Some finiteness properties of generalized graded local cohomology modules

TL;DR

This paper investigates the finiteness properties, specifically Artinianess, of certain generalized graded local cohomology modules over a Noetherian homogeneous ring, extending understanding of their structural behavior.

Contribution

It provides new results on the Artinianess of various generalized graded local cohomology modules, broadening the theoretical framework in local cohomology.

Findings

Proves Artinianess of specific local cohomology modules.

Establishes conditions for finiteness properties of modules.

Extends known results to generalized graded settings.

Abstract

Let $R = \bigoplus_{n \in \mathbb{N}_0} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak{m}_0)$ and let $M$ and $N$ be finitely generated graded $R$-modules. Let $i,j\in\mathbb{N}_0$. In this paper we will study Artinianess of $\Gamma_{\mathfrak m_0R}(H_{R_+}^i(M,N)), H_{\mathfrak m_0R}^1(H_{R_+}^i(M,N)), H_{R_+}^i(M,N)/{\mathfrak m_0}H_{R_+}^i(M,N), H_{R_+}^j(M,H_{\mathfrak m_0R}^i(N)), H_{\mathfrak m_0R}^j(M,H_{R_+}^i(N))$, where $R_+$ denotes the irrelevant ideal of $R$.

Figures

Equations

$$\dots \stackrel{x.}{\to }{H}_{R_{+}}^i(M,N)\to H_{R_{+}}^i(M,N/xN)\to H_{R_{+}}^{i+1}(M,N)(-1)\stackrel{x.}{\to }{H}_{R_{+}}^i(M,N).$$

$${\Gamma }_{{\mathfrak{m}}_{0}R}(H_{R_{+}}^i(M,N/xN))\to {\Gamma }_{{\mathfrak{m}}_{0}R}(0{:}_{H_{R_{+}}^i(M,N)}x)(-1)\to H_{{\mathfrak{m}}_0}(U^i)✠.$$

$$\cdots \to H_{R_{+}}^{i-1}(M_1,N)\to H_{R_{+}}^i(M_2,N)\to H_{R_{+}}^i(M,N)\to H_{R_{+}}^i(M_1,N)\to \dots .$$

$$0\to {\mathrm{Hom}}_R(M_2,{\Gamma }_{R_{+}}(\mathcal{I}))\to {\mathrm{Hom}}_R(M,{\Gamma }_{R_{+}}(\mathcal{I}))\to {\mathrm{Hom}}_R(M_1,{\Gamma }_{R_{+}}(\mathcal{I}))\to 0.$$

$$\cdots \to H^{i-1}({\mathrm{Hom}}_R(M_1,{\Gamma }_{R_{+}}(\mathcal{I})))\to H^i({\mathrm{Hom}}_R(M_2,{\Gamma }_{R_{+}}(\mathcal{I})))$$

$$\to H^i({\mathrm{Hom}}_R(M,{\Gamma }_{R_{+}}(\mathcal{I})))\to H^i({\mathrm{Hom}}_R(M_1,{\Gamma }_{R_{+}}(\mathcal{I})))\to \dots$$

$$H_{R_{+}}^{n+c-1}(K,N)\to H_{R_{+}}^{n+c}(M,N)\to H_{R_{+}}^{n+c}(R^t,N).$$

$$H_{R_{+}}^{n+c-1}(K,N)/{\mathfrak{m}}_0{H}_{R_{+}}^{n+c-1}(K,N)\to H_{R_{+}}^{n+c}(M,N)/{\mathfrak{m}}_0{H}_{R_{+}}^{n+c}(M,N)\to 0.$$

$$H_{R_{+}}^i(M,N)\stackrel{x.}{\to }{H}_{R_{+}}^i(M,N)\to H_{R_{+}}^i(M,N/xN)\to H_{R_{+}}^{i+1}(M,N).$$

$$H_{R_{+}}^{i-1}(M,N/xN)\to H_{R_{+}}^i(M,N)\stackrel{x.}{\to }{H}_{R_{+}}^i(M,N).$$

$$H_{R_{+}}^{i-1}(K,N)\stackrel{\alpha }{\to }{H}_{R_{+}}^i(M,N)\to H_{R_{+}}^i(R^t,N)\stackrel{\beta }{\to }{H}_{R_{+}}^i(K,N).$$

$$E_2^{p,q}:=H_{R_{+}}^p(M,H_{{\mathfrak{m}}_{0}R}^q(N))\underset{p}{⟹}{H}_{\mathfrak{m}}^{p+q}(M,N).$$

$$E_2^{p+1,0}\to H_{\mathfrak{m}}^{p+1}(M,N)\to E_2^{p,1}\to E_2^{p+2,0}\to H_{\mathfrak{m}}^{p+2}(M,N).$$

$$H_{{\mathfrak{m}}_{0}R}^{j-1}(K,H_{R_{+}}^i(N))\to H_{{\mathfrak{m}}_{0}R}^j(M,H_{R_{+}}^i(N))\to H_{{\mathfrak{m}}_{0}R}^j(R^t,H_{R_{+}}^i(N)).$$

