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)

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′′.

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半粘合誘導的t-結構

尹幼奇
(上海交通大學數學系,上海 200240)
(紹興文理學院數學系,浙江紹興 312000)

本文研究了對于給定的一個三角范疇的上(下)粘合(C′,C,C′′),如何由C的一個t-結構誘導C′和C′′的t-結構的問題.利用左(右)t-正合函子的概念,給出了由C的一個t-結構可誘導出C′和C′′的t-結構的充分條件.將粘合的一些相關結果推廣到了上(下)粘合的情形.

三角范疇;上(下)粘合;穩定t-結構

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.