Torsion classes generated by silting modules

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by  Simion Breaz

TORSION CLASSES GENERATED BY SILTING MODULES
6 SIMIONBREAZANDJANZˇEMLICˇKA
1
0
2 Abstract. Westudytheclassesofmoduleswhicharegenerated byasilting
module. In the case of either hereditary or perfect rings it is proved that
r these are exactly the torsion T such that the regular module has a special
a
T-preenvelope. In particular every torsion enveloping class in Mod-R are of
M
theformGen(T)foraminimalsiltingmoduleT. Forthedualcaseweobtain
for general rings that the covering torsion-freeclasses of modules are exactly
7 theclassesoftheformCogen(T),whereT isacosiltingmodule.

]
T
R
. 1. Introduction
h
t Silting modules were introduced in [3] in order to extend the τ-tilting theory,
a
m developed in [1] and [11] (for finitely generated modules over artin algebras), to
infinitelygeneratedmodules. Thedualnotion,i.e. cosiltingmodules,wasstudiedin
[
[8]. AmoduleT ∈Mod-Risa(co)silting module ifthe classofallT-(co)generated

2
modules is a torsion (torsion-free) class of a special form.
v
Itwasprovedin[1,Proposition1.1andSection2]thatinthecaseoffinitelygen-
5
5 erated modules over artin algebras the classes of the form gen(T) (i.e. epimorphic
6 images of finite direct sums of copies of T) induced by a τ-tilting module coincide
6 to the torsion classes which are enveloping. We refer to [6, Section 5] for similar
0
characterizations in the (co)tilting cases. As in the case of infinitely generated
.
1 (co)tilting modules, [5], a naturalquestionis to ask for characterizationsof torsion
0 classeswhichareoftheformGen(T)(respectivelyCogen(T))fora(co)siltingmod-
6
uleT. HereGen(T)(Cogen(T))denotesthetheclosuretoisomorphismsoftheclass
1
of all quotients (submodules) of direct sums (products) of copies of T. We recall
:
v that silting modules are in bijection with silting objects in the derived category of
i
X Mod-Rwhichcanberepresentedbycomplexesoftheform0→P−1 →P0 →0with
P andP projectives. Therefore,theyarealsoincorrespondencewithimportant
r −1 0
a concepts as (co-)t-structures or simply-minded collections of objects (see [12] and
[3]). It was proved recently that for some classes of rings (e.g. hereditary or com-
mutative ring), they can be parametrized by universal localisations, [13], Gabriel
topologiesoffinitetype,[2],orwidesubcategoriesoffinitelypresentedmodules[4].
For other correspondences and constructions we refer to [14] and [15]. For various
correspondences in the cosilting case, we refer to [18] and [19].
In this note we provide a general characterization (Proposition 2.1) for silting
classes, as torsion classes which are generated via some special pushout construc-
tions. In the case when R is right perfect (Theorem 2.4), respectively right heredi-
tary (Theorem2.6) it leads to characterizationswhich can be viewed as extensions
ofthe correspondingresultfortilting classes,[5,Theorem2.1]. Inparticular,every
2010 Mathematics Subject Classification. 16E30,18G15.
Keywords and phrases. Silting,cosilting,torsiontheory,preenvelopingclass,precoveringclass.
1
2 SIMIONBREAZANDJANZˇEMLICˇKA
enveloping torsion class of modules over a perfect ring or over an hereditary ring
is generated by a silting module (Corollary 2.9). The case of perfect rings extends
the corresponding results proved for finitely generated modules over artin algebra
in [16] and [1, Theorem 2.7].
The lastsection ofthe paper is devotedto the dualsetting, namely, we consider
torsion-freeclasses whichare of the form Cogen(T), where T is a cosilting module.
Since injective modules form an enveloping class over a general ring, we obtain
using dual tools that for every ring R torsion-free covering classes in Mod-R are
exactly the classes which are cogenerated by cosilting modules (Theorem 3.5).
2. Silting classes
In this paper R is a unital ring, and Mod-R will denote the category of all right
R-modules. Moreover, P will be the class of all projective modules in Mod-R and
P→ will denote the class of all homomorphisms σ :P →P with P ,P ∈P.
