代数表示论是近三十多年来代数学的一个新的重要分支.目前,代数表示论发展的特点之一就是与代数几何的交叉和渗透.其中,沟通代数表示论和代数几何的桥梁是三角范畴(导出范畴)的理论体系.在这一理论体系中,Beilinson,Bernstain和Deligne在上个世纪八十年代初提出的三角范畴的recollement和t-结构起着关键性的作用.本学位论文围绕recollement与t-结构展开研究.全文共分成四章.在第一章,我们对与论文有关的研究背景起源及发展动态作简要介绍,并阐述本文的主要工作与已有成果的关系.在第二章,我们考虑在已知三个三角范畴和四个正合函子的情况下,构造新的recollement....The theory of representations of algebras is one of the mostimportant branches of algebra during the last thirty years.Recently, one of the developement characters of the theory ofrepresentations of algebras is the intersection and penetration withthe algebraic geometry. And, the bridge establishing the relationbetween the theory of representations of algebras and the algebraicgeometry is the the...学位：理学硕士院系专业：数学科学学院数学与应用数学系_基础数学学号：20042300

刘宏锦

Xiamen University Institutional Repository

m`^710384 - 8m200423005 UDCI F n S 5 VA~go recollementRecollement of Triangulated Categories4 , 2uRZdj^ [ m _ Zev n ^ O W P f l\igXba 2007 ` 5 r\iQNba 2007 ` rlhepba 2007 ` rm_Q[<  ?2007 ; 5 厦门大学博硕士论文摘要库'YaVVPWIaKIzmWQÆT%$iD,i%{8	Hm\H<%$Q?Y7>mr!$Y_|m\H<Æ7jbN-b%$e2}CMd{ZQBBm6C&&$dw  )厦门大学博硕士论文摘要库'YaVVPBXJxa%$	A(ZbW}2IKyWQm8w(ZbW}I36:i4:ZY,w[.izQm.4CsMÆ}wWQyt6kmm5?23QQ.W9[5#AÆ}wWQmq+*.}2a%GonÆ}wWQm-zCaT*S\mWQ\IUy%8w%WQ℄  1\ Æ   w\IUy%X[  27\j40s'Aq_ ” ”Y!d      )  w  )iDd      )  w  )厦门大学博硕士论文摘要库 s`.OOS0Ev)`Uld Bl<`'j`.OOlwr)dS`DlyB?6+~`.OOB`Dl<S0{	P (hR	P) l0O{""d0O{"6 Beilinson, Bernstain B Deligne 3 QhEvQxRl0{	Pl recollement B t-  -J1rHlWw$UO! recollement t-  -[*Ggqd[O|1l["`xtWp`62C_$l?`&Wh|G;l1"q[Mh(0 0{	PBg %E?Ll&$-Bl recollement.q0[[0{	P recollement $ AR- 0{lNHy&`: {D′ ,D,D′′ ; i∗, i∗ = i!, i!, j!, j∗ = j!, j∗} Sd recollement, /D | AR- 0{ D′ B D′′ b| AR- 0{qg[?`vO0{	P| t-  -lC#Rl OS: (D≤0,D≥0) S0{	P D l| t-  -/ D 6%lld 6!7 X, RR X ∈ D≤0, X X ∈ D≥1, Y t-  -lCSfVll 0{	P recollement; AR- 0{ t-  -i厦门大学博硕士论文摘要库AbstractThe theory of representations of algebras is one of the most impor-tant branches of algebra during the last thirty years. Recently, one of thedevelopement characters of the theory of representations of algebras is theintersection and penetration with the algebraic geometry. And, the bridgeestablishing the relation between the theory of representations of algebrasand the algebraic geometry is the theory systems of triangulated categories(derived categories). In the theory systems, the recollement and t-structureof triangulated category play the main roles, which were first introduced byBeilinson, Bernstain and Deligne in the early eighty of last century. Thisdesertation concentrates the recollement, which includes four chapters alto-gether.In the first chapter, we give a brief introduction of the