ATOMICAL GROTHENDIECK CATEGORIES

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ATOMICAL GROTHENDIECK CATEGORIES

C. N˘AST˘ASESCU and B. TORRECILLAS Received 11 September 2002

Motivated by the study of Gabriel dimension of a Grothendieck category, we intro- duce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories.

2000 Mathematics Subject Classification: 18E15, 16S90.

1. Introduction. Given a Grothendieck categoryᏭ, we can associate with it the lattice of all localizing categories ofᏭand denote it by Tors(Ꮽ). We will show (Theorem 3.3) that ifᏭhas Gabriel dimension, then the lattice Tors(Ꮽ⁾ is semi-Artinian. Moreover, the Gabriel dimension ofᏭis exactly the Loewy length of this lattice. Example 3.4 proves that the converse statement does not hold. (Therefore, the properties of the lattice Tors(Ꮽ⁾do not determine the properties of the categoryᏭ.) These facts suggest introducing the con- cept of atomical Grothendieck category. Precisely,Ꮽwill be called atomical if the lattice Tors(Ꮽ⁾has only two elements, that is, Ꮽhas only two localizing subcategories, namely,{0}andᏭ. The classification of atomical Grothendieck categories is given inSection 4.

2. Preliminaries. Throughout this paper,Ꮽwill denote a Grothendieck cat- egory, that is, an abelian category with a generator, such that colimits exist and direct limits are exact.

It is well known that in a Grothendieck category each objectXhas an injec- tive hull, denoted in the sequel byE(X).

IfᏭis a category, then by a subcategoryᏮofᏭwe will always mean a full subcategory ofᏭ^.

A subcategoryᏯofᏭis calledclosed(or hereditary pretorsion class) if it is closed with respect to kernels, cokernels, and direct sums.

Byσ [X]we denote the full subcategory ofᏭwhose objects are subobjects ofX-generated objects. These objects are said to be subgenerated byX, and Xis a subgenerator ofσ [X]. This is the smallest closed full subcategory ofᏭ containingX.

By definition, the objects ofσ [X]form a closed subcategory inᏭ. On the other hand, every closed subcategory᐀ ⁱⁿ Ꮽis of the form σ [X]for some objectX; for example, forXequals the direct sum of all (nonisomorphic) cyclic objects in᐀.

Following Goldman [2], a functorτ:Ꮽ→Ꮽis called a kernel functor if (1) it is a subfunctor of the identity functor, that is,τ(M)⊆Mandf:M→

Mimpliesf (τ(M))⊆τ(M);

(2) N⊆Mimpliesτ(N)=N∩τ(M).

The trivial kernel functors 0 and∞are defined by setting 0(X)=0, and∞(X)= X, for every objectX∈Ꮽ^.

For any kernel functorτ,X is called aτ-torsion module ifτ(X)=X, and aτ-torsion-free module ifτ(X)=0. The collection of᐀τ of all theτ-torsion module is a closed subcategory ofᏭ. Conversely, for any closed subcategory Ꮿ, there exists a unique kernel functorτsuch thatᏯ=᐀τ.

Lemma2.1. LetGbe a generator ofᏭandᏯa closed subcategory. Then Ꮿ=σ

⊕{G/X|G/X∈Ꮿ}. (2.1)

Corollary2.2. The closed subcategories (and hence, the kernel functors) form a set.

Proposition2.3. The set of all closed subcategories ofᏭis a complete lat- tice. For a family{Xλ}Λof objects ofᏭ,

Λ

σX_λ

=σ

⊕ΛX_λ,

Λ

σ Xλ

=

Λ

σ Xλ

. (2.2)

Remark2.4. (1) (cf. [4]). For a coalgebraC, the lattice of all closed subcate- gories of the category of comodules overCis anti-isomorphic to the lattice of subcoalgebras ofC.

(2) The Serre subcategories ofᏭ(i.e., the subcategories᏿^of Ꮽ^satisfying that for any exact sequence fromᏭ,

0 →X →X →X →0, (2.3)

whereXis in᏿if and only ifXandXare in᏿) do not form a set. For example, we consider the Grothendieck categoryᐂof vector spaces over a division ring.