−1 0 −1 0
If σ : P → P is a homomorphism from P→ then we can associate to σ the
−1 0
class
D ={X ∈Mod-R|Hom (σ,X) is an epimorphism}.
σ R
If T is a rightR-module then Gen(T) denotes the class of allepimorphic images of
direct sums of copies of T.
If T is a class of modules, we will use the following classes:
• ◦T ={X ∈Mod-R|Hom(X,T)=0 for all T ∈T},
• T◦ ={X ∈Mod-R|Hom(T,X)=0 for all T ∈T},
• ⊥T ={X ∈Mod-R|Ext1(X,T)=0 for all T ∈T},
• T⊥ ={X ∈Mod-R|Ext1(T,X)=0 for all T ∈T},
• (cid:3)T ={α∈P→ |T ⊆D }, and
α
• ⋄T ={Coker(α)|α∈(cid:3)T}.
Recall from [3] that a module T is presilting if there exists a projective presen-
tation
σ
P →P →T →0
−1 0
such that D is a torsion class and T ∈ D . Then Gen(T) ⊆ D ⊆ T⊥ and
σ σ σ
(Gen(T),T◦) is a torsion pair. If D =Gen(T) then T is called a silting module.
σ
Let T be a class of modules. Then a homomorphism ǫ : X → T with T ∈ T is
a T-preenvelope if Hom(ǫ,T′) is surjective for all T′ ∈ T, i.e. all homomorphisms
X →T′withT′ ∈T factorizethroughǫ. TheT-preenvelopeǫisaT-envelopeifev-
eryhomomorphismα:T →T with the propertyǫ=αǫ hasto be anisomorphism.
IfallmodulesX ∈Mod-RhaveaT-preenvelope(envelope)thenT ispreenveloping
(resp. enveloping). A preenveloping class T is special if for every X ∈ Mod-R we
can find a T-preenvelope ǫ which is monic and Coker(ǫ)∈⊥T. The corresponding
dual notions are that of (special) precover/precoveringand cover/covering,respec-
tively.
Tiltingclasses,i.e. thetorsionclassesoftheformGen(T)withT atiltingmodule,
can be characterized by the fact that they are exactly the special preenveloping
torsion classes in Mod-R (cf. [6, Section 5], [5, Theorem 2.1]). We refer to [9]
for a recent study of this kind of special preenveloping situation which involves
homomorphismsinsteadofobjects. Eventhe orthogonalityusedinthis paperdoes
not cover the (co)silting case (cf. [7, Remark 39]), we can characterize torsion
TORSION CLASSES GENERATED BY SILTING MODULES 3
classesoftheformGen(T)withT asiltingmodulebytheexistenceofpreenvelopes
with some special properties.
Proposition 2.1. The following are equivalent for a class T of R-modules:
(1) There exists a silting module T such that T =Gen(T);
(2) (a) T is a torsion class,
(b) there exists a T-preenvelope ǫ:R→M which can be obtained as a pushout
ρ // // //
L L K 0
−1✤ 0
❴
δ
(cid:15)(cid:15) (cid:15)(cid:15)
ǫ // // //
R M K 0
such that ρ∈(cid:3)T.
(3) (a) T is a torsion class,
(b) for every R-module X there exists an T-preenvelope ǫ′ :X →T′ which can
0
be obtained as a pushout
ρ // // //
L L T 0
−1✤ 0 1
❴
(cid:15)(cid:15) (cid:15)(cid:15)
// // //
X T T 0
0 1
such that ρ∈(cid:3)T.
Ifwehaveadiagram asin(2)thenM⊕K isasiltingmoduleandT =Gen(T)=
D .
ρ
Proof. (1)⇒(3) Let σ : P → P be a homomorphism from P→ such that T =
−1 0
Coker(σ), and T is silting with respect to σ. Hence T =D .
