A classification of some thick subcategories in locally noetherian Grothendieck categories

Abstract

Let $\mathcal{A}$ be a locally noetherian Grothendieck category. We classify all full subcategories of $\mathcal{A}$ which are thick and closed under taking arbitrary direct sums and injective envelopes by injective spectrum. This result gives a generalization from the commutative noetherian ring to the locally noetherian Grothendieck category.

1. Introduction

The classification of various types of subcategories is a fundamental problem in many different contexts. It will help us to better understand the structure of categories. Let $R$ be a commutative Noetherian ring, and Spec $R$ be the set of prime ideals $\textsf{p}$ in $R$ . For $\textsf{p}\in \text{Spec}R$ , set $S=R\backslash \textsf{p}$ and denote by $R_{\textsf{p}}=R[S^{-1}]$ the localization. The support of a $R$ -module $X$ is the set of points $ \textsf{p}$ in $ \text{Spec}R$ such that $X_{\textsf{p}}\;:\!=\;X\otimes R_{\textsf{p}}\neq 0$ . Using this support, Gabriel [4] showed that there exist one-to-one correspondences between the set of localizing subcategories of ${{\mathrm{Mod}}}R$ , Serre subcategories of ${{\mathrm{mod}}}R$ , and the specialization-closed subsets of the prime spectrum Spec $A$ . For complexes of modules, Foxby [3] introduced useful notions of support, which are defined as follows: Let $k(\textsf{p})$ be the residue field $R_{\textsf{p}}/\textsf{p}R_{\textsf{p}}$ of the local ring $R_{\textsf{p}}$ . The $support$ of a complex $X$ of $R$ -modules is the subset

\begin{equation*}{{\mathrm {Supp}}}_{R}X=\left\{ \textsf {p}\in \text {Spec}R|X\otimes _{R}^{\mathbb {L}}k(\textsf {p})\neq 0 \right\}.\end{equation*}

Using this support of the complex, Hopkins [6] and Neeman [13] classified all localizing subcategories of the unbounded derived category $D(R)$ . Kanda [9] classified the Serre subcategories in an abelian category using the atom spectrum consisting of equivalence classes of monoform objects. Many authors have classified various types of subcategories, see [1, 2, 5, 7, 8, 10, 14–17] and so on.

Let $R$ be a commutative Noetherian ring, and $M$ be a $R$ -module. Consider $M$ to be a stalk complex. Following Foxby [3], the $support$ of $M$ is

\begin{equation*}{{\mathrm {Supp}}}_{R}M=\left\{ \textsf {p}\in \text {Spec}R|{{\mathrm {Tor}}}^{R}_{*}(M,k(\textsf {p}))\neq 0 \right\}.\end{equation*}

Using this support, Krause [11] classified all full subcategories of $R$ -modules which are thick (i.e. closed under taking kernels, cokernels, and extensions) and closed under taking arbitrary direct sums. In this paper, we want to generalize Krause’s result from commutative noetherian rings to locally noetherian Grothendieck categories. So we need to find a new support in abelian category. Foxby [3, 2.8,2.9] proved that a point $\textsf{p}$ is in the support of a complex $X$ with ${{\mathrm{H}}}^{n}(X)=0$ for $n\ll 0$ if and only if the injective envelope of $R/\textsf{p}$ appears in the minimal semi-injective resolution of $X$ . When $X$ is a $R$ -module, the minimal semi-injective resolution of $X$ is just the minimal injective resolution of $X$ . So we get the equivalent characterization of ${{\mathrm{Supp}}}_{R}X$ : $\textsf{p}\in{{\mathrm{Supp}}}_{R}X$ if and only if the injective envelope of $R/\textsf{p}$ appears in the minimal injective resolution of $X$ . Let ${{\mathrm{Sp(Mod}}}A)$ denote the representative set of the isomorphism classes of indecomposable injective objects in ${{\mathrm{Mod}}}A$ . Then, there is a bijection between $\text{Spec}R$ and ${{\mathrm{Sp(Mod}}}A)$ [12, Corollary 2.4.15]. We use the equivalent characterization to define the support of objects in locally noetherian Grothendieck category and get the following main result.

Theorem 1.1. Let $ \mathcal{A}$ be a locally noetherian Grothendieck category. Then, the assignment $\mathcal{X}\mapsto{{\mathrm{Supp}}}\mathcal{X}$ induces an inclusion-preserving bijection between the set of thick subcategories of $ \mathcal{A}$ , which are closed under taking arbitrary direct sums and injective envelopes, and the set of coherent subsets of ${{\mathrm{Sp}}}(\mathcal{A})$ , with inverse ${{\mathrm{Supp}}}^{-1}$ .