For any cardinalα, the subcategory of all vector spaces of dimension less than or equal toαis a Serre subcategory. Thus, the Serre subcategories ofᐂ^{are not} a set.

We now recall the notion of semi-Artinian lattice. LetLbe an upper contin- uous and modular lattice. An atom ofLis a nonzero elementa∈Lsuch that wheneverb∈Landb < a, thenb=0, that is, the interval[0,a]has exactly two elements, 0 anda. Ifa,b∈Landx < y, then the interval[x,y]is simple ifyis an atom in the sublattice[x,y]ofL. The lattice is called semiatomic if 1 is a joint of atoms, andLis calledsemi-Artinianif for everyx∈L,x≠1, the sublattice[x,1]ofLcontains an atom.

The(ascending) Loewy series ofL,

s0(L) < s1(L) <···< s_λ(L)(L), (2.4) is defined recursively as follows:s0(L)=0,s1(L)is the socle Soc(L)ofL(i.e., the join of all atoms ofL), and if the elementss_β(L)ofLhave been defined for all ordinalsβ < α, thens_α(L)=

β<αs_β(L)ifαis a limit ordinal ands_α= Soc([sγ(L),1])ifα=γ+1.

TheLoewy lengthλ(L)ofLis the least ordinal such thats_λ(L)=s_λ+1(L).

3. Gabriel dimension and localizing subcategories. A subcategory᐀⊆Ꮽ is a localizing subcategory if it is closed under subobjects, quotient objects, extensions, and coproducts. IfᏮ⊆Ꮽis an arbitrary subcategory, we denote by᐀⁽Ꮾ⁾the smallest localizing subcategory containingᏮ^.

Examples3.1. (i) An objectA∈Ꮽ^issingular if there exists a short exact sequence

0 →A →A →A →0, (3.1)

where the monomorphism is essential.

In any Grothendieck category, we can always consider theGoldie localizing subcategory, denoted byᏳ, as the smallest localizing subcategory containing the singular objects.

(ii) We can associate to any injective objectE∈Ꮽa localizing subcategory

᐀E=

A∈Ꮽ|Hom_Ꮽ(A,E)=0

. (3.2)

This localizing subcategory is said to be cogenerated byE.

(iii) For a projective objectP∈Ꮽ, we can define

᐀P=A∈Ꮽ|Hom_Ꮽ(P,A)=0. (3.3) It is clear that᐀P is a localizing subcategory closed under direct product.

(iv) If S is a simple object in Ꮽ, we denote byᏭS the smallest localizing subcategory containingS. In fact,

ᏭS= {M∈Ꮽ|N⊂M, M/Ncontains a simple object isomorphic toS}. (3.4) The objects in this localizing subcategory are calledS-primary.

Let ᐀ be a localizing subcategory. The corresponding torsion functor or idempotent kernel functor is denoted by

t_᐀:Ꮽ →᐀. (3.5)

This functor assigns to an object A∈Ꮽ the maximal subobject t_᐀(A)⊆A in᐀. An objectX∈Ꮽis᐀-torsion-free (resp.,᐀-torsion) ift_᐀(X)=0 (resp.,

t_᐀(X)=X). LetH¹t_᐀ denote the first higher derived functor of the left exact functort_᐀. A᐀-torsion-free objectE∈Ꮽis᐀-closed ifH¹t_᐀=0.

If᐀is a localizing subcategory ofᏭ, we can consider the quotient category Ꮽ/᐀. We denote byT_᐀:Ꮽ→Ꮽ/᐀the canonical functor and byS_᐀:Ꮽ/᐀→Ꮽ the right adjoint functor ofT_᐀.

It is well known that the categoryᏭ^/᐀is equivalent to the full subcategory ofᏭof᐀-closed objects.