background andrecent developments about the fields we study in the dissertation. And welist some concepts and theorems which are closely related to this disserta-tion.In the second chapter, we construct a new recollement in the case ofthe given three triangulated categories and four exact functors which satisfythe certain conditions.In the third chapter, we study the problem of preservation of AR-triangler under the recollement. And we get the following conclusion. Let{D′,D,D′′; i∗, i∗ =i!, i!, j!, j∗ = j!, j∗} be a recollement, and D has AR-triangles. Then D′and D′′also have AR-triangles.In the fouth chapter, we discuss that the heart of bounded t-structure.And we get the following conclusion. Let (D≤0,D≥0) be bounded t-structureon D. Suppose that for any indecomposable object X in D, X belongs toD≤0, or X belongs to D≥1. Then the heart of the t-structure is hereditary.Key words: triangulated categories; recollement; AR-triangles; t-strutureii厦门大学博硕士论文摘要库: 3U	r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i{U	r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iirt l4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§1.1 }f/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§1.2 }fn recollement tx\j . . . . . . . . . . . . . . . . . . . . . . . . . . . 3§1.3 }f recollement ntap)d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7§1.4 }f t- !ta^p)d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8§1.5 \mR4Ud' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10rx [D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12§2.1 }f recollement n1{ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12§2.2 }f recollement nen{ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13§2.3 }fn recollement Yt t- ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14r Recollement  AR-  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17§3.1 }fn AR-  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17§3.2 }fn Recollement  AR-  . . . . . . . . . . . . . . . . . . . . . . . . . . 18rK }fn t- ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21§4.1 }f|$ t- !ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21§4.2 }fn|$ t- !vj Abel }f . . . . . . . . . . . . . . . . . . . . . . . . . 22b+U\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26vHEmR=y`Odn4U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30i厦门大学博硕士论文摘要库ContentsAbstract(in Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAbstract(in English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiChapter  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§1.1 Triangulated categories and examples . . . . . . . . . . . . . . . . . . . . . . . 