Full text

1. Introduction

Throughout this paper, let $R=\bigoplus_{n\in\mathbb{N}_{0}}R_{n}$ be a Noetherian homogeneous ring with local base ring $(R_{0},\mathfrak{m}_{0})$. So $R_{0}$ is a Noetherian ring and there are finitely many elements $l_{1},\dots,l_{r}\in R_{1}$ such that $R=R_{0}[l_{1},\dots,l_{r}]$. Let $R_{+}:=\bigoplus_{n\in\mathbb{N}}R_{n}$ denote the irrelevant ideal of $R$ and let $\mathfrak{m}:=\mathfrak{m}_{0}\oplus R_{+}$ denote the graded maximal ideal of $R$. Finally let $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$ and $N=\bigoplus_{n\in\mathbb{Z}}N_{n}$ be finitely generated graded $R$-modules.

Herzog introduced a generalization of local cohomology so called generalized local cohomology, denoted by $H_{R_{+}}^{i}(M,N)$ which is isomorphic to $\underset{\underset{n}{\rightarrow}}{\mbox{lim}}{\rm{Ext}}_{R}^{i}(M/{{R_{+}}^{n}}M,N)$. We note that if $M=R$, then $H_{R_{+}}^{i}(R,N)=H_{R_{+}}^{i}(N)$ is the usual local cohomology. As is well known, the finiteness of the local cohomology modules have an important role in commutative algebra and algebraic geometry. Many of mathematicians work on finiteness of local cohomology. One of approaches in finiteness is Artinianess. Authors, in [BFT], [BRS], [RS] and [S] studied Artinianess of the graded modules $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(N)$, $H_{R_{+}}^{i}(N)/{\mathfrak{m}_{0}}H_{R_{+}}^{i}(N)$ and $H_{\mathfrak{m}_{0}R}^{1}(H_{R_{+}}^{i}(M))$. In the recent paper we will study the Artinianess of Generalized graded local cohomology. Here, we briefly mention some our results which have been proved in this paper. Let $f$ be the least non-negative integer such that the graded module $H_{R_{+}}^{f}(M,N)$ is not finitely generated. We prove that $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N))$ is Artinian for each $i\leq f$. We also prove that if $M$ is of finite projective dimension with ${\rm pd}_{R}M=n$ and $c=c_{R_{+}}(N)$ is the largest non-negative integer $i$ such that $H_{R_{+}}^{i}(N)\neq 0$, then $H_{R_{+}}^{n+c}(M,N)/{\mathfrak{m}_{0}}H_{R_{+}}^{n+c}(M,N)$ is Artinian. In particular if $c_{R_{+}}(M,N)$ is the largest non-negative integer $i$ such that $H_{R_{+}}^{i}(M,N)\neq 0$, then $c_{R_{+}}(M,N)\leq{\rm pd}_{R}M+c_{R_{+}}(N)$. We prove a similar result for invariant $a_{R_{+}}(N)$ which shows the largest non-negative integer $i$ such that $H_{R_{+}}^{i}(N)$ is not Artinain. Moreover, we show that if $H_{R_{+}}^{i}(M,N)$ is $R_{+}$-cofinite, then $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N))$ is Artinian. Finally we study Artinianess of generalized local cohomology when ${\rm dim}(R_{0})\leq 1$. In this case, we prove that $H_{R_{+}}^{i}(M,N)/{\mathfrak{m}_{0}}H_{R_{+}}^{i}(M,N),\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N)),H_{\mathfrak{m}_{0}R}^{1}(H_{R_{+}}^{i}(M,N)),H_{R_{+}}^{j}(M,H_{\mathfrak{m}_{0}R}^{1}(N))$ and $H_{\mathfrak{m}_{0}R}^{j}(M,H_{R_{+}}^{i}(N))$ are Artinian for all $i,j\in\mathbb{N}_{0}$.

2. The results

It should be noted that local flat morphism of local Noetherian rings is faithfully flat. So, if $R^{\prime}_{0}$ is flat over $R_{0}$ and $\mathfrak{m}_{0}^{\prime}=\mathfrak{m}_{0}R^{\prime}_{0}$, then $R^{\prime}_{0}$ is faithfully flat over $R_{0}$. Moreover, it follows from [K, Theorem 1] that if $(R^{\prime}_{0},\mathfrak{m}_{0}^{\prime})$ is a faithfully flat local $R_{0}$-algebra, then $A$ is a graded Artinian $R$-module if and only if $A^{\prime}:=R^{\prime}_{0}\otimes_{R_{0}}A$ is a graded Artinian module over $R^{\prime}:=R^{\prime}_{0}\otimes_{R_{0}}R$. In view of this argument, we have the following proposition.

