Lecture 14 : Localization Motivation I . Let R be a Noetherian be a ring . Let S central closed set in R multiplicatively . Definite ) Ms( R) E MCR to be the full sub category of finitely generated - S torsion R - modules Then MCR) is an abelian . category and MSCR) is an abelian subcategory that is closed under sub a-d extensions . objects, quotients, A Serre of an abelian Def . subcategory B category It is a sub abelian category that is closed under subobjects quotients and extensions . I 1 Mlk We can Then form a quotient category )/mgCR) - ' and it with Mls R) More identify . generally given a Serre subcategory BEA we can form a 's quotient category A/B using Gabriel of calculus fractions . Construction we define A/B to with the same objects be a category ' in f : An as We a A A . say map - if It is a B isomorphism kerf in The and Coker f are objects B . classes morphisms in A/B are equivalence ' of A spans ) ¥ Az ' - - where A' → A is a B isomorphism class in save equivalence Two spans are the if there is a commuting diagram ' A Lt → A- ← -1 A , A 2 Tati ' ' ' - → where both of As A a d A- A are - B isomorphisms . of two spans Composition " 's and , A AEA Yaz AYA , back is defined by the pull A t y "ba ay ! As . There is an evident functor Ioc : A LA IAIB ' A 1- A A k ↳ ' Az A- n Az 1- Al - abelian The A/B is a Prof . construction a-d the factor category Ioc : An A/B is exact . " " See K - If . Swan Algebraic theory . p . 44 If for example , E Rene . If B B then to 0 BE A/B is isomorphic since Bi f - the unique P B " i , " - o od XB Yo o " " " .' T.is?i...H . BxB by d u Y B B c- BxB → T di B similarly in are sent to , maps B in A CB which are isomorphisms , ' A- where both legs exactly maps ya , - A,¢ are B ios Localization) The . ( of a small Let B be a Serre subcategory Then there is a abelian category A . fiber hot op y sequence KEB) ikCA)- KQIAIB) t howto induces a LES py groups a ) - KFCB) a kitAt Koi LAIB K¥iCBlikiECHt#BH . : be a a Cer Let R Noetherian rig and S Then central multiplicatively closed subset . -K thereis a homotopy fiber segue Q 5k) KCMSCRD a GCR) → GC The construction has a Rene . A/B F factors At universal property that 8 where I Danube 1.2- category ad F (B)Io for all B E B and i - B F is a - ( g) isomorphism in factor as for all maps g B A → A/B t Y b . In there factorization particular, MLK) - MCRKMSIR) -' →MCs R) check that and it is easy to of This fine for is an equivalence categories . domain be a Dedekind Core : Let R Then with field of fractions F . fiber there is a homotopy sequence - u KCF) IT K CRIP) KCR) . per prime ideal the there . corollary If By previous , is a fiber sequence ) TGIF) KIM R)) → GCR . ,↳of By the Resolution theorem t ) ) k CF . GCR) e KCR) and GIF Resolution By DE vissage and ) CR)) - GC Rho KQCM - ,z so, ppp see KIND ( = . eY;m±YeIE . ) Iptr Proof of Localization - we will Quiller's theorem B apply , to so we need show 1) BQBE B toe) : u Q (A/B) where Qloc QA . and I 2) B ( iQ Ioc) - BlyQloc) ' for every map Le L in QAIB into we break this down further we consider several . First steps , E E LIQ toe full subcategories . be The steps will then to Show the following . = Step . BQB B to all in . For L QAIB step , Q = 10C BFL Bu . The categories en- Step . FL - it be written as a filtered coli Colin EN ET N - L of categories EN has a full steps . Each Ew subcategory ' EN EEN such that ' BEN = BQB . a 4 The inclusion induces Step . - ' = BEN homotopy equivalence BEN . that in such a we will do this way the homotopy equivalence BQB - Blanco! En)' BFL is compatible with the maps BE - BE ' . Is A BQB - - = Ioc ) We write It : Ioc CA ) and I : Ch . We start by defining EE Latos to be the full subcategory consisting of objects ( AFF :L - A C- QAIB) ' ' A L ' where & y BiH 43-iso ALw has the property B -isomorphisms . that all ups are Then when 2=0 Fo E QB , 's theorem A to Step we use Quillen Q Ioc induces prove