1§1.2 Definement and basical properties of recollement of triangulatedcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3§1.3 Some study achievements of recollements of triangulatedcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7§1.4 T-structures of triangulated categories and some related studyachievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8§1.5 The brief introduction on the desertation . . . . . . . . . . . . . . . . . . . 10Chapter  Construction of recollement . . . . . . . . . . . . . . . . . . 12§2.1 Two methods of construction of recollement of triangulatedcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12§2.2 New methods of construction of recollement of triangulatedcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13§2.3 Recollement of triangulated categories and stable t-structure . . . 14Chapter  Recollement and AR-triangles . . . . . . . . . . . . . . .17§3.1 AR-triangles in the triangulated categories . . . . . . . . . . . . . . . . . . 17§3.2 Recollement of triangulated categories and AR-triangles . . . . . . . 18ii厦门大学博硕士论文摘要库Chapter 	 T-structures on triangulated categories . . . . . .21§4.1 The heart of the bounded t-structures . . . . . . . . . . . . . . . . . . . . . .21§4.2 The bounded t-structures and hereditary abelian categories . . . . 22References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26Papers published and finished by the auther . . . . . . . . . . . . . 29Acknowledges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30iii厦门大学博硕士论文摘要库#DJ B5 1Tot k℄3 QhEvQ Beilinson,Bernstain B Deligne xR0{	Plrecollement B0{	Pl t-  -lxB0O{"`.OOB`Dl:"x(Bl<`&$[|1[5l"`xtWp`S02$UOl?`&WVp`6§1.1 Bh00{	Plx Verdier l&W ([29]),  [2], [12] 6b|6: C Sd j	PT : C → C Sd -+R[F) g (X shift)?Lud T−1. C 3lKS (X, Y, Z, µ, ν, ω) z C 6l7 X, Y, Z i`t8 µ : X → Y, ν : Y → Z, ω : Z → TX -G[DKS (X, Y, Z, µ, ν, ω)d X µ→ Y ν→ Z ω→ TX.KS (X, Y, Z, µ, ν, ω)jKS (X ′, Y ′, Z ′ , µ′, ν ′, ω′)lt8vmIk$!yRlt8 (f, g, h):Xµ→ Yν→ Zω→ TX↓ f ↓ g ↓ h ↓ TfX′ µ′→ Y′ ν′→ Z′ ω′→ TX′,[FRR3_yRlt8 (f, g, h) 0{t8/ f, g, h yS-F/rl= KS-tx 1.1.1 : S S C 6d;KSl^E,;$Bg }sRRF(C,S) X C S0{	P S 6ljF0{(TR1) V 0{-lKSS0{ C 6V t8 µ : X → Y y!i- C 6ld 0{ X µ→ Y ν→ Z ω→ TX; X 1X→ X → 0 → TX S0{(TR2) ,; X µ→ Y ν→ Z ω→ TX S0{ Y ν→ Z ω→ TX −Tµ→ TY bS0{(TR3) ,;$SF0{lyRXµ→ Yν→ Zω→ TX↓ f ↓ g ↓ TfX′ µ′→ Y′ ν′→ Z′ ω′→ TX′,厦门大学博硕士论文摘要库#DJ B5 2^t8 h : Z → Z ′ Ik3(yR(TR4) (^{)0) /|0{ (X, Y, Z ′, µ, i, i′), (Y, Z, X ′, ν, j, j ′) B (X, Z,Y′, νµ, k, k′), ^ f : Z ′ → Y ′, g : Y ′ → X ′ Ik$!yRq0F0{T−1Y′ T−1k′−→ X1X−→ X↓ T−1g ↓ µ ↓ νµT−1Y′ T−1j′−→ Yν−→ Zj−→ X′ j′−→ TY↓ i ↓ k ↓ 1X′ ↓ T iZ′ f−→ Y′ g−→ X′ T ij′−→ TZ′↓ i′↓ k′TX1TX−→ TX. 1.1.2 $o9`jl0{	Pl%lH3!