SPECTRAL SEQUENCES: FRIEND OR FOE? - Stanford University

1 SPECTRALSEQUENCES:FRIENDORFOE?RAVIVAKILS pectralsequencesare a introducedbyLerayinthe1940' reputationforbeingabstruseanddif hasbeensuggestedthatthename` SPECTRAL 'was givenbecause,likespectres,spectralsequen cesare terrifying,evil, haveheard noonedisagreewiththisinterpretation,whic his perhapsnotsurprisingsinceI justmadeit ,thegoalofthisnoteistotellyouenoughthaty oucanusespectralse-quenceswithouthesitat ionorfear, andwhyyoushouldn'tbefrightenedwhentheyco meupina seminar. Whatis differentinthispresentationis thatwewillusespectralsequencetoprovethin gsthatyoumayhavealreadyseen, specialcase ofdoublecomplexes(whichistheversionbyfar themostoftenusedinalgebraicgeometry),and notinthegeneralformusuallypresented( lteredcomplexes,exactcouples,etc.)

2 Seechapter5 ofWeibel'smarvelousbookformore detailedinformationif youwanttobecomecomfortablewithspectralse quences, ,weworkinthecategoryvectorspacesovera given ,everythingwesaywillapplyinanyabeliancat egory, rst-quadrantdoublecomplexis a collectionofvectorspacesEp;q(p; q2Z), whichare zero unlessp; q 0, and rightward morphismsdp;q>:Ep;q!Ep;q+1and upward morphismsdp;q^:Ep;q!Ep+1;q. Inthesuperscript,the rstentrydenotestherownumber,andthesecond entrydenotesthecolumnnumber, inkeepingwiththeconventionformatrices,bu toppositetohowthe(x; y)-planeis willalwayswritetheseasd>andd^andignore require thatd>andd^satisfying(a)d2>=0, (b)d2^=0, andonemorecondition:(c)eitherd>d^=d^d>(a llthesquarescommute)ord>d^+d^d>=0(theyal lanticommute).

3 Bothcomeupinnature,andyoucanswitchfromon etotheotherbyreplacingdp;q^withdp^(-1)q. SoI'llassumethatallthesquaresanticommute ,butthatyouknowhowtoturnthecommutingcase intothisone.(Youwillseethatthere isnodifferenceintherecipe,basicallybecau setheimageandkernelofa homomorphismfequaltheimageandkernelrespe ctivelyof-f.)Date: Tuesday, March12, +1;qdp+1;q>//Ep+1;q+1anticommutesEp;qdp; q^OOdp;q>//Ep;q+1dp;q+1^OOThere are variationsonthisde nition,where forexampletheverticalarrowsgodown-wards, orsomedifferentsubsetoftheEp;qare requiredtobezero,butI'llleavethesestraig htforward corresponding(single)complexE withEk= iEi;k-i, withd=d>+d^.

4 Inotherwords,whenthere is asinglesuperscriptk, wemeana Notethatd2= (d>+d^)2=d2>+ (d>d^+d^d>) +d2^=0, soE is indeeda willinsteadusethephrase cohomologyofthedoublecomplex .Ourinitialgoalwillbeto a won' , thisfragmentarybitofinformationis suf >Ep;q0,>Ep;q1,>Ep;q2, .. (p; q2Z), where>Ep;q0=Ep;q, alongwitha differential>dp;qr:>Ep;qr!>Ep+r;q-r+1wit h>dp;qr >dp;qr=0, alongwithanisomorphismofthecohomologyof> drat>Ep;q( >dp;qr=im>dp-r;q+r-1r) with>Ep;qr+ one:d0=d>. Theleftsubscript > is bestunderstoodvisually:(1) d0//d1 OOd2WW//////////////d3ZZ5555555555555555 55555555 (themorphismseachapplytodifferentpages). Noticethatthemapalwaysis degree1 inthegradingofthesinglecomplexE.

5 Theactualde nitiondescribeswhatE ; randd ; ractuallyare, intermsofE ; . We willdescribed0,d1, andd2below, ;qris alwaysa subquotientofthecorrespondingtermonthe0t hpageEp;q0=Ep;q. Inparticular, ifEp;q=0, thenEp;qr=0forallr, soEp;qr=0unlessp; q2Z 0. Noticealsothatforany xedp; q, oncerissuf cientlylarge,Ep;qr+1iscomputedfrom(E ; r; dr)usingthecomplex0Ep;qrdp;qrYY333333333 33330dp+r;q-r-1rYY3333333333333andthuswe havecanonicalisomorphismsEp;qr =Ep;qr+1 =Ep;qr+2 = We denotethismoduleEp; nowdescribethe rstfewpagesofthespectralsequenceexplicit ly. Asstatedabove,thedifferentiald0onE ; 0=E ; is de nedtobed>. Therowsare complexes: // // The0thpageE0: // // // // nowverticalmapsdp;q1:Ep;q1!

6 Ep+1;q1oftherowcohomologygroups,inducedb yd^, andthatthesemakethecolumnsintocomplexes. (We have usedupthehorizontalmorphisms ,but theverticaldifferentialsliveon .) The1stpageE1: OO OO OO OO OO OOWe takecohomologyofd1onE1, givingusa newtable,Ep;q2. Itturnsoutthatthereare naturalmorphismsfromeachentrytotheentryt woaboveandonetotheleft,andthatthecomposi tionofthesetwois0. (Itisa veryworthwhileexercisetoworkouthowthisna turalmorphismd2shouldbede shortexactsequenceofcomplexes, ) The2ndpageE2: thebeginningofa isa theoremthatthere isa ltrationofHk(E )byEp;q1wherep+q=k. (Wecan'tyetstateit asanof cialTheorembecausewehaven'tpreciselyde nedthepagesanddifferentialsinthespectral sequence .)

7 More precisely, there is a ltration(2)E0;k1 E1;k-11//? E2;k-21// Ek;0//Hk(E )where thequotientsare displayedaboveeachinclusion.(I alwaysforgetwhichwaythequotientsare supposedtogo, ;0orE0; is byhavingsomeideaofhowtheresultis proved.)We saythatthespectralsequence>E ; convergestoH (E ). We oftensaythat>E ; 2(oranyotherpage)abutstoH (E ).Althoughthe ltrationgivesonlypartialinformationabout H (E ), sometimesonecan ndH (E )precisely. Oneexampleis if allEi;k-i1are zero,orif allbutoneofthemare zero( ;k-irhaspreciselyonenon-zero roworcolumn,inwhichcaseonesaysthatthespe ctralsequencecollapsesattherthstep,altho ughwewillnotusethisterm).

8 Anotherexampleisinthecategoryofvectorspa cesovera eld,inwhichcasewecan ndthedimensionofHk(E ). Also,inluckycircumstances,E2(orsomeother smallpage) :INFORMATIONFROMTHESECONDPAGE. ShowthatH0(E ) =E0;01=E0;02and0//E0;12//H1(E )//E1;02d1;02//E0;22//H2(E ) , (compare to(1)).(3) OO//''OOOOOOOOOOOOOO%%JJJJJJJJJJJJJJJJJJ JJJJJJ Thisspectralsequenceis denoted^E ; ( withtheupwardsorientation ).Thenwewouldagaingetpiecesofa ltrationofH (E )(where wehavetobea bitcarefulwiththeorderwithwhich^Ep;q1cor respondstothesubquotients it intheoppositeordertothatof(2)for>Ep;q1). Warning:ingeneralthere is noisomorphismbetween>Ep;q1and^Ep; , (H (E )), andusuallywedon'tcare aboutthe nalanswer weoftencare abouttheanswerwegetinoneway, andwegetatit 're nowreadytoseehowthisis ,youmayhaveprovedvariousfactsinvolvingva rioussortsofdiagrams, , you'lljustplugthemintoa spectralsequence, OOB// OOC OO//05where therowsare exactandthesquarescommute.

9 (NormallytheSnakeLemmaisde-scribedwithth everticalarrowspointingdownwards,butI wantto tthisintomyspec-tralsequenceconventions. )We wishtoshowthatthere is anexactsequence(4)0!ker !ker !ker !im !im !im !0:We plugthisintoourspectralsequencemachinery . We rstcomputethecohomologyusingtherightward sorientation, (1).Thenbecausetherowsare exact,Ep;q1=0, sothespectralsequencehasalreadyconverged :Ep;q1= nextcomputethis 0 inanotherway, ^E ; 1(withitsdifferentials)is:0//im //im //im //00//ker //ker //ker //0:Then^E ; 2is oftheform:0''NNNNNNNNNNNNNN0''NNNNNNNNNN NNNNN0''NNNNNNNNNNNNNNN??''NNNNNNNNNNNNN N?''NNNNNNNNNNNNNNN?00??''NNNNNNNNNNNNNN N?

10 ?''NNNNNNNNNNNNNN000We seethatafter^E2, allthetermswillstabilizeexceptforthedoub le-question-marks allmapstoandfromthesinglequestionmarksar e ^E3, ^E2, alltheentriesmustbezero,exceptforthetwod ouble-question-marks, !ker !ker !ker andim !im !im !0are bothexact(thatcomesfromthevanishingofthe single-question-marks),andcoker(ker !ker ) =ker(im !im )isanisomorphism(thatcomesfromtheequalit yofthedouble-question-marks).Takentogeth er, wehaveprovedtheexactnessof(4),andhenceth eSnakeLemma!Spectralsequencesmakeit easytoseehowtogeneralizeresultsfurther. Forexample,ifA!Bis nolongerassumedtobeinjective,howwouldthe conclusionchange? (5)F//G//H//I//JA// OOB// OOC OO//D// OOE OOwhere therowsare , , , are 'llshowthat is rstcomputethecohomologyofthetotalcomplex usingtherightwardsorientation(1).