Width and Local Homology Dimension for Triangulated Categories

Abstract

Let $\mathcal{T}$ be a compactly generated triangulated category. In this paper, the width and local homology dimension of an object X with respect to a homogeneous ideal $\mathfrak{a}$, ${\mathrm{width}}_R(\mathfrak{a},X)$ and $\mathrm{hd}(\mathfrak{a},X)$, respectively, are introduced. The local nature and some basic properties of ${\mathrm{width}}_R(\mathfrak{a},X)$ and $\mathrm{hd}(\mathfrak{a},X)$ are provided. In addition, we give an upper bound and lower bound of ${\mathrm{width}}_R(\mathfrak{a},X)$. What is more, we give the relationship between the local homology dimension $\mathrm{hd}(\mathfrak{a},X)$ and the arithmetic rank of $\mathfrak{a}$ and $\dim R$.

1. Introduction

In [1], the notion of local cohomology functors for categories of modules is introduced, alongside complexes that have garnered significant attention from numerous scholars, as evidenced in [1,2,3]. Over the past six decades, the theory of local cohomology has flourished, emerging as a pivotal tool in the domains of algebraic geometry and commutative algebra. Nevertheless, its counterpart, the theory of local homology, has remained relatively underdeveloped. Initial investigations into local homology functors were conducted by Matlis in [4,5], focusing on ideals generated by regular sequences. Subsequently, Greenlees and May’s work in [6], along with Tarrĺo, López and Lipman’s contributions in [7], highlighted a profound connection between local homology and local cohomology. This linkage becomes particularly apparent (as shown in [7]) when formulated within the derived category $D\left(R\right)$ of the R-module category, specifically, before delving into homology.

The concept of support serves as a cornerstone, offering a geometric framework for delving into diverse algebraic formations. Benson, Iyengar and Krause [8] introduced the concept of support for entities within any compactly generated triangulated category that permits small coproducts. Their methodology is grounded in the establishment of local cohomology functors “$\Gamma$” within triangulated categories, relative to a central ring of operators. Since then, numerous academics have pursued their vision through various endeavors (see [9,10,11,12]). For instance, Asadollahi, Salarian and Sazeedeh [13] allocated to each entity in a category a distinct subset of $\mathrm{Spec}\left(R\right)$, also known as the Support (big support). Their investigation into this support revealed that it adheres to principles such as exactness, orthogonality and separation. Using this support, they analyzed the dynamics of local cohomology functors and demonstrated that these triangulated functors adhere to boundedness. Liu and coauthors [14] introduced the depth concept for objects within such triangulated categories and discovered that when $(R,\mathfrak{m})$ is a graded commutative Noetherian local ring, the depth of any cohomologically bounded and cohomologically finite object does not exceed its dimension.

To classify colocalized subcategories, Benson, Iyengar and Krause [15] developed the local homology functor “$\Lambda$” and a cosupport for triangulated categories, which is based on their work on local cohomology and support. Since then, many scholars have begun to study the cosupport and related theories (see [16,17,18,19]). Motivated by these studies, the aim of this paper is to study the invariants (see Definitions 1 and 2)

$${\mathrm{width}}_R(\mathfrak{a},X)=\inf \left\{{\inf }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right|C\in {\mathcal{T}}^c\}$$

$$\mathrm{hd}(\mathfrak{a},X)=\sup \left\{{\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right|C\in {\mathcal{T}}^c\}$$

for some object X in $\mathcal{T}$. Clearly, ${\mathrm{width}}_R(\mathfrak{a},X)⩽\mathrm{hd}(\mathfrak{a},X)$ for any $X\in {\mathcal{T}}^b$. We give some basic properties and the following boundness of these two invariants for an ideal $\mathfrak{a}$.

Theorem 1. 

Let $\mathfrak{a}$ be a homogeneous ideal of Noetherian graded ring R. For any $X\in {\mathcal{T}}^{+}$, the following hold:

$$\inf \left\{{\inf }_{C}X\right|C\in {\mathcal{T}}^c\}⩽{\mathrm{width}}_R(\mathfrak{a},X)⩽\inf \left\{{\inf }_C{\Gamma }_{\mathcal{V}\left(\mathfrak{a}\right)}X\right|C\in {\mathcal{T}}^c\}.$$

Theorem 2. 

For any non-zero object X of ${\mathcal{T}}^{-}$

$\left(1\right)$ let $\mathfrak{a}$ be a homogeneous ideal of Noetherian graded ring R. Then

$$\mathrm{hd}(\mathfrak{a},X)⩽\mathrm{ara}\left(\mathfrak{a}\right)+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}.$$

$\left(2\right)$ Let $\mathfrak{a}$ be a homogeneous ideal of Noetherian ${\mathbb{N}}_0$-graded ring $R={⨁}_{n\in {\mathbb{N}}_0}{R}_n$. Then

$$\mathrm{hd}(\mathfrak{a},X)⩽\dim R+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}.$$

2. Preliminaries

In this section, we revisit some fundamental concepts and terminology pertinent to the current paper. Our terminology will adhere to the standards set forth in [8,13,15,20].

Compact generation. Let $\mathcal{T}$ be a triangulated category, and $\Sigma$ the suspension on $\mathcal{T}$. An object C in $\mathcal{T}$ is considered $compact$ if it admits set-indexed coproducts and the functor ${\mathrm{Hom}}_{\mathcal{T}}(C,-)$ preserves all coproducts. We write ${\mathcal{T}}^c$ for the full subcategory of compact objects in $\mathcal{T}$. A set $\mathcal{G}$ of objects of $\mathcal{T}$ is called a $generating$ $set$ for $\mathcal{T}$ if for each non-zero object $X\in \mathcal{T}$ there exists an object G in $\mathcal{G}$ such that ${\mathrm{Hom}}_{\mathcal{T}}(G,X)\ne 0$. The category $\mathcal{T}$ is $compactly$ $generated$ if it is generated by a set of compact objects.

For any two objects X and Y in $\mathcal{T}$, ${\mathrm{Hom}}_{\mathcal{T}}^{*}(X,Y)$ is defined as the graded abelian group of morphisms given by ${\mathrm{Hom}}_{\mathcal{T}}^{*}(X,Y)={⨁}_{i\in \mathbb{Z}}{\mathrm{Hom}}_{\mathcal{T}}(X,{\Sigma }^{i}Y)$. Additionally, ${\mathrm{End}}_{\mathcal{T}}^{*}\left(X\right)={\mathrm{Hom}}_{\mathcal{T}}^{*}(X,X)$ forms a graded ring, and ${\mathrm{Hom}}_{\mathcal{T}}^{*}(X,Y)$ serves as a right ${\mathrm{End}}_{\mathcal{T}}^{*}\left(X\right)$ and left ${\mathrm{End}}_{\mathcal{T}}^{*}\left(Y\right)$ bimodule.