2.1 Proposition**.**

Let $f:=f_{R_{+}}(M,N)$ be the least non-negative integer such that $H_{R_{+}}^{f}(M,N)$ is not finitely generated. Then $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N))$ is Artinian for each $i\leq f$.

Proof.

We prove the assertion by induction on $i$. If $i=0$, then $\Gamma_{R_{+}}(M,N)$ is finitely generated; and hence $\Gamma_{\mathfrak{m}_{0}R}(\Gamma_{R_{+}}(M,N))$ is finitely generated. Thus there exists some positive integer $n$ such that ${\mathfrak{m}}^{n}\Gamma_{\mathfrak{m}_{0}R}(\Gamma_{R_{+}}(M,N))=0$ and so the result follows in this case. Now, suppose inductively that the result has been proved for all values smaller than or equal to $i$ with $i<f$ and we prove it for $i+1$, where $i+1\leq f$. Consider the exact sequence $0\rightarrow\Gamma_{R_{+}}(N)\rightarrow N\rightarrow N/\Gamma_{R_{+}}(N)\rightarrow 0.$ Application of the functor $H_{R_{+}}^{i}(M,-)$ to the above exact sequence induces the following exact sequence

$H_{R_{+}}^{i+1}(M,\Gamma_{R_{+}}(N))\rightarrow H_{R_{+}}^{i+1}(M,N)\stackrel{{\scriptstyle\phi}}{{\rightarrow}}H_{R_{+}}^{i+1}(M,N/\Gamma_{R_{+}}(N))\rightarrow H_{R_{+}}^{i+2}(M,\Gamma_{R_{+}}(N))\hskip 14.22636pt{\ddagger}.$ Set $K_{1}={\rm Ker}\phi$ and $K_{2}={\rm Coker}\phi$. We note that for each $i$, there is an isomorphism $H_{R_{+}}^{i}(M,\Gamma_{R_{+}}(N))\cong{\rm Ext}^{i}(M,\Gamma_{R_{+}}(N))$ and then this module is an $R_{+}$-torsion finitely generated graded $R$-module. Therefore $K_{1}$ and $K_{2}$ are $R_{+}$-torsion finitely generated graded $R$-modules. Consider $\Gamma_{\mathfrak{m}_{0}R}(K_{1})$ and $\Gamma_{\mathfrak{m}_{0}R}(K_{2})$. By the previous argument these modules are finitely generated and $R_{+}$-torsion and then there exists some positive integer number $n$ such that ${\mathfrak{m}}^{n}\Gamma_{\mathfrak{m}_{0}R}(K_{1})={\mathfrak{m}}^{n}\Gamma_{\mathfrak{m}_{0}R}(K_{2})=0$ and this implies that $\Gamma_{\mathfrak{m}_{0}}(K_{1})$ and $\Gamma_{\mathfrak{m}_{0}R}(K_{2})$ are Artinian. Now, in view of the sequence ${\ddagger}$, we can conclude that $\Gamma_{\mathfrak{m}_{0}}(H_{R_{+}}^{i+1}(M,N))$ is Artinian if and only if $\Gamma_{\mathfrak{m}_{0}}(H_{R_{+}}^{i+1}(M,N/\Gamma_{R_{+}}(N))$ is Artinian. So we may assume that $\Gamma_{R_{+}}(N)=0$. Let $\mathbf{x}$ be an indeterminate and let $R^{\prime}_{0}:=R_{0}[\mathbf{x}]_{\mathfrak{m}_{0}R_{0}[\mathbf{x}]}$, $\mathfrak{m}_{0}^{\prime}:=\mathfrak{m}_{0}R^{\prime}_{0},R^{\prime}=R^{\prime}_{0}\otimes_{R_{0}}R$, $M^{\prime}:=R^{\prime}_{0}\otimes_{R_{0}}M$, and $N^{\prime}:=R^{\prime}_{0}\otimes_{R_{0}}N$. Then by the flat base change property of local cohomology, for each $i$ we have $R^{\prime}_{0}\otimes_{R_{0}}\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i+1}(M,N))\cong\Gamma_{\mathfrak{m}_{0}R^{\prime}_{0}}(H_{(R^{\prime}_{0}\otimes_{R_{0}}R)_{+}}^{i+1}(R^{\prime}_{0}\otimes_{R_{0}}M,R^{\prime}_{0}\otimes_{R_{0}}N))$ and using the above argument $\Gamma_{\mathfrak{m}_{0}}(H_{R_{+}}^{i+1}(M,N))$ is Artinian if and only if $\Gamma_{\mathfrak{m}_{0}R^{\prime}_{0}}(H_{(R^{\prime}_{0}\otimes_{R_{0}}R)_{+}}^{i+1}(R^{\prime}_{0}\otimes_{R_{0}}M,R^{\prime}_{0}\otimes_{R_{0}}N))$ is Artinian. So we may assume that the residue field $k$ of $R_{0}$ is infinite. As $\Gamma_{R_{+}}(N)=0$, there exists an element $x\in R_{1}$ which is a non-zerodivisor with respect to $N$ and so there is an exact sequence $0\rightarrow N(-1)\stackrel{{\scriptstyle x.}}{{\rightarrow}}N\rightarrow N/xN\rightarrow 0$ of graded $R$-modules. Application of the functor $H_{R_{+}}^{i}(M,-)$ to this exact sequence induces the exact sequence

