MUTATION PAIRS AND TRIANGULATED QUOTIENTS

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1 MUTATION PAIRS AND TRIANGULATED QUOTIENTS ENGQIANG LIN AND MINXIONG WANG rxi: [mth.ct] 1 Jn 2014 Astrt. We define muttion pir in pseudo-trinulted teory. We proe tht under ertin onditions, for muttion pir in pseudo-trinulted teory, the orrespondin quotient teory rries nturl trinulted struture. This result unifies mny preious onstrutions of quotient trinulted teories. 1. Introdution Trinulted teory s introdued y Grothendiek nd lter y Verdier in the sixties of lst entury. It is n importnt struture in oth eometry nd ler. Quotient teories ie y to produe trinulted teories. As shonyhppel [5], if ereiennextteory(b,s) stisfyinfroenius ondition, tht is, (B, S) hs enouh S-injeties nd enouh S-projeties nd the S-injeties oinide ith the S-projeties, then the quotient teory B/I rries trinulted struture, here I is the full suteory of S-injeties. Beliinnis [2] otined similr result y replin B ith trinulted teory C nd replin S ith proper lss of trinles E. Bsed on [3], Jørensen [7] shoed tht if X is funtorilly finite suteory of trinulted teory C ith Auslnder-Reiten trnsltion τ, nd if τx = X, then the quotient C/X is trinulted teory. As enerliztion of muttion of luster tiltin ojets in luster teories, Iym nd oshino [6] defined muttion pir in trinulted teory. They shoed tht if (,) is D-muttion pir in trinulted teory C nd is n extension-losed suteory of C, here D nd D is riid, then the quotient /D is trinulted teory. Reently Liu nd hu [8] defined D-muttion pir in riht trinulted teory, nd ot similr result, hih unifies the onstrutions of Iym-oshino in [6] nd Jørensen in [7]. To unify the onstrutions of quotient trinulted strutures in [5] nd [6], Nkok [9] defined Froenius ondition on pseudo-trinulted teory, hih is similr to tht on n ext teory, nd onstruted quotient trinulted teory. Pseudo-trinulted teories re nturl enerliztion of elin teories nd trinulted teories, hose riht trinles nd left trinles ehe muh etter thn those on pretrinulted teories. Unfortunetly Nkok pointed out tht his onstrution nnot oer Beliinnis s result in [2] Mthemtis Sujet Clssifition. 18E10, 18E30. Key ords nd phrses. Quotient teory; muttion pir; pseudo-trinulted teory; trinulted teory. Supported y the Ntionl Nture Siene Foundtion of Chin (Grnts No , ), nd the Nturl Siene Foundtion of Fujin Proine (Grnts No. 2013J05009). 1

2 2 ENGQIANG LIN AND MINXIONG WANG The im of this rtile is to ie y to unify the different onstrutions of quotient trinulted teory. We define D-muttion pir (, ) in pseudotrinulted teory C, nd proe tht the quotient teory /D rries trinulted struture under ertin resonle onditions. As pplitions, Our result unifies the quotient trinulted teories onsidered y Iym-oshino [6], y Hppel [5], y Beliinnis [2], y Jørensen [7] nd y Nkok [9] respetiely, ut not [8]. The pper is ornized s follos. In Setion 2, e reie the definition of pseudo-trinulted teory nd ie some preliminries. In Setion 3, e define D-muttion pir in pseudo-trinulted teory nd proe our min result Theorem In Setion 4, e ie fie exmples. 2. pseudo-trinulted teories We rell some sis on pseudo-trinulted teories from [9]. Let C e n dditie teory nd Σ : C C n dditie funtor. A sextuple (A,B,C,f,,h) in C is of the form A f B C h ΣA. A morphism of sextuples from A f B C h ΣA to A f B C h ΣA is triple (,,) of morphisms suh tht the folloin dirm ommutes: A f B C h ΣA A f B C h ΣA. Σ Ifin ddition,nd reisomorphismsin C, then (,,)is lledn isomorphism of sextuples. Definition 2.1. (f. [3], [9]) Let C e n dditie teory nd Σ : C C n dditie funtor. Let e lss of sextuples, hose elements re lled riht trinles. The triple(c, Σ, ) is lled pseudo riht trinulted teory, nd Σ is lled suspension f untor, if the folloin xioms re stisfied: (RTR0) is losed under isomorphisms. (RTR1) For ny A C, 0 A 1A A 0 is riht trinle. And for ny morphism f : A B in C, there exists riht trinle A f B C h ΣA. (RTR2) If A f B C h ΣA is riht trinle, then so is B C h ΣA f ΣB. (RTR3) For ny to riht trinles A f B C h ΣA nd A f B C h ΣA, nd ny to morphisms : A A, : B B suh tht f = f, there exists : C C suh tht (,,) is morphism of riht trinles. (RTR4) Let A f B C h ΣA, A l M m B n ΣA nd A l M m B n ΣA e riht trinles stisfyin m l = f. Then there exist C(B,C) nd h C(C,ΣA ) suh tht the folloin dirm is ommuttie nd the third