Assume that R is a graded commutative Noetherian ring, satisfying the property that for any homogeneous elements $r,s$ in R, the multiplication follows the rule $r·s={(-1)}^{\left|r\right|\left|s\right|}s·r$, where $\left|r\right|$ denote the degree of r. We refer to a triangulated category $\mathcal{T}$ as R-linear if there exists a homomorphism of graded rings $R\to Z^{*}\left(\mathcal{T}\right)$, where $Z^{*}\left(\mathcal{T}\right)$ represents the graded center of $\mathcal{T}$. Consequently, for each object X, there arises a homomorphism of graded rings ${\phi }_X:R\to {\mathrm{End}}_{\mathcal{T}}^{*}\left(X\right)$, such that, for all objects $X,Y\in \mathcal{T}$, the R-module structures on ${\mathrm{Hom}}_{\mathcal{T}}^{*}(X,Y)$ induced by ${\phi }_X$ and ${\phi }_Y$ agree, up to the usual sign rule.

We denote the set of homogeneous prime ideals of R as SpecR. Given a point $\mathfrak{p}$ that belongs to $\mathrm{Spec}R$, we represent the homogeneous localization of R with respect to $\mathfrak{p}$ as $R_{\mathfrak{p}}$; this localization is a graded local ring, as defined by Bruns and Herzog [21] (1.5.13), with maximal ideal $\mathfrak{p}{R}_{\mathfrak{p}}$. We set

$$\mathcal{Z}\left(\mathfrak{p}\right)=\{\mathfrak{q}\in \mathrm{Spec}R|\mathfrak{q}⊈\mathfrak{p}\}.$$

Given a homogeneous ideal $\mathfrak{a}$ in R, we set

$$\mathcal{V}\left(\mathfrak{a}\right)=\{\mathfrak{p}\in \mathrm{Spec}R|\mathfrak{a}\subseteq \mathfrak{p}\}.$$

Let $\mathcal{U}$ be a subset of SpecR. The $specialization$ $closure$ of $\mathcal{U}$ is the set

$$\mathrm{cl}\mathcal{U}=\{\mathfrak{p}\in \mathrm{Spec}R|\mathrm{there}\mathrm{is}\mathfrak{q}\in \mathcal{U}\mathrm{with}\mathfrak{q}\subseteq \mathfrak{p}\}.$$

The subset $\mathcal{U}$ is $specialization$-$closed$ if $\mathrm{cl}\mathcal{U}=\mathcal{U}$. Note that the subsets $\mathcal{V}\left(\mathfrak{a}\right)$ and $\mathcal{Z}\left(\mathfrak{p}\right)$ are specialization-closed.

Throughout this paper, we shall assume that R is a graded commutative Noetherian ring and $\mathcal{T}$ is a compactly generated R-linear triangulated category that possesses set-indexed coproducts.

Koszul object. Let r be a homogeneous element of R, and let X be an object in the category $\mathcal{T}$. We refer to any object that arises in an exact triangle

$$X\stackrel{r}{\to }{\Sigma }^{\mid r\mid }X\to X//r⇝,$$

as a $Koszul$ $object$ of r acting on X, denoted by $X//r$. This notation indicates a well-defined module (non-unique) isomorphism. For a given homogeneous ideal $\mathfrak{a}$ in R, we denote $X//\mathfrak{a}$ as any Koszul object obtained by repeatedly applying the aforementioned construction using a finite sequence of generators for $\mathfrak{a}$. It is worth noting that this object may vary depending on the choice of the minimal generating sequence for $\mathfrak{a}$.

Local cohomology and homology. An exact functor $L:\mathcal{T}\to \mathcal{T}$ is referred to as $localization$ when there exists a morphism $\eta :{\mathrm{Id}}_{\mathcal{T}}\to L$ that satisfies the condition that the composition $L\eta :L\to L^2$ is invertible and also fulfills the equality $L\eta =\eta L$. Given a specialization-closed subset $\mathcal{V}$ of SpecR, $L_{\mathcal{V}}$ is the corresponding localization functor. According to [8] (Definition 3.2), there arises an exact functor ${\Lambda }^{\mathcal{V}}$ on $\mathcal{T}$, which for each object X induces an exact triangle

$${\Gamma }_{\mathcal{V}}X\to X\to L_{\mathcal{V}}X⇝.$$

We refer to ${\Gamma }_{\mathcal{V}}$ as the $local$ $cohomology$ of X supported on $\mathcal{V}$. According to [8] (Corollary 6.5), the functors $L_{\mathcal{V}}$ and ${\Gamma }_{\mathcal{V}}$ on $\mathcal{T}$ preserve coproducts, thereby possessing right adjoints due to Brown representability. The right adjoints of ${\Gamma }_{\mathcal{V}}$ and $L_{\mathcal{V}}$ are denoated as ${\Lambda }^{\mathcal{V}}$ and $V^{\mathcal{V}}$, respectively. They induce a functorial exact triangle

$$V^{\mathcal{V}}X\to X\to {\Lambda }^{\mathcal{V}}X⇝.$$

This triangle leads to the definition of an exact $local$ $homology$ functor ${\Lambda }^{\mathcal{V}}:\mathcal{T}\to \mathcal{T}$.

Cosupport. The concept of the $cosupport$ of an object X in $\mathcal{T}$ is introduced in [15] (Section 4) as follows:

$${\mathrm{cosupp}}_{R}X=\{\mathfrak{p}\in \mathrm{Spec}R|{\Lambda }^{\mathfrak{p}}X=V^{\mathcal{Z}\left(\mathfrak{p}\right)}{\Lambda }^{\mathcal{V}\left(\mathfrak{p}\right)}X\ne 0\}.$$

One has the following statements by [15] (Theorem 4.5) and [15] (Proposition 4.7t):

$\left(1\right)$ ${\mathrm{cosupp}}_{R}X=\varnothing \iff X=0$;

$\left(2\right)$ ${\mathrm{cosupp}}_R{\Lambda }^{\mathcal{V}}X=\mathcal{V}\cap {\mathrm{cosupp}}_{R}X$;

$\left(3\right)$ ${\mathrm{cosupp}}_R{V}^{\mathcal{V}}X=(\mathrm{Spec}R\setminus \mathcal{V})\cap {\mathrm{cosupp}}_{R}X$.

We refer to the exact triangles in the subsequent text as the $\mathrm{Mayer}$–$\mathrm{Vietoris}$ triangles corresponding to $\mathcal{V}$ and $\mathcal{W}$, which is the dual of [8] (Theorem 7.5) and is a useful tool in this article.

