局部域

Introduction

Leitfaden

PartOne

LOCALFIELDS(BASICFACTS)

ChapterI

DiscreteValuationRingsandDedekindDomains

§1.DefinitionofDiscreteValuationRing

§2.CharacterisationsofDiscreteValuationRings

§3.DedekindDomains

§4.Extensions

§5.TheNormandInclusionHomomorphisms

§6.Example:SimpleExtensions

§7.GaloisExtensions

§8.FrobeniusSubstitution

ChapterII

Completion

§1.AbsoluteValuesandtheTopologyDefinedbyaDiscreteValuation

§2.ExtensionsofaCompleteField

§3.ExtensionandCompletion

§4.StructureofCompleteDiscreteValuationRingsI:EqualCharacteristicCase

§5.StructureofCompleteDiscreteValuationRingsII:UnequalCharacteristicCase

§6.WittVectors

PartTwo

RAMIFICATION

ChapterIII

DiscriminantandDifferent

§1.Lattices

§2.DiscriminantofaLatticewithRespecttoaBilinearForm

§3.DiscriminantandDifferentofaSeparableExtension

§4.ElementaryPropertiesoftheDifferentandDiscriminant

§5.UnramifiedExtensions

§6.ComputationofDifferentandDiscriminant

§7.ADifferentialCharacterisationoftheDifferent

ChapterIV

RamificationGroups

§1.DefinitionoftheRamificationGroups;FirstProperties

§2.TheQuotientsGi/Gi+1,i≥0

§3.TheFunctionsφandψ;HerbrandsTheorem

§4.Example:CyclotomicExtensionsoftheFieldQp

ChapterV

TheNorm

§1.Lemmas

§2.TheUnramifiedCase

§3.TheCyclicofPrimeOrderTotallyRamifiedCase

§4.ExtensionoftheResidueFieldinaTotallyRamifiedExtension

§5.MultiplicativePolynomialsandAdditivePolynomials

§6.TheGaloisTotallyRamifiedCase

§7.Application:ProofoftheHasse-ArfTheorem

ChapterVI

ArtinRepresentation

§1.RepresentationsandCharacters

§2.ArtinRepresentation

§3.Globalisation

§4.ArtinRepresentationandHomology(forAlgebraicCurves)

PartThree

GROUPCOHOMOLOGY

ChapterVII

BasicFacts

§1.G-Modules

§2.Cohomology

§3.ComputingtheCohomologyviaCochains

§4.Homology

§5.ChangeofGroup

§6.AnExactSequence

§7.SubgroupsofFiniteIndex

§8.Transfer

Appendix

Non-abelianCohomology

ChapterVIII

CohomologyofFiniteGroups

§1.TheTateCohomologyGroups

§2.RestrictionandCorestri&ion

§3.CupProducts

§4.CohomologyofFiniteCyclicGroups.HerbrandQuotient

§5.HerbrandQuotientintheCyclicofPrimeOrderCase

ChapterIX

TheoremsofTateandNakayama

§1.p-Groups

§2.SylowSubgroups

§3.InducedModules;CohomologicallyTrivialModules

§4.Cohomologyofap-Group

§5.CohomologyofaFiniteGroup

§6.DualResults

§7.ComparisonTheorem

§8.TheTheoremofTateandNakayama

ChapterX

GaloisCohomology

§1.FirstExamples

§2.SeveralExamplesof"Descent"

§3.InfiniteGaloisExtensions

§4.TheBrauerGroup

§5.ComparisonwiththeClassicalDefinitionoftheBrauerGroup

§6.GeometricInterpretationoftheBrauerGroup:Severi-BrauerVarieties

§7.ExamplesofBrauerGroups

ChapterXI

ClassFormations

§1.TheNotionofFormation

§2.ClassFormations

§3.FundamentalClassesandReciprocityIsomorphism

§4.AbelianExtensionsandNormGroups

§5.TheExistenceTheorem

Appendix

ComputationsofCupProducts

PartFour

LOCALCLASSFIELDTHEORY

ChapterXII

BrauerGroupofaLocalField

§1.ExistenceofanUnramifiedSplittingField

§2.ExistenceofanUnramifiedSplittingField(DirectProol)

§3.DeterminationoftheBrauerGroup

ChapterXIII

LocalClassFieldTheory

§1.TheGroupZandItsCohomology

§2.Quasi-FiniteFields

§3.TheBrauerGroup

§4.ClassFormation

§5.DworksTheorem

ChapterXIV

LocalSymbolsandExistenceTheorem

§1.GeneralDefinitionofLocalSymbols

§2.TheSymbol(a,b)

§3.ComputationoftheSymbol(a,b)vintheTamelyRamifiedCase

§4.ComputationoftheSymbol(a,b)vfortheFieldQp(n=2)

§5.Thesymbols[a,b)

§6.TheExistenceTheorem

§7.Example:TheMaximalAbelianExtensionofQp

Appendix

TheGlobalCase(StatementofResults)

ChapterXV

Ramification

§1.KernelandCokernelofanAdditive(resp.Multiplicative)Polynomial

§2.TheNormGroups

§3.ExplicitComputations

Bibliography

SupplementaryBibliographyfortheEnglishEdition

Index

　　Thegoalofthisbookistopresentlocalclassfieldtheoryfromthecohomologicalpointofview,followingthemethodinauguratedbyHochschildanddevelopedbyArtin-Tate.Thistheoryisaboutextensions--primarilyabelian--of"local"(i.e.,completeforadiscretevaluation)fieldswithfiniteresiduefield.Forexample,suchfieldsareobtainedbycompletinganalgebraicnumberfield;thatisoneoftheaspectsof"localisation".　　Thechaptersaregroupedin"parts".Therearethreepreliminaryparts:thefirsttwoonthegeneraltheoryoflocalfields,thethirdongroupcohomology.Localclassfieldtheory,strictlyspeaking,doesnotappearuntilthefourthpart.