3 MUTATION PAIRS AND TRIANGULATED QUOTIENTS 3 olumn is riht trinle. A A A A l f l M m B n ΣA f B m n ΣA C h ΣA h ΣA Σl ΣM Remrk 2.2. In prtiulr, if suspension funtor Σ is n equilene, then C is trinulted teory. Pseudo-left trinulted teories (C, Ω, ) re defined dully, here Ω : C C is lled loop funtor, nd is the lss of left trinles. Remrk 2.3. Condition (RTR4) is slihtly different from tht in [3]. But the folloin to lemms re lso true in this sense. Lemm 2.4. ([1, Lemm 1.3]) Let C e pseudo-riht trinulted teory, nd A f B C h ΣA riht trinle. (1) is pseudookernel of f, nd h is pseudookernel of. (2) If Σ is fully fithful, then f is pseudokernel of, nd is pseudokernel of h. Lemm 2.5. ([8, Proposition 2.13]) Let (,, ) e morphism of riht trinles in pseudo-riht trinulted teory C: A f If nd re isomorphisms, so is. B C h ΣA A f B C h ΣA. Definition 2.6. ([9, Definition 3.1]) Let (C, Σ, ) e pseudo-riht trinulted teory, nd f C(A,B). (1) f is lled Σ-null if it ftors throuh some ojet in ΣC. (2) f is lled Σ-epi if for ny C(B,B ), f = 0 implies is Σ-null. For pseudo-left trinulted teory(c, Ω, ), e n define Ω-null morphisms nd Ω-moni morphisms dully. Definition 2.7. ([9, Definition 3.3]) The sextuple (C,Σ,Ω,,,ψ) is lled pseudo-trinulted teory if (C, Σ, ) is pseudo-riht trinulted teory, (C,Ω, ) is pseudo-left trinulted teory, (Ω,Σ) is n djoint pir ith the djunt ψ : hom(ωc,a) hom(c,σa), hih stisfy the folloin luin onditions (G1) nd (G2): (G1) If C(B,C) is Σ-epi, nd ΩC e A f B C, then A f B C ψ(e) ΣA. (G2) Iff C(A,B) is Ω-moni, nd A f B C h ΣA, then ΩC ψ 1 (h) A f B C. Σl Σ

4 4 ENGQIANG LIN AND MINXIONG WANG Exmple 2.8. ([9, Exmple 3.4]) Let (C,Σ,Ω,,,ψ) e pseudo-trinulted teory. (1) C is n elin teory if nd only if Σ = Ω = 0. (2) C is trinulted teory if nd only if Σ is the qusi-inerse of Ω. Definition 2.9. ([9, Definition 4.1]) Let (C,Σ,Ω,,,ψ) e pseudo-trinulted teory. A sequene in C ΩC e A f B C h ΣA is lled n extension if A f B C h ΣA, ΩC e A f B C, nd h = ψ(e). A morphism of extensions from ΩC e A f B C h ΣA to ΩC e A f B C h ΣA is triple (,,) suhtht the folloin dirmis ommuttie ΩC e A f B C h ΣA Ω ΩC e A f B C h ΣA. Note tht e = e Ω is equilent to Σ h = h, thus morphism of extensions is the sme s morphism in or. Exmple (f. [9, Proposition 4.6]) Let C e pseudo-trinulted teory. (1) For ny A,B C, ΩB 0 A ia A B pb B 0 ΣA is n extension, here i A nd p B re the injetion nd projetion respetiely. (2) IfC iselin, thennextensionisnothinotherthnshortextsequene. (3) If C is trinulted teory, then n extension is nothin other thn distinuished trinle. The folloin lemm ill e frequently used in the next setion. Lemm ([9, Proposition 4.7]) Let ΩC e A f B C h ΣA, ΩB k A l M m B n k ΣA nd ΩB A l M m B n ΣA e extensions stisfyin m l = f. Then there exist C(B,C) nd h C(C,ΣA ) suh tht the folloin dirm is ommuttie nd the fourth olumn is n extension. ΩB Ω ΩC Σ A k A ψ 1 (h ) ΩB k A ΩC e A l f l M m B n ΣA f B m n ΣA C h ΣA h Σl ΣA Σl ΣM