[TABLE]

If we consider $U^{i}:=H_{R_{+}}^{i}(M,N)/xH_{R_{+}}^{i}(M,N)$, then we have the exact sequence $0\rightarrow U^{i}\rightarrow H_{R_{+}}^{i}(M,N/xN)\rightarrow(0:_{H_{R_{+}}^{i+1}(M,N)}x)(-1)\rightarrow 0.$ Applying the functor $\Gamma_{\mathfrak{m}_{0}R}(-)$ to this sequence, we get the following exact sequence

[TABLE]

We note that $U^{i}$ is finitely generated and $R_{+}$-torsion and so $H_{\mathfrak{m}_{0}R}^{i}(U^{i})\cong H_{\mathfrak{m}}^{i}(U^{i})$ is Artinian. On the other hand, one can easily show that $f(M,N/xN)\geq f(M,N)-1$ and so $i+1\leq f(M,N)$ implies that $i\leq f(M,N)-1\leq f(M,N/xN)$; and hence by induction hypothesis $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N/xN))$ is Artinian. Therefore by the sequence $\maltese$, the module $\Gamma_{\mathfrak{m}_{0}}(0:_{H_{R_{+}}^{i}(M,N)(-1)}x)=(0:_{\Gamma_{\mathfrak{m}_{0}}(H_{R_{+}}^{i}(M,N))(-1)}x)$ is Artinian. Now, since $\Gamma_{\mathfrak{m}_{0}}(H_{R_{+}}^{i}(M,N))$ is $x$-torsion, using Melkersson’s Lemma this module is Artinain. ∎

2.2 Lemma**.**

Let $0\rightarrow M_{1}\rightarrow M\rightarrow M_{2}\rightarrow 0$ be a short exact sequence of finitely generated graded $R$-modules. Then for any $R$-module $N$, there is the following long exact sequence

[TABLE]

Proof.

Let $\mathcal{I}:=0\rightarrow I^{0}\rightarrow I^{1}\rightarrow I^{2}\rightarrow\dots$ be an injective resolution of $N$. We note that for any finitely generated $R$-module $M$ and any $i$, there is an isomorphism $H_{R_{+}}^{i}(M,N)=H^{i}({\rm Hom}_{R}(M,\Gamma_{R_{+}}(\mathcal{I})))$ and by the basic properties of section functor each $\Gamma_{R_{+}}(I^{i})$ in $\Gamma_{R_{+}}(\mathcal{I})$ is injective. Thus there is an exact sequence of complexes

[TABLE]

Now, by using a basic theorem in homology theory, there is the following long exact sequence of $R$-modules

[TABLE]

[TABLE]

and this completes the proof. ∎

2.3 Theorem**.**

*Let $M$ be of finite projective dimension with pd${}_{R}(M)=n$ and $c:=c_{R_{+}}(N)$ be the largest positive integer $i$ such that $H_{R_{+}}^{i}(N)$ is not zero. Then the following condition hold.

(i) The graded module $H_{R_{+}}^{n+c}(M,N)/\mathfrak{m}_{0}H_{R_{+}}^{n+c}(M,N)$ is Artinian.

(ii) If $c_{R_{+}}(M,N)$ is the largest positive integer $i$ such that $H_{R_{+}}^{i}(M,N)$ is not zero, then $c_{R_{+}}(M,N)\leq{\rm pd}_{R}M+c_{R_{+}}(N)=n+c$.*

Proof.

