HE Dong-lin,LI Yu-yan

(Department of Mathematics,Longnan Teachers College,Longnan,742500,China)

Abstract:In this paper,we consider some generalizations of tilting torsion classes and cotilting torsion-free classes,give the definition and characterizations of n-tilting torsion classes and n-cotilting torsion-free classes,and study n-tilting preenvelopes and n-cotilting precovers.

Key words:n-tilting torsion classes;n-cotilting torsion-free classes;preenvelopes;precovers

§1.Introduction

Tilting theory plays an important role in the representation of Artin algebra.The classical tilting modules were first considered in the early eighties by Brenner-Bulter[1],Bongartz[2]and Happle and Ringel[3]etc.Beginning with Miyashita[4],the defining conditions for a classical tilting module were extented to arbitary rings or Abel categories by many authors,Wakamatsu[5],Colby and Fuller[6],Colpi and Trifaj[7],and recently,Angeleri Hgel and coelho[8],Bazzoni[9],Wei[10],Colpi and Fuller[11],and Di et al[12].Among them,Miyashita[4]considered finitely generated tilting modules of finite projective dimension,while generalizations of tilting modules of projective dimension one over arbitary rings.In[7]an(not necessarily finitely generated)module T is said to be tilting(simplely,1-tilting module)if GenT=T⊥1,where GenT is the class of modules which are epimorphic images of direct sums of copies of T and T⊥1is the class of modules M such that Ext1R(T,M)=0.And it is proved that t? R-Mod is a tilting class if and only if t=GenP for a faithful, fi nendo,and t-projective module P.Angeleri and Trlifaj[13]discussed tilting preenvelopes and cotilting precovers.Meanwhile,tilting torsion classes(resp,cotilting torsion-free classes)were characterized as those pretorsion classes(resp,pretorsion-free classes)which provided special preenvelopes(resp,special precovers)for all modules.Bazzoin[9]considered(not necessarily finitely generated)tilting modules of projective dimension≤n(simply n-tilting modules),and proved that T is n-tilting module if and only if PresnT=T⊥i≥1.Dually,U is n-cotilting module if and only if CopresnT=⊥i≥1T.It is nutural to consider n-tilting torsion classes and n-cotilting torsion-free classes,and to investigate ntilting preenvelopes and n-cotilting precovers which are generalizations of tilting preenvelopes and cotilting precovers in[7].

The contents of this paper are summarized as follows.In section 2,we collect some known notions and results.In section 3,we introduce n-tilting torsion classes and discuss n-tilting preenvelopes.Furthermore,we give some characterizations of n-tilting torsion classes,and prove that:if t is n-tilting torsion classes,then t is envelope class.Section 4 is devoted to n-cotilting torsion-free classes and n-cotilting precovers.

§2.Preliminaries

Throughout this paper,R will be an associative ring with nonzero identity and all modules are unitary.Let R-Mod be the category of left R-modules and T∈R-Mod.We denote by T⊥1≤i≤n:={M ∈ R-Mod|ExtiR(T,M)=0 for all 1 ≤ i≤ n},T⊥i≥1:={M ∈ R-Mod(T,M)=0 for all i≥ 1},and T⊥1:={M ∈ R-Mod|Ext1R(T,M)=0}.Dually,⊥1≤i≤nT,⊥i≥1T,and⊥1T are defined similarly.Denote by AddT the class of modules isomorphic to direct summands of direct sums of copies of T and by PresnT:={M∈R-Mod|there exists an exact sequence Tn→ ···→T2→T1→M →0 with Ti∈AddT for all 1≤i≤n}.It is clear that Presn+1TPresnT and Pres1T=GenT,where GenT denotes the class of all left R-modules generated by T.Dually,denote by ProdT the class of modules isomorphic to direct summands of direct products of copies of T,and by CopresnT:={M∈R-Mod|there exists an exact sequence 0→M →C1→C2→ ···→Cnwith Ci∈ProdT for all 1≤i≤n}.It is clear that Copresn+1TCopresnT and Copres1T=CogenT,where CogenT denotes the class of all left R-modules cogenerated by T.T is a tilting module[7]provided that GenT=T⊥1.T is a cotilting module[14]provided that CogenT=⊥1T.Let xR-Mod,then x is a pretorsion class provided that x is closed under direct sums and factors.Moreover,x is a tilting torsion class provided that x=GenT for a tilting module T.Dually,x is pretorsion-free class provided that x is closed under direct products and submodules.x is a cotilting torsion-free class provided that x=CogenU for a cotilting module U.