Let $\mathcal{V}\subseteq \mathcal{W}$ be specialization-closed subsets within R. Consider the following exact triangles:

$$V^{\mathcal{V}}X\to X\to {\Lambda }^{\mathcal{V}}X⇝\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}\hspace{1em}{V}^{\mathcal{W}}X\to X\to {\Lambda }^{\mathcal{W}}X⇝.$$

Apply the functors ${\Lambda }^{\mathcal{W}}(-)$ and $V^{\mathcal{V}}(-)$ to the above triangles, respectively. By [15] (4.2), one has natural morphisms:

$${\Lambda }^{\mathcal{W}}X\stackrel{{\theta }_{\mathcal{W},\mathcal{V}}}{\to }{\Lambda }^{\mathcal{V}}X\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}\hspace{1em}{V}^{\mathcal{W}}X\stackrel{{\eta }_{\mathcal{W},\mathcal{V}}}{\to }{V}^{\mathcal{V}}X.$$

Lemma 1. 

Given that $\mathcal{V}$ and $\mathcal{W}$ are specialization-closed subsets of $\mathrm{Spec}R$, there exist natural exact triangles for each X in $\mathcal{T}$:

$${\Lambda }^{\mathcal{V}\cup \mathcal{W}}X\stackrel{{({\theta }_{\mathcal{V}\cup \mathcal{W},\mathcal{V}},{\theta }_{\mathcal{V}\cup \mathcal{W},\mathcal{W}})}^{\top }}{\to }{\Lambda }^{\mathcal{V}}X⨿{\Lambda }^{\mathcal{W}}X\stackrel{({\theta }_{\mathcal{V},\mathcal{V}\cap \mathcal{W}},-{\theta }_{\mathcal{W},\mathcal{V}\cap \mathcal{W}})}{\to }{\Lambda }^{\mathcal{V}\cap \mathcal{W}}X⇢,$$

$$V^{\mathcal{V}\cup \mathcal{W}}X\stackrel{{({\eta }_{\mathcal{V}\cup \mathcal{W},\mathcal{V}},{\eta }_{\mathcal{V}\cup \mathcal{W},\mathcal{W}})}^{\top }}{\to }{V}^{\mathcal{V}}X⨿V^{\mathcal{W}}X\stackrel{({\eta }_{\mathcal{V},\mathcal{V}\cap \mathcal{W}},-{\eta }_{\mathcal{W},\mathcal{V}\cap \mathcal{W}})}{\to }{V}^{\mathcal{V}\cap \mathcal{W}}X⇢.$$

Arithmetic rank and dimension. Let $\mathfrak{a}$ be a graded ideal of R. The $arithmetic$ $rank$ of $\mathfrak{a}$, abbreviated as $\mathrm{ara}\left(\mathfrak{a}\right)$, is defined as follows:

$$\mathrm{ara}\left(\mathfrak{a}\right)=\min \{n\in \mathbb{N}:\exists \mathrm{homogeneous}\mathrm{elements}{b}_1,\cdots b_n\in R\mathrm{with}\sqrt{(b_1,\cdots b_n)}=\sqrt{\mathfrak{a}}\}.$$

Note that $\mathrm{ara}\left(0R\right)=0$.

Let X be an object of $\mathcal{T}$. The dimension of X is defined as ${\dim }_{R}X=\sup \left\{{\dim }_R{\mathrm{H}}_C^{*}\left(X\right)\right|C\in {\mathcal{T}}^c\}.$ Obviously, for any object X of $\mathcal{T}$, the equality ${\dim }_{R}X={\dim }_R(\Sigma X)$ holds true.

For any object C belonging to ${\mathcal{T}}^c$ and X in $\mathcal{T}$, we define ${\mathrm{Hom}}_{\mathcal{T}}(C,X)={\mathrm{H}}_C^{*}\left(X\right)$. We denote

$${\inf }_{C}X=\inf {\mathrm{H}}_C^{*}\left(X\right)=\inf \{n\in \mathbb{Z}|{\mathrm{H}}_C^n\left(X\right)\ne 0\},$$

$${\sup }_{C}X=\sup {\mathrm{H}}_C^{*}\left(X\right)=\sup \{n\in \mathbb{Z}|{\mathrm{H}}_C^n\left(X\right)\ne 0\}.$$

An object X in $\mathcal{T}$ is said to be cohomologically bounded above (resp., bounded below) if, for any compact object C, there exists a positive integer $n\left(C\right)$ such that ${\sup }_{C}X⩽n\left(C\right)$ (resp., ${\inf }_{C}X⩾-n\left(C\right)$). X is termed cohomologically bounded if it satisfies both conditions. We denote by ${\mathcal{T}}^{-}$ (resp., ${\mathcal{T}}^{+}$, ${\mathcal{T}}^b$) the full subcategory of $\mathcal{T}$, consisting of all cohomologically bounded above (resp., bounded below) objects.

The following lemma is used very frequently in this article, which recovers part of [13] (Lemma 4.5) and [13] (Lemma 4.8).

Lemma 2. 

Let r be a homogeneous element from the set R, and let C represent a compact object of $\mathcal{T}$. Then, for any $X\in \mathcal{T}$,

$\left(1\right)$ ${\sup }_C(X//r)={\sup }_{C}X-1$, if $\left|r\right|>0$; ${\sup }_C(X//r)⩽{\sup }_{C}X-\left|r\right|$, if $\left|r\right|⩽0$.

$\left(2\right)$ ${\inf }_C(X//r)={\inf }_{C}X-\left|r\right|$, if $\left|r\right|>0$; ${\inf }_C(X//r)⩾{\inf }_{C}X-1$, if $\left|r\right|⩽0$.

Proof. 

By [13] (Lemma 4.5) and [13] (Lemma 4.8), we know that ${\sup }_C(X//r)={\sup }_{C}X-1$ and ${\inf }_C(X//r)={\inf }_{C}X-\left|r\right|$ for any $\left|r\right|>0$. We only need to prove $\left|r\right|⩽0$. To demonstrate the desired inequality $\left|r\right|⩽0$, we shall assume the contrary: $\left|r\right|=-d⩽0$, where $d⩾0$.

$\left(1\right)$ Consider the exact triangle

$$(†)\hspace{1em}\hspace{1em}\hspace{1em}X\stackrel{r}{\to }{\Sigma }^{-d}X\to X//r⇝,$$

This induces the exact sequence

$$\cdots \to {\mathrm{H}}_C^{i+d}X\to {\mathrm{H}}_C^{i+d-d}X\to {\mathrm{H}}_C^{i+d}(X//r)\to {\mathrm{H}}_C^{i+d+1}X\to \cdots .$$

Clearly, ${\sup }_C(X//r)⩽{\sup }_{C}X+d={\sup }_{C}X-\left|r\right|$.