σ
For every module X we consider the canonical homomorphism δ : P(I) → X,
−1
where I =Hom (P ,X), and we construct the pushout diagram
R −1
P(I) σ(I) //P(I) //T(I) //0
−1✤ 0
❴
δ δ0
(cid:15)(cid:15) (cid:15)(cid:15)
X µ //T ν // T(I) //0
0
Then, as in the proof of [3, Theorem 3.12] we obtain that T ∈D =T.
0 σ
Moreover, for every Y ∈ T and every homomorphism α : X → Y there exists
β :P(I) →Y suchthatδα=βσ(I). Bythe pushoutuniversalpropertythereexists
0
γ :T →Y such that α=γµ, hence µ is a T-preenvelopingmap. Since σ(I) ∈(cid:3)T,
0
the proof is complete.
(3)⇒(2) is obvious.
(2)⇒(1) If X ∈ D then every homomorphism R → X can be lifted to a ho-
ρ
momorphism M → X. Therefore every element of X is in the image of a homo-
morphism M → X, hence X ∈ Gen(M). It follows that D ⊆ Gen(M) = T.
ρ
(cid:3)
But T ⊆ D since ρ ∈ T, and it follows that D = T is a torsion class. More-
ρ ρ
over, K ∈ T = D , hence K is presilting with respect to ρ. By the proof of
ρ
[3, Theorem 3.12] it follows that T = M ⊕ K is a silting module T such that
Gen(T)=D =T. (cid:3)
ρ
4 SIMIONBREAZANDJANZˇEMLICˇKA
Inordertoapplytheabovepropositionwewillusethefollowingcharacterization
for pushout diagrams.
Lemma 2.2. In a commutative diagram
// ι // ρ // // //
0 V L L T 0
−1 0 1
α δ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
// υ // ξ // // //
0 U X T T 0
0 1
the middle square is a pushout if and only if α is an epimorphism.
Proof. Supposethatthemiddlesquareisapushout. Ifαisnotanepimorphism,we
considerπ :U →U/Im(α)thecanonicalepimorphism,andµ:U/Im(α)→E isthe
embedding of U/Im(α) into its injective envelope. There exists a homomorphism
ν :X →E such that νυ =µπ, hence νδι=0. Then νυ factorizesthroughρ. Since
themiddlesquareisapushout,ν factorizesthroughξ. Itfollowsthatµπ =νυ =0.
Since µ is monic, we obtain π =0, hence α is an epimorphism.
Conversely,ifαisanepimorphismandwehavetwohomomorphismsβ :X →Y
1
and β : L → Y such that β δ = β ρ then β υα = 0, hence β υ = 0. It follows
2 0 1 2 1 1
that there exists a unique homomorphism β :Im(ξ)→Y such that β =βυ.
1
Let δ : Im(ρ) → Im(ξ) be the homomorphism induced by δ. If ι : Im(ρ) → L
ρ 0
and ι : Im(ξ) → T are the canonical inclusions, then βδ = β ι . Since the first
ξ 0 2 ρ
square in the commutative diagram
0 // Im(ρ) ιρ // L // T // 0
0 1
δ
(cid:15)(cid:15) (cid:15)(cid:15)
0 //Im(ξ) ιξ //T // T // 0
0 1
is a pushout, there exists a unique homomorphismβ∗ :T →Y suchthat β (hence
0
β ) and β factorize through β∗, and the proof is complete. (cid:3)
1 2
Let Y be a submodule of a module P, and consider a canonical projection π :
P → P/Y. Recall that Y ≪ P means that Y is a superfluous submodule of a
module P, i.e. that for every ϕ ∈ Hom(M,P) if πϕ is an epimorphism then ϕ is
an epimorphism.
Make a well-known easy observation:
Lemma 2.3. Let X,P,T be modules over a ring R such that X ≪ P and α ∈
Hom(P,T). Then α(X)≪α(P). If, furthermore, α(X)=α(P) then α=0.
Now we are ready to characterize torsion classes generated by silting modules
over right perfect rings.
Theorem 2.4. Let R be a right perfect ring and T ⊆Mod-R a torsion class. The
following are equivalent:
(1) T =Gen(T) for a silting module T;
(2) There exists a T-preenvelope ǫ:R→M such that M ∈T ∩⊥T.
