graded $p

Let 𝑅 =⊕𝑛𝜖ℕ0 𝑅𝑛 be a standard graded Noetherian ring, i.e. 𝑅0 is a commutative Noetherian ring and 𝑅 is generated (as an 𝑅0 -algebra) by finitely many elements of degree one, 𝑅+ =⊕𝑛𝜖ℕ 𝑅𝑛 be the irrelevant ideal of 𝑅 and 𝒂 stands for a homogeneous ideal of 𝑅. Also, 𝑀 denotes a finitely generated graded 𝑅-module. For 𝑖𝜖ℕ0 and 𝑛𝜖ℤ let 𝐻𝒂 𝑖 (𝑀)𝑛 denotes the n-th component of the i-th graded local cohomology module 𝐻𝒂 𝑖 (𝑀) of 𝑀 with support in 𝒂 (our terminology on local cohomology comes from [3]). It is well-known that 𝐻𝑅+ 𝑖 (𝑀)𝑛 is a finitely generated 𝑅0 -module for all 𝑛𝜖ℤ and it vanishes for all sufficiently large values of n ([3, 15.1.5]). In spite of the case 𝑛 → +∞, the asymptotic behavior of 𝐻𝑅+ 𝑖 (𝑀)𝑛 when 𝑛 → −∞ is so complicated, see for example [4], and it attracts lots of interest, see [6] and [1], for a good survey on this topic. Also, in [5] the authors studied the graded components of 𝐻𝒂 𝑖 (𝑀) in the case where 𝒂 is a homogeneous ideal of 𝑅 containing the irrelevant ideal. In the case where 𝑅0 is an Artinian ring, for all 𝑖𝜖ℕ0 , ℎ𝑀 𝑖 : ℤ → ℤ(𝑛 → 𝑙𝑒𝑛𝑔𝑡ℎ𝑅0 (𝐻𝑅+ 𝑖 (𝑀)𝑛 ), is called the i-th Hilbert cohomological function of 𝑀. In [2], Brodmann et. all. studied the problem of finiteness of the set of Hilbert cohomological functions of some classes of graded modules. In this paper, we also consider this problem. Another reason of this paper, is to study the support of graded local cohomology modules 𝐻𝒂 𝑖 (𝑀) of 𝑀 with support in an arbitrary homogeneous ideal 𝒂 of 𝑅. Let cd𝒂(𝑀) ≔ sup{iϵℤ | 𝐻𝒂 𝑖 (𝑀) ≠ 0}, denotes the cohomological dimension of 𝑀 with respect to 𝒂. In [1, 3.7], it is shown that the set Supp𝑅0 (𝐻𝑅+ cd𝑅+ (𝑀) (𝑀)𝑛) is eventually stable when 𝑛 → −∞, i.e. there exists 𝑋 ⊆ Spec(𝑅0) such that Supp𝑅0 (𝐻𝑅+ 𝑐𝑑𝑅+ (𝑀) (𝑀)𝑛) = 𝑋, for all 𝑛 ≪ 0. In this paper, we study the asymptotic behavior of the set Supp𝑅0 (𝐻𝒂 𝑖 (𝑀)𝑛 ) when 𝑛 → −∞.

First, in a special case we describe 𝐻𝒂 𝑖 (𝑀) in terms of some homologies of the minimal graded free resolution of 𝑀. Then, as a consequence, we find a class of graded modules with a finite set of Hilbert cohomological functions.

We show that, in a special case, the support and dimension of 𝐻𝑅+ 𝑖 (𝑀)𝑡 have upper bound which doesn't depend on i and 𝑀 for sufficiently small values of t. Also, it is shown that, in some cases, the set {Supp𝑅0 (𝐻𝒂 𝑐𝑑𝒂(𝑀) (𝑀)𝑛)} 𝑛𝜖ℤ is eventually increasing, in the sense that Supp𝑅0 (𝐻𝒂 𝑐𝑑𝒂(𝑀) (𝑀)𝑛) ⊆ Supp𝑅0 (𝐻𝒂 𝑐𝑑𝒂(𝑀) (𝑀)𝑛−1) for all 𝑛 ≪ 0.

It is worth to find cases for the existence of a non-zero divisor x on the module M in the ideal a for which cd𝒂 (𝑀/𝑥𝑀) = cd𝒂(𝑀)− 1. It helps us to study the last non-vanishing local cohomology module of M with support in a.

‎In this paper‎, ‎we provide a new difference scheme on a graded mesh for solving the time-space fractional diffusion problem‎. ‎In ‎this ‎equation ‎the ‎time ‎derivative ‎is ‎the ‎Caputo ‎of ‎order ‎‎$‎‎gammain(0,1)$ ‎and ‎the ‎space ‎derivative ‎is ‎the ‎Riesz ‎of ‎order ‎‎$‎‎alphain(1,2]$.‎ ‎The stability and convergence of the difference scheme are discussed which provides the theoretical basis of the proposed‎ ‎schemes‎. W‎e prove that the new difference scheme is unconditionally stable‎. Also, ‎we find that the difference scheme is convergent with order $min{2-gamma,rgamma}$ in time for all $gammain (0,1)$ and $alpha in (1,2]$‎. ‎A test example is given to verify the efficiency and accuracy of the difference scheme‎.