$\left(2\right)$ For any compact object $C\in {\mathcal{T}}^c$, $(†)$ induces the long exact sequence

$$\cdots \to {\mathrm{H}}_C^{i-1-d}X\to {\mathrm{H}}_C^{i-1}(X//r)\to {\mathrm{H}}_C^{i}X\stackrel{r}{\to }{\mathrm{H}}_C^{i-d}X\to \cdots .$$

Clearly, ${\inf }_C(X//r)>{\inf }_{C}X-2$. If $d=0$, then ${\inf }_C(X//r)⩾{\inf }_{C}X-1$. If $d>0$, assume that ${\mathrm{H}}_C^{{\inf }_{C}X-1}(X//r)=0$, then we have ${\mathrm{H}}_C^{{\inf }_{C}X}X=0$ since ${\mathrm{H}}_C^{{\inf }_{C}X-d}X=0$, which is impossible. Therefore, ${\inf }_C(X//r)={\inf }_{C}X-1$. □

3. Width in Triangulated Categories

In this section, we define the width for objects relative to a homogeneous ideal $\mathfrak{a}$ utilizing the local homology functors on triangulated categories. Additionally, we provide a bound and discuss some fundamental properties of it.

Definition 1. 

Let $\mathfrak{a}$ represent a homogeneous ideal in the ring R and let X be an object belonging to the triangulated category $\in {\mathcal{T}}^{+}$. We define the $\mathfrak{a}$-$width$ of X as follows:

$${\mathrm{width}}_R(\mathfrak{a},X)=\inf \left\{{\inf }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right|C\in {\mathcal{T}}^c\}.$$

We set ${\mathrm{width}}_R(\mathfrak{a},X)=\infty$ whenever ${\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X=0$.

Remark 1. 

$\left(1\right)$ If $X\cong Y$, then ${\mathrm{width}}_R(\mathfrak{a},X)={\mathrm{width}}_R(\mathfrak{a},Y);$

$\left(2\right)$ ${\mathrm{width}}_R(\mathfrak{a},X)=\infty$ if $\mathcal{V}\left(\mathfrak{a}\right)\cap {\mathrm{cosupp}}_{R}X=\varnothing$;

$\left(3\right)$ For any $X\in \mathcal{T}$, by Lemma 2, one has that

$${\mathrm{width}}_R(\mathfrak{a},X//r)={\mathrm{width}}_R(\mathfrak{a},X)-\left|r\right|if\left|r\right|>0;$$

$${\mathrm{width}}_R(\mathfrak{a},X//r)⩾{\mathrm{width}}_R(\mathfrak{a},X)-1if\left|r\right|⩽0.$$

For any $X\in \mathcal{T}$, the following theorem gives ${\mathrm{width}}_R(\mathfrak{a},X)$ an upper bound and a lower bound.

Theorem 3. 

For any object $X\in {\mathcal{T}}^{+}$, the following hold:

$$\inf \left\{{\inf }_{C}X\right|C\in {\mathcal{T}}^c\}⩽{\mathrm{width}}_R(\mathfrak{a},X)⩽\inf \left\{{\inf }_C{\Gamma }_{\mathcal{V}\left(\mathfrak{a}\right)}X\right|C\in {\mathcal{T}}^c\}.$$

Proof. 

The second inequality is simpler; we just have to prove ${\inf }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X⩽{\inf }_C{\Gamma }_{\mathcal{V}\left(\mathfrak{a}\right)}X$ for any compact object C. We validate this assertion for the specific case of $\mathfrak{a}=\left(r\right)$, and by a straightforward inductive argument, we derive the generalized result. If $\left|r\right|⩽0$, then ${\inf }_C\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)⩽{\inf }_C({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X//r)+1={\inf }_C(X//r)+1$ by Lemma 2 and [15] (Corollary 4.8). Therefore, according to [19] (Proposition 2.2(1)), it follows that ${\inf }_C\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)⩽{\inf }_C\left({\Gamma }_{\mathcal{V}\left(\right(r\left)\right)}X\right)$. If $\left|r\right|>0$, then ${\inf }_C\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)={\inf }_C({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X//r)+\left|r\right|={\inf }_C(X//r)+\left|r\right|$ by Lemma 2 and [15] (Corollary 4.8). Hence [19] (Proposition 2.2(2)) implies that ${\inf }_C\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)={\inf }_C\left({\Gamma }_{\mathcal{V}\left(\right(r\left)\right)}X\right)$.

For the first inequality, if $X//\mathfrak{a}=0$, then ${\mathrm{cosupp}}_R(X//\mathfrak{a})=\varnothing$ by [15] (Theorem 4.5), so we have $\mathcal{V}\left(\mathfrak{a}\right)\cap {\mathrm{cosupp}}_{R}X=\varnothing$ by [15] (Lemma 4.12). The conclusion follows from Remark 1(2).

Assume $X//\mathfrak{a}\ne 0$; we can employ induction on the number of generators of $\mathfrak{a}$ for any compact object C in $\mathcal{T}$ to demonstrate that ${\inf }_C\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)⩾{\inf }_{C}X$. To initiate this, let us assume $\mathfrak{a}=\left(r\right)$. The triangle $X\stackrel{r}{⟶}{\Sigma }^{\left|r\right|}X⟶X//r⇝$ induces the triangle

$$\left(\S \right):\hspace{1em}\hspace{1em}\hspace{1em}{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\stackrel{r}{⟶}{\Sigma }^{\left|r\right|}{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X⟶{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X//r⇝.$$

If $\left|r\right|>0$, then in light of Lemma 2, we can establish the following equivalences:

$${\inf }_{C}X-\left|r\right|={\inf }_{C}X//r={\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X//r={\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X-\left|r\right|.$$

Here, the second equality follows from ${\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X//r\cong X//r$ by [15] (Lemma 4.12) and [15] (Corollary 4.8). Therefore, ${\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X={\inf }_{C}X$.

If $d=0$, the (§) induces the following exact sequence of $R_0$-modules:

$$\to {\mathrm{H}}_C^{i-1}(X//r)\to {\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\stackrel{r}{\to }{\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\to {\mathrm{H}}_C^i(X//r)\to \cdots .$$

Obviously, ${\mathrm{H}}_C^i(X//r)=0$ for all $i<{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X-1$. If ${\mathrm{H}}_C^{{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X-1}(X//r)=0$, we find that the morphism ${\mathrm{H}}_C^{{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\stackrel{r}{⟶}{\mathrm{H}}_C^{{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)$ is injective. Utilizing [15] (Lemma 6.1), we infer that any element of ${\mathrm{H}}_C^{{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)$ is annihilated by a power of r, leading to the conclusion that ${\mathrm{H}}_C^{{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)=0$, which contradicts our previous assumption. Therefore, ${\inf }_{C}X//r={\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X-1$. Finally, considering Lemma 2, we deduce that ${\inf }_{C}X⩽{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X$.