5 MUTATION PAIRS AND TRIANGULATED QUOTIENTS 5 In the rest of this setion, e ie some properties on Σ-epi morphisms nd Ω-moni morphisms in pseudo-trinulted teory C. Lemm (1)Let ΩC e A f B C e left trinle. Then the folloin sttements re equilent. () is Σ-epi. () A f B C ψ(e) ΣA is riht trinle. () ΩC e A f B C ψ(e) ΣA is n extension. (2) Let A f B C h ΣA e riht trinle. Then the folloin sttements re equilent. () f is Ω-moni. () ΩC ψ 1 (h) A f B C is left trinle. () ΩC ψ 1 (h) A f B C h ΣA is n extension. Proof. We only sho (1). () () follos from luin ondition (G1). By definition of extension e et () (). It remins to sho () (). Sine B C ψ(e) ΣA Σf ΣB is riht trinle, is Σ-epi y definition. Lemm The folloin sttements re equilent. (1) f : A B is Σ-epi. (2) For ny riht trinle A f B C h ΣA, there exists n ojet C C suh tht C = ΣC. (3) There exists riht trinle A f B ΣC h ΣA. Proof. (1) (2). Let ΩB d C e f A B e left trinle. Sine f is Σ-epi, C e f ψ(d) A B ΣC is riht trinle. By (RTR2), A f B ψ(d) ΣC Σe ΣA is riht trinle. So C = ΣC y Lemm 2.5. (2) (3) nd (3) (1) re ler. Lemm Let f : A B, : B C nd h : A C e morphisms in C suh tht h = f. (1) If h is Σ-epi, then so is ; (2) If h is Ω-moni, then so is f; (3) If f nd re Σ-epi, then so is h. Proof. For (1) nd (2) e see [9, Lemm 4.4]. No proe (3). Sine : B C is Σ-epi, there exists riht trinle L f B C h ΣL y Lemm By (RTR4), e et the folloin ommuttie dirm L L f A f B M h ΣA A h C N ΣA h ΣL ΣL

6 6 ENGQIANG LIN AND MINXIONG WANG here the middle to olumns nd middle to ros re riht trinles. Sine f : A B is Σ-epi, there exits n isomorphism m : M ΣM ith M C y Lemm Thus m : L ΣM is Σ-epi y definition, nd there exists riht trinle L m ΣM m ΣN n ΣL y Lemm By (RTR3) nd Lemm 2.5 there exists n isomorphism n : N ΣN suh tht the folloin dirm is ommuttie. L M N ΣL L m ΣM m ΣN n ΣL Aordin to Lemm 2.13 in, h is Σ-epi. m n 3. Min result Let D e suteoryof n dditie teoryc. A morphism f : A B in C is lled D-epi, if for ny D D, the sequene C(D,A) C(D,f) C(D,B) 0 is ext. A riht D-pproximtion of X in C is D-epi mp f : D X, ith D in D. If for nyojet X C, there exists rihtd-pproximtionf : D X, then D is lled ontrrintly finite suteory of C. Dully e he the notions of D-moni mp, left D-pproximtion nd orintly finite suteory. The suteory D is lled funtorilly finite if D is oth ontrrintly finite nd orintly finite. Definition 3.1. Let D nd e to suteories of pseudo-trinulted teory C, nd D. The pir (,) is lled D-muttion pir if it stisfies (1) For ny X, there exists n extension Ω e X f D h suh tht, f is left D-pproximtion nd is riht D-pproximtion. (2) For ny, there exists n extension Ω e X f D h suh tht X, f is left D-pproximtion nd is riht D-pproximtion. Definition 3.2. Let C e pseudo-trinulted teory. A suteory of C is lled extension-losed if for ny extension in C X, implies. Ω e X f h, Let Ω e X f h e n extension in C nd X,,, e simply sy the extension is in. Assumption 3.3. Assume tht C is pseudo-trinulted teory stisfyin the folloin onditions: Let ΩC e A f B C h ΣA nd ΩC e A f B C h ΣA e ny to extensions in C. (A1) If C(C,C ) stisfies h = 0 nd = 0, then there exists C(C,B ) suh tht =. (A2) If C(A,A ) stisfies f = 0 nd e = 0, then there exists C(B,A ) suh tht f =. Remrk 3.4. If C is trinulted teory, it is esy to see tht Assumption 3.3 is triil. If C is n elin teory, then is epi so tht = 0 implies tht

