YIN You-qi

(Department of Mathematics,Shanghai Jiao Tong University,Shanghai 200240,China)

(Department of Mathematics,Shaoxing College of Arts and Sciences,Shaoxing 312000,China)

T-STRUCTURES INDUCED BY HALF RECOLLEMENTS

YIN You-qi

(Department of Mathematics,Shanghai Jiao Tong University,Shanghai 200240,China)

(Department of Mathematics,Shaoxing College of Arts and Sciences,Shaoxing 312000,China)

LetC′,CandC′′be triangulated categories.In this paper,we consider how to inducet-structures onC′andC′′from at-structure onCgiven an upper(resp.lower)recollement ofCrelative toC′andC′′.By the concept of left(right)t-exact,we give a sufficient condition such that at-structure onCmay inducet-structures onC′andC′′,which generalizes some results concerning recollements to upper(resp.lower)recollements.

triangulated category;upper(lower)recollement;stablet-structure

1 Introductio n

Recollements of triangulated categories play an important role in algebraic geometry(see[1]),representation theory(see[2–5]),etc.A recollement(C′,C,C′′)of triangulated categories provides a platform for various questions concerning the three terms in arecollement.For examples,given arecollement of a triangulated categoryCrelative toC′andC′′,t-structures(C′≤0,C′≥0)and(C′′≤0,C′′≥0)ofC′andC′′,respectively,Beilinson,Bernstein and Deligne[1]proved thatCalso has at-structure(C≤0,C≥0),where

On the other hand,Lin[6]proved that certaint-structure onCmay inducet-structures onC′andC′′.Chen[7]studied the relationship of cotorsion pairs among three triangulated categories in arecollement.She proved the following results:cotorsion pairs onCmay be obtained from cotorsion pairs onC′andC′′and certain cotorsion pairs onCmay induce cotorsion pairs onC′andC′′.More relevant results can be seen in[8–11],etc.

In a viewpoint of Beilinson,Ginsburg and Schechtman(see[12]),upper and lower recollements are more fundamental than arecollement(upper and lower recollements arecalled steps in[8]).For a given upper(lower)recollement ofCrelative toC′andC′′,a sufficient condition thatt-structures onC′andC′′may be induced by at-structure onCis given in this paper.

2 Preliminaries

Recall the following de finitions.

De finition 2.1LetC′,CandC′′be triangulated categories.

(1)[1]A recollement ofCrelative toC′andC′′is a diagram of triangle functors

such that

(R1)(i∗,i∗),(i∗,i!),(j!,j∗)and(j∗,j∗)are adjoint pairs;

(R2)i∗,j!andj∗are fully faithful;

(R3)j∗i∗=0;

(R4)for eachX∈C,there are distinguished triangles

where∈Xis the counit of(j!,j∗),ηXis the unit of(i∗,i∗),ωXis the counit of(i∗,i!),andζXis the unit of(j∗,j∗).

(2)[5,12,13]LetC′,CandC′′be triangulated categories.An upper recollement ofCrelative toC′andC′′is a diagram of triangle functors

such that the conditions involvedi∗,i∗,j!,j∗in(1)are satisfied.

(3)[5,12,13]LetC′,CandC′′be triangulated categories.An lower recollement ofCrelative toC′andC′′is a diagram of triangle functors

such that the conditions involvedi∗,i!,j∗,j∗in(1)are satisfied.

For short,we denote respectively the recollement(2.1),upper recollement(2.2)and lower recollement(2.3)by(C′,C,C′,i∗,i∗,i!,j!,j∗,j∗),(C′,C,C′,i∗,i∗,j!,j∗)and(C′,C,C′,i∗,i!,j∗,j∗),or uniformly by(C′,C,C′′).

We need the following fact.

Lemma 2.2(see[14])Let(C′,C,C′′)be an upper recollement.Then there exists a triangle-equivalencesuch thatwhereV:C→C/i∗C′is the Verdier functor.

The subcategories in this section are full subcategories closed under isomorphisms.

De finition 2.3[1]LetCbe a triangulated category with the shift functor[1].Atstructure onDis a pair of full subcategories(D≤0,D≥0)with the following properties:

If we putD≤n:=D≤0[−n]andD≥n:=D≥0[−n],∀n∈Z,we have

(t1)HomD(X,Y)=0,∀X∈D≤0,Y∈D≥1;

(t2)D≤0⊆D≤1andD≥1⊆D≥0;

(t3)For eachX∈D,there is a distinguished triangle

whereA∈D≤0,B∈D≥1.

