ttructnres
play

ttructnres structures - t 2. Co - - t towards structure 3. t - PDF document

CO THE FIRST DECADE STRUCTURES T : - - - 0 Overview . I . ttructnres structures - t 2. Co - - t towards structure 3. t skewed Categories co - or - . The and of 4 bijection Yang Kcimg lyamatheiti . The of Aihara 5


  1. CO THE FIRST DECADE STRUCTURES T : - - - 0 Overview . I . ttructnres structures - t 2. Co - - t towards structure 3. t skewed Categories co - or - . The and of 4 bijection Yang Kcimg lyamatheiti . The of Aihara 5 mutation and lyama sitting - structures ( after Adachi - t Intermediate 6 a - .

  2. 0.Overviewt-structurefyiltingot@6m-ya.tg .mu#gsjhnbujn9 t.tt#itrsionde@ AnAwtiltingob € , i

  3. : Beilin due to Notion Bernstein and son , [ BBD ) ( 1982 ) Deligne = . Dbfhod R ) Let R the be bounded derived a - ring , category tener xe Db( Mod R ) For truncation soft ofivo - a , DBMODRI £9ModN=A 0d → d → J°d tnang D%ModRI=B htttrncthres (f) in in Dbtbdr ) ( a ,B ) that is is - a : , ' (1) [ BSB [ As A , , , B) (2) Hand 0 = , At B (3) = , each deDbµodR) permit ( 4 a triangle ie .

  4. Tied that i. . The [ BBD ] H= An of a t structure ZB - . short with abelian exact is sequences , by the with terms X. in triangles given ( In the . ) Mod ( R ) N examples . theorem , B) - structure ( A tf bet t is a - T in , = µ UEA EB , banded In fact , H then the heart T generate . = µ EF 'N* ? [ Pat Etse :* . . ( stated stability in 's Bridgeland manifolds on paper . ) Remand It the definition of follows from - t that the and terms of the structure triangle - 6) d. functionally depend on 3 .

  5. 2*+7 Notion to due Bondarko and independently - R ) Let Kblprj R the be homotopy category a - ring , of bounded of projective complexes kekblprj R ) hard truncation For gives a - Panksztello , thongs 0k → k → o9k ( tt ) attractive In � 1 � Kbcpri * ay , under there devote closures RI isomorphism in Kb '%lPjR) Kbiilprj R ) and of ( t.gl in Kblkj R ) , that is is a - , ( 0 ) and A direct closed under Y are ( structures but t not automatic for - shmmhhds and Y needed for determine and here to E other ) , ' each (1) [ AEA EYEY , , , Hour ( Hey ) (2) 0 = , KBCPRJR ke KYPRJRI (3) ) HKY = . triangle 6th ( i.e. each permit a 4.

  6. [ hf The C themark not An is in = - it abelian but does satisfy , . ) Ottom , C0E ) ( In ( C the 0 E=PrjlR ) examples = . general If At ,4 ( Bondarko ) Theory is a - . - t - structure T that is in , , KEY Et µ T cheat = = , � 2 � In fact , then wheat E 7 the generate Inkigayo . boundedw = µ ' e Ete [ T :* * . � 1 � and � 2 � that Reward a Properties is a - T.i.ie?thftom(e , 200=0 of EIT " " thick The term and sitting was . - Vossieok Keller that Note sub coined tilting by - . special are are a categories of the triangle fttd Reward The terms end do not - k necessarily functionally depend on . 5 .

  7. ↳ A with bounded triangulated a category : " like - structure t " derived while is category a , " like " - t bounded structure with a W is a one - homotopy category e- linear then Definition If set T is is - , Es ) ±Ekµ(e4 if T(s with called d- in cohowdogical e degree otephenna - Yang - Zhou ) ( Keller 3.tk#dowEdtxAruitnres-Ph1pky theorem and de Z Let . t.CH/K2 ) DQA consider with E as a d di ff erential and oohomohogial in degree zero . = D4A ) Eliwar the Then 4 Hour is unique finite which is triangulated category algebraic " " thick d- spherical of object a . theorem ( Ham F Yangl If T has then d I > - - , . - t - structures trivial but E indexed non one w - , - structures nontrivial t family of . For ds 0 via versa , . 6 .

  8. ifectnnrtfknmg 4.11 aiding finite theorem Let ^ dimensional be - a t the There E bijection following are to , . algebra su@eaia.aw.mm Htpflt 5 9 simple HFtytihjsminddjej@ihDblmodwobjt.ca .info#tiesaamPY*nEigfm*s**i.Miii:awmwMiiiwiinia BftTea-Boundedw-t-tructntfiMdrwthwhwlinkblpr@tHrara-lyama.edu classes of Isomorphism addlnl st is a . sitting subncatyony where ( A 9 , B) An the H EB H in = , 5 additive basic e of a -% generator

  9. 5.lt#g*mofAil*md_yama T Let Elinor . finite be with Hour split - left idempotent Let basic be mootm , mute m sitting object a - = with indecomposable mo mont Let not be Mi distinguished - a - - left with minimal addcmt triangle a µ approximation . the Theory and not is indecomposable t motion at of is sitting a m mo , object . There's theorem of right notion symmetric a - mutation Motion mutation takes Right of . , back to MO0M , us m = . 8 .

  10. ↳ ↳ ↳ ↳ ↳ ↳ ↳ ↳ T Definition has of - a , class for sitting book of vertex each isomorphism m → M* of T and object an arrow AR quiver if M$ left mutation of is m a . Kbcprjctz ) T= has the Etxmpk following - , 0¥ • • 0 × Xq Xo Xz • 3 5 ' . ' 7 X 7 quiver 7 T T × y × × , . . . tucking × X. Xp , . -80 6h the Here where of silting part quiver , is ->2n E → h mlaw in M - . circa ii IxtI ⇐ mut . at x mut at x , . , 3 × 5 × 7 . no .* × × gives the - t structures tore to ' × i¥a÷ × o0 × eF÷÷ × o0 × a are ' corresponding co the blue first two A red object in in CY , , , the coheart .tl ↳ %t ° • 0 0 • • T 7 . T y . . 7 . . . • • • • × 4 XO 0 • 1 7 ' ' . T 7 7 . . 0 • • • • g.

  11. ate#runr±( 6¥ E algebra a finite : Let dimensional ^ be Definition Let tewodn be basic - . ) = 0 t called and is tnti@kgifttomtt.tt the t number summands of direct equals in a afpctnekEg Kdmod N rank . a.pe#AlacIlyamaRei4- ' tilting t if called it is is t 1/4 ) eeh for idempotent over some t.AM#.YinKbCprjN_ A Definition co - - , intermediate if " Kb Kbisocprj N Cprj N a- He called ' is overlines devote under where closure isomorphism A Definition Kbcprj N is sitting object in - m the twoWm= if has it called form → p , 0 .

  12. There the theorem bijection following are - Kbcpnj N in , obj Etf %o two term sitting amazon.LT#*mama }imrphimdasw÷] lettres where Htpflt additive 9 basic e= of a generator *n 5 ( Pet po ) taken d. .

Recommend


More recommend