r-modules such that, pdr(m ), the projective dimension of m , is finite. under some conditions on hi r+(m, n ), it will be shown that there exists x ∈ m0 such that `r0 ((0 :hi r+(m,n )n x)) is of reverse polynomial type of degree less than i; i.e., there exists a polynomial p (x) ∈ q[x] of degree less than i such that `r0 ((0 :hi r+(m,n )n x)) = p (n), for all n  0. also, it is proved that, `r0 (γm0rhi r+(m, n/xn )n) and `r0 (h1 m0rhi r+(m, n )n), for a parameter element x ∈ m0, are of reverse polynomial type of degree less than i. finally, for an m0-primary ideal q0, we study the asymptotic behaviour of the multiplicities ed(q0, hi r+ (m, n )n) for n  0, where d = dim̂r (∗d(hi r+(m, n ) ⊗r0̂ r0))