Let(U,V)be at-structure onC.We call(U,V)a stablet-structure,ifUandVare triangulated subcategories ofC(see[15,De finition 0.2]).

Here are basic properties of stablet-structures.

Lemma 2.4(see[15])LetDbe a triangulated category,Ca thick subcategory ofD,andQ:D→D/Cthe canonical quotient.For a stablet-structure(U,V)onD,the following are equivalent.

(i)(Q(U),Q(V))is a stablet-structure onD/C,whereQ(U)(resp.Q(V))is the full subcategory ofD/Cconsisting of objectsQ(X)forX∈U(resp.Q(Y)forY∈V);

(ii)(U∩C,V∩C)is a stablet-structure onC.

De finition 2.5[1] LetCandDbe two triangulated categories witht-structures(C≤0,C≥0)and(D≤0,D≥0).An triangle functorF:C−→Dis

(i)leftt-exact ifF(C≥0)⊂D≥0;

(ii)rightt-exact ifF(C≤0)⊂D≤0.

3 t-Structure Induced by Upper Recollement

This section aims to prove the main result of this paper.LetC′,CandC′′be triangulated categories.Given a upper recollement ofCrelative toC′andC′′,at-structure onCinducest-structures onC′andC′′under some conditions.

Proposition 3.6LetC′,CandC′′be triangulated categories,let diagram(2.2)be an upper recollement ofCrelative toC′andC′′,and let(C≤0,C≥0)be at-structure onC.Ifi∗i∗is leftt-exact andj!j∗is rightt-exact,then

(i)(i∗(C≤0),i∗(C≥0))is at-structure onC′;

(ii)(j∗(C≤0),j∗(C≥0))is at-structure onC′′;

(iii)If(C≤0,C≥0)and(i∗(C≤0),i∗(C≥0))are stablet-structures onCandC′,respectively,then(j∗(C≤0),j∗(C≥0))is a stablet-structure onC′.

Proof(i)ForX∈C≤0,Y∈C≥1,since(i∗,i∗)is an adjoint pair andi∗i∗is leftt-exact,we have HomC′(i∗X,i∗Y)≌HomC(X,i∗i∗Y)=0.Thus(t1)hold.

Condition(t2)follows from the closure ofC≤0andC≥0under the shifts[1]and[-1],respectively.

LetX′ ∈C′.There is a distinguished triangleA→i∗X′→B→A[1]inC,whereA∈C≤0,B∈C≥1.Applyingi∗to this triangle,we havei∗A→i∗i∗X′→i∗B→i∗A[1],wherei∗A∈i∗(C≤0),i∗B∈i∗(C≥1).Sincei∗is fully faithful and(i∗,i∗)is an adjoint pair,we havei∗i∗X′≌X′.Therefore,the distinguished trianglei∗A→X′→i∗B→i∗A[1]is thet-decomposition ofX′.We have condition(t3).

(ii)Similarly,we obtain argument(ii).

(iii)We prove the last statement by three steps.

Step 1j!j∗is rightt-exact⇒i∗i∗is rightt-exact.

LetX∈C≤0,forY∈C≥1.Applying cohomological functor HomC(−,Y)to the distinguished triangle

we get an exact sequence

Since HomC(X,Y)=HomC(X[1],Y)=0,we get HomC(i∗i∗X,Y)≌HomC(j!j∗X[1],Y)=0.

Step 2We claimi∗i∗(C≤0)=i∗C′∩C≤0andi∗i∗(C≥0)=i∗C′∩C≥0.

By Step 1 we havei∗i∗is rightt-exact,i.e.i∗i∗(C≤0)⊆C≤0.Therefore,i∗i∗(C≤0)⊆i∗C′∩C≤0.Conversely,forX∈i∗C′∩C≤0,there exists a distinguished trianglej!j∗X→X→i∗i∗X→(j!j∗X)[1].SinceX∈i∗C′,it followsj!j∗X=0.SinceXis inC≤0,we haveX≌i∗i∗X⊆i∗i∗(C≤0).

Similarly we havei∗i∗(C≥0)=i∗C′∩C≥0.

Therefore,(i∗C′∩C≤0,i∗C′∩C≥0)=(i∗i∗(C≤0),i∗i∗(C≥0)).