Precovers and preenvelopes were first defined in[15]in the following manner:if y is a class of modules(closed under isomorphisms),a y-precover of R-module M is a morphism φ from Y(Y ∈ y)to M,such that HomR(Y′,φ)is surjective for every Y′∈ y.If in addition,any morphism α :Y → Y verifying φ ?α = φ is automorphism,then φ is said to be an y-cover.y is a precover(resp,cover)class provided that each R-module has a y-precover(resp,y-cover).y-preenvelope and y-envelope,preenvelope class and envelope class can be defined dually.An R-module M is y-projective(resp,y-injective)provided that the functor HomR(M,-)(resp,HomR(-,M))is exact on short exact sequence of the form 0→U→V→W→0,where U,V,W∈y.Denote by y⊥={M∈R-Mod|Ext1R(Y,M)=0 for any Y∈y.⊥y is defined dually.

definition 2.1[9]An R-module T is said to be n-tilting module if the following conditions are satis fi ed:

(1)pdRT≤n(here pdRT denotes the projective dimension of T).

(2)ExtiR(T,T(λ))=0 for all i≥ 1 and all cardinal λ.

(3)There is a long exact sequence 0→R→T0→T1→···→Tr→0 with Ti∈AddT for every 0≤i≤r.

Dually,an R-module U is said to be n-cotilting module if it satisfy the following conditions:

(1)idRU≤n(here idRU denotes the injective dimension of U).

(2)ExtiR(Uλ,U)=0 for all i≥ 1 and all cardinal λ.

(3)There is a long exact sequence 0→Ur→···→U1→U0→E→0 with Ui∈ProdT for every 0≤i≤r.

Lemma 2.1[9]An R-module T is said to be n-tilting module if and only if PresnT=T⊥i≥1.Dually,an R-module Uis said to be n-cotilting module if and only if CopresU=⊥i≥1U.

Remark 2.1[9]Tilting modules in[7]are exactly 1-tilting modules,cotilting modules in[7]are exactly 1-cotilting modules.

Proposition 2.1 The following conditions are hold:

(1)If T is n-tilting module,then T is m-tilting module for any non-negative integer m≥n.

(2)If T is n-tilting module,then PresnT=Presn+1T=Presn+2T= ···.

Proof (1)Asumme that T is n-tilting module,then PresnT=T⊥i≥1by lemma 2.1.It is sufficient to prove that PresnT=PresmT for any non-negative integer m≥n.If m=n,then it is clear that PresnT=PresmT.If m＞n,then PresmT?PresnT and PresnT=Presn+1T by[16,theorem 4.3].For any M∈PresnT=Presn+1T,there exists an exact sequence

with Ti∈AddT for all 1≤i≤n+1.Note that K1=Kerf1∈PresnT,we can get M∈Presn+2T.Repeat the process,and so on,it is easy to get PresnT=PresmT.Therefore,if T is n-tilting module,then T is m-tilting module for any non-negative integer m≥n.

(2)It is obvious following(1).

We can obtain the following proposition dually.

Proposition 2.2 For any R-module U and any non-negative integer n,the following conditions are hold:

(1)If U is n-cotilting module,then U is m-cotilting module for any non-negative integer m≥n.

(2)If U is n-cotilting module,then CopresnU=Copresn+1U=Copresn+2U=....