=, [12]./ 1.1.3 : A S Abel 	PKb(A)  A 3l|ElN	PG TS A lE	P C(A) 3l5U gd8l g?La (TX .)n = Xn+1, ∀n ∈Z, dTX. = −dX., aX . : · · · → X0d0→ X1 → · · · → Xndn→ Xn+1 → · · · ,TX . : · · · → X1−d1→ X2 → · · · → Xn+1−dn+1→ Xn+2 → · · · ./ µ : X . → Y .,  Tµ : TX . → TY ., 6 (Tµ)n = (µ)n+1, ∀n ∈ Z. ST : Kb(A) → Kb(A) SN-d X .[1] := TX ., µ[1] := Tµ.: µ : X . → Y . S A 3l|E Cb(A) lEt8vmu8H Con.(µ),$Con.(µ) := X .[1] ⊕ Y ., lL  −dX. 0−µ dY ..G S S Kb(A) 6"_d;0{l^E";0{ (X .′, Y .′, Z .′, µ′, ν ′, ω′) -0{ (X ., Y ., Con.(µ), µ, (0, 1)τ , (1, 0), a^-l0{t8 (f, g, h) Ik厦门大学博硕士论文摘要库#DJ B5 3$!yR (NyR).X .′ µ′−→ Y .′ ν′−→ Z .′ ω′−→ X .′[1]↓ f ↓ g ↓ h ↓ TfX .µ−→ Y .(0,1)τ−→ Con.(µ)(1,0)−→ X .[1],6 f, g, h SNnl (Kb(A),S) S0{	P: f : X . → Y . S Cb(A) 6t8/ Hn(f) : Hn(X .) ∼= Hn(Y .), ∀n ∈ Z, "2 Hn(X .) = Kerdn/Imdn+1 S X . lq n [3u7F f St-GS = {f |f S Kb(A) 6lt8St- }, S S Kb(A) ld /*J"\!kj Kb(A) 3ld 9O	PS−1Kb(A),F Al|hR	Pd Db(A), !i&` Db(A)S0{	P§1.2 Bh recollement puy ℄kRecollement lxSz Grothendieck T)xRl ([11]), qrFp"NPM$n X 3l? X 3'L$n` X l5hRl?)nl1"YFl recollement Q-vmhR	P3W7 Beilinson,Bernstain BDeligne  1982 vvm0{	Pl recollement([2]). [e1```l.O0O6R+lT	Py\Al Kazhdan-LusztigB Lusztig ;1 Cline,Parshall B Scott 5whR	Pl recollement le1p-tfV`lx ([6]), 2ItfV`G.OOl<`[' (,q, [6], [8], [25] n). YH0{	P recollement lG`.OO[l"X#yV)[S0d$0{	P recollement lvmtx 1.2.1[2] : D,D′ B D′′ S0{	P D 1 D′ B D′′ ld rec-ollement, dW {D′,D,D′′; i∗, i∗ = i!, i!, j!, j∗ = j!, j∗} X D′ ←⇄ D ←⇄ D′′ , S+0{	PlK %E?Li∗ = i! : D′→ D; j∗ = j! : D → D′′; i∗, i! : D → D′; j∗, j! : D′′→ D,RR$Bg }s厦门大学博硕士论文摘要库#DJ B5 4(R1) (i∗, i∗, i!) B (j!, j∗, j∗) Sm0S(R2) j!i∗ = j∗i! = 0;(R3) i∗, j! B j∗ SR-l?L (\ i∗i∗ ∼= idD′ , j∗j∗ = j∗j! ∼= idD′′ );(R4)  D 6l%ld7 X, !iv= 0{(i) i!i!X → X → j∗j∗X → i!i!X[1] B (ii) j!j!X → X → i∗i∗X → j!j!X[1],"2lt8 i!i!X → X B j!j!X → X S;}t8X → j∗j∗X B X → i∗i∗XSH;}t8w/p$ [23], / D′,D,D′′; i∗, i∗, j!, j! RRR+vm 1.2.16lo|}sF {D′,D,D′′; i∗, i∗, j!, j!} Sd U recollement; / D′,D,D′′; i!, i!, j∗, j∗RRR+vm 1.2.1 6lo|}sF {D′,D,D′′; i!, i!, j∗, j∗} Sd  rec-ollement. 1.2.2 vm 1.2.1 6	P i∗D′ S Gabriel lm$l9OB9OL	PMiyachi B Beligannis n, [20],[3] 6/vO recollement b9O TTF- 0Ol1" 1.2.3 $i3U recollement B recollement lvm*kR {D′, D , D′′; i∗, i∗ , j! , j!} SU recollement gg {D′′ , D , D′ ;j! , j!, i∗, i∗} S recollement. 	d8p/ {D′,D,D′′; i∗, i∗ = i!, i!, j!, j∗ = j!, j∗} Sd recollement,  {D′ , D , D′′ ; i∗, i∗ , j! , j!} SU recollement, {D′ , D, D′′; i! , i! , j∗ , j∗} S recollement. )/ {D′,D,D′′; i∗, i∗, j!, j!