SpectralSequences

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1、SpectralSequencesDanielMurfetOctober5,2006InthisnotewegiveaminimalpresentationofspectralsequencesfollowingEGA.Wecoveressentiallyonlythatpartofthetheoryneededinalgebraicgeometry.InSection2westartwithafiltrationofacomplex,andshowhowthevariouspiecesofaspec

2、tralsequencearise.ApplyingtheseobservationsinSection3totwonaturalfiltrationsofthetotalcomplexofabicomplex,wededucetwospectralsequencesanddiscoverthattheirfirstthreepagesarenotmysteriousatall:theyconsistofverynaturalinvariantsofthebicomplex,arrangedintheo

3、bviousway.WiththisbackgroundwecanstudytheGrothendieckspectralsequenceinSection4.Contents1Definitions12TheSpectralSequenceofaFiltration32.1FirstQuadrantFiltration................................73TheSpectralSequencesofaDoubleComplex84TheGrothendieckSpect

4、ralSequence124.1Examples.........................................144.1.1TheLeraySpectralSequence..........................144.1.2TheLocal-to-GlobalExtSpectralSequence..................145HyperderivedFunctors151DefinitionsAsusualweassumethatourabeliancat

5、egoriesallcomewithcanonicalstructures,allowingustodefinecokernels,kernels,imagesanddirectsumsinacanonicalway.IfwehavetwosubobjectsX,YofanobjectAthenX⊆YmeansthatXprecedesYasasubobject(i.e.themorphismX−→AfactorsthroughY−→A).Weusethenotationandconventionso

6、fournotesonDerivedFunctors(DF)andAbelianCategories(AC).InparticularwetendtodenotethedifferentialofanycomplexXby∂n:Xn−→Xn+1.ThedefinitionsinthissectionfollowEGAIIICh.0§11.1.Definition1.LetAbeanabeliancategoryandXanobjectofA.Afiltration(ordecreasingfiltration

7、)orXisasequenceofsubobjectsofX···⊇F0(X)⊇F1(X)⊇···⊇Fp(X)⊇···Iftheyexist,wewriteinf(Fp(X))fortheintersection∩Fp(X)andsup(Fp(X))fortheunionp∪Fp(X).Wesaythatthefiltrationisseparatedifinf(Fp(X))=0andcoseparatedorexhaustivepifsup(Fp(X))=X.Wesaythatthefiltratio

8、nisdiscreteifthereexistsp∈ZwithFp(X)=0,andcodiscreteifthereexistsp∈ZwithFp(X)=X.Definition2.LetAbeanabeliancategoryanda≥0aninteger.AspectralsequenceinAstartingonpageaconsistsofthefollowingelements:1(a)AnobjectEpqofAforeveryp,q∈Zandr≥a.r(