The paper is organized as follows. In Section 2, we give some definitions and notations. In Section 3, we prove the main result.

2. Preliminaries

In this section, we introduce some basic properties and definitions which we will use later.

A category $\mathcal{A}$ is a $ Grothendieck$ $category$ if

1. (Gr1) the category $\mathcal{A}$ is abelian,

2. (Gr2) for every set of objects $(X_{i})_{i\in I}$ there is a coproduct $\coprod _{i\in I}X_{i}$ in $\mathcal{A}$ ,

3. (Gr3) there is a $generator$ $G$ , that is, for every object $X$ the canonical morphism $\coprod _{\phi \in{{\mathrm{Hom}}}_{\mathcal{A}}(G,X)}G\to X$ is an epimorphism, and

4. (Gr4) every filtered colimit of exact sequences is exact.

A monomorphism $\phi \;:\;X\to Y$ is $essential$ if any morphism $\alpha \;:\; Y\to Y^{\prime}$ is a monomorphism provided that the composite $\alpha \phi$ is a monomorphism. This condition can be rephrased as follows: if $U\subseteq Y$ is a subobject with $U\bigcap{{\mathrm{Im}}} \phi$ = 0, then $U$ = 0. A monomorphism $\phi \;:\;X\to I$ is an $injective$ $ envelope$ of $X$ if $I$ is injective and $\phi$ is essential. For $X\in \mathcal{A}$ , an $injective$ $resolution$ of $X$ is an exact sequence $0 \to X\to I^{0}\stackrel{i_{1}}\to I^{1} \stackrel{i_{2}}\to I^{2} \stackrel{i_{3}}\to \ldots$ with all $ I^{i}$ are injective. In addition, the above injective resolution is called $minimal$ if $I^{n}$ is the injective envelope of ${{\mathrm{Ker}}}i_{n+1}$ , for $n\in \mathbb{N}$ . Every object in a Grothendieck category admits an injective envelope [12, Corollary 2.5.4].

Let $\mathcal{A}$ be a Grothendieck category. $\mathcal{A}$ is called $locally$ $noetherian$ if $\mathcal{A}$ has a generating set of noetherian objects. If $\mathcal{A}$ is locally noetherian, then the full subcategory of injective objects is closed under coproducts [12, Theorem 11.2.12].

Let $\mathcal{A}$ be a locally noetherian Grothendieck category. A full subcategory $\mathcal{C}$ of $\mathcal{A}$ is called $thick$ if for each exact sequence $M_{1}\to M_{2} \to M_{3} \to M_{4} \to M_{5}$ with $M_{i}$ in $\mathcal{C}$ for $i = 1, 2, 4, 5$ , the module $M_{3}$ belongs to $\mathcal{C}$ . It is clear that $\mathcal{C}$ is thick if and only if $\mathcal{C}$ is closed under taking kernels, cokernels, and extensions. A thick subcategory is an abelian category, and the inclusion functor is exact. A full subcategory $\mathcal{C}$ of $\mathcal{A}$ is called $closed$ $under$ $taking$ $injective$ $envelopes$ if for each $X\in \mathcal{C}$ , the injective envelope $I(X)$ of $X$ is also in $\mathcal{C}$ .

3. Main result

In this section, $\mathcal{A}$ is always a locally noetherian Grothendieck category and Sp $\mathcal{A}$ is a representative set of the isomorphism classes of indecomposable injective objects in $\mathcal{A}$ (the spectrum of $\mathcal{A}$ ), unless otherwise specified.

We wish to classify all thick subcategories of $\mathcal{A}$ which are closed under taking arbitrary direct sums and taking injective envelopes. Firstly, we list some definitions which will be used later.

Definition 3.1. Let $M\in \mathcal{A}$ and $0 \to M\to I^{0}\stackrel{i_{1}}\to I^{1} \stackrel{i_{2}}\to I^{2} \stackrel{i_{3}}\to \ldots$ be the minimal injective resolution of $M$ . The support of $M$ is defined by

\begin{equation*}{{\mathrm {Supp}}}M=\{I\in {{\mathrm {Sp}}}\mathcal {A}| I \ arises \ as \ a \ direct \ summand\ of\ I^{n}\ for \ some\ n\in \textbf {N}\} .\end{equation*}

For $\mathcal{X}\subseteq \mathcal{A}$ ,

\begin{equation*}{{\mathrm {Supp}}}\mathcal {X}=\bigcup _{X\in \mathcal {X}}{{\mathrm {Supp}}}X.\end{equation*}

Denote ${{\mathrm{Ind}}}X$ be the set of the indecomposable direct summand of object $X$ .