It is well known thatᏭhas a set of localizing subcategories Tors(Ꮽ). Given a family of localizing subcategories(Ꮿi)_i∈I, we define the meet by _i∈IᏯi= i∈IᏯi, and the join by

i∈IᏯi, as the smallest localizing subcategory contain- ing the union of theᏯi. Notice that Tors(Ꮽ)is not a sublattice of the lattice of all closed subcategories ofᏭ. It is also known that this set is a frame (i.e., it is a complete latticeLsuch thata∧(X)=

{a∧x|x∈X}for each elementa and subsetXofL). Frames are also known as local lattices, complete Heyting algebras, or complete Brouwerian lattices. The lattice of closed subcategories is not a frame in general.

We need the following preliminary result.

Proposition3.2. LetᏭbe a Grothendieck category and letᏯ⊆Ꮽbe a lo- calizing subcategory. There exists a bijective correspondence between the local- izing subcategories ofᏭ^/Ꮿand the localizing subcategoriesᏮ^ofᏭ^containing Ꮿ. Moreover,Tors(Ꮽ/Ꮿ)is a subframe of Tors(Ꮽ)

Proof. LetT:Ꮽ→Ꮽ/Ꮿbe the canonical functor. ConsiderᏮ, a localizing subcategory ofᏭcontainingᏯ, thenT (Ꮾ⁾= {Z∈Ꮽ^/Ꮿ|ZT (X),X∈Ꮾ}is a localizing subcategory ofᏭ^/Ꮿ. In fact, it is clear thatT (Ꮾ⁾is closed under subobjects, quotients, and direct sums. It remains to show thatT (Ꮾ)is closed under extensions. First, we observe thatT (Ꮾ)= {Z∈Ꮽ/Ꮿ|S(Z)∈Ꮾ}. To see this, consider the exact sequence

0 →Kerf →X →ST (X)S(Z) →Cokerf →0, (3.6) where Kerf, Cokerf∈Ꮿ. Therefore, Kerf, Cokerf∈Ꮾ, andX∈Ꮾif and only ifS(Z)∈Ꮾ. Now if

0 →Z →Z →Z →0 (3.7)

is an exact sequence inᏭ^/Ꮿ, withZ,Z ∈T (Ꮾ), we apply the functor S to obtain

0→S(Z) →S(Z) →S(Z). (3.8)

Thus,S(Z)∈ᏮandZ∈T (Ꮾ).

LetᏰbe a localizing subcategory inᏭ/Ꮿ. we defineT⁻¹Ᏸ= {X∈Ꮽ|T (X)∈ Ᏸ}. SinceTis an exact functor which commutes with direct sums, thenT⁻¹(Ᏸ⁾ is a localizing subcategory which containsᏯ. It is not difficult to see that these

two operations establish a bijection between the localizing subcategories of Ꮽ/Ꮿand the localizing subcategoriesᏮofᏭcontainingᏯ.

We now recall the notion of Gabriel dimension of a Grothendieck category Ꮽ. For any ordinalα, we will denote byᏯαthe localizing subcategory defined in the following way:Ꮿ0is the zero subcategory;Ꮿ1is the smallest localizing subcategory containing all simple objects; ifα=β+1, an objectXofᏭ^will be contained inᏯα if and only ifTβ(X)∈Ob(Ꮽ/Ꮿβ)1, whereTβ:Ꮽ→Ꮽ/Ꮿβ

is the canonical functor; and ifαis a limit ordinal, thenᏯαis the localizing subcategory generated by all localizing subcategoriesᏯβ, withβ≤α.

It is clear that ifα≤α, thenᏯα⊆Ꮿα. Hence, there exists an ordinalτsuch thatᏯτ=Ꮿαfor any ordinalα≥τ. We defineᏯτ= ∪αCα.

The set of localizing subcategoriesᏯα is called theGabriel filtration ofᏭ^. We say that an objectX ofᏭhasGabriel dimension ifX is inᏯτ. Then the smallest ordinalαverifyingXinᏯαis called the Gabriel dimension ofX.

We say thatᏭhas Gabriel dimension ifᏭ=Ꮿτor, equivalently, any object of Ꮽhas Gabriel dimension. We are now ready for the main result of this section.

Theorem3.3. LetᏭbe Grothendieck category. IfᏭhas Gabriel dimension α, thenTors(Ꮽ⁾is a semi-Artinian lattice with Loewy lengthα.