In these conditions, if K = Coker(ǫ) then M ⊕ K is a silting module, and
T =Gen(M ⊕K).
TORSION CLASSES GENERATED BY SILTING MODULES 5
Proof. (1)⇒(2) This is a consequence of Proposition 2.1 (see also [3, Proposition
3.11]).
ǫ ρ
(2)⇒(1) We consider the exact sequence 0 → U → R → M → K → 0. If
ǫ : R/U → M is the homomorphism induced by ǫ then for every T ∈ T the
homomorphism Hom(ǫ,T) is an epimorphism. Since M ∈⊥T we obtain K ∈⊥T.
For an epimorphism γ : P → M with P projective, we have a commutative
0 0
diagram
σ // // //
P P K 0
−1 0
πσ
(cid:15)(cid:15)
// σ // // //
0 Z P K 0
0
δ γ
(cid:15)(cid:15) (cid:15)(cid:15)
// ǫ // // //
0 R/U M K 0,
where σ and ǫ are the canonical homomorphisms induced by σ and ǫ, respectively,
and π :P →Z is the projective cover of Z.
σ −1
Since P is projective we can construct
−1
0 //X ιX //P σ // P // K //0
−1 0
υ δ γ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // U ιU //R ǫ // M //K //0
such that such that π δ = δπ , where π : R → R/U is the canonical projection.
U σ U
The maps ι and ι are the inclusion maps. Moreover, X ≪P .
X U −1
We will show that Hom(σ,T) is onto for all T ∈T. Let us consider the induced
short exact sequence
0→Z →P →K →0,
0
where Z = Im(σ) ∼= P /X, and note that for every T ∈ T we have a short exact
−1
sequence
Hom(σ,T)
(∗) 0→Hom(K,T)→Hom(P ,T) −→ Hom(Z,T)→0
0
since Ext1(K,T)=0. Fix an arbitrary T ∈T and an arbitrary ϕ∈Hom(P ,T).
−1
Let us denote by π : P → P /X and π : T → T/ϕ(X) the canonical
X −1 −1 T
projections. Then we can find a homomorphism ϕ∈Hom(P /X,T/ϕ(X)) which
−1
satisfies ϕπ = π ϕ. As T/ϕ(X) ∈ T, there exists ψ ∈ Hom(P ,T/ϕ(X)) for
X T 0
which ψσ = ϕπ by the exactness of (∗). Since P is projective and π is an
X 0 T
epimorphism, ψ factorizes through π , i.e. there exists ψ ∈ Hom(P ,T) such that
T 0
π ψ =ψ. Hence
T
π ψσ =ψσ =ϕπ =π ϕ.
T X T
Put α := ϕ − ψσ ∈ Hom(P ,T). From π α = 0 we have α(P ) ⊆ ϕ(X).
−1 T −1
Furthermore, α| =ϕ| since ψσ(X)=0, whichimplies that α(P )⊆α(X). By
X X −1
Lemma 2.3 we obtain α=0, so T ∈D .
σ
6 SIMIONBREAZANDJANZˇEMLICˇKA
Using the pushout of σ and δ we obtain a commutative diagram
// // σ // // //
0 X P P K 0
−1✤ 0
❴
υ′ δ γ′
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
// // ǫ′ // ρ′ // //
0 V R L K 0
υ γ
(cid:15)(cid:15) (cid:15)(cid:15)
// // ǫ // ρ // //
0 U R M K 0,
suchthatυυ′ =υ andγ =γγ′. Inordertosimplify thepresentation,letusremark
that υ′ is surjective and υ is injective, hence V can be identified to the image of
υ. In this case the equality υυ′ = υ represents the canonical decomposition of υ
through its image.
We will prove that γ is a T-preenvelope for L. Similar techniques were also
used in [17]. For every T ∈ T and every homomorphism α : L → T there exists
α:M →T suchthatαǫ′ =αǫ. Then(αγ−α)ǫ′ =0,hencethereexistsβ :K →T
such that αγ −α = βρ′ = βργ. Then α = (α−βρ)γ, and the claim is proved.