Definition 3.2. Let $\Phi$ be a subset of ${{\mathrm{Sp}}}(\mathcal{A})$ . We call $\Phi$ coherent if each morphism $I^0 \to I^1$ with ${{\mathrm{Ind}}}(I^0\oplus I^1) \in \Phi$ can be completed to an exact sequence $I^0 \to I^1 \to I^2$ with ${{\mathrm{Ind}}}I^2$ in $\Phi$ .

Let $\Phi$ be a subset of ${{\mathrm{Sp}}}(\mathcal{A})$ . Denote ${{\mathrm{Supp}}}^{-1}\Phi \;:\!=\;\{M \in \mathcal{A} |{{\mathrm{Supp}}} M \subseteq \Phi \}.$ The following lemma is obvious.

Lemma 3.3. Let $\Phi$ be a coherent subset of ${{\mathrm{Sp}}}(\mathcal{A})$ , and $0 \to M\to I^{0}\stackrel{i_{1}}\to I^{1} \stackrel{i_{2}}\to I^{2} \stackrel{i_{3}}\to \ldots$ be the minimal injective resolution of $M$ . Then, $M\in{{\mathrm{Supp}}}^{-1}\Phi$ if and only if ${{\mathrm{Ind}}}( I^{0}\oplus I^{1})\subseteq \Phi$ .

Lemma 3.4. Let $\Phi$ be a coherent subset of ${{\mathrm{Sp}}}(\mathcal{A})$ . Then, ${{\mathrm{Supp}}}^{-1}\Phi$ is a thick subcategory of $ \mathcal{A}$ .

Proof. Taking $M$ , $N\in{{\mathrm{Supp}}}^{-1}\Phi$ , we have ${{\mathrm{Supp}}} M,{{\mathrm{Supp}}} N\subseteq \Phi$ . Thus, ${{\mathrm{Supp}}} (M\oplus N)\subseteq \Phi$ . Hence, ${{\mathrm{Supp}}}^{-1}\Phi$ is an additive category. In the following, we prove that ${{\mathrm{Supp}}}^{-1}\Phi$ is closed under taking kernels, cokernels, and extensions.

(a) ${{\mathrm{Supp}}}^{-1}\Phi$ is closed under taking kernels. For any $M\stackrel{f}\to N$ with $M, N \in{{\mathrm{Supp}}}^{-1}\Phi$ , suppose $g:K\to M$ is the kernel of $f:M\to N$ . Take the minimal injective resolutions of $K$ , $M$ , $N$ as follows:

\begin{equation*}0\to K \stackrel {i_{K}^{0}}\to I_{K}^{0}\stackrel {i_{K}^{1}}\to I_{K}^{1}\to \ldots,\end{equation*}

\begin{equation*}0\to M \stackrel {i_{M}^{0}}\to I_{M}^{0}\stackrel {i_{M}^{1}}\to I_{M}^{1}\to \ldots,\end{equation*}

\begin{equation*}0\to N \stackrel {i_{N}^{0}}\to I_{N}^{0}\stackrel {i_{N}^{1}}\to I_{N}^{1}\to \ldots .\end{equation*}

Since $M,$ $N\in{{\mathrm{Supp}}}^{-1}\Phi$ , ${{\mathrm{Ind}}}(\oplus _{i\in \textbf{N}}(I_{M}^{i}\oplus I_{N}^{i}))\subseteq \Phi$ . To prove that ${{\mathrm{Supp}}}^{-1}\Phi$ is closed under taking kernels, it is enough to prove that ${{\mathrm{Ind}}}(I_{K}^{0}\oplus I_{K}^{1})\subseteq \Phi$ by $\Phi$ coherent. Since $I_{M}^{0}$ is injective, there is a morphism $g^{0}\;:\;I_{K}^{0}\to I_{M}^{0}$ such that the following diagram is commutative:

Since $g^{0} i_{K}^{0}$ is monic and $i_{K}^{0}$ is essential, we have $g^{0}$ is monic. Thus, $ I_{K}^{0}$ is a direct summand of $I_{M}^{0}$ . Hence, ${{\mathrm{Ind}}}I_{K}^{0}\subseteq \Phi$ .