§3.n-tilting Torsion Classes and n-tilting Preenvelopes

We start with the following definition.

definition 3.1 Let y?R-Mod.y is said to be an n-tilting torsion class,if there exists an n-tilting module T∈R-Mod,such that y=PresnT.

y is a 1-tilting torsion class,if and only if there exists a 1-tilting module T such that y=Pres1T=GenT.It is clear that 1-tilting torsion classes are exactly tilting torsion classes in[7],1-tilting torsion classes are n-tilting torsion classes.n-tilting torsion classes are generalizations of tilting torsion classes.According to[13,theorem2.1],tilting torsion classes are characterized as follows.

Lemma 3.1 Let R be a ring and y?R-Mod be a pretorsion class.Then the followings are equivalent:

(1)y is tilting torsion class;

(2)every module has a special y-preenvelope;

(3)there is a special y-preenvelope of R;

(4)there is a y-preenvelope of R,b:R→B,such that b is injective and B is y-projective.

We now can state one of our main results by lemma 3.1 as follows.

Theorem 3.1 Let y?R-Mod be a pretorsion class.Then the followings are equivalent:

(1)y is n-tilting torsion class;

(2)for any R-module M,there is an exact sequence

with Ti∈y and Imdi∈⊥y(i=1,2,...,n);

(3)there exists an exact sequence

with Ti∈ y and Imdi∈⊥y(i=1,2,...,n);

(4)there exists an exact sequence

with Ti∈y and Imd1∈⊥y and Tiis y-projective(i=1,2,...,n).

Proof (1)?(2)Suppose y is n-tilting torsion class,then there exists an n-tilting module T ∈ R-Mod,such that y=PresnT=T⊥i≥1.For any cardinal k,we have ExtiR(T,T(k))=0 by Tk∈PresnT.According to[17,lemma6.8],for any module M,there is a y-torsion resolution of M of the form 0→ M → T1→ T(λ1)→ 0,where T1∈ y,λ1is a cardinal,ExtiR(T(λ1),N)=0 for all N ∈ y.Repeat the process of M for T(λ1),and so on,we can get an exact sequence

in which Ti∈y and Imdi∈⊥y(i=1,2,...,n).

(2)?(3)It is obvious.

(3)?(4)Assume that there exists an exact sequence

with Ti∈ y and Imdi∈⊥y(i=1,2,...,n).It is only to prove Tiis y-projective(i=1,2,...,n).Consider the short exact sequence 0→Imdn-1→Tn→I→0,since I～=Imdn∈⊥y and Imdn-1∈⊥y,We can get Tn∈⊥y,which shows that HomR(Tn,-)is exact on any epimorphism with kernal in y.In particular,Tnis y-projective.Repeat the process to the short exact sequence 0→Imdn-2→Tn-1→Imdn-1→0,and so on,it is not difficult to obtain that Tiis y-projective(i=1,2,...,n).

(4)?(1)Assume that there exists an exact sequence

with Ti∈y and Imdi∈⊥y(i=1,2,...,n).Consider the short exact sequence 0→R→T1→Imd1→0,since Imd1∈⊥y,so R→T1is a y-preenvelope of R.Note that T1is y-projective and R→T1is injective,we can obtain that y is 1-tilting torsion class by lemma 3.1.Therefore,y is n-tilting torsion class.

Theorem 3.2 Suppose y is an n-tilting torsion class in R-Mod.If there exists x?R-Mod which is closed under extensions,such that AddPx? x?⊥y and Px⊥1=y for some Px∈y.Then is an envelope class.

Proof Assume that there exists x?R-Mod which is closed under extensions,such that Px⊥1=y and AddPx?x?⊥y for some Px∈y.For any M∈R-Mod,since y is an n-tilting torsion class,we have y=PresnT for some n-tilting module T.Note that PresnT is closed under direct sums and Px⊥1=y,so Pxλ∈y and Ext1R(Px,P(xλ))=0 for all cardinals λ.Therefore,we obtain an exact sequence ε:0→ M → Y → Px(α)→ 0 by[17,lemma 6.8],where Y ∈ y,α is a cardinal.Then ε is a generator for Ext1R(Y,M)in the sense of[18,proposition2.2.1].According to our assumption and[18,theorem2.2.6],we know that M has an x⊥-envelope.Since y=(⊥y)⊥? x⊥? (AddPx)⊥? P⊥x1=y,Then the conclusion is proved.