Moreover,since M ∈⊥T, if we use the exact sequence 0→Ker(γ)→L→M →0
it follows that Hom(Ker(γ),T)=0 for all T ∈T.
Wesplitthebottomrectangleinthepreviouscommutativediagramintwocom-
mutative diagrams with short exact sequences,
0 // V // R ǫ′ //Im(ǫ′) //0
υ ζ
(cid:15)(cid:15) (cid:15)(cid:15)
// // ǫ // //
0 U R Im(ǫ) 0,
and
0 // Im(ǫ′) //L ρ′ //K // 0
ζ γ
(cid:15)(cid:15) (cid:15)(cid:15)
// // ρ // //
0 Im(ǫ) M K 0,
where ζ can be identified to the canonical surjection R/V →R/U. Applying Ker-
Coker Lemma, we observe that C =Coker(υ)∼=Ker(γ).
Ifπ :P →C isaprojectivecoverforC andα:P →T isa homomorphismwith
T ∈T then the induced homomorphism
α:P/Ker(π)→T/α(Ker(π)), α(x+Ker(π))=α(x)+α(Ker(π)),
is zero. Therefore α(P) = α(Ker(π)). Since Ker(π) is superfluous, it follows that
α=0, hence Hom(P,T)=0.
We lift π to a homomorphismπ :P →U. By Lemma 2.2, υ′ is anepimorphism,
and it is easy to see that Im(υ)+Im(π) = Im(υ)+Im(π) = U. Therefore, the
canonical map (π,υ):P ⊕X →U induced by π and υ is an epimorphism.
TORSION CLASSES GENERATED BY SILTING MODULES 7
Now we construct the commutative diagram
0 // P ⊕X 1P⊕ιX// P ⊕P (0,σ) //P //K // 0
−1 0
(π,υ) (ιUπ,δ) γ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //U ιU //R ǫ // M //K // 0.
(cid:3)
Since (0,σ)∈ T, it remains to apply Lemma 2.2 and Proposition2.1 to complete
the proof. (cid:3)
The followingclassofexamples,usedalsoin[2],showsthatthatthe implication
(2)⇒(1) does not hold in general.
Example 2.5. Let R be a local ring with the unique maximal ideal J such that
J2 = J 6= 0 and put S = R/J. Since idempotency of J implies that extensions
of semisimple modules by semisimple modules are semisimple over R as well, we
get that T = Gen(S) = {S(κ)| κ} is a torsion class and Ext1(T,U) = 0 for each
T,U ∈T. ItiseasytoseethatthenaturalprojectionR→S formsaT-envelopeof
R. Clearly S ∈T ∩⊥T and we will show that S(κ) is not silting for every cardinal
κ>0.
Consider an exact sequence P →σ P →ρ S(κ) → 0. Since ρ factorizes through
−1 0
the canonical projection π : R(κ) → S(κ) we may suppose that P = R(κ) and
0
ρ= π. Then P 6=0 and Im(σ) =J(κ) = P J. Hence for every T ∈T and every
−1 0
homomorphism ϕ ∈ Hom(P ,T) we have Im(σ) = P J ⊆ ker(ϕ), and so ϕσ = 0.
0 0
As Hom(σ,T) = 0 while Hom(P ,T) 6= 0 for all nonzero T ∈ T, we obtain that
−1
no generator of T is silting.
Inthe caseofhereditaryrings,siltingtorsionclassescanbe characterizedbythe
existence of a special long exact sequence.
Theorem 2.6. Let R be a right hereditary ring and T ⊆ Mod-R a torsion class.
The following are equivalent:
(1) T =Gen(T) for a silting module T;
(2) There exists a T-preenvelope ǫ:R→M such that M ∈T ∩⊥T.
(3) There exists an exact sequence 0→U →R→M →K →0 such that M ∈T,
U ∈◦T and K ∈⊥T.
Proof. (2)⇒(3) As in the proof of Theorem 2.4 we obtain K ∈⊥T.