7 MUTATION PAIRS AND TRIANGULATED QUOTIENTS 7 = 0, thus e n tke = 0 in (A1). Therefore, Assumption 3.3 is triil for oth trinulted teory nd elin teory. Fromnoontothe endofthissetion, essumethtc ispseudo-trinulted teory stisfyin Assumption 3.3, nd ssume tht (, ) is D-muttion pir. Lemm 3.5. Let Ω Ωz i e X f x i y i z i h Ω e X f D h Σx i (i=1,2) e morphisms of extensions in, here D D nd f is D-moni, i = 1,2. Then in quotient teory /D, x 1 = x 2 implies tht z 1 = z 2. Proof. Sine x 1 = x 2, there exist 1 : X D, 2 : D X suh tht x 1 x 2 = 2 1, here D D. Sine f is D-moni, there exists 3 : D suh tht 1 = 3 f. Thus (x 1 x 2 )e = 2 1 e = 2 3 fe = 0. Then Σ(x 1 x 2 ) h = Σ(x 1 x 2 )(ψ(e)) = ψ((x 1 x 2 )e) = 0. Note tht (y 1 y 2 f 2 3 )f = (y 1 y 2 )f f (x 1 x 2 ) = 0, there exists d : D suh tht y 1 y 2 f 2 3 = d. No (z 1 z 2 d) = (y 1 y 2 ) (y 1 y 2 f 2 3 ) = 0, nd h (z 1 z 2 d) = h (z 1 z 2 ) = Σ(x 1 x 2 ) h = 0. By (A1), there exists d : D suh tht z 1 z 2 d = d. So z 1 z 2 = (d+d ) nd z 1 = z 2. Lemm 3.6. Let Ω e X f D h nd Ω e X f D h e extensions in, here f,f re left D-pproximtions. Then nd re isomorphi in /D. Proof. Sine f,f re left D-pproximtions, e otin the folloin ommuttie dirm Ω e X f D h Ωy Ω e d X f D h y Ωy e Ω X f D h. By Lemm 3.5, e et y y = 1. Similrly, e n sho tht yy = 1. Hene nd re isomorphi in /D. d For ny X, y definition of D-muttion pir, there exists n extension ΩTX e X f D TX h, here D D, TX, f is left D- pproximtion nd is riht D-pproximtion. By Lemm 3.6, T X is unique up to isomorphism in the quotient teory /D. So for ny X, e fix n extension s oe. For ny x (X,X ), Sine f is left D-pproximtion, e n omplete the folloin ommuttie dirm: ΩTX e X f D TX h Ωy x d Σx ΩTX e X f D TX h. y y

8 8 ENGQIANG LIN AND MINXIONG WANG We define funtor T : /D /D y T(X) = TX on the ojets X of /D nd y Tx = y on the morphisms x : X X of /D. By Lemm 3.5, Tx is ell defined nd T is ndditie funtor. Sometimes edenote y y Txfor oneniene. Lemm 3.7. The funtor T : /D /D is n equilene. Proof. For ny, e fix n extension Ω e T f D h ΣT, here D D, T, f is left D-pproximtion nd is riht D-pproximtion. We n similrly define n dditie funtor T : /D /D y T () = T. It is esy to hek tht T T = id nd TT = id. Thus T is n equilene. Let Ω u X x e n extension in nd is D-moni. Sine X, e he n extension ΩTX e X f D TX h. Then e n otin the folloin ommuttie dirm sine is D-moni. Ω u X x Ωz ΩTX e X f D TX h We ll the sequene X z TX stndrd trinle in /D. We define the lss of distinuished trinle s the sextuples hih re isomorphi to stndrd trinles. Lemm 3.8. Let y z Ω u X x Ω Ω u X x e morphism of extensions in, nd, e D-moni. Then e he the morphism of stndrd trinles in /D: Σ X z TX T X z TX. Proof. By the definition of stndrd trinle nd definition of the funtor T e he the folloin ommuttie dirms Ω u X x Ωz ΩTX e X f D y z TX h ΩT ΩTX e X f D TX h, d T Σ