In the case where $\left|r\right|=-d<0$ with $d>0$, the exact triangle $\left(\S \right)$ yields the following exact sequence of $R_0$-modules:

$$\to {\mathrm{H}}_C^{i-1}(X//r)\to {\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\stackrel{r}{\to }{\mathrm{H}}_C^{i-d}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\to {\mathrm{H}}_C^i(X//r)\to \cdots .$$

Clearly, ${\mathrm{H}}_C^i(X//r)=0$ for all $i<{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X-1$. If ${\mathrm{H}}_C^{{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X-1}(X//r)=0$, we get that ${\mathrm{H}}_C^{{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)=0$ since ${\mathrm{H}}_C^{{\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X-d}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)=0$, which is a contradiction. Thus ${\inf }_{C}X//r={\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X-1$. Lastly, in view of Lemma 2, we obtain

$${\inf }_{C}X-1⩽{\inf }_{C}X//r={\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X//r={\inf }_C{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X-1.$$

Therefore, the desired conclusion holds in this case. Now, let us assume inductively that the result has been established for all values less than n. Given $\mathfrak{a}=(r_1,\dots ,r_n)$. let us define $\mathfrak{b}=(r_2,\dots ,r_n)$. Based on the inductive step, we obtain the following inequalities:

$${\inf }_{C}X⩽{\inf }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{b}\right)}X⩽{\inf }_C{\Lambda }^{\mathcal{V}\left(r_1\right)}\left({\Lambda }^{\mathcal{V}\left(\mathfrak{b}\right)}X\right)={\inf }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X.$$

This complete the proof. □

The subsequent proposition encapsulates the manner in which width behaves in exact triangles.

Proposition 1. 

Given a homogeneous ideal $\mathfrak{a}$ of R, for any exact triangle $X\to Y\to Z⇝$ in $\in \mathcal{T}$, the following hold:

$\left(1\right)$ ${\mathrm{width}}_R(\mathfrak{a},Y)⩾\min \{{\mathrm{width}}_R(\mathfrak{a},X),{\mathrm{width}}_R(\mathfrak{a},Z)\};$

$\left(2\right)$  ${\mathrm{width}}_R(\mathfrak{a},X)⩾\min \{{\mathrm{width}}_R(\mathfrak{a},Y),{\mathrm{width}}_R(\mathfrak{a},Z)+1\};$

$\left(3\right)$  ${\mathrm{width}}_R(\mathfrak{a},Z)⩾\min \{{\mathrm{width}}_R(\mathfrak{a},X)-1,{\mathrm{width}}_R(\mathfrak{a},Y)\}.$

Proof. 

The triangle $X\to Y\to Z⇝$ yields the induced triangle

$${\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\to {\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}Y\to {\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}Z⇝.$$

For any compact object $C\in {\mathcal{T}}^c$, by applying the functor ${\mathrm{H}}_C^{*}(-)$ to the above triangle, we obtain the long exact sequence:

$$\cdots \to {\mathrm{H}}_C^{i-1}\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}Z\right)\to {\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)\to {\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}Y\right)\to {\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}Z\right)\to \cdots .$$

It is noteworthy that ${\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}Y\right)=0$ if ${\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)$ and ${\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}Z\right)$ both vanish. Consequently, the first inequality arises from the definition of the width of an object in triangulated categories. The same reasoning applies to the other inequalities. □

Proposition 2. 

Given a homogeneous ideal $\mathfrak{a}$ of R and an object X of $\in \mathcal{T}$, the following holds:

$${\mathrm{width}}_R(\mathfrak{a},{\Sigma }^{n}X)={\mathrm{width}}_R(\mathfrak{a},X)+n.$$

Proof. 

According to the definition of width, we can establish the following series of equalities:

$$\begin{array}{cc}\hfill {\mathrm{width}}_R(\mathfrak{a},{\Sigma }^{n}X)& =\inf \left\{{\inf }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}\left({\Sigma }^{n}X\right)\right|C\in {\mathcal{T}}^c\}\hfill \\ \hfill & =\inf \left\{\inf {\mathrm{H}}_C^{*}\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}\left({\Sigma }^{n}X\right)\right)\right|C\in {\mathcal{T}}^c\}\hfill \\ \hfill & =\inf \{\inf {\mathrm{H}}_C^{*}\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)+n|C\in {\mathcal{T}}^c\}\hfill \\ \hfill & ={\mathrm{width}}_R(\mathfrak{a},X)+n.\hfill \end{array}$$

This complete the proof. □

4. Local Homology Dimension at a Homogeneous Ideal

Given a homogeneous ideal $\mathfrak{a}$ of R and an object X in $\mathcal{T}$, this section introduces an invariant denoted as $\mathrm{hd}(\mathfrak{a},X)$, which is defined as $\mathrm{hd}(\mathfrak{a},X)=\sup \left\{{\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right|C\in {\mathcal{T}}^c\}$. The main results of this section, Theorems 4 and 5, establish the relationship between the local homology dimension $\mathrm{hd}(\mathfrak{a},X)$ and the arithmetic rank of $\mathfrak{a}$ and $\dim R$.

Definition 2. 

Given a homogeneous ideal $\mathfrak{a}$ of R and an object X in ${\mathcal{T}}^{-}$, we define the homological dimension of X with respect to $\mathfrak{a}$, denoted as $\mathrm{hd}(\mathfrak{a},X)$, as 

$$\mathrm{hd}(\mathfrak{a},X)=\sup \left\{{\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right|C\in {\mathcal{T}}^c\}.$$

We set $\mathrm{hd}(\mathfrak{a},X)=-\infty$ whenever ${\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X=0$.

Remark 2. 

Let X be an object in ${\mathcal{T}}^b$. By the definitions of width and homological dimension of X, we have ${\mathrm{width}}_R(\mathfrak{a},X)⩽\mathrm{hd}(\mathfrak{a},X).$

Lemma 3 

([19] (Proposition 2.5)). Let $r\in R$ be a homogeneous element and X an object in $\mathcal{T}$.

$\left(1\right)$ If $\left|r\right|⩽0$, then ${\sup }_C{\Lambda }^{\mathcal{V}\left(r\right)}X={\sup }_C(X//r)+\left|r\right|$ for any $C\in {\mathcal{T}}^c$;

$\left(2\right)$ If $\left|r\right|>0$, then ${\sup }_C{\Lambda }^{\mathcal{V}\left(r\right)}X={\sup }_C(X//r)+1$ for any $C\in {\mathcal{T}}^c$.

The above lemma gives the formula for computing ${\sup }_C{\Lambda }^{\mathcal{V}\left(r\right)}X$, which recovers part of [22] (Theorem 4.1). It is very useful for proving the following proposition.

Proposition 3. 

Let $r_1,r_2$ be homogeneous elements in R. For any object X in $\mathcal{T}$, we observe that

$${\sup }_C{\Lambda }^{\mathcal{V}(r_1,r_2)}X⩽{\sup }_C{\Lambda }^{\mathcal{V}\left(r_1\right)}X\mathrm{for}\mathrm{any}C\in {\mathcal{T}}^c.$$

More specifically, for any homogeneous ideals $\mathfrak{b}\subseteq \mathfrak{a}$ of R and any $C\in {\mathcal{T}}^c$, one finds that ${\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X⩽{\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{b}\right)}X$.