Since ǫ is a T-preenvelope, every homomorphism R →T with T ∈ T factorizes
through R/U. Therefore, for every T ∈ T we have that Hom(π,T) is an isomor-
phism, where π : R → R/U is the canonical epimorphism. Then first natural
homomorphism from the exact sequence
0→Hom(R/U,T)→Hom(R,T)→Hom(U,T)→Ext1(R/U,T)
is an isomorphism. Moreover,using the exact sequence 0→R/U →M →K →0,
weobtainExt1(R/U,T)=0forallT ∈T. ThereforeHom(U,T)=0forallT ∈T.
(3)⇒(1) Since R is hereditary, there exists a projective resolution
σ
0→P →P →K →0.
−1 0
Using the hypothesis K ∈ ⊥T, it follows that σ ∈ (cid:3)T. If U = Ker(ǫ) we can
construct, as in the proof of Theorem 2.4, the projectivity of P , a commutative
0
8 SIMIONBREAZANDJANZˇEMLICˇKA
diagram
// // σ // // //
0 U U ⊕P P K 0
−1 0
(ι,δ) γ
(cid:15)(cid:15) (cid:15)(cid:15)
// // ǫ // // //
0 U R M K 0,
where ι: U →R is the inclusion map. Since U is projective, by U ∈ ◦T is follows
(cid:3)
that (0,σ) ∈ T. From Proposition 2.1 we conclude that Gen(T)= T is a silting
class. (cid:3)
The class of silting modules is an intermediate class between the class of tilting
modulesandthatofquasi-tiltingmodules. UsingatheoremofWei,[20],itisproved
in [3, Proposition 3.15] that in the case of finitely generated modules over finitely
dimensional algebras the silting finitely generated modules coincide to (finendo)
quasi-tilting modules. In the case of hereditary or right perfect rings we obtain a
similar result for the classes generated by silting, respectively finendo quasi-tilting
modules.
Corollary 2.7. Let R be a right hereditary or right perfect ring. The following are
equivalent for a torsion class T ⊆Mod-R:
(1) T =Gen(T) for a silting module T;
(2) T =Gen(T) for a finendo quasi-tilting module T.
Proof. This follows from the previous theorems by using [3, Proposition 3.10] and
[3, Theorem 3.2(3)]. (cid:3)
We recall from [4] that a silting module T is minimal if there exists a Gen(T)-
envelope for the regular module R. In order to apply the above results to minimal
silting modules we need a lemma whose proof is a simple exercise.
Lemma 2.8. Let T a class of modules. If ǫ:R→M is an T-envelope then every
epimorphism α:N →M with N ∈T splits.
Consequently, if T is a class closed under extensions and ǫ : R → M is an
T-envelope then M ∈⊥T.
Corollary 2.9. The following are equivalent for a torsion class T of modules over
a right hereditary or right perfect ring R:
(1) T =Gen(T) for a minimal silting module T;
(2) There exists a T-envelope ǫ:R→M.
In particular, all enveloping torsion classes over hereditary or right perfect rings
are generated by silting modules.
Moreover,a half ofSalce’s Lemma [10, Lemma 5.20]is valid for silting modules:
Proposition 2.10. Let T be a silting module. If T = Gen(T) then for every
R-module X there exists a short exact sequence
υ
0→L→U →X →0
such that υ is a ⋄T-precover for X and L∈T.
Consequently, ⊥T is a special precovering class.
TORSION CLASSES GENERATED BY SILTING MODULES 9
Proof. If X is an R-module, we consider a pushout diagram
// υ // // //
0 Y P X 0
α β
(cid:15)(cid:15) (cid:15)(cid:15)
// γ // // //
0 L U X 0,
where P is an projective module and α is a T-preenvelope for Y obtained as a
pushout
ζ // // //
P P Z 0
−1 0
υ′
(cid:15)(cid:15) (cid:15)(cid:15)
α // // //
Y L Z 0
(cid:3)
for some ζ ∈ T. Then we have a pushout square
υυ′ //
P P
−1
ζ β
(cid:15)(cid:15) (cid:15)(cid:15)
γγ′ //
P U,
0
henceU isthecokernelofthehomomorphismδ :P →P ⊕P inducedbyυυ′ and
−1 0
ζ. Since every homomorphism f : P → T with T ∈ T can be writen as f = gζ
−1
for some g : P → T, it follows that f = g′δ, where g′ : P ⊕P → T is defined by
0 0
g′ =g and g′ =0. Then δ ∈(cid:3)T, so U ∈⋄T.
|P0 |P
Now, for every V ∈⋄T we have T ⊆V⊥, and it follows that γ is a ⋄T-precover
for X.