9 MUTATION PAIRS AND TRIANGULATED QUOTIENTS 9 nd Ω u X x Ω Ω u X x Ωz ΩTX e X f D TX h. y z Σ By Lemm 3.5 e et T z = z. Lemm 3.9. Let X X l X l M m f X f m e ommuttie dirm, here m is the pseudo-okernel of l nd m l = 0. If l nd f re D-moni, then f is lso D-moni. Proof. For ny : X D, here D D. We need to sho tht ftors throuh f. Sine l is D-moni, there exists morphism : M D suh tht = l. Sine f is D-moni, there exists morphism : D suh tht l = f. Thus ( m )l = l f = 0. There exists morphism d : D suh tht m = dm. Hene = l = l m l = dml = df. Theorem Let C e pseudo-trinulted teory stisfyin Assumption 3.3, nd e n extension-losed suteory of C. If (,) is D-muttion pir, then (/D, T, ) is trinulted teory. Proof. We ill hek tht the distinuished trinles in stisfy the xioms of trinulted teory. (TR1) Let (X,). Tke = (,f) T : X D nd p D = (0,1 D ) : D D, here f : X D is left D-pproximtion of X. Then p D = f. Sine f is D-moni nd Ω-moni, e et is D-moni, nd is lso Ω-moni y Lemm 2.14(2). No there re three extensions ΩTX e X f D TX h, ΩD 0 i D pd D 0 Σ nd Ω u X D x. By Lemm 2.11, there exist C(,TX) nd h C(TX,Σ) suh tht the folloin

10 10 ENGQIANG LIN AND MINXIONG WANG dirm is ommuttie nd the fourth olumn is n extension. ΩD 0 Ω ΩTX ψ 1 (h ) Ω u ΩTX e X X i D f D p D 0 Σ f x (3.1) TX h Σ h Σ Σi Σ( D) Sine,TX nd is extension-losed, e et. Thus X D TX is stndrd trinle. So X f TX is distinuished trinle. It is esy to see tht Ω0 X 1X X 0 is n extension nd 1 X is D-moni. Thus X 1 X X 0 TX. (TR2) Let X z TX (3.2) e distinuished trinle indued y the extension Ω u X x, here is D-moni. By (TR1) e suppose tht X f TX (3.3) is distinuished trinle indued y. Sine is D-moni, there exists y : D suh tht f = y. Thus e he the folloin ommuttie dirm Ω u X x Ωz Ω u X (1,y) T D z x Ωz Ω u X p z x. Sine p (1,p) T = 1, e et z z is n isomorphism y Lemm 2.5. On the other hnd, let = (f, ). Sine is pseudookernel of nd (y, 1 D ) = (y, 1 D )(,f) T = 0, there exists : D suh tht (y, 1 D ) = = (f, ). Thus y = f, = 1 D. By Lemm 2.11, e he the folloin ommuttie

11 MUTATION PAIRS AND TRIANGULATED QUOTIENTS 11 dirm Ω D 0 Ω Ω D ψ 1 ( ) Ω u X i D D x Ω u X p z x 0 ΣD Σ ΣD ΣiD Σ( D), here the fourth olumn is n extension. Note tht z is pseudookernel of nd (1 + ) = = 0, there exists z : suh tht z z = 1 +. So z z = 1. Therefore, z is n isomorphism in /D, nd the distinuished trinle (3.2) nd (3.3) re isomorphi y Lemm 3.8. No e n reple the trinle (3.2) ith (3.3). Aordin to the proof of (TR1), e et n extension ΩTX ψ 1 (h ) f TX h Σ in. Sine i nd f re D-moni, so is f y Lemm 3.9. Tke n extension ΩT e f h D T Σ, here f is left D-pproximtion nd is riht D-pproximtion. Then e et the folloin ommuttie dirm ΩTX ψ 1 (h ) f TX h Σ Ωz ΩT e y f D T z h Σ. Thus f TX z T is distinuished trinle. To end the proofof(tr2) e only need to sho tht z = T. The ommuttie dirm (3.1) implies the folloin ommuttie dirm ΩTX e X f D TX h 1 ΩTX ΩTX ψ 1 (h ) f 1 TX TX h Σ Σ. In ft, f = i D f = i D p D = ( i p ) = i p = f, = i D = p D i D = nd Σ h = Σp Σ h = Σp Σi h = h. Composin the oe to ommuttie dirms, e otin the folloin