Take the minimal injective resolution of ${{\mathrm{Im}}}f$ :

\begin{equation*}0\to {{\mathrm {Im}}}f \to I_{f}^{0}\stackrel {i_{f}^{1}}\to I_{f}^{1}\to \ldots .\end{equation*}

Similarly to the proof of $g_{0}$ , we can get the following commutative diagram

where $k^{0}$ is monic. Hence, ${{\mathrm{Ind}}}I_{f}^{0}\subseteq \Phi$ . Consider the exact sequence $0\to K\to M\to{{\mathrm{Im}}}f \to 0$ . By the Horseshoe Lemma, we have the following commutative diagram with exact rows and columns

Since $I_{M}^{0}$ is an injective envelope of $M$ , we can construct an isomorphism $\phi \;:\;I_{M}^{0}\oplus E \stackrel{\sim }\to I_{K}^{0}\oplus I_{f}^{0}$ such that the following diagram commutative

Hence, ${{\mathrm{Ind}}}E\subseteq \Phi$ and $\psi ^{-1}\omega \;:\;\text{Coker} i_{K}^{0}\to \text{Coker} i_{M}^{0}\oplus E$ is monic. Consider the following commutative diagram

Similarly to the proof of ${{\mathrm{Ind}}}I_{K}^{0}\subseteq \Phi$ , we can get ${{\mathrm{Ind}}}I_{K}^{1}\subseteq \Phi$ . Therefore, ${{\mathrm{Supp}}}^{-1}\Phi$ is closed under taking kernels.

(b) ${{\mathrm{Supp}}}^{-1}\Phi$ is closed under taking extensions. This follows from the Horseshoe Lemma.

(c) ${{\mathrm{Supp}}}^{-1}\Phi$ is closed under taking cokernels. For any $M\stackrel{f}\to N$ with $M,N\in{{\mathrm{Supp}}}^{-1}\Phi$ , suppose $g:N\to C$ is the cokernel of $f\;:\;M\to N$ . Take the minimal injective resolutions of $M$ , $N$ , then we can get the following commutative diagram with exact rows:

Consider the following push-out square

(**)

Taking the injective envelope $ I$ of $D$ , then we have the following exact sequence

where $g^{0}=kb$ . Since $M,$ $ N\in{{\mathrm{Supp}}}^{-1}\Phi$ and $\Phi$ is coherent, we have ${{\mathrm{Ind}}}I\subseteq \Phi$ .

Claim that there is a monomorphism $i^{0}\;:\;C\to I$ such that $i^{0}g=g^{0}i_{N}^{0}.$ In fact, by the properties of the push-out square, we have the following exact commutative diagram:

where $\tilde{f^{0}}$ is epic. Then, we have the following exact commutative diagram:

Since $\tilde{f^{0}}$ is epic and $i_{N}^{0}$ is monic, $h$ is monic. Taking $i^{0}\;:\!=\;k\circ h$ , obviously $i^{0}\;:\;C \to I$ is monic and $i^{0}g=g^{0}i_{N}^{0}.$

Consider the following push-out square

Taking the injective envelope $ I^{\prime}$ of $D^{\prime}$ , then we have the following exact sequence

where $g^{1}=k^{\prime}a^{\prime}$ , $i^{1}=k^{\prime}b^{\prime}$ . Since ${{\mathrm{Ind}}}(I_{N}^{0}\oplus I_{N}^{1} \oplus I )\subseteq \Phi$ and $\Phi$ is coherent, we have ${{\mathrm{Ind}}}I^{\prime}\subseteq \Phi$ . Based on the above discussion, we get the following commutative diagram

Next, we prove that the second row in the above commutative diagram is exact. Since both $i^{0}$ and $k^{\prime}$ are monic, it is enough to prove that $C\simeq \text{ker}b^{\prime}$ . By the properties of push-out, we can get the following exact commutative diagram such that $\tilde{g^{0}}\;:\;N\to \text{ker}b^{\prime}$ is epic.

(**)

Since $b^{\prime}i^{0}$ =0 and $i^{0}$ is monic, there is a monomorphism $j\;:\;C\to \text{ker}b^{\prime}$ such that $i^{0}=i^{\prime}j$ . It’s easy to get that $j$ is also epic by the following commutative diagram.

Hence, $C\simeq \text{ker}b^{\prime}\simeq \text{ker}i^{1}$ . Since $\Phi$ is coherent, $C\in{{\mathrm{Supp}}}^{-1}\Phi$ .