(i) We proceed by induction on pd${}_{R}(M)=n$. If $n=0$, the the result is clear by [RS, Theorem 2.1]. Now, suppose inductively that the result has been proved for all values smaller that $n>0$ and so we prove this for $n$. Since pd${}_{R}(M)=n$, there exists a positive integer $t$ and an exact sequence of graded modules $0\rightarrow K\rightarrow R^{t}\rightarrow M\rightarrow 0$ such that pd${}_{R}(K)=n-1$. In view of Lemma 2.2, if we apply the functor $H_{R_{+}}^{n+c}(-,N)$ to this exact sequence, we have the following exact sequence

[TABLE]

We note that $n+c>c=c_{R_{+}}(N)$ and so $H_{R_{+}}^{n+c}(N)=0$. Now, application of the functor $R_{0}/{\mathfrak{m}_{0}}\otimes_{R}-$ to the above exact sequence induces the following epimorphisms

[TABLE]

By using induction hypothesis, the graded module $H_{R_{+}}^{n+c-1}(K,N)/{\mathfrak{m}_{0}}H_{R_{+}}^{n+c-1}(K,N)$ is Artinian. Thus the result follows by the above epimorphism. (ii) In this part, similar to (i), we can apply an easy induction on pd${}_{R}M=n$. ∎

2.4 Theorem**.**

*Let $M$ be of finite projective dimension and let $a_{R_{+}}(M,N)$ be the largest non-negative integer $i$ such that $H_{R_{+}}^{i}(M,N)$ is not Artinian. Then we have the following conditions.

(i) $a_{R_{+}}(M,N)\leq{\rm pd}_{R}M+a_{R_{+}}(N)$, where $a_{R_{+}}(N)$ is the largest non-negative integer $i$ such that $H_{R_{+}}^{i}(N)$ is not Artinian.

(ii) $H_{R_{+}}^{a+n}(M,N)/\mathfrak{m}_{0}H_{R_{+}}^{a+n}(M,N)$ is Artinian, where $a=a_{R_{+}}(N)$ and pd${}_{R}M=n$.*

Proof.

(i) Let pd${}_{R}M=n$ and $a=a_{R_{+}}(N)$. We prove the assertion by induction on pd${}_{R}M=n$. If $n=0$, the result is clear. Suppose, inductively that $n>0$ and the result has been proved for all values smaller than $n$ and so we prove it for $n$. As pd${}_{R}M=n$, there exists a positive integer $t$ and an exact sequence $0\rightarrow M_{1}\rightarrow R^{t}\rightarrow M\rightarrow 0$ of $R$-modules such that pd${}_{R}M_{1}=n-1$. In view of Lemma 2.2, if we apply the functor $H_{R_{+}}^{i}(-,N)$ to the above exact sequence, we get the following exact sequence of $R$-modules $H_{R_{+}}^{i-1}(M_{1},N)\rightarrow H_{R_{+}}^{i}(M,N)\rightarrow H_{R_{+}}^{i}(N)^{t}.$ Now, consider $i>a+n$. We note that $i-1>a+n-1={\rm pd}_{R}M_{1}+a_{R_{+}}(N)$ and $i>a+n>a$. Thus, using induction hypothesis, $H_{R_{+}}^{i-1}(M_{1},N)$ and $H_{R_{+}}^{i}(N)$ are Artinian. Now, in view of the above exact sequence the result follows. (ii) To prove this part, we again proceed by induction on pd${}_{R}M=n$. If $n=0$, then the result follows by [S, Theorem 2.3]. Now, suppose, inductively that $n>0$ and the result has been proved for all values smaller than $n$ and so we prove it for $n$. By a similar proof which mentioned in (i), there exists an exact sequence of $R$-modules $H_{R_{+}}^{a+n-1}(M_{1},N)\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}H_{R_{+}}^{a+n}(M,N)\stackrel{{\scriptstyle\beta}}{{\rightarrow}}H_{R_{+}}^{a+n}(N)^{t}.$ Consider $X={\rm Im}(\alpha)$ and $Y={\rm Im}(\beta)$. Since $a+n>a$, the module $Y/\mathfrak{m}_{0}Y$ is Artinian. On the other hand, by using induction hypothesis, one can easily see that $X/\mathfrak{m}_{0}X$ is Artinian. Now, the result follows easily.

∎

2.5 Lemma**.**

Let $i\in\mathbb{N}_{0}$ and $H_{R_{+}}^{i}(M,N)$ is $R_{+}$-cofinite. Then $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N))$ is Artinian.

Proof.

Since $H_{R_{+}}^{i}(M,N)$ is $R_{+}$-cofinite, ${\rm Hom}_{R}(R/R_{+},H_{R_{+}}^{i}(M,N))$ is finitely generated and $R_{+}$-torsion. Thus $\Gamma_{\mathfrak{m}_{0}R}({\rm Hom}_{R}(R/R_{+},H_{R_{+}}^{i}(M,N)))\cong\Gamma_{\mathfrak{m}_{0}R}((0:_{H_{R_{+}}^{i}(M,N)}R_{+}))=(0:_{\Gamma_{\mathfrak{m}_{0}}(H_{R_{+}}^{i}(M,N))}R_{+})$ is finitely generated and $\mathfrak{m}$-torsion. It implies that the last term is Artinian. On the other hand, since $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N))$ is $R_{+}$-torsion, it is Artinian. ∎

2.6 Proposition**.**

Let $M$ be of finite projective dimension and let $R_{+}$ be a principal graded ideal of $R$. Then $\Gamma_{\mathfrak{m}_{0}}(H_{R_{+}}^{i}(M,N))$ is Artinian for each $i$.