Proof. We will show this result by transfinite induction. IfG- dimᏭ=1, thenᏭ=Ꮿ1, the localizing subcategory generated by the simple objects ofᏭ. Hence,Ꮽ=

ᏯS.

Now, we assume that the result is true for any Grothendieck category of Gabriel dimensionβ < α. Ifα=γ+1 is not a limit ordinal, then any objectX∈ Ꮽbelongs toᏯαor, equivalently,T_γ(X)∈(Ꮽ^/Ꮿγ)1. Now,Ꮿγ=s_γ(Tors(Ꮿγ)).

IfX∈Ꮽsatisfies thatT_γ(X)is a simple object inᏭ^/Ꮿγ, then(Ꮽ^/Ꮿγ)_Tγ(X) is an atom in Tors(Ꮽ/Ꮿγ). ByProposition 3.2, T_γ⁻¹((Ꮽ/Ꮿγ)Tγ(X))is an atom in [Ꮿγ,Ꮽ]. We will see thatᏭ=T_γ⁻¹(Ꮽ^/Ꮿγ)_Tγ(X). LetA∈Ꮽ and considerA→ A→0, withA≠0. Applying the functorT_γ, we obtainT_γ(A)→T_γ(A)→0.

If T (A)=0, the proof is finished; otherwise Tγ(A)contains a simple ob- jectT_γ(X). Therefore, we haveK→X→AandAcontainsX/K which is in T_γ⁻¹(Ꮽ^/Ꮿγ)_Tγ(X). Ifαis a limit ordinalᏭ=

β<αᏯβ, thenᏭ=

β<αᏯβ. The next example shows that the converse ofTheorem 3.3is not true.

Example 3.4. Let R be a commutative nondiscrete valuation domain of Krull dimension 1, with maximal idealM. Then

(i) M²=M,

(ii) ifx∈M, then

n≥0Rxⁿ=0, (iii) Tors(R- Mod)has four elements:

{0} ⊆(R- Mod)_R/M⊆᐀⊆R- Mod, (3.9) where᐀is the usual torsion theory in a domain and(R- Mod)_R/M is a semisimple category,

(iv) the quotient categoryR- Mod/(R- Mod)R/M has no simple objects, (v) R- Mod has no Gabriel dimension.

Proof. (i) Takex∈M. Since the valuation is not discrete, we can find an elementy∈M such thatv(y²)=2v(y) < v(x). Hence,x∈(y²)⊆M²and M=M².

(ii) Letᏽ=

n≥0Rxⁿ. We claim thatᏽis a prime ideal. Leta,b∈Rwitha∈ᏽ andb∈ᏽ. Hence, there existnandmsuch thata∈Rxⁿandb∈Rx^m. Thus, Rxⁿ⊂RaandRx^m⊂Rb. ThenRx^n+m⊂Rxⁿb⊂Rabandab∈ᏽ. Therefore, ᏽ=0.

(iii) LetᏯbe a localizing subcategory properly containing(R- Mod)_R/M and letIbe a nonzero ideal. We takeJMwithR/J∈Ꮿ. Thus, there existx∈M\J.

By (ii),

n≥0Rxⁿ=0, and it follows thatRxⁿ⊆Ifor somenandR/I∈Ꮿ. (iv) Any simple object ofR- Mod/(R- Mod)_R/M is given by an(R- Mod)_R/M- critical ideal, but this kind of ideals is prime. This prime is 0. So the cocritical module is isomorphic toR. Therefore,R/I is semisimple for every nonzero idealIofR—a contradiction.

(v) The proof follows from (iv).

4. Atomical Grothendieck categories. We have proved inTheorem 3.3that if a Grothendieck category has Gabriel dimension, then the lattice of localiz- ing subcategories is semi-Artinian.Example 3.4shows that the converse is not true. This fact suggests the study of Grothendieck categoriesᏭwith the prop- erty that the categoryᏭis an atom in the lattice Tors(Ꮽ), that is,Ꮽ^{has only} two localizing subcategories{0}andᏭ.