§4.n-cotilting Torsion-free Classes and n-cotilting Precovers

We start with the following definition.

definition 4.1 Let w?R-Mod.w is said to be an n-cotilting torsion-free class,if there exists an n-cotilting module U∈R-Mod,such that w=CopresnU.

w is a 1-cotilting torsion-free class,if and only if there exists a 1-cotilting module U such that w=Copres1U=CogenU.It is clear that 1-cotilting torsion-free classes are exactly cotilting torsion-free classes in[7],1-cotilting torsion-free classes are n-cotilting torsion-free classes.ncotilting torsion-free classes are generalizations of cotilting torsion-free classes.According to[13,theorem2.5],cotilting torsion-free classes are characterized as follows.

Lemma 4.1 Let R be a ring and w?R-Mod be a pretorsion-free class.Then the followings are equivalent:

(1)w is cotilting torsion-free class;

(2)every module has a special w-precover;

(3)there is a special w-precover of an injective cogenerator of R-Mod;

(4)there is a w-precover,π :P → E,of an injective cogenerator E of R-Mod such that π is surjective and P is faithful and w-injective.

We now can state one of our main results by lemma 4.1 as follows.

Theorem 4.1 Let w?R-Mod be a pretorsion-free class.Then the followings are equivalent:

(1)w is n-cotilting torsion-free class;

(2)for any R-module M,there is an exact sequence

with Wi∈w and Kerfi∈w⊥(i=1,2,...,n);

(3)there exists an exact sequence

with Wi∈w and Kerfi∈w⊥(i=1,2,...,n),where E is an injective cogenerator of R-Mod;

(4)there exists an exact sequence

with Wi∈ w and Kerf1∈ w⊥,W1is faithful and Wiis w-injective(i=1,2,...,n),where E is an injective cogenerator of R-Mod;

Proof (1)?(2)Suppose w is n-cotilting torsion-free class,then there exists an ncotilting module U ∈ R-Mod,such that w=CopresnU=⊥i≥1U.By[19,lemma 2.14],for any module M,there is a w-torsion-free resolution of M of the form 0→Uλ1→W1→M→0,where W1∈ w,λ1is a cardinal,Ext1R(N,Uλ1)=0 for all N ∈ w.

Repeat the process of M for Uλ1,and so on,we can get an exact sequence

with Wi∈ w and Kerfi=Uλi∈ w⊥(i=1,2,...,n).

(2)?(3)It is clear.

(3)?(4)Assume that there exists an exact sequence

with Wi∈w and Kerfi∈w⊥(i=1,2,...,n),where E is an injective cogenerator of R-Mod.Consider the short exact sequence 0→K→Wn→Kerfn-1→0,since KKerfn∈w⊥and Kerfn-1∈ w⊥,We can get Wn∈ w⊥,which shows that HomR(-,Wn)is exact on any monomorphism with cokernal in w.In particular,Wnis w-injective.Repeat the process to the short exact sequence 0→ Kerfn-1→ Wn-1→Kerfn-2→ 0,and so on,it is not difficult to obtain that Wiis w-injective(i=1,2,...,n).Meanwhile.According to our assumption and lemma 4.1,we can get Wiis faithful.

(4)?(1)Assume that there exists an exact sequence

with Wi∈w and Kerf1∈w⊥,W1is faithful and Wiis w-injective(i=1,2,...,n),where E is an injective cogenerator of R-Mod.Consider the short exact sequence 0→Kerf1→W1→E→0,since Kerf1∈w⊥,so W1→E is a w-precover of R.Note that W1is w-injective and faithful,and W1→E is surjective,we can obtain that w is 1-cotilting torsion-free class by lemma 4.1.Therefore,w is n-cotilting torsion-free class.

Theorem 4.2 Suppose w is an n-cotilting torsion-free class in R-Mod.If w is closed under direct limits.Then w is an cover class.

Proof It is easy to prove by theorem 2.5 and[18,theorem 2.2.8].