Proof. 

First, assume that $|r_1|>0$. If $|r_2|>0$, by Lemma 2, we find that ${\sup }_C(X//(r_1,r_2))={\sup }_C(X//r_1)-1$. Hence, we have following equalities by Lemma 3,

$$\begin{array}{cc}\hfill {\sup }_C\left({\Lambda }^{\mathcal{V}(r_1,r_2)}X\right)& ={\sup }_C(X//(r_1,r_2))+2\hfill \\ \hfill & ={\sup }_C(X//r_1)+1\hfill \\ \hfill & ={\sup }_C\left({\Lambda }^{\mathcal{V}\left(r_1\right)}X\right).\hfill \end{array}$$

If $|r_2|⩽0$, by Lemma 2, we find that ${\sup }_C(X//(r_1,r_2))⩽{\sup }_C(X//r_1)-\left|r_2\right|$, and then

$$\begin{array}{cc}\hfill {\sup }_C\left({\Lambda }^{\mathcal{V}(r_1,r_2)}X\right)& ={\sup }_C({\Lambda }^{\mathcal{V}\left(r_2\right)}X//r_1)+1\hfill \\ \hfill & ={\sup }_C{\Lambda }^{\mathcal{V}\left(r_2\right)}(X//r_1)+1\hfill \\ \hfill & ={\sup }_C(X//(r_1,r_2))+\left|r_2\right|+1\hfill \\ \hfill & ⩽{\sup }_C(X//r_1)+1\hfill \\ \hfill & ={\sup }_C\left({\Lambda }^{\mathcal{V}\left(r_1\right)}X\right).\hfill \end{array}$$

Now, assume that $|r_1|⩽0$. If $|r_2|>0$, by Lemma 2, we find that ${\sup }_C(X//(r_1,r_2))={\sup }_C(X//r_1)-1$. Hence, we have following equalities by Lemma 3,

$$\begin{array}{cc}\hfill {\sup }_C\left({\Lambda }^{\mathcal{V}(r_1,r_2)}X\right)& ={\sup }_C({\Lambda }^{\mathcal{V}\left(r_1\right)}X//r_2)+1\hfill \\ \hfill & ={\sup }_C{\Lambda }^{\mathcal{V}\left(r_1\right)}(X//r_2)+1\hfill \\ \hfill & ={\sup }_C(X//(r_1,r_2))+\left|r_1\right|+1\hfill \\ \hfill & ⩽{\sup }_C(X//r_1)+\left|r_1\right|\hfill \\ \hfill & ={\sup }_C\left({\Lambda }^{\mathcal{V}\left(r_1\right)}X\right).\hfill \end{array}$$

If $|r_2|⩽0$, by Lemma 2, we find that ${\sup }_C(X//(r_1,r_2))⩽{\sup }_C(X//r_1)-\left|r_2\right|$, and then

$$\begin{array}{cc}\hfill {\sup }_C\left({\Lambda }^{\mathcal{V}(r_1,r_2)}X\right)& ={\sup }_C({\Lambda }^{\mathcal{V}\left(r_2\right)}X//r_1)+\left|r_1\right|\hfill \\ \hfill & ={\sup }_C{\Lambda }^{\mathcal{V}\left(r_2\right)}(X//r_1)+\left|r_1\right|\hfill \\ \hfill & ={\sup }_C(X//(r_1,r_2))+|r_2|+|r_1|\hfill \\ \hfill & ⩽{\sup }_C(X//r_1)+\left|r_1\right|\hfill \\ \hfill & ={\sup }_C\left({\Lambda }^{\mathcal{V}\left(r_1\right)}X\right).\hfill \end{array}$$

This complete the proof. □

Corollary 1. 

Given a local ring $(R,\mathfrak{m})$ and a homogeneous ideal $\mathfrak{a}$ of R, for any object X in $\mathcal{T}$, there exists an inequality stating that $\mathrm{hd}(\mathfrak{m},X)⩽\mathrm{hd}(\mathfrak{a},X)$.

Lemma 4. 

Let $\mathfrak{a}$ be a graded ideal of R and $X\ne 0$ be an object of ${\mathcal{T}}^{-}$. If ${\dim }_{R}X=0$, then $\mathrm{hd}(\mathfrak{a},X)⩽\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}$.

Proof. 

By [13] (Corollary 4.4), we have that if ${\dim }_{R}X=0$, then ${\sup }_C{\Gamma }_{\mathcal{V}\left(\mathfrak{a}\right)}X⩽{\sup }_{C}X$ for any $C\in {\mathcal{T}}^c$. So, we just have to prove that ${\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X⩽{\sup }_C{\Gamma }_{\mathcal{V}\left(\mathfrak{a}\right)}X$ for any $C\in {\mathcal{T}}^c$. To validate the assertion for $\mathfrak{a}=\left(r\right)$, a straightforward iterative approach leads to the generalized conclusion.

Utilizing [20] (Lemma 2.6) and [8] (Corallary 5.7), we deduce that ${\Gamma }_{\mathcal{V}\left(\right(r\left)\right)}X//r\cong X//r$. Consequently, let us consider the exact triangle

$${\Gamma }_{\mathcal{V}\left(\right(r\left)\right)}X\stackrel{r}{\to }{\Sigma }^{\left|r\right|}{\Gamma }_{\mathcal{V}\left(\right(r\left)\right)}X\to X//r⇝.$$

If $\left|r\right|⩽0$, then ${\sup }_C(X//r)⩽{\sup }_C\left({\Gamma }_{\mathcal{V}\left(\right(r\left)\right)}X\right)-\left|r\right|$. Consequently, Lemma 3 yields that

$${\sup }_C\left({\Gamma }_{\mathcal{V}\left(\right(r\left)\right)}X\right)⩾{\sup }_C\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right).$$

On the other hand, if $\left|r\right|>0$, then ${\sup }_C(X//r)={\sup }_C\left({\Gamma }_{\mathcal{V}\left(\right(r\left)\right)}X\right)-1$. Therefore, Lemma 3 implies that

$${\sup }_C\left({\Gamma }_{\mathcal{V}\left(\right(r\left)\right)}X\right)={\sup }_C\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right).$$

This complete the proof. □

Let $\mathfrak{a}$ be a homogeneous ideal of R. The following theorem gives the relationship between the homology dimension with respect to $\mathfrak{a}$ and the arithmetic rank of $\mathfrak{a}$.

Theorem 4. 

Given a homogeneous ideal $\mathfrak{a}$ of R and an object $X\ne 0$ of ${\mathcal{T}}^{-}$, we find that $\mathrm{hd}(\mathfrak{a},X)⩽\mathrm{ara}\left(\mathfrak{a}\right)+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}$.

Proof. 

Suppose $\mathrm{ara}\left(\mathfrak{a}\right)=n$. We shall employ induction on n to establish the desired result.