Proof.

As $R_{+}$ is principal, using [DS, Theorem 2.8], $H_{R_{+}}^{i}(M,N)$ is $R_{+}$-cofinite for each $i$. Now, the assertion follows by the previous lemma. ∎

2.7 Lemma**.**

Let $M$ and $N$ be finitely generated graded $R$-modules and $N$ be $\mathfrak{m}_{0}$-torsion. Then $H_{R_{+}}^{i}(M,N)$ is Artinian for all $i$.

Proof.

Since $N$ is $\mathfrak{m}_{0}$-torsion, there exists an injective resolution $\mathcal{I}:=0\rightarrow I^{0}\rightarrow I^{1}\rightarrow\dots$ such that each $I^{i}$ is $\mathfrak{m}_{0}$-torsion. By the definition of generalized local cohomology there are the following isomorphisms

$H_{R_{+}}^{i}(M,N)\cong H^{i}(\Gamma_{R_{+}}(M,\mathcal{I}))\cong H^{i}(\Gamma_{R_{+}}(M,\Gamma_{\mathfrak{m}_{0}R}(\mathcal{I})))\cong H^{i}({\rm Hom}(M,\Gamma_{\mathfrak{m}}(\mathcal{I}))\cong H_{\mathfrak{m}}^{i}(M,N).$ By the basic properties of generalized local cohomology, the last term is Artinian; and hence the result follows. ∎

2.8 Proposition**.**

Let ${\rm dim}(R_{0})\leq 1$. Then for every $i\in\mathbb{N}_{0}$, the module $H_{R_{+}}^{i}(M,N)/\\ \mathfrak{m}_{0}H_{R_{+}}^{i}(M,N)$ is Artinian

Proof.

If ${\rm dim}(R_{0})=0$, then $N$ is $\mathfrak{m}_{0}$-torsion and so in view of Lemma 2.7, the graded module $H_{R_{+}}^{i}(M,N)$ is Artinian for each $i$. By using [BFT, Lemma 2.2 ] we can get the assertion. Now, suppose that dim$(R_{0})=1$ and consider the short exact sequence $0\rightarrow\Gamma_{\mathfrak{m}_{0}R}(N)\rightarrow N\rightarrow N/\Gamma_{\mathfrak{m}_{0}R}(N)\rightarrow 0$ of graded $R$-modules. Application of the functor $H_{R_{+}}^{i}(M,-)$ to this exact sequence induces the following exact sequence $H_{R_{+}}^{i}(M,\Gamma_{\mathfrak{m}_{0}R}(N))\rightarrow H_{R_{+}}^{i}(M,N)\rightarrow H_{R_{+}}^{i}(M,N/\Gamma_{\mathfrak{m}_{0}R}(N))\rightarrow H_{R_{+}}^{i+1}(M,\Gamma_{\mathfrak{m}_{0}R}(N)).$ In view of Lemma 2.7 and using [BFT, Lemma 2.2], one can easily show that for each $i$, the module $R_{0}/{\mathfrak{m}_{0}}\otimes_{R_{0}}H_{R_{+}}^{i}(M,N)$ is Artinian if and only if $R_{0}/{\mathfrak{m}_{0}}\otimes_{R_{0}}H_{R_{+}}^{i}(M,N/\Gamma_{\mathfrak{m}_{0}}(N))$ is Artinian. So we may assume that $\Gamma_{\mathfrak{m}_{0}R}(N)=0$. Now, this fact implies that there exists an element $x\in\mathfrak{m}_{0}$ which is a non-zerodivisor of $M$ and then there exists a short exact sequence $0\rightarrow N\stackrel{{\scriptstyle x.}}{{\rightarrow}}N\rightarrow N/xN\rightarrow 0$ of graded $R$-modules. Application of the functor $H_{R_{+}}^{i}(M,-)$ to the above exact sequence induces the following exact sequence

[TABLE]

Since $R_{0}$ is of dimension one, $N/xN$ is $\mathfrak{m}_{0}$-torsion and so Lemma 2.7 implies that $H_{R_{+}}^{i}(M,N/xN)$ is Artinian for each $i$. Now, using [BFT, Lemma 2.2], $R_{0}/\mathfrak{m}_{0}\otimes_{R_{0}}H_{R_{+}}^{i}(M,N)/xH_{R_{+}}^{i}(M,N)$ is Artinian. On the other hand, application of the functor $R_{0}/\mathfrak{m}_{0}\otimes_{R_{0}}-$ to the above long exact sequence implies the following isomorphism $R_{0}/\mathfrak{m}_{0}\otimes_{R_{0}}H_{R_{+}}^{i}(M,N)\cong R_{0}/\mathfrak{m}_{0}\otimes_{R_{0}}H_{R_{+}}^{i}(M,N)/xH_{R_{+}}^{i}(M,N)$; and hence the assertion follows. ∎

2.9 Proposition**.**

*Let dim$(R_{0})\leq 1$. Then we have the following conditions.