} SUrecollement, {D′,D,D′′; i!, i!, j∗, j∗} S recollement, 2 i∗ = i!, j∗ = j!, {D′,D,D′′; i∗, i∗ = i!, i!, j!, j∗ = j!, j∗} S recollement. 1.2.4 (1): U ,V Sj	P C l7.,;%ll U ∈ U , V ∈ Vy| HomD(U, V ) = 0, [d HomD(U ,V) = 0.(2) : F : C → D Sj	Pn?Ld ImF = {Y ∈ D|∃X ∈ C, Y ∼=F (X)}, KerF = {X ∈ C|F (X) ∼= 0}.(3) : X S0{	P D lRL	Pd X⊥ , ⊥X /.O X lX*	PBUX*	P"2X⊥ = {M ∈ D|HomD(X , M) = 0},⊥X = {N ∈ D|HomD(N,X ) = 0}.厦门大学博硕士论文摘要库#DJ B5 5$^[#R0{	P recollement ld;Z$H3z. 1.2.5: {D′,D,D′′; i∗, i∗, j!, j!}Sd U recollement,  Kerj! = Imi∗,Keri∗ = Imj!.&`z j!i∗ = 0, ( Imi∗ ⊆ Kerj!. F%ll X ∈ Kerj!, ^d 0{ j!j!X → X → i∗i∗X → j!j!X[1]. o j!X ∼= 0, oi X ∼= i∗i∗X ∈ Imi∗,\ Kerj! ⊆ Imi∗. oi Kerj! = Imi∗. 0!& Keri∗ = Imj!. 2z. 1.2.6 : {D′,D,D′′; i∗, i∗, i!, j!, j∗, j∗} Sd recollement, (1) i∗Z = 0 gg Z ∈ Imj!;(2) j∗Y = 0, gg Y ∈ Imi∗;(3) i!X = 0, gg X ∈ Imj∗.&`o {D′,D,D′′; i∗, i∗, j!, j∗} Sd U recollement, $p0 1.2.5, ((1) B (2) G7 {D′,D,D′′; i∗, i!, j∗, j∗} Sd  recollement,  {D′′,D,D′;j∗, j∗, i∗, i!} SU recollement, zp0 1.2.5, ( (3) G7 2 1.2.7,; {D′,D,D′′; i∗, i∗, i!, j!, j∗, j∗}Sd recollement, $, [3],(Imj!, Imi∗, Imj∗) -G0{	P D ld TTF- 0OzY!k(i) (Imi∗)⊥ = Imj∗,  (Imj!)⊥ = Imi∗;(ii) Imi∗ =⊥ (Imj∗),  Imj! =⊥ (Imi∗).z. 1.2.8[2] : {D′,D,D′′; i∗, i∗, i!, j!, j∗, j∗} Sd recollement, ^N+R η : j! → j∗, %ll X ∈ D′′,j∗(ηX) S-9M 1.2.9: {D′,D,D′′; i∗, i∗, i!, j!, j∗, j∗}Sd recollement,η Sp0 1.2.86lN+R$B}snl(1)i!j! = 0;(2)η : j! → j∗ SNnl(3)i∗j∗ = 0;(4) 0{	P D !.OG D l0{RL	P Imi∗ B Imj∗ l0{*BlEM&` (1)⇒(2). : X ∈ D′′, v ηX : j!X → j∗X -j D 6l0{d(∆):j!XηX→ j∗X → Y → j!X[1], w j∗ WwY0{$p0 1.2.8  j∗(ηX)厦门大学博硕士论文摘要库#DJ B5 6S-( j∗Y = 0, zp0 1.2.6, Y ∈ Imi∗. ,; i!j! = 0, w i! Ww0{(∆), z i!j!X = 0 ` i!j∗X = 0, [kj i!Y = 0, $p0 1.2.6 `E 1.2.7,Y ∈ Imj∗ ∩ Imi∗ = {0}. oi ηX S-oY η SNnl(2)⇒(3). %ll X ∈ D′′, ,; ηX : j!X → j∗X S- i∗j∗X ∼=i∗j!X = 0. oiz X l%lH( i∗j∗ = 0.(3)⇒(2).  (1)⇒(2) l=K.h(2)⇒(4). : η : j! → j∗ SNnl Imj! = Imj∗, zE 1.2.7, [|HomD(Imi∗, Imj∗) = HomD(Imj∗, Imi∗) = 0.V)%ll X ∈ D, M0{i∗i!X α→ X β→ j∗j∗X γ→ i∗i!X[1], "2l α, β S;}t8z γ ∈ HomD(Imj∗, Imi∗) = 0, oi X ∼= i∗i!X ⊕ j∗j∗X,a D 6l%ld 7y!i.OG Imi∗ B Imj∗ 6l7l*BlEM[ D 6l%ld 0{ (Ω):X µ→ Y ν→ Z ω→ X[1], zi3Y0{!i>G,$EM- - -X1 ⊕ X2 Y1 ⊕ Y2 Z1 ⊕ Z2 X1[1] ⊕ X2[1](µ1µ2) (ν1ν2) (ω1ω2),6 X1, Y1, Z1 \ Imi∗, X2, Y2, Z2 \ Imj∗. v µ1 : X1 → Y1, µ2 : X2 → Y2/-j Imi∗ B Imj∗ 6l0{d (∆1) : X1 µ1→ Y1 ν′1→ Z ′1 ω′1→ X1[1] B(∆2) : X2µ2→ Y2ν′2→ Z′2ω′2→ X2[1], v0{ (∆1)  (∆2) W*B|,$yR- - -X1 ⊕ X2 Y1 ⊕ Y2 Z1 ⊕ Z2 X1[1] ⊕ X2[1](µ1µ2) (ν1ν2) (ω1ω2)?- - -hX1 ⊕ X2 Y1 ⊕ Y2 Z′1 ⊕ Z′2 X1[1] ⊕ X2[1](µ1µ2) (ν′1ν′2) (ω′1ω′2), h S-\ (Ω) !.OG Imi∗ B Imj∗ 6l0{l*BlEMoY0{	P D !.OG D l0{RL	P Imi∗ B Imj∗ l0{*BlEM(4)⇒(3). / D = Imi∗ ⊕ Imj∗,  HomD(Imj∗, Imi∗) = 0, a| Imj∗ ⊆厦门大学博硕士论文摘要库#DJ B5 7⊥(Imi∗). $E 1.2.7, Imj∗ ⊆ Imj!. \%ll X ∈ D′′, ^ X ′ ∈ D′′, Ik j!X ′ ∼= j∗X. oi i∗j∗X ∼= i∗j!X ′ = 0, a| i∗j∗ = 0. 