Definition4.1. A Grothendieck categoryᏭis calledatomicalif it has only two localizing subcategories, namely,{0}andᏭ.

A maximal localizing category᐀is a maximal element of Tors(Ꮽ)−Ꮽ. By Proposition 3.2,Ꮽis a maximal localizing category ofᏭif and only ifᏭ^/᐀^is an atomical Grothendieck category.

Recall that an object C in Ꮽ is called a cogenerator if for each nonzero morphismf:X→YinᏭ, there exists a morphismg:Y →Csuch thatgf≠0.

This is equivalent to the existence of a monomorphismA→C^Ifor some index set I, for every objectA∈Ꮽ. It is clear that an injective objectE of Ꮽis a cogenerator if and only if for each nonzero objectA∈Ꮽ, there exists a nonzero morphismf:A→E.

Proposition 4.2. If Ꮽis a Grothendieck category, thenᏭ is an atomical category if and only if every nonzero injective object ofᏭis a cogenerator.

Moreover, if the category has enough projectives, thenᏭis an atomical cat- egory if and only if every nonzero projective object ofᏭis a generator.

Proof. Assume thatᏭis atomical, then any nonzero injective object co- generates a nonzero torsion-free class. Hence, this torsion-free class must be

the whole category and this injective is a cogenerator. Since any localizing subcategory ofᏭis cogenerated by an injective object, the converse is clear.

It is clear that for an atomical Grothendieck categoryᏭ, we have that the Goldie torsion theory has to be either{0}orᏭ. In the first case, we say that Ꮽ is a nonsingular Grothendieck category and we characterize this type of simple Grothendieck categories. Recall that a Grothendieck categoryᏭis called spectral if any short exact sequence splits and a spectral Grothendieck category is called discrete if every object is semisimple.

Proposition4.3. LetᏭbe a Grothendieck category. The categoryᏭis non- singular atomical if and only ifᏭis a spectral category which is equivalent to R-Mod/Ᏻ^{, whereR}is a regular prime self-injective ring andᏳis the Goldie lo- calizing subcategory. Moreover,Ꮽcontains a simple object if and only ifR is isomorphic to the ring of all linear transformations of a left vector space over a division ring.

Proof. SupposeᏭ is nonsingular and atomical. Since Ꮽ is nonsingular, thenᏳ=0. Hence,X⊆E(X) withE(X)/X singular, a matter which implies thatX=E(X)and any object is injective. Thus,Ꮽis a spectral Grothendieck category.

LetU be a generator ofᏭandR=Hom_Ꮽ(U,U), by the Gabriel-Oberst the- orem [5, Chapter XII, Theorem 1.3]Ꮽis equivalent toR- Mod/Ᏻ, whereRis a regular self-injective ring andᏳis the Goldie’s localizing subcategory. Since R- Mod/Ᏻis atomical, thenᏳis maximal. Hence, by [1, Theorem 2.2], 0=t_Ᏻ(R) is a prime ideal.

Conversely, assume thatRis a prime self-injective regular ring. SinceR is prime, then it is nonsingular. Thus,t_Ᏻ(R)=0 is a prime ideal, whereᏳ^{is a} maximal localizing subcategory by [1, Theorem 2.2]. Therefore,R- Mod/Ᏻ is an atomical Grothendieck category.

Assume thatᏭcontains a simple object, thenᏭcoincides with the localizing subcategory generated by this simple object. Hence, as an object inR- Mod/Ᏻ, Rcontains a simple object. Therefore, there exists aᏳ-cocritical left idealCof R. IfC is not simple as a leftR-module, then we can find a finitely generated left idealI≠0 contained inC. SinceRis regular, there exists a left idealJsuch thatI⊕J=R. Thus,C=I⊕(J∩C), which is a contradiction sinceIis essential inC. Therefore,Cis a simple left ideal and Soc(R)≠0. By [3, Theorem 9.12],R is the ring of all linear transformations of any left vector space over a division ring.

Conversely, ifR is the ring of all linear transformations of any left vector space over a division ring, then Soc(R)is not zero. Any simple left ideal will produce a simple object in the quotient category.