If $n=0$, ${\Lambda }^{\mathcal{V}\left(0\right)}X\cong X$ as ${\mathrm{cosupp}}_{R}X\subseteq R=\mathcal{V}\left(0\right)$ by [15] (Corollary 4.8), then $\mathrm{hd}(\mathfrak{a},X)=\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}$. Therefore, the conclusion holds.

Assume that $n=1$ and $\mathfrak{a}=\left(r\right)$. By [15] (Corollary 4.8) and [15] (Lemma 4.12), we have that ${\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X//r\cong X//r$. So, we have the triangle

$$(‡)\hspace{1em}\hspace{1em}\hspace{1em}{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\stackrel{r}{\to }{\Sigma }^{\left|r\right|}{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\to X//r⇝.$$

If $\left|r\right|⩾0$, consider the induced long exact sequence of $R_0$-modules of (‡):

$$\cdots \to {\mathrm{H}}_C^i(X//r)\to {\mathrm{H}}_C^{i+1}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\stackrel{r}{\to }{\mathrm{H}}_C^{i+1+\left|r\right|}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\to \cdots .$$

Due to Lemma 2, we know that ${\mathrm{H}}_C^i(X//r)=0$ for all $i⩾{\sup }_{C}X+1$. Therefore, the homomorphism ${\mathrm{H}}_C^{i+1}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\stackrel{r}{\to }{\mathrm{H}}_C^{i+1+\left|r\right|}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)$ is injective for all $i⩾{\sup }_{C}X+1$. This, in turn, implies that ${\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)=0$ for all $i⩾{\sup }_{C}X+2$, since any element of ${\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)$ is annihilated by some power of r according to [15] (Lemma 6.1). Hence,

$$\mathrm{hd}(\mathfrak{a},X)⩽\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}+1=\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}+\mathrm{ara}\left(\left(r\right)\right).$$

If $\left|r\right|=-d<0$, where $d>0$, consider the induced long exact sequence of $R_0$-modules of (‡):

$$\cdots \to {\mathrm{H}}_C^{i+1+d}(X//r)\to {\mathrm{H}}_C^{i+2+d}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\stackrel{r}{\to }{\mathrm{H}}_C^{i+2}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\to \cdots .$$

As stated in Lemma 2, ${\mathrm{H}}_C^i(X//r)=0$ for all $i⩾{\sup }_{C}X$. Consequently, the homomorphism ${\mathrm{H}}_C^{i+2+d}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)\stackrel{r}{\to }{\mathrm{H}}_C^{i+2}\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)$ is injective for all $i⩾{\sup }_{C}X$. This injection guarantees that ${\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)=0$ for all $i⩾{\sup }_{C}X+2$, due to the fact that any element of ${\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\right)$ is annihilated by some power of r. Therefore, $\mathrm{hd}(\mathfrak{a},X)⩽\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}+1=\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}+\mathrm{ara}\left(\left(r\right)\right)$.

Assuming inductively that the integer $n>1$ and the desired result holds for all ideals with an arithmetic rank less than n, let us take an ideal $\mathfrak{a}$ defined as $\mathfrak{a}=(r_1,\dots ,r_n)$, where $r_1,\dots ,r_n$ are homogeneous elements of R. Next, we consider the ideal $\mathfrak{b}$, which is given by $\mathfrak{b}=(r_2,\dots ,r_n)$. Now, turning our attention to the Mayer–Vietoris triangle

$${\Lambda }^{\mathcal{V}\left(\mathfrak{b}\right)\cup \mathcal{V}\left(\left(r_1\right)\right)}X\to {\Lambda }^{\mathcal{V}\left(\mathfrak{b}\right)}X⨿{\Lambda }^{\mathcal{V}\left(\left(r_1\right)\right)}X\to {\Lambda }^{\mathcal{V}\left(\mathfrak{b}\right)\cap \mathcal{V}\left(\left(r_1\right)\right)}X⇝.$$

In light of the fact that $\mathcal{V}\left(\mathfrak{b}\right)\cup \mathcal{V}\left(\left(r_1\right)\right)=\mathcal{V}\left(\mathfrak{b}{r}_1\right)$ and $\mathcal{V}\left(\mathfrak{b}\right)\cap \mathcal{V}\left(\left(r_1\right)\right)=\mathcal{V}\left(\mathfrak{a}\right)$, we obtain the following triangle

$${\Lambda }^{\mathcal{V}\left(\mathfrak{b}{r}_1\right)}X\to {\Lambda }^{\mathcal{V}\left(\mathfrak{b}\right)}X⨿{\Lambda }^{\mathcal{V}\left(\left(r_1\right)\right)}X\to {\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X⇝.$$

For any $C\in {\mathcal{T}}^c$, this induces a long exact sequence

$$\cdots \to {\mathrm{H}}_C^i({\Lambda }^{\mathcal{V}\left(\mathfrak{b}\right)}X⨿{\Lambda }^{\mathcal{V}\left(\left(r_1\right)\right)}X)\to {\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)\to {\mathrm{H}}_C^{i+1}\left({\Lambda }^{\mathcal{V}\left(\mathfrak{b}{r}_1\right)}X\right)\to \cdots ,$$

since $\mathrm{ara}\left(\mathfrak{b}{r}_1\right)⩽\mathrm{ara}\left(\mathfrak{b}\right)⩽\mathrm{ara}\left(\mathfrak{a}\right)$; hence, if $i>\mathrm{ara}\left(\mathfrak{a}\right)+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}$, we find that ${\mathrm{H}}_C^i({\Lambda }^{\mathcal{V}\left(\mathfrak{b}\right)}X⨿{\Lambda }^{\mathcal{V}\left(\left(r_1\right)\right)}X)={\mathrm{H}}_C^{i+1}\left({\Lambda }^{\mathcal{V}\left(\mathfrak{b}{r}_1\right)}X\right)=0$ and this implies that ${\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)=0$ for all $i>\mathrm{ara}\left(\mathfrak{a}\right)+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}$. Hence, $\mathrm{hd}(\mathfrak{a},X)⩽\mathrm{ara}\left(\mathfrak{a}\right)+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}$. □

In the following section of the paper, we assume that $R={⨁}_{n\in {\mathbb{N}}_0}{R}_n$ is a Noetherian ${\mathbb{N}}_0$-graded ring.

Proposition 4. 

Let $\mathfrak{a}$ be a homogeneous ideal of R, with r being a homogeneous element belonging to $\mathfrak{a}$, and let C be a compact object of $\mathcal{T}$. Then, for any $X\in \mathcal{T}$,

$${\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X-1⩽{\sup }_C({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X//r)⩽{\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X.$$

Proof. 

