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 [Reference Krause12, 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 [Reference Krause12, 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 $X$ .

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.

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, $C\in{{\mathrm{Supp}}}^{-1}\Phi$ .

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, $X\in \mathcal{Y}$ .

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 [Reference Krause11, Lemma 3.5]). Take $ \mathcal{A}={{\mathrm{Mod}}}A$ . Applying Theorem 3.6 to ${{\mathrm{Mod}}}A$ , we can get the following corollary.

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.