The last statement follows from the inclusion ⋄T ⊆⊥T. (cid:3)
3. Cosilting classes
For the dual results, let us recall from [8] that we can associate to every homo-
morphism σ :Q →Q between injective modules the class
0 1
B ={X ∈Mod-R|Hom (X,σ) is an epimorphism},
σ R
and a module T is partial cosilting if there exists an injective presentation
σ
0→T →Q →Q
0 1
such that B is a torsion-free class and T ∈ B . Then Cogen(T) ⊆ B ⊆ ⊥T. If
σ σ σ
B =Cogen(T) then T is called cosilting.
σ
LetI betheclassofallinjectivemodules,andI→theclassofallhomomorphisms
between injective modules. If F is a class of right R-modules then we associate to
F the following classes
• F(cid:3) ={σ :S →S |σ ∈I→, and F ⊆B }, and
0 1 σ
• F⋄ ={Ker(σ)|σ ∈F(cid:3)}.
Inorderto dualize Theorem2.4andCorollary2.9letus formulatedual versions
of Propositions 2.1.
Proposition 3.1. The following are equivalent for a class F of R-modules:
(1) There exists a cosilting module T such that F =Cogen(T);
(2) (a) F is a torsion-free class,
10 SIMIONBREAZANDJANZˇEMLICˇKA
(b) If E is a fixed injective cogenerator for Mod-R then there exists an F-
precovering ǫ:M →E which can be obtained as a pullback
// // ǫ //
0 K M E
ν
(cid:15)(cid:15) (cid:15)(cid:15)
0 // K //Q′ ζ′ // Q′
0 1
such that ζ′ ∈F(cid:3).
(3) (a) F is a torsion-free class,
(b) for every R-module X there exists an F-precovering α:M →X which can
be obtained as a pullback
// // α //
0 S M X
(cid:15)(cid:15) (cid:15)(cid:15)
// // σ //
0 S S S
0 1
such that σ ∈F(cid:3).
If we have a diagram as in (2) then K ⊕M is a cosilting modules and F =
Cogen(K⊕M).
If Y is a submodule of a module P with the canonical embedding ν : Y → P,
then Y is an essential submodule of P, Y E P, if an arbitrary homomorphism
ϕ∈Hom(P,N) is a monomorphism whenever ϕν is a monomorphism.
Lemma 3.2. Let Y,Q,F be modules over a ring R such that Y E Q and α ∈
Hom(F,Q). Then β(F)∩Y Eβ(F). If, furthermore, β(F)∩Y =0, then β =0.
Lemma 3.3. If F is a class closed under extensions and ǫ:M →E is an F-cover
of an injective module E then M ∈F⊥.
As in the (co)tilting theory, from [10, Theorem 5.31] and [8, Corollary 4.8] we
obtain the following:
Lemma 3.4. If T is a cosilting module then Cogen(T) is a covering class.
Since every module has an injective envelope overan arbitraryring, application
of dual techniques to that applied in the silting case gives us the dual result to
Corollary 2.9 that cosilting classes are exactly that torsion-free classes which are
covers over general rings.
Theorem 3.5. Let R be a ring and E a fixed injective cogenerator for Mod-R. If
F is a torsion-free class in Mod-R, the following are equivalent:
(1) F =Cogen(T) for a cosilting module T;
(2) F is a covering class;
(3) there exists an F-cover ǫ:M →E;
(4) There exists an F-precover ǫ:M →E such that M ∈F ∩F⊥.
Moreover, if R is hereditary, then the above conditions are equivalent to:
(5) There exists an exact sequence 0→K →M →E →V →0 such that M ∈F,
V ∈F◦ and K ∈F⊥.
In these conditions, if K = Ker(ǫ) then M ⊕K is a cosilting module and F =
Cogen(M ⊕K).