ATOM SPECTRA OF GROTHENDIECK CATEGORIESRYO KANDA

Abstract. This paper explains recent progress on the study of Grothendieck categories using
the atom spectrum, which is a generalization of the prime spectrum of a commutative ring.
As a part, we give a classification of localizing subcategories which can be applied to both
locally noetherian schemes and noncommutative noetherian rings. It is shown that the atom
spectrum of a Grothendieck category can have a rich poset structure compared with the prime
spectrum of a commutative ring. We also show some properties on minimal elements of theatom spectrum for noncommutative noetherian rings.

1. Introduction
The aim of this paper is to explain recent progress on the study of Grothendieck categories.
We investigate a Grothendieck category by using a kind of spectrum, which we call the atom
spectrum. A typical example of a Grothendieck category is the category of modules over a ring.
In the case where the ring is commutative, the atom spectrum of the module category coincides
with the prime spectrum of the commutative ring. Therefore this theory can be regarded as
an attempt to generalize the notion of the prime spectrum to noncommutative rings. It seems
possible to understand and reformulate some classical noncommutative ring theory from thecategorical viewpoint.
The theory of atom spectrum is not only for the study of noncommutative rings. Anotherexample of a Grothendieck category is the category of quasi-coherent sheaves of a scheme. We
can show that the atom spectrum of the category of quasi-coherent sheaves of a locally noetherian
scheme coincides with the set of points of the scheme, and as a consequence, we can show aclassification of localizing subcategories in a general setting including both the case of locallynoetherian schemes and the case of noncommutative noetherian rings.
The reader may find the details of this paper in [Kan12a], [Kan12b], [Kan13], and [Kan14].
The reader who is unfamiliar with terms of abelian categories may be referred to [Pop73] or[Ste75].
We start with the definition of a Grothendieck category.
Definition 1.1. An abelian category A is called a Grothendieck category if it satisfies the following conditions.
(1) A admits arbitrary direct sums (and hence arbitrary direct limits), and for every direct
system of short exact sequences in A, its direct limit is also a short exact sequence.
(2) A has a generator G, that is, every object in A is isomorphic to a quotient object of thedirect sum of some copies of G.
As we mentioned, the category Mod Λ of right modules over a ring Λ and the category QCoh X
of quasi-coherent sheaves on a scheme X (see [Con00, Lemma 2.1.7]) are Grothendieck categories.
2010 Mathematics Subject Classification. 18E15 (Primary), 16D90, 14A22, 13C60 (Secondary).
Key words and phrases. Grothendieck category; atom spectrum; localizing subcategory.
This paper contains an announcement of our results. The detailed version concerning the results will be
submitted for publication elsewhere.
The author is a Research Fellow of Japan Society for the Promotion of Science. This work is supported by
Grant-in-Aid for JSPS Fellows 25·249.
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RYO KANDA

One might think the notion of Grothendieck categories is quite an abstract setting given inorder to include module categories. However, it is shown that every Grothendieck category is apart of some module category.
Theorem 1.2 (Gabriel and Popescu [PG64, Proposition]). Let A be a Grothendieck category.
Then there exist a ring Λ and a localizing subcategory X of Mod Λ such that A is equivalent to(Mod Λ)/X .
In this paper, we adopt Grothendieck categories as main objects to study. Recall that for acommutative ring R, the prime spectrum Spec R plays an fundamental role. For a Grothendieck
category A, we will consider the atom spectrum ASpec A.
2. Atom spectrum
From now on, let A be a Grothendieck category. The atom spectrum of a Grothendieck
category is defined by using the notion of monoform objects.
Definition 2.1. A nonzero object H in A is called monoform if for each nonzero subobject L
of H, no nonzero subobject of H is isomorphic to a subobject of H/L.
In the case of a commutative ring, the following result shows how monoform objects are related
to prime ideals.
Proposition 2.2 ([Sto72, Lemma 1.5]). Let R be a commutative ring and a an ideal of R. Then
R/a is a monoform object in Mod R if and only if a is a prime ideal of R.
We state basic properties of monoform objects.
Proposition 2.3. Let H be a monoform object in A.
(1) ([Kan12a, Proposition 2.2]) Every nonzero subobject of H is again monoform.
(2) ([Kan12a, Proposition 2.6]) H is uniform, that is, for every nonzero subobjects L1 andL2 of H, we have L1 ∩ L2 6= 0.
Even in the case of a commutative ring R, the collection of monoform objects is quite different
from the set of prime ideals. Indeed, it is known that the residue field k(p) = Rp /pRp is amonoform object in Mod R for each prime ideal p of R ([Sto72, p. 626]). Hence all its submodules
are monoform. See [Kan12a, Example 8.3] for an example of a noncommutative ring. In order toobtain a generalization of the prime spectrum of a commutative ring, we introduce an equivalence
relation between monoform objects.
Definition 2.4. We say that monoform objects H1 and H2 in A are atom-equivalent (denotedby H1 ∼ H2 ) if there exists a nonzero subobject of H1 isomorphic to a subobject of H2 .
Definition 2.5. The atom spectrum ASpec A of A is defined by{ monoform objects in A }ASpec A =.
∼Each element of ASpec A is called an atom in A. For each monoform object H in A, theequivalence class of H is denoted by H.
The notion of atoms was originally introduced by Storrer [Sto72], and the generalization toabelian categories was stated in [Kan12a].
The following result shows that the atom spectrum is a generalization of the prime spectrumof a commutative ring.
Theorem 2.6 (Storrer [Sto72, p. 631]). Let R be a commutative ring. Then the map
Spec R → ASpec(Mod R)
given byp 7→ R/pis bijective.

ATOM SPECTRA OF GROTHENDIECK CATEGORIES3

For a locally noetherian scheme X, the atom spectrum of QCoh X coincides with the set ofpoints of X.
Theorem 2.7 ([Kan14, Theorem 7.6]). Let X be a locally noetherian scheme. Then the map|X| → ASpec(QCoh X)
given byx 7→ jx ∗ k(x)is bijective, where k(x) is the residue field of x, and jx : Spec OX,x → X is the canonical morphism.
Matlis’ correspondence between the prime ideals and the indecomposable injective modules
can be generalized to a wide class of Grothendieck categories including the category Mod Λ for aright noetherian ring Λ.
For an object M in a Grothendieck category A, the injective envelope E(M ) of M always
exists and it is unique up to isomorphism (see [Pop73, Theorem 10.10]).
We recall the statement of Matlis’ correspondence.
Theorem 2.8 (Matlis [Mat58, Proposition 3.1]). Let R be a commutative noetherian ring. Thenthe map{ indecomposable injective R-modules }Spec R →∼=given byp 7→ E(R/p)
is bijective.
In order to generalize Matlis’ correspondence, we need to consider some noetherianness of aGrothendieck category. The notion of the locally noetherianness is well-investigated one.
Definition 2.9. A Grothendieck category A is called locally noetherian if there exists a generating set GLof A consisting of noetherian objects, that is, A admits a set G of noetherian objects
such that G∈G G is a generator of A.
For a ring Λ, the Grothendieck category Mod Λ is locally noetherian if and only if Λ is rightnoetherian. Therefore the following generalization can be applied to right noetherian rings.
Theorem 2.10 ([Kan12a, Theorem 5.9]; see also [Sto72, Corollary 2.5]). Let A be a locallynoetherian Grothendieck category. Then the map{ indecomposable injective objects in A }ASpec A →∼=given by
H 7→ E(H)is bijective.
3. Classification of localizing subcategories
In this section, we state a classification of localizing subcategories.
Definition 3.1. A full subcategory X of A is called a localizing subcategory if the following
conditions are satisfied.
(1) X is closed under subobjects, quotient objects, and extensions. In other words, for every
exact sequence
0→L→M →N →0
in A, we have M ∈ X if and only if L, N ∈ X .
(2) X
L is closed under arbitrary direct sums, that is, for every set S of objects in X , we have
M ∈S M ∈ X .

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We recall a classification of localizing subcategories for a commutative noetherian ring. This
classification given by [Gab62] is regarded as an origin of many kinds of classification of subcategories.
For a commutative ring R, we say that a subset Φ of Spec R is closed under specialization if
for every p ⊂ q in Spec R, the assertion p ∈ Φ implies q ∈ Φ.
Theorem 3.2 (Gabriel [Gab62, Proposition VI.4]). Let R be a commutative noetherian ring.
Then the map{ localizing subcategories of Mod R } → { specialization-closed subsets of Spec R }given by
X 7→[

Supp MM ∈X

is bijective. The inverse map is given byΦ 7→ { M ∈ Mod R | Supp M ⊂ Φ }.
The key notion to generalize Gabriel’s classification is the “support” of an object in a
Grothendieck category. It is defined in terms of atoms as follows.
Definition 3.3. For each object M in A, define the subset ASupp M of ASpec A by
∼ L0 /L for some L ⊂ L0 ⊂ M }.
ASupp M = { H ∈ ASpec A | H =This is called the atom support of M .
Proposition 3.4 ([Kan13, Proposition 3.2]). The set{ ASupp M | M ∈ A }satisfies the axioms of open subsets of ASpec A.
This simple proposition is quite impressive from the viewpoint of ring theory. For a commutative ring R, the set of subsets of the form Supp M is exactly the set of specialization-closed
subsets, and hence it is also closed under infinite intersection. However, this is not necessarily
true for a Grothendieck category. Indeed, Example 4.3 gives a counter-example.
We call the topology on ASpec A defined by Proposition 3.4 the localizing topology.
We define maps which will be used in the generalized classification of localizing subcategories.
Definition 3.5.
(1) For a full subcategory X of A, define the subset ASupp X of ASpec A by[
ASupp X =ASupp M.
M ∈X

(2) For a subset Φ of ASpec A, define the full subcategory ASupp−1 Φ of A by
ASupp−1 Φ = { M ∈ A | ASupp M ⊂ Φ }.
We introduce a class of Grothendieck categories, which includes all locally noetherian
Grothendieck categories, in particular Mod Λ for a right noetherian ring Λ, and QCoh X fora locally noetherian scheme X (which is not necessarily a locally noetherian Grothendieck category. See [Har66, p. 135, Example]).
Definition 3.6. We say that a Grothendieck category A has enough atoms if A satisfies thefollowing conditions.
(1) Every injective object in A has an indecomposable decomposition.
(2) Each indecomposable injective object in A is isomorphic to E(H) for some monoform
object H in A.
See [Kan14] for more details on Grothendieck categories with enough atoms. It is shown in[Kan14, Theorem 7.6] that the Grothendieck category QCoh X has enough atoms for every locallynoetherian scheme X.

ATOM SPECTRA OF GROTHENDIECK CATEGORIES

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Theorem 3.7 ([Kan14, Theorem 6.8]; see also [Her97, Theorem 3.8], [Kra97, Corollary 4.3], and[Kan12a, Theorem 5.5]). Let A be a Grothendieck category with enough atoms. Then the map{ localizing subcategories of A } → { specialization-closed subsets of ASpec A }given by
X 7→[

ASupp MM ∈X

is bijective. The inverse map is given byΦ 7→ { M ∈ A | ASupp M ⊂ Φ }.
For a localizing subcategory X of A, it is known that the categories A and A/X are
Grothendieck categories (see [Pop73, Corollary 4.6.2]). It is natural to ask how their atom
spectra are related to each other.
Proposition 3.8 ([Kan13, Proposition 5.12 and Theorem 5.17]; see also [Kra97, Corollary 4.4]and [Her97, Proposition 3.6]). Let X be a localizing subcategory.
(1) ASpec X is homeomorphic to the open subset ASupp X of ASpec A.
(2) ASpec(A/X ) is homeomorphic to the closed subset ASpec A \ ASupp X of ASpec A.
In particular, under the identifications by these homeomorphisms, we haveA
ASpec A = ASpec X q ASpec
X
as a set.
4. Partial order
In this section, we introduce a partial order on the atom spectrum and investigate its structure.
Definition 4.1. Let α, β ∈ ASpec A. We write α ≤ β if α belongs to the topological closure {β}of β with respect to the localizing topology.
In fact, the relation ≤ is a partial order on ASpec A (see [Kan13, Proposition 3.5]). The
following result shows that this is a generalization of the inclusion relation between prime ideals
of a commutative ring.
Proposition 4.2 ([Kan13, Proposition 4.3]). For a commutative ring R, the bijection in Theorem 2.6 gives an isomorphism(Spec R, ⊂) ∼= (ASpec(Mod R), ≤)of posets.
For a commutative ring R, the open subsets of Spec R with respect to the localizing topology
is exactly the specialization-closed subsets. Therefore the localizing topology on Spec R and theposet (partially ordered set) structure of Spec R can be recovered from each other. However, as
the next example shows, the localizing topology cannot necessarily be recovered from the poset
structure for a Grothendieck category.
Example 4.3 ([Pap02, Example 4.7]). Let k be a field. We consider the graded ring k[x]with deg x = 1. The category GrMod k[x] of Z-graded k[x]-modules with degree-preserving
homomorphisms is a locally noetherian Grothendieck category. For each object M in GrMod k[x]and i ∈ Z, the object M (i) in GrMod k[x] is defined by M (i)j = Mi+j . Let S := k[x]/(x). Then
we have
ASpec(GrMod k[x]) = {k[x]} ∪ { S(i) | i ∈ Z }.
Note that k[x] = k[x](i) for each i ∈ Z and that S(i) = S(j) if and only if i = j.
/ Φ or there exists n ∈ Z such
A subset Φ of ASpec(GrMod k[x]) is open if and only if k[x] ∈that Φn ⊂ Φ, whereΦn := {k[x]} ∪ { S(i) | i ≤ n }.

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RYO KANDA

Although all Φn are open, their intersection\

Φn = {k[x]}n∈Z

is not open. Since every element of ASpec(GrMod k[x]) is a closed point, we have α ≤ β in
ASpec(GrMod k[x]) if only if α = β.
Since we have the naturally defined partial order on ASpec A, it is expected to investigate its
general property. Let us recall the case of commutative rings. For every commutative ring R, theposet Spec R has a maximal element and a minimal element. Some other properties of Spec R
were also known (see for example, [Kap74, Theorem 11]). The next theorem, essentially shownby Hochster [Hoc69], states all general properties of the poset Spec R. The precise statement wasgiven by Speed [Spe72].
Theorem 4.4 (Hochster [Hoc69, Proposition 10] and Speed [Spe72, Corollary 1]). Let P be aposet. Then the following assertions are equivalent.
(1) There exists a commutative ring R such that P ∼= Spec R as a poset.
(2) P is an inverse limit of finite posets.
We establish the same type of result for Grothendieck categories, but the conclusion is quite
different from the case of commutative rings.
Theorem 4.5 ([Kan13, Theorem 7.27]). For every poset P , there exists a Grothendieck category
A such that P ∼= ASpec A as a poset.
The construction uses colored quivers. See [Kan13] for the details.
Theorem 4.5 shows that there are quite various kinds of Grothendieck categories comparedwith commutative rings. By combining this theorem with Theorem 1.2 and Proposition 3.8, we
also realize a diversity of noncommutative rings.
Corollary 4.6 ([Kan13, Corollary 5.19]). For every poset P , there exists a ring Λ such that P
is homeomorphic to a closed subset of ASpec(Mod Λ).
From now on, we state some results on the poset structure of the atom spectrum of aGrothendieck category with some noetherian property.
Proposition 4.7 ([Kan13, Proposition 4.6]). Let A be a locally noetherian Grothendieck category.
Then ASpec A satisfies the ascending chain condition.
The previous proposition is expected from the analogous result on commutative noetherianrings. On the other hand, we will see a different phenomenon about minimal elements. Denote
by AMin A the set of minimal elements of ASpec A.
Proposition 4.8 ([Kan13, Proposition 8.2]). There exists a locally noetherian Grothendieck
category A such that AMin A = ∅.
We regard this proposition as a consequence of the weakness of the condition of the locallynoetherianness. Instead of this condition, we consider a Grothendieck category having a noetherian generator. Note that for every right noetherian ring Λ, the Grothendieck category Mod Λ has
the noetherian generator Λ. We obtain the following result with an impressive proof.
Theorem 4.9. Let A be a Grothendieck category with a noetherian generator.
(1) ([Kan13, Proposition 4.7]) For each β ∈ ASpec A, there exists α ∈ AMin A such thatα ≤ β.
(2) ([K]) AMin A is a finite set.
Sketch of proof. (2) It can be shown that Φ := ASpec A \ AMin A is an open subset of ASpec A.
Let X := ASupp−1 Φ. Then we have ASpec(A/X ) = AMin A. Let G be a noetherian generatorof A and G0 its image in A/X . Then G0 is a generator of A/X which is of finite length. Therefore
ASpec(A/X ) = ASupp G0 is a finite set.


ATOM SPECTRA OF GROTHENDIECK CATEGORIES

7

Note the following result on Grothendieck categories.
Theorem 4.10 (Năstăsescu [Nas81, Theorem 3.3]). Let A be a Grothendieck category with anartinian generator. Then there exists a right artinian ring Λ such that A ∼= Mod Λ.
For a given right noetherian ring Λ, the category Mod Λ is a Grothendieck category with
the noetherian generator Λ. By the above argument, there exists a right artinian ring Λ0 suchthat A/X ∼= Mod Λ0 , where X = ASupp−1 (ASpec A \ AMin A). In particular, AMin(Mod Λ) =ASpec(Mod Λ0 ). Consequently, we obtain a right artinian ring (unique up to Morita equivalence)from a right noetherian ring in a categorical way.
5. Acknowledgement
The author would like to express his deep gratitude to Osamu Iyama for his elaborated guidance. The author thanks Shiro Goto, Mitsuyasu Hashimoto, S. Paul Smith, Ryo Takahashi, andYuji Yoshino for their valuable comments.
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Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, Aichi-ken,
464-8602, Japan
E-mail address: kanda.ryo@a.mbox.nagoya-u.ac.jp