12 12 ENGQIANG LIN AND MINXIONG WANG ommuttie dirm ΩTX e X f D TX h Ωz ΩT e y f D T z h Σ Σ, hih implies tht T = z. (TR3) Suppose there is ommuttie dirm X z TX T X z TX, here the ros re distinuished trinles. We my ssume tht, re D-moni nd the distinuished trinles re risin from the extensions Ω u X x nd Ω u X x respetiely. Sine =, there exist 1 : X D, 2 : D suh tht = 2 1, here D D. Sine is D-moni, there exists 3 : D suh tht 1 = 3. Thus ( 2 3 ) = 2 1 =. So y (RTR3) there exists : suh tht the folloin dirm is ommuttie Ω u X 2 3 x Ω Ω u X x. Σ Hene (TR3) follos from Lemm 3.8. (TR4) Let X z TX, X z TX nd X t q TX e distinuished trinles. Assume ( tht nd)( re) D-moni. ( Let ) 0 f : X D e left D-pproximtionofX. Sine = 0 1 D f f ( ) ( ) ( ) is D-moni, replin y, f y, 0 0 y, nd djustin 0 1 D the orrespondin distinuished trinles, e n ssume tht is D-moni too. Thus e n ssume tht the oe three distinuished trinles re stndrd, hih re indued y the folloin three extensions Ω u X x, Ω u X x nd Ω n X q r. By

13 MUTATION PAIRS AND TRIANGULATED QUOTIENTS 13 Lemm 2.11, e et the folloin ommuttie dirm Ω X u Ωq Ω X n Ω u Ωq X Ω n X q x p q x r r Σ Σ Σ, here the third ro, the fourth ro, the third olumn nd the fourth olumn re ll extensions, moreoer,,,,p re ll D-moni y Lemm 3.9. Thus y Lemm 3.8 e et the folloin ommuttie dirm of distinuished trinles: X X X p z TX X q q t TX z t TX TX. To end the proof, it remins to sho tht the folloin dirm is ommuttie: t t TX T (3.4) TX T T We first lim tht there exist morphisms of extensions nd Ω n X Ω ΩT e q f D T Ω n X p r q h Σ Σ (3.5) r Ω ΩT e f D T h Σ Σ,

14 14 ENGQIANG LIN AND MINXIONG WANG suh tht f = +, = T t nd = T t. In ft, sine is D-moni, there exists : D suhtht f =, then y (RTR3) there exists : T suh tht (,,) is morphism of extensions. Note tht (f ) = 0, there exists : D suh tht = f. Thus f = + nd f = + = p, then y (RTR3) there exists : T suh tht (,, ) is morphism of extensions. By the onstrution of stndrd trinle, e he the folloin ommuttie dirm Ω n X q r Ωt ΩTX e X f D TX h. On the other hnd, y the definition of T, e he the folloin ommuttie dirm ΩTX e ΩT ΩT e X f D s d f D T t TX h T h Σ Σ. Composin the lst to dirms, e immeditely otin the folloin ommuttie dirm Ω n X Ω(T t) ΩT e ds q f D T T t r h Σ. Σ (3.6) Comprin dirm (3.5) nd dirm (3.6), e et = T t y Lemm 3.5. Similrly e n sho = T t. Noenshodirm(3.4)isommuttie. Ononehnd, h (+ ) = Σ r+ Σ r = 0. Ontheotherhnd,(+ )q = q + q = + = f = 0. Sine,q re Σ-epi, so is q y Lemm 2.14(3). Thus e otin n extension Ω α β q γ Σ y Lemm So + ftors throuh D y (A1). Then T t+t t = + = 0. Remrk In the proof of (TR4), e see [4] for the reson hy e n djust the morphisms nd. 4. Exmples Sine Assumption 3.3 is triil for oth trinulted teories nd elin teories, e n pply Theorem 3.10 to seerl situtions. Exmple 4.1. ([6, Theorem 4.2]) Let D e riid suteory of trinulted teory C nd e suteory of C suh tht D. We ll (,) D-muttion pir in the sense of Iym-oshino, if it stisfies (1) C(,ΣD) = C(D,Σ) = 0. (2) For ny X, there exists distinuished trinle X f D h ith D D nd.