By Lemma 2, we only show that ${\sup }_C({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X//r)⩾{\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X-1$ if $\left|r\right|=0$. Considering the exact triangle

$$X\stackrel{r}{\to }X\to X//r⇝,$$

applying the exact functor ${\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}$ to the above triangle, we find that

$${\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\stackrel{r}{\to }{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\to {\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}(X//r)⇝.$$

For any $C\in {\mathcal{T}}^c$, this leads to the exact sequence

$$\cdots \to {\mathrm{H}}_C^{i-1}({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X//r)\to {\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)\stackrel{r}{\to }{\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)\to \cdots .$$

If ${\mathrm{H}}_C^{{\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X-1}({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X//r)=0$, then the morphism ${\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)\stackrel{r}{\to }{\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)$ has to be injective, which is impossible as ${\mathrm{H}}_C^{*}\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)$ is $\left(r\right)$-torsion by [15] (Lemma 6.1); then, any element of ${\mathrm{H}}_C^{{\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X}\left({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X\right)$ is annihilated by a certain power of r. Consequently,

$${\mathrm{H}}_C^{{\sup }_C{\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X-1}({\Lambda }^{\mathcal{V}\left(\mathfrak{p}\right)}X//r)\ne 0.$$

This complete the proof. □

Theorem 5. 

Given a homogeneous ideal $\mathfrak{a}$ of R and an object $X\ne 0$ of ${\mathcal{T}}^{-}$, we get the inequality $\mathrm{hd}(\mathfrak{a},X)⩽\dim R+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}$.

Proof. 

If $\dim R=\infty$, the conclusion clearly holds.

If $\dim R=0$, the result is evident since ${\dim }_{R}X=0$ for any object X of $\mathcal{T}$, and so the result follows from Lemma 4.

If $\dim R=d>0$, let us assume there exists $\mathfrak{q}\in \min R$ such that $\mathfrak{a}\subseteq \mathfrak{p}$. Arrange the minimal prime ideals in $\mathrm{cosupp}R$ in a manner such that $\mathfrak{a}\subseteq {\mathfrak{p}}_i$ for $i=1,\dots ,n$ and $\mathfrak{a}⊈{\mathfrak{p}}_i$ for $i=n+1,\dots ,t$. Choose $r\in {\bigcap }_{i=n+1}^t{\mathfrak{p}}_i\setminus {\bigcup }_{i=1}^n{\mathfrak{p}}_i$. Define ${\mathfrak{a}}_1=\mathfrak{a}\cap Rr$ and ${\mathfrak{a}}_2=\mathfrak{a}\cup Rr$. Clearly, ${\mathfrak{a}}_1\subseteq {\bigcap }_{\mathfrak{p}\in \min R}\mathfrak{p}$ and ${\mathfrak{a}}_2⊈{\bigcup }_{\mathfrak{p}\in \min R}\mathfrak{p}$. Utilizing Lemma 1, we investigate the Mayer–Vietoris triangle

$${\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)\cup \mathcal{V}\left(\right(r\left)\right)}X\to {\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X⨿{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\to {\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)\cap \mathcal{V}\left(\right(r\left)\right)}X⇝.$$

In view of the facts that $\mathcal{V}\left(\mathfrak{a}\right)\cup \mathcal{V}\left(\left(r\right)\right)=\mathcal{V}\left({\mathfrak{a}}_1\right)$ and $\mathcal{V}\left(\mathfrak{a}\right)\cap \mathcal{V}\left(\left(r\right)\right)=\mathcal{V}\left({\mathfrak{a}}_2\right)$, we get the following triangle

$${\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_1\right)}X\to {\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X⨿{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X\to {\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X⇝.$$

For any $C\in {\mathcal{T}}^c$, this induces a long exact sequence

$$\cdots \to {\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_1\right)}X\right)\to {\mathrm{H}}_C^i({\Lambda }^{\mathcal{V}\left(\mathfrak{a}\right)}X⨿{\Lambda }^{\mathcal{V}\left(\right(r\left)\right)}X)\to {\mathrm{H}}_C^i\left({\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X\right)\to \cdots .$$

Since ${\mathrm{cosupp}}_{R}X\subseteq \mathcal{V}\left({\mathfrak{a}}_1\right)$, we have ${\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_1\right)}X\cong X$ by [15] (Corollary 4.8), we then find that

$$\mathrm{hd}({\mathfrak{a}}_1,X)⩽\dim R+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}.$$

Hence, we obtain the following equivalence relationship:

$$\mathrm{hd}(\mathfrak{a},X)⩽\dim R+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}\iff \mathrm{hd}({\mathfrak{a}}_2,X)⩽\dim R+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}.$$

Let $r_1\in {\mathfrak{a}}_2\setminus {\bigcup }_{\min R}\mathfrak{p}$ be a homogeneous element. Therefore, $\dim R/r_{1}R<\dim R$. If ${\mathfrak{a}}_2\subseteq {\bigcap }_{\mathfrak{p}\in \min (R/r_{1}R)}\mathfrak{p}$, then since $r_1\in {\mathfrak{a}}_2$, we find that $\sqrt{{\mathfrak{a}}_2}=\sqrt{r_{1}R}$, and so $\mathrm{ara}\left({\mathfrak{a}}_2\right)=1$. Hence the result follows from Theorem 4. Otherwise, we may find $r_2\in {\mathfrak{a}}_2$ such that $\dim R/(r_1,r_2)<\dim R/r_{1}R$. Continuing in this way, after d steps, we may deduce that $\dim R/(r_1,\dots ,r_d)=0$. Therefore, ${\dim }_{R}X//(r_1,\dots ,r_d)=0$. Consequently, according to Lemma 4 and Proposition 4, for any compact object C of T, we obtain

$${\sup }_C({\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X//(r_1,\dots ,r_d))⩽{\sup }_C(X//(r_1,\dots ,r_d))⩽{\sup }_{C}X.$$

This implies that ${\mathrm{H}}_C^{{\sup }_{C}X+1}({\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X//(r_1,\dots ,r_d))=0$. Now, considering the long exact sequence of cohomology modules arising from the exact triangle

$${\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X//(r_1,\dots ,r_{d-1})\stackrel{r_d}{\to }{\Sigma }^{|r_d|}{\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X//(r_1,\dots ,r_{d-1})\to {\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X//(r_1,\dots ,r_d)⇝.$$

we infer that ${\mathrm{H}}_C^{{\sup }_{C}X+2}({\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X//(r_1,\dots ,r_{d-1}))=0$. This is due to the fact that $r_d\in a$ and any element of ${\mathrm{H}}_C^{*}({\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X//(r_1,\dots ,r_{d-1}))$ vanishes by a power of $r_d$. Following back from this argument gives us $H_C^{{\sup }_{C}X+d+1}\left({\Lambda }^{\mathcal{V}\left({\mathfrak{a}}_2\right)}X\right)=0$. In other words:

$$\mathrm{hd}({\mathfrak{a}}_2,X)⩽\dim R+\sup \left\{{\sup }_{C}X\right|C\in {\mathcal{T}}^c\}.$$

The proof is complete because C was arbitrary. □