ie : E- ↳ : BE-eBuQkx. a homotopy equivalence Bic = * we need to show Billa u) , Note all IA u) in u Qloc . for pairs , that a -up in Qb B , ④ Jy A B is the Sae data as a sequence B - B see B devoted ( Bz ,) Bas , , , have a- and if we fix B then these , ordering by if thereis a ( B ) E ( Bc Bg! Bz , , → X B >→ B Bz Bz , Sequence Byte of mono morphisms . for i Consequently example tha u) , , ordered set forms a partially since she- and it is even filtered and CA , Az) I Az Ay) , , there exits n Az + Ay) FA , Az , such that Z ( As Au) Ac) , . E IA , nA3 Azt , , CA , , Az) are contractible Since filtered categories , ad e - u, Biya , loc . BE E B LIQ with Note that this is a-petiole ' in Q A/B . morph's - s Luc We define categories Step . Ew for N in A to be the category I of ( A h : Aun EA) where pairs , in A/B . Amorphism is an isomorphism a - between pairs is a spa ' ' A- ← A' ← A in QA such that the two composites " u A N agree . We define kN : Ew - QB ( A h ht kerh by , ) ' to be the We define Ew E En such h) on CA full subcategory pairs , h . Write that is an epimorpsnism ' ' : ↳ -t.Q.is Kw Ew n . we further define a category Ic where L C- A/B with N IN a : DEL) where objects , nd D Es L is an - i- A is a object The isomorphism i - A/B . morphine ' in are B- isomorphisms IN,a)elN , B) Iu ' in such that g. Nun A - 5 - i N - N commutes a) Ldp we check This is a filtered category . just this : one of the properties to illustrate - ( ) Given g :(ou L) Nip , , go , so -Sz) EB then imlg , 5 , -52=0 - - E ' ' → = N' ad N Let N' - - s.) gu) Miles, lines , Then pY[ 8 " ' car a) ICN ) - IN 8) is a , , p , co equalizer . we define afuetor of grey ( category - Cat ) IL categories 2) 1- EN IN , - a) ) n Ew Eni LN , ucnip ' ' rt N : New n n: Ann) ( goh) g ( , , B- do we also define futons : En - E Pen as , and CN a) C- IN for fixed L arbitrary , by " ni : LEGE ) h : M -N) rt (m IIa IN , , and for : CN at u IN p) naps g , , = 's . Pen; pyo . Playa) So a there is map → FL Colin Ew . (NATIN Step The map 1 Colin Ent =L WINE IN of categories . is a isomorphism because the we just check on objects is basically aregvnert on morphisms same the . , idm) ) =p ! o:< Eat Note that IM ,m o .mg , , Cm o) in So this for FL . any , Given nap is surjective on objects . ' N' ) = ( Nih) in Pen a) Payal , ' ' I -_I then N =N and . Letting we W' =im/n - hi) produce nap ' ') ( N a) u CN ) S.t. g ( Nih) , , p , =gp(w,h ' ' So as i - Colin Ew IN h) - CN h ) objects , , . (WHEN We constructed Steps . already " ' : - QB Ew nd tutors Kal Ew . A . we again apply Quillen's theorem ) ere It suffices to show BIKNYB C- that for all objects B B . Note ' ' is fibered over kN # Ew and it her ( ) u) objects ( min , ' where Im h nd u : Ben Kern , ) EEN in be a . Let is nap QB 8 ' the full subcategory of kN 1B consisting such that u is of of M h) u) (( , , " the form ! : BB j B kerh . Since in QB is of every map u ! and The form = i for some a j , epi j mono i , Ker e Given ( M h) ul w ( , , HIB e t, we write w , u : Keri IB ne .pk?hbBYBd ! so u = oi j : and define o M W N - on her h - " . 11 ' I L t u v U N 0 - h_ u- Bo im T ) ! ) hi nie m j Then (cm ( (i. , , , , d- elusion is adjoint to the I Kehl y B . jpg :B → 0 ! Moreover 4N idw) ) , , Ig , in BE = * is initial b so . - Valerie This is hohotopyeq is capable - I go 1gal " B EN I , BQB 9 t B ; → ' ; {w = " We now prove EN ↳ EN 't BEN Blw hero . induces a top y equivalence of sub Let I be the ordered set objects t EB . of N s . NII . I sun observe a We regarded as category . - is - Tutto that there Pw : Ew -I inn M h) 1- ( , - ' is with fiber CI) which fibered Pn . The base change functor - I - ' : CI) ) pj' LIST) p n up NIJ 11 11 l l EI EJ (m m →It IN , HIM , J¥m→sJ) with This clearly commutes ' the inclusions EI → Ej ↳Eat . By the previous result ' so we can apply BEI → BEI Quillen 's theorem ↳ Y Lg BQB B . I is a filtered Also BI to since , ordered set.