15 MUTATION PAIRS AND TRIANGULATED QUOTIENTS 15 (3) For ny, there exists distinuished trinle X f D h ith D D nd X. Inondition(2)nd(3),f isleftd-pproximtionnd isrihtd-pproximtion immeditely follos from ondition (1). Thus (, ) is D-muttion pir in the sense of Definition 3.1. In this se, if is extension-losed, then /D is trinulted teory y Theorem Exmple 4.2. ([5, Theorem 2.6]) Let C e n elin teory. Let (B,S) e Froenius suteory of C nd I the suteory of B onsistin of ll S-injeties. Then B/I is trinulted teory. Proof. Aordin to Theorem 3.10, e need to sho tht (B,B) is I-muttion pir. In ft, for ny X B, sine B hs enouh S-injeties, there exists short ext sequene 0 X f I 0 in S, here I I. It is ler tht f is left I-pproximtion of X. Sine the S-projeties oinide ith the S-injeties, is riht I-pproximtion of y definition. The seond ondition n e shoed similrly. Exmple 4.3. ([2, Theorem 7.2]) Let C e trinulted teory nd E proper lss of trinles on C. An ojet I C is lled E-injetie, if for ny distinuished trinle A B C ΣA in E, the indued sequene 0 C(C,I) C(B,I) C(A,I) 0 is ext. Denote y I the full suteory of E-injetie ojets of C. We sy tht C hs enouh E-injeties if for ny ojet A C there exists distinuished trinle A I C ΣA in C ith I I. If C hs enouh E-injeties nd enouh E-projeties nd I = P, here P is the suteory of E-projeties, then e n sho tht (C,C) is I-muttion pir, thus C/I is trinulted teory y Theorem We remrk tht C(I,ΣI) is not zero euse I is losed under Σ. So (C,C) is not I-muttion pir in the sense of Iym-oshino. Exmple 4.4. ([7, Theorem 2.3]) Let C e trinulted teory ith Auslnder- Reiten trnsltion τ, nd X funtorilly finite suteory ith τx = X. Let X f D h e trinle in C, then e n sho tht f is left X- pproximtion if nd only if is riht X-pproximtion (see [7, Lemm 2.2]). Thus (C,C) is X-muttion pir, nd C/X is trinulted teory y Theorem We remrk tht (C,C) my e not X-muttion pir in the sense of Iym- oshino. For exmple, if C is luster teory, then C(X,) is not zero. Exmple 4.5. ([9, Theorem 6.17]) Let C e pseudo-trinulted teory, if (C,,D) is Froenius (see [9, Definition 5.9]), then (,) is I D -muttion pir, here I D is the full suteory of D onsistin of injetie ojets. If is extension-losed nd C stisfies Assumption 3.3, then /I D is trinulted teory y Theorem Aknoledements The uthors thnk Professor Xiou Chen for useful omments nd dies. Referenes [1] I. Assem, A. Beliinnis, N. Mrmridis. Riht Trinulted Cteories ith Riht Semiequilenes. Cnndin Mthemtil Soiety Conferene Proeedin. Volume 24, 17-37, 1998.

16 16 ENGQIANG LIN AND MINXIONG WANG [2] A. Beliinnis. Reltie homoloil ler nd purity in trinulted teories. J. Aler. 227, no.1, , [3] A. Beliinnis, N. Mrmridis. Left trinulted teories risin from ontrrintly finite suteories. Communitions in ler. 22(12), , [4] X. Chen. To lemms on xiom (TR4), ust. edu.n/ xhen/ pulition.htm. [5] D. Hppel. Trinulted Cteories in the Representtion Theory of Finite Dimensionl Alers. London Mthemtil Soiety, LMN 119, Cmride, [6] O. Iym,. oshino. Muttion in trinulted teories nd riid Cohen-Muly modules. Inent. Mth. 172, no. 1, , [7] P. Jørensen. Quotients of luster teories. Proeedins of the Royl Soiety of Edinuih. 140A, 65-81, [8]. Liu, B. hu. Trinulted quotient teories. to pper in Communitions in Aler, [9] H. Nkok, Froenius ondition on pretrinulted teory, nd trinultion on the ssoited stle teory. rxi: , Shool of Mthemtil sienes, Huqio Uniersity, Qunzhou , Chin. E-mil ddress: Shool of Mthemtil sienes, Huqio Uniersity, Qunzhou , Chin. E-mil ddress: