Section3.8-Higher Direct Image Of Sheaves .pdf

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1、Section3.8-HigherDirectImagesofSheavesDanielMurfetOctober5,2006InthisnotewestudythehigherdirectimagefunctorsRif(−)andthehighercoinverseimage∗functorsRif!(−)whichwillplayaroleinourstudyofSerreduality.ThemaintheoremistheproofthatifFisquasi-coherentthensoisRif(F),whichweprovefirstfornoeth

2、erianschemes∗andthenmoregenerallyforquasi-compactquasi-separatedschemes.MostproofsarefromeitherHartshorne’sbook[1]orKempf’spaper[2],withsomeelaborations.Contents1Definition12ModuleStructure33DirectImageandQuasi-coherence64HigherCoinverseImage85DirectImageandQuasi-coherence(Generalcase)

3、105.1LocalisationsofSheaves.................................105.2LocalisationasRestriction(Invertiblesheaves)....................135.3LocalisationasRestriction................................155.4TheProof.........................................176UniquenessofCohomology201DefinitionDefi

4、nition1.Letf:X−→Ybeacontinuousmapoftopologicalspaces.ThenwedefinethehigherdirectimagefunctorsRif:Ab(X)−→Ab(Y)tobetherightderivedfunctorsofthe∗directimagefunctorf∗fori≥0.Sincef∗isleftexactthereisacanonicalnaturalequivalenceR0f∼=f.ForanyshortexactsequenceofsheavesofabeliangroupsonX∗∗0−→F

5、0−→F−→F00−→0thereisalongexactsequenceofsheavesofabeliangroupsonY0/f0)/f(F)/f(F00)/R1f(F0)/···∗(F∗∗∗···/Rif∗(F00)/Ri+1f∗(F0)/Ri+1f∗(F)/Ri+1f∗(F00)/···Remark1.Ifthefunctorf∗:Ab(X)−→Ab(Y)isexact,thenfori>0thehigherdirectimagefunctorRifiszero.Inparticularthisisthecaseiffisaclosedembedding

6、(SGR,Definition17).∗Remark2.LetXbeatopologicalspace.In(COS,Section1.3)wedefinedforeveryi≥0anadditivefunctorHi(−):Ab(X)−→Ab(X)whichmapsasheafofabeliangroupsFtothepresheafofcohomologydefinedbyΓ(U,Hi(F))=Hi(U,F).HereHi(U,−)denotestheith1rightderivedfunctorofΓ(U,−):Ab(X)−→Ab.By(COS,Lemma12)t

7、hegroupHi(U,F)iscanonicallyisomorphictotheusualcohomologygroupHi(U,F

8、)ofthesheafF

9、.UUProposition1.Letf:X−→YbeacontinuousmapofspacesandFasheafofabeliangroupsonX.Foreveryi≥0thereisacanonicalisomorphismofsheavesofabeliangroupsonYnaturalinFν:afHi(F)−→Rif(F)∗∗Inotherwords,Rif(F)isthesheafifi

10、catio