2§1.3 Bh recollement pubq*eÆS.Koenigp-<Elx#RP3lhR	Pl recollement^l|vv0+[F) Koenig v0RG3Koenig v0!WShR	P Ricard v0l7 ([32]). {p*,$Koenig t.[16] : A SP A lhR	P D−(Mod − A) P|d 1P B, C lhR	P D−(Mod − B), D−(Mod − C) l recollementD−(Mod − B)←−⇄ D−(Mod − A)←−⇄ D−(Mod − C)gg^= <E M . ∈ Kb(Proj − A), N . ∈ Kb(PA) RR(1) EndA(M.) ∼= B;(2) EndA(N.) ∼= C;(3) Hom.A(N., M .) = 0(a HomA(N ., M .[n]) = 0, %l n ∈ Z G7);(4) (M .)⊥ ∩ (N .)⊥ = 0("2 (M .)⊥ = {Y . ∈ D−(Mod−A)|Hom.A(M ., Y .) =0}, F M . lX*	P)."2 Proj − A .Oo| 8f-Gl	P PA .Oo||-AG 8f-Gl	PKb(Proj −A) .O Proj −A 3l|ElN	PKb(PA).O PA 3l|ElN	P[i3l recollement SFl,;Q| Hom.A(M ., N .) = 0  M . ∈ Kb(PA). YFQ| recollementD−(Mod − C)←−⇄ D−(Mod − A)←−⇄ D−(Mod − B).Miyachi 5w<E P . kjyRP3l 8` A lhR	PD−(A) P|d recollement {KP , D−(A), D−(B); i∗, i∗ = i!, i!, j!, j∗ = j!, j∗},6 B = EndD−(A)(P .), KP = {X . ∈ D−(A)|HomD−(A)(P ., X .[i]) = 0, ∀i ∈ Z}.Miyachi QMz 8` A l℄n e hRl recollement {D−(A/AeA),D−(A), D−(eAe)} lnl1"{!=q, [21].EL - DYm$ Koenig v0(`l recollement #Rd;l ;厦门大学博硕士论文摘要库#DJ B5 8t. 1.3.1[5] : AS|- k-`,; AlhRf	P D−(Mod−A) P|d 1|- k- ` B lhRf	P D−(Mod − B) B C lhRf	P D−(Mod − C) lFl recollementD−(Mod − B)←−⇄ D−(Mod − A)←−⇄ D−(Mod − C), (1)A l< (repetitive) ` Â lhRf	P D−(Mod − Â) bP|,$lFl recollementD−(Mod − B̂)←−⇄ D−(Mod − Â)←−⇄ D−(Mod − Ĉ).(2)A l(` RmA lhRf	P D−(Mod−RmA ) bP|,$lFlrecollementD−(Mod − RmB )←−⇄ D−(Mod − RmA )←−⇄ D−(Mod − RmC ). 1.3.2 Franjou, Pirashvili v recollement lx7j Abel 	P3-Mvvm 1.1.1 6 (R4) t{!=, [10]. Macpherson B Vilonen([22])#R Abel 	Pl recollement lZ$- (+F MV -). DYm - AD+5w BB- <f!iv TTF- .\!ikj Abel 	P3l recolle-ment([19]).§1.4 Bh t- "	 ub_q*eÆV)[S00{	P t-  -lvm`d;$UO/1l t-  -lH3tx 1.4.1[2] : D S0{	P D 3l t-  -Sd jRL	P(D≤0,D≥0), RRi$0 }s(1) D≤0 ⊆ D≤1,D≥0 ⊇ D≥1;(2) HomD(X, Y ) = 0, %ll X ∈ D≤0,Y ∈ D≥1;(3)  D 6l%ld 7 X, ^0{ A → X → B → A[1], 6A ∈ D≤0, B ∈ D≥1."2 D≤n := D≤0[−n],D≥n := D≥0[−n], ∀n ∈ Z.厦门大学博硕士论文摘要库Degree papers are in the “Xiamen University Electronic Theses and Dissertations Database”. Fulltexts are available in the following ways: 1. If your library is a CALIS member libraries, please log on http://etd.calis.edu.cn/ and submitrequests online, or consult the interlibrary loan department in your library. 2. For users of non-CALIS member libraries, please mail to etd@xmu.edu.cn for delivery details.厦门大学博硕士论文摘要库

Recollement of Triangulated Categories

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Recollement of Triangulated Categories

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