Lemma 3.5. Let $\mathcal{X}$ , $\mathcal{Y}$ be two thick subcategories closed under taking direct sums and injective envelopes. Then, $\mathcal{X}\subseteq \mathcal{Y}$ if and only if ${{\mathrm{Supp}}}\mathcal{X}\subseteq{{\mathrm{Supp}}}\mathcal{Y}$ .

Proof. The “only if” part. It is clear that ${{\mathrm{Supp}}}\mathcal{X}\subseteq \mathcal{X} \subseteq \mathcal{Y}$ . So ${{\mathrm{Supp}}}\mathcal{X}\subseteq{{\mathrm{Supp}}}\mathcal{Y}$ .

The “if” part. For any $X\in \mathcal{X}$ . Taking the minimal injective resolution of $X$ ,

\begin{equation*}0\to X \stackrel {i^{0}}\to I^{0}\stackrel {i^{1}}\to I^{1}\to \ldots,\end{equation*}

we have ${{\mathrm{Ind}}}I^{i}\subseteq{{\mathrm{Supp}}}\mathcal{X}\subseteq{{\mathrm{Supp}}}\mathcal{Y}\subseteq \mathcal{Y}$ . Since $\mathcal{Y}$ is closed under taking kernel and direct sums, $X\in \mathcal{Y}$ .

Theorem 3.6. Let $ \mathcal{A}$ be a locally noetherian Grothendieck category. Then, the assignment $\mathcal{X}\mapsto{{\mathrm{Supp}}}\mathcal{X}$ induces an inclusion-preserving bijection between the set of thick subcategories of $ \mathcal{A}$ , which are closed under taking arbitrary direct sums and injective envelopes, and the set of coherent subsets of ${{\mathrm{Sp}}}(\mathcal{A})$ , with inverse ${{\mathrm{Supp}}}^{-1}$ .

Proof. For any coherent subset of ${{\mathrm{Sp}}}(\mathcal{A})$ , ${{\mathrm{Supp}}}^{-1}\Phi$ is a thick subcategory of $ \mathcal{A}$ by Lemma 3.4. Since $ \mathcal{A}$ is locally noetherian, it is clear that ${{\mathrm{Supp}}}^{-1}\Phi$ is closed under taking arbitrary direct sums. Obviously, ${{\mathrm{Supp}}}^{-1}\Phi$ is closed under taking injective envelopes. Claim that ${{\mathrm{Supp}}}({{\mathrm{Supp}}}^{-1}\Phi )=\Phi$ . In fact, for any $I\in{{\mathrm{Supp}}}({{\mathrm{Supp}}}^{-1}\Phi )$ , there is an object $X\in{{\mathrm{Supp}}}^{-1}\Phi$ , that is ${{\mathrm{Supp}}}X\subseteq \Phi$ , such that $I\in{{\mathrm{Supp}}}X$ . So ${{\mathrm{Supp}}}({{\mathrm{Supp}}}^{-1}\Phi )\subseteq \Phi$ . Conversely, for any $I^{\prime}\in \Phi$ , ${{\mathrm{Supp}}}I^{\prime}=\{I^{\prime}\}\subseteq \Phi$ . Hence, $I^{\prime}\in{{\mathrm{Supp}}}({{\mathrm{Supp}}}^{-1}\Phi )$ . Therefore, $\Phi \subseteq{{\mathrm{Supp}}}({{\mathrm{Supp}}}^{-1}\Phi )$ .

Now let $\mathcal{X}$ be a thick subcategory of $ \mathcal{A}$ , which is closed under taking arbitrary direct sums and injective envelopes. Let $\Phi ={{\mathrm{Supp}}}\mathcal{X}$ . Firstly, $\Phi \subseteq \mathcal{X}$ since $\mathcal{X}$ is closed under taking injective envelopes. We claim that $\Phi$ is coherent. In fact, each morphisms $I^{0}\stackrel{f}\to I^{1}$ with ${{\mathrm{Ind}}}(I^{0}\oplus I^{1})\subseteq \Phi$ can be completed to an exact sequence

\begin{equation*} I^{0} \quad \stackrel {f}{\longrightarrow } \quad I^{1} \quad \longrightarrow \quad I^{2} \end{equation*}

where $I^{2}$ is the injective envelope of ${{\mathrm{Coker}}}f$ . Since $\mathcal{X}$ is closed under taking cokernels and injective envelopes, we have $I^{2}\in \mathcal{X}$ . Hence, ${{\mathrm{Ind}}}I^{2}={{\mathrm{Supp}}}I^{2}\subseteq \Phi$ . So $\Phi$ is coherent. Next, we prove that ${{\mathrm{Supp}}}^{-1}({{\mathrm{Supp}}}\mathcal{X})=\mathcal{X}$ . For any $X\in{{\mathrm{Supp}}}^{-1}({{\mathrm{Supp}}}\mathcal{X})$ , we have ${{\mathrm{Supp}}}X\subseteq{{\mathrm{Supp}}}\mathcal{X}$ . Take the minimal injective resolution of $X$

\begin{equation*}0\to X \stackrel {i^{0}}\to I^{0}\stackrel {i^{1}}\to I^{1}\to \ldots,\end{equation*}

we have ${{\mathrm{Ind}}}I^{i}\subseteq{{\mathrm{Supp}}}\mathcal{X}\subseteq \mathcal{X}$ . Since $\mathcal{X}$ is closed under taking direct sums and kernels, $X\in \mathcal{X}$ . Conversely, for any $X^{\prime}\in \mathcal{X}$ , we have ${{\mathrm{Supp}}}X^{\prime}\subseteq{{\mathrm{Supp}}}\mathcal{X}$ , that is, $X^{\prime}\in{{\mathrm{Supp}}}^{-1}({{\mathrm{Supp}}}\mathcal{X})$ .

In conclusion, the map sending a subcategory $\mathcal{X}$ to ${{\mathrm{Supp}}}\mathcal{X}$ induces a bijection between the set of thick subcategories of $ \mathcal{A}$ , which are closed under taking arbitrary direct sums and injective envelopes, and the set of coherent subsets of ${{\mathrm{Sp}}}(\mathcal{A})$ . By Lemma 3.5, this bijection is inclusion-preserving.

Let $A$ be a commutative noetherian ring. Then, ${{\mathrm{Mod}}}A$ is a locally noetherian Grothendieck category. Let $\mathcal{X}$ be a thick subcategory of ${{\mathrm{Mod}}}A$ , which is closed under taking arbitrary direct sums. Then, $\mathcal{X}$ is closed under taking injective envelopes (see [11, Lemma 3.5]). Take $ \mathcal{A}={{\mathrm{Mod}}}A$ . Applying Theorem 3.6 to ${{\mathrm{Mod}}}A$ , we can get the following corollary.

Corollary 3.7. Let $A$ be a commutative noetherian ring. Then, we have an inclusion-preserving bijection between the set of thick subcategories of ${{\mathrm{Mod}}}A$ , which are closed under taking arbitrary direct sums and the set of coherent subsets of ${{\mathrm{Spec}}}(A)$ .

Finally, we give an example where every subset of ${{\mathrm{Sp}}}(\mathcal{A})$ is coherent. Let $\mathcal{A}$ be an abelian category. For a pair of objects $X$ , $Y$ and $n\geqslant 1$ , let ${{\mathrm{Ext}}}^{n}_{\mathcal{A}}(X,Y)$ denote the abelian group of $n$ - $extensions$ in the sense of Yoneda, that is equivalence classes of exact sequences $\xi \;:\; 0\to Y\to E_{n}\to \cdots \to E_{1}\to X\to 0$ . The $injective$ $dimension$ of $X$ is by definition

\begin{equation*}{{\mathrm {inj.dim}}}X={{\mathrm {inf}}}\{n\geqslant 0|{{\mathrm {Ext}}}^{n+1}_{\mathcal {A}}(-,X)=0\}.\end{equation*}

Recall that an abelian category $\mathcal{A}$ is $hereditary$ provided that the functor ${{\mathrm{Ext}}}^{2}_{\mathcal{A}}(-,-)$ vanishes. If $\mathcal{A}$ has enough injective objects, that is for every objects $X\in \mathcal{A}$ , $X$ has injective envelope, $\mathcal{A}$ is hereditary if and only if for any $X\in \mathcal{A}$ , ${{\mathrm{inj.dim}}}X\leqslant 1$ , if and only if for any $X\in \mathcal{A}$ , there is an exact sequence $0\to X\to I_{0}\to I_{1}\to 0$ such that $I_{0},$ $I_{1}$ are injective objects. Then we have the following result.

Corollary 3.8. Let $\mathcal{A}$ be a locally noetherian Grothendieck category. If $\mathcal{A}$ is hereditary, then every subset of ${{\mathrm{Sp}}}(\mathcal{A})$ is coherent.