(i) The graded $R$-module $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N))$ is Artinian for each $i\in\mathbb{N}_{0}$.

(ii) The graded $R$-module $H_{\mathfrak{m}_{0}R}^{1}(H_{R_{+}}^{i}(M,N))$ is Artinian for each $i\in\mathbb{N}_{0}$.*

Proof.

If dim$(R_{0})=0$, then $R_{0}$ is Artinian. In this case any finitely generated $R$-module is Artinian, and then for each $i$, the module $H_{R_{+}}^{i}(M,N)$ is Artinian. Thus $\Gamma_{\mathfrak{m}_{0}R}(H_{\mathfrak{m}_{0}}^{1}(H_{R_{+}}^{i}(M,N)))$ is Artinian and $H_{\mathfrak{m}_{0}}^{1}(H_{R_{+}}^{i}(M,N))=0$ for each $i$ and so (i) and (ii) are clear in this case. Now, assume that dim$(R_{0})=1$. (i). By applying the functor $H_{R_{+}}^{i}(M,-)$ to the exact sequence $0\rightarrow\Gamma_{\mathfrak{m}_{0}R}(N)\rightarrow N\rightarrow N/\Gamma_{\mathfrak{m}_{0}R}(N)\rightarrow 0$, and applying the functor $\Gamma_{\mathfrak{m}_{0}R}(-)$ to the induced exact functor, we can conclude that $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N))$ is Artinian if and only if $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N/\Gamma_{\mathfrak{m}_{0}R}(N)))$ is Artinian and so we may assume that $\Gamma_{\mathfrak{m}_{0}R}(N)=0$. Now, let $x\in\mathfrak{m}_{0}$ be a non-zerodivisor of $N$. Then there is an exact sequence $0\rightarrow N\stackrel{{\scriptstyle x.}}{{\rightarrow}}N\rightarrow N/xN\rightarrow 0$ of graded $R$-modules. Application of the functor $H_{R_{+}}^{i}(M,-)$ to this sequence induces the following exact sequence

[TABLE]

We note that $N/xN$ is $\mathfrak{m}_{0}$-torsion and so by Lemma 2.7, $H_{R_{+}}^{i-1}(M,N/xN)$ is Artinian and then $(0:_{H_{R_{+}}^{i}(M,N)}x)$ is Artinian. This implies that $\Gamma_{\mathfrak{m}_{0}}((0:_{H_{R_{+}}^{i}(M,N)}x)=(0:_{\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N))}x)$ is Artinian. Now, since $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(M,N))$ is $x$-torsion, by using Melkesson’s Lemma, it is Artinian. (ii). We proceed the assertion by induction on $i$. If $i=0$, then $H_{\mathfrak{m}_{0}R}^{1}(\Gamma_{R_{+}}(M,N))=H_{\mathfrak{m}}^{1}(\Gamma_{R_{+}}(M,N)).$ We note that the last term is Artinian because $\Gamma_{R_{+}}(M,N)$ is finitely generated. Suppose, inductively that the result has been proved for all values smaller than $i$ and so we prove it for $i$. Let $y\in\mathfrak{m}_{0}$ be a system of parameter of $\mathfrak{m}_{0}$. As $M$ is finitely generated, for some positive integer $t$ there exists a short exact sequence $0\rightarrow K\rightarrow R^{t}\rightarrow M\rightarrow 0$ of $R$-modules. In view of Lemma 2.2, if we apply the functor $H_{R_{+}}^{i}(-,N)$ to the above exact sequence, we get the following exact sequence

[TABLE]

Consider $A:={\rm Im}(\alpha)$, $B:={\rm Ker}(\beta)$ and $C:={\rm Im}(\beta)$. Application of the functor $H_{yR}^{i}$ to the above exact sequence gives the epimorphism $H_{yR}^{1}(H_{R_{+}}^{i-1}(K,N))\twoheadrightarrow H_{yR}^{1}(A)$, the monomorphism $\Gamma_{yR}(C)\rightarrowtail\Gamma_{yR}(H_{R_{+}}^{i}(K,N))$, and the exact sequence $\Gamma_{yR}(C)\rightarrow H_{yR}^{1}(B)\rightarrow H_{yR}^{1}(H_{R_{+}}^{i}(R^{t},N))\hskip 8.5359pt({\ddagger}).$ By using induction hypotheses $H_{yR}^{1}(H_{R_{+}}^{i-1}(K,N))=H_{\mathfrak{m}_{0}R}^{1}(H_{R_{+}}^{i-1}(K,N))$ is Artinian and so is $H_{yR}^{1}(A)$. On the other hand since, by (i), the module $\Gamma_{yR}(H_{R_{+}}^{i}(K,N))$ is Artinain, the module $\Gamma_{yR}(C)$ is Artinian. Using [BFT, Theorem 2.5], the module $H_{yR}^{1}(H_{R_{+}}^{i}(R^{t},N))$ is Artinian and then the exact sequence ${\ddagger}$ and the previous arguments imply that $H_{yR}^{1}(B)$ is Artinian. Now, since both $H_{yR}^{1}(A)$ and $H_{yR}^{1}(B)$ are Artinian, one can easily deduce that $H_{yR}^{1}(H_{R_{+}}^{i}(M,N))=H_{\mathfrak{m}_{0}R}^{1}(H_{R_{+}}^{i}(M,N))$ is Artinian. ∎

