LNT. R Mod. Rtdd. Stlod. next. the. two. Rtlod. fenctov of the MQR LNTH. Lectured MORP. Mar, Torre 1. exact. for. any. natural. point. claim.

Transcription

1 We hve defied dditive fu dor Tor exmple of fix D# T Give left LT fuctio left derived ext od derived fuctio Tory T set d exmie y Throughout put # R lectures two fectov of ot ecessrily commuttive R S Rtlod AI Rtdd Stlod right R od : I which rig R left derived : deote d by LTH ROI : Ats Lectured tht so where P Exmie Po Exmple Tory The P P tesor H exct product 22 Torre sub vhig 72 r Po O true P so Po or poit of projective P turl exct projective resolutio of ORP s tke my right t th beli groups ExmpI re hece Torre projective ORP projective resolutio of PO To o y B KI 72 sice turlly clim over OR y i oly y Th beig projective PID sme of R hs i cosequece fct reso th

2 Hece By Th mes PID R eed oly worry bout Tore strt with f g i fct sice d Tor commutes with direct limits d turlly direct limit y of its f sub g th essetilly eough to compute Tov over y PID but will ot use th clssifictio of f g over R d fct tht EI Prove tht if T : R_od57 dditive TC Oil i I TCi y collectio { i iei of R Lemm For y rig R d o zero divor re R re turl omorphm Tory R / r { xe I rx Roof The sequece R r R Rhr exct sice r o Rkr divor compute Tor from d gives projective resolutio of o ORR # RR h o " 2 R R i e Tory Rkx Kev 3 s climed D tosiouof od Th explis ottio : To # Z computes beli group

3 3 Exmie Withottiooflemm re re coicl Rtievmps y : Rkry 7 cr Y : Rkr Rkry y 5 5 YCJ rswhichlifttompso resolutios RIJR %z o RJR rt t t it O RI R Rkr o R : 4 Rkr O R R/rzO dheuidumpso R of resolutios 2 TorE R/rz e Torlm wmupodsudercz Torl fr { xer2xo x Tor I r f t r { x rx rx to Tom Yr o Tore Rki etorlmitlwmespodsto l Tor 4 { xeml rx x t I rz { xe r2x x Thres turl questio : uhtr ify rdo hve TORIY r { xemlrix osomeizo?

4 But 4 We will sr th 22 but sme thig works y PID d eve more geerlly with suitble tks first eed geerl orem which will orem For y rig right R d exct sequece of left R " hve log exct sequece of beli groups Tory Tore Tork " Torre pioeer Torre To # " Toe Tore Tort? " r r OR " orem th commuttive Torre Tory ll d R Exmpte R commuttive d F flt Torre F Torre F H For P For o sice For exct

5 SO Ext s flt 7L d flt R exct cosider 2 Tork prticulr I lz Tors i Lets { from which exct d p sequece deduce hece t prime d of Tor groups gist th uique tk 245 write 2 { Pttk with Sylowp subgroup of { ] XE 745 ] I idetified pkx th of 2 7<45 c S Torsp K some pportobioof tese omorphm 2472 Tork lz Z? Tor Z i lz < 9e exct sequece mp Toil obti some x flt lz e Are coicl / xe s { 33 Toe TORE Ylmtm where fiite IES ie sequece 2 ez Toe There R closed correspodig log exct sequece d Exmie multiplictively O Sice S 22 Ex_mpe Tke stes tes SER commuttive d R with ]/z

6 But 6 Ex3_ Rove tht 2 [ FYIZ Zpo ijective evelopeof Zp my uite Tore El Zp { xemlpkx some k 745k so Ex4_ Rove tht y set { i iei of R tht Torie I i OITORFC il From bove deduce tht decompositio /z + ECZP ppvime from erlier i lectures gives re to decompositio of Tors every beli group Tors To # lz Tory PEIZP Otp Tore Etkp Ap Torsp Of course th c be see much more esily by direct clcultio Hu Tor lguge llows s to deduce exmple tht if " exct re exct sequece Torspm Torsp Torsp " * Pithy CFY " [ P Y " which less immeditely obvious