Step 3Assume that(i∗(C≤0),i∗(C≥0))is a stablet-structure onC′.Sincei∗is fully faithful,(i∗i∗(C≤0),i∗i∗(C≥0))is a stablet-structure oni∗C′.By Step 2,(i∗C′∩C≤0,i∗C′∩C≥0)is a stablet-structure oni∗C′.Hence(Q(C≤0),Q(C≥0))is a stablet-structure onC/i∗C′by Lemma 2.4.There exists a triangle-equivalenceej∗:C/i∗C′≌C′′such thatj∗=ej∗Q,so(j∗(C≤0),j∗(C≥0))is a stablet-structure onC′′.The proof is completed.

By the similar argument we have statements for lower recollements.

Corollary 3.7LetC′,CandC′′be triangulated categories,let diagram(2.3)be a lower recollement ofCrelative toC′andC′′,and let(C≤0,C≥0)at-structure onC.Ifi∗i!is rightt-exact andj∗j∗is leftt-exact,then

(i)(i!(C≤0),i!(C≥0))is at-structure onC′;

(ii)(j∗(C≤0),j∗(C≥0))is at-structure onC′′;

(iii)If(C≤0,C≥0)and(i!(C≤0),i!(C≥0))are stablet-structures onCandC′,respectively,then(j∗(C≤0),j∗(C≥0))is a stablet-structure onC′′.

[1]Beilinson A,Bernstein J,Deligne P.Faisceaux pervers[J].Astérisque,1982,100:5–171.

[2]Cline E,Parshall B,Scott L.Algebraic stratification in representation categories[J].J.Alg.,1988,117:504–521.

[3]Cline E,Parshall B,Scott L.Finite dimensional algebras and highest weight categories[J].J.Reine Angew.Math.,1988,391:85–99.

[4]Jφrgensen P.Recollement for differential graded algebras[J].J.Alg.,2006,299:589–601.

[5]König S.Tilting complexes,perpendicular categories and recollements of derived module categories of rings[J].J.Pure Appl.Alg.,1991,73:211–232.

[6]Lin Zengqiang.t-structure and recollement of hearts[J].J.Huaqiao Univ.(Nat.Sci.),2010,31(3):356–360.

[7]Chen Jianmin.Cotorsion pairs in arecollement of triangulated categories[J].Comm.Alg.,2013,41:2903–2915.

[8]Wiedemann A.On stratifications of derived module categories[J].Canad.Math.Bull.,1991,34(2):275–280.

[9]Happel D.Reduction techniques for homological conjectures[J].Tsukuba J.Math.,1993,17(1):115–130.

[10]Han Yang.Recollement and Hochschild theory[J].J.Alg.,2014,197:535–547.

[11]Lin Ji,Yao Yunfei.Torsion theory of triangulated categories and abelian categories[J].J.Math.,2014,34(6):1134–1140.

[12]Beilinson A,Ginsburg V,Schechtman V.Koszul duality[J].J.Geom.Phys.,1998,5(3):317–350.

[13]Parshall B.Finite dimensional algebras and algebraic groups[J].Contemp.Math.,1989,82:97–114.

[14]Zhang P.Triangulated categories and derived categories[M].Beijing:Science press,2015.

[15]Iyama O,Kato K,Miyachi J.Recollement on homotopy categories and Cohen-Macaulay modules[J].J.K-Theory,2011,8(3):507–542.

半粘合诱导的t-结构

尹幼奇
(上海交通大学数学系,上海 200240)
(绍兴文理学院数学系,浙江绍兴 312000)

本文研究了对于给定的一个三角范畴的上(下)粘合(C′,C,C′′),如何由C的一个t-结构诱导C′和C′′的t-结构的问题.利用左(右)t-正合函子的概念,给出了由C的一个t-结构可诱导出C′和C′′的t-结构的充分条件.将粘合的一些相关结果推广到了上(下)粘合的情形.

三角范畴;上(下)粘合;稳定t-结构

O153.3

18A40;18E35;18E30

A

0255-7797(2017)06-1215-05

date:2015-11-11Accepted date:2016-02-18

Supported by National Natural Science Foundation of China(11271251;11431010;11571239);Zhejiang Provincial Natural Science Foundation(LY14A010006).

Biography:Yin Youqi(1979–),female,born at Shengzhou,Zhejiang,lecturer,major in represent theory of algebras.