2.10 Proposition**.**

Let dim$(R_{0})\leq 1$. Then $H_{R_{+}}^{p}(M,H_{\mathfrak{m}_{0}R}^{1}(N))$ is Artinian for each $p\in\mathbb{N}_{0}$.

Proof.

If dim$(R_{0})=0$, then $H_{\mathfrak{m}_{0}R}^{1}(N)=0$ and so the result is clear in this case. Now, assume that dim$(R_{0})=1$. By the Grothendieck spectral sequence (see [R, Theorem 11.38]), for each $p,q\in\mathbb{N}_{0}$, there is

[TABLE]

As dim$(R_{0})=1$, we have $H_{\mathfrak{m}_{0}R}^{q}(N)=0$ for all $q>1$ and then $E_{2}^{p,q}=0$ for all $q\neq 0,1$. Thus we can apply the dual of [W, Ex. 5.2.2] to get the following exact sequence

[TABLE]

It is easy to see that $H_{\mathfrak{m}}^{p+1}(M,N)$ and $E_{2}^{p+2,0}=H_{R_{+}}^{p+2}(M,\Gamma_{\mathfrak{m}_{0}R}(N))=H_{\mathfrak{m}}^{p+2}(M,\Gamma_{\mathfrak{m}_{0}R}(N))$ are Artinian. Thus the above exact sequence implies that $E_{2}^{p,1}=H_{R_{+}}^{p}(M,H_{\mathfrak{m}_{0}R}^{1}(N))$ is Artinian. ∎

2.11 Proposition**.**

Let dim$(R_{0})\leq 1$. Then $H_{\mathfrak{m}_{0}R}^{j}(M,H_{R_{+}}^{i}(N))$ is Artinian for each $j,i\in\mathbb{N}_{0}$.

Proof.

If dim$(R_{0})=0$, then each finitely generated $R$-module is $\mathfrak{m}_{0}$-torsion. Thus $H_{R_{+}}^{i}(N)$ is Artinian and so is $\mathfrak{m}_{0}$-torsion. Then for each $j$, there is an isomorphism $H_{\mathfrak{m}_{0}R}^{j}(M,H_{R_{+}}^{i}(N))\cong{\rm Ext}_{R}^{j}(M,H_{R_{+}}^{i}(N))$. One can easily show that the last module is Artinian. Now, assume that dim$(R_{0})=1$. We proceed by induction on $j$. If $j=0$, then we have $H_{\mathfrak{m}_{0}R}^{0}(M,H_{R_{+}}^{i}(N))={\rm Hom}_{R}(M,\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(N)))$. By using [BFT, Theorem 2.5], the module $\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(N))$ is Artinian and so one can easily show that ${\rm Hom}_{R}(M,\Gamma_{\mathfrak{m}_{0}R}(H_{R_{+}}^{i}(N)))$ is Artinian. Now, we assume that $j>0$ and the result has been proved for all values smaller than $j$ and we prove it for $j$. Since $M$ is finitely generated, for some positive integer $t$, there is an exact sequence $0\rightarrow K\rightarrow R^{t}\rightarrow M\rightarrow 0$ of $R$-module. In view of Lemma 2.2, if we apply the functor $H_{\mathfrak{m}_{0}R}^{j}(-,H_{R_{+}}^{i}(N))$ to the above exact sequence, we have the following exact sequence

[TABLE]

We note that by induction hypotheses, the module $H_{\mathfrak{m}_{0}R}^{j-1}(K,H_{R_{+}}^{i}(N))$ is Artinian and $H_{\mathfrak{m}_{0}R}^{j}(R^{t},H_{R_{+}}^{i}(N))=0$ for all $j>1$ and also $H_{\mathfrak{m}_{0}R}^{1}(R^{t},H_{R_{+}}^{i}(N))$ is Artinian by [BFT, Theorem 2.5]. Now, in view of the above exact sequence, we get our assertion. ∎