We will now consider the case where the Goldie torsion theory coincides with the whole category. When the Grothendieck category contains simple objects,

we have the following characterization. Recall that a Grothendieck categoryᏭ is called semi-Artinian if every nonzero object ofᏭcontains a simple object.

Proposition4.4. LetᏭbe a singular Grothendieck category. IfᏭis atomi- cal, and it has simple objects, thenᏭis a semi-Artinian Grothendieck category with a unique isomorphic class of simple objects.

Proof. SinceᏭis atomical, the localizing subcategory generated by a sim- ple object coincides with categoryᏭ. Hence, the result follows.

Proposition4.5. LetᏭ be a locally finitely generated Grothendieck cate- gory. ThenᏭis atomical if and only if any object ofᏭ^isS-primary, andᏭ^is semisimple or singular.

We now specialize our discussion to the module category R- Mod. In this case, we have the following result.

Proposition4.6. R-Modis an atomical category if and only if the ringRis local right perfect.

Proof. If R is local right perfect, then R- Mod is clearly atomical. Con- versely, if R- Mod is atomical andR is nonsingular, then the Goldie torsion theory is trivial. Hence, any module is injective and R is semisimple. Since there is only an isomorphic class of simple modules,Ris simple Artinian. We only need to consider the case whenRis singular. But thenR- Mod=(R- Mod)_S for some simple left R-moduleS and there is only an isomorphic class of left simpleR-modules. Thus,Ris semi-Artinian andJ=ann(S). We will see that R/J is a simple Artinian ring. In fact, consider Soc(R/J)=A/J ≠0. If A≠R, then A⊆M for some maximal left ideal M. Therefore,A(R/M)=0 andA⊆ann(S)=J, a contradiction. Hence,A=RandR/Jis simple Artinian.

SinceRis semi-Artinian,JisT-nilpotent. SinceR/J is simple Artinian andJ isT-nilpotent, thenRis a local right perfect ring.

Now, we consider the case of closed subcategories ofR- Mod.

Corollary4.7. LetM be a leftR-module. Thenσ [M]is an atomical cat- egory if and only if eitherM is semisimple orM isS-primary withS a simple singular leftR-module.

Finally, we present an example of a singular atomical Grothendieck category without simple objects.

Example4.8. We consider the same ring as inExample 3.4. Then the quo- tient category᐀/(R- Mod)(R/M)is an atomical singular Grothendieck category without simple objects.

Proof. We have proved thatR- Mod/(R- Mod)(R/M)has no simple objects, then᐀^{/(R- Mod)}(R/M) has no simple objects. We also know fromExample 3.4 that this category is atomical. We will denote byT:᐀→᐀/(R- Mod)(R/M) the

canonical functor. Let 0≠I⊂M be an ideal of Rsuch thatI≠M. It is clear that R/I∈᐀, and R/I∉(R- Mod)(R/M). LetJ/I be the torsion part ofR/I∈ (R- Mod)_(R/M). SinceM²=M, thenJ⊂MandJ≠M. By the exact sequence

0→J/I →R/I →R/J →0, (4.1)

it follows thatT (R/I)T (R/J). SinceRis a valuation ring, we have thatR/Jis a uniform (coirreducible)R-module, soT (R/J)is still uniform in the quotient category. DenoteX=T (R/J)T (R/I). ThenX is uniform and contains no simple objects (because the category does not have nonzero simple objects).

Then we can considerY as a nonzero subobject ofX such thatY ≠X. It is clear thatX/Y belongs to the Goldie torsion theory (of the quotient category) andX/Y≠0. As the quotient category is an atomical category, it must be the same as the Goldie torsion theory.

Acknowledgment. The research of the second author was partially sup- ported by Grant BFM2002-02717 from MCT.

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C. N˘ast˘asescu: Faculty of Mathematics, University of Bucharest, RO 70109 Bucharest 1, Romania

E-mail address:[email protected]

B. Torrecillas: Departmento de Álgebra y Análisis Matemático, Universidad de Alme- ría, 04071 Almería, Spain

E-mail address:[email protected]