Homology of Analytic Sheaves and Duality Theorems

222 Pages·1989·11.701 MB·English

by  Viktor Dmitrievich Golovin

Homology
of ‘ *
Analytic
Sheaves
and '
_ Duality
‘ Theorems
v D, Golovin

Homology of
Analytic Sheaves
and Duality Theorems
CONTEMPORARY SOVIET MATHEMATICS
Series Editor: Revaz Gamkrelidze, Steklou Institute, Moscow, USSR
ASYMPTOTICS OF OPERATORAND PSEUDO-DIFFERENTIAL
EQUATIONS
V. P. Maslov and V. E. Nazaikinskii
COHOMOLOGY OF INFINITE-DIMENSIONAL LIE ALGEBRAS
D. B.Fuks
DIFFERENTIAL GEOMETRY AND TOPOLOGY
A. T. Fomenko
HOMOLOGY OF ANALYTIC SHEAVES AND DUALITY
THEOREMS
V. D. Golovin
LINEAR DIFFERENTIAL EQUATIONS OF PRINCIPAL TYPE
Yu. V. Egorov
THE OBLIQGE DERIVATIVE PROBLEM OF POTENTIAL THEORY
A. l. Yanushauskas'
OPTIMAL CONTROL
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin
THEORY OF SOLITONS: The Inverse Scattering Method
8. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov
TOPICS IN MODERN MATHEMATICS: Petrovskii Seminar No. 5
Edited by O. A. Oleinik
Homology of
Analytic Sheaves
and Duality Theorems
V. D. Golovin
A. M. GorkyKharkovStateUniversity
Kharkov, USSR
Translated from Russian by
Norman Stein
Consultants Bureau 0 New York and London
LibraryofCongressCataloging inPublicationData
Golovin,V.D.(ViktorDmitrievich)
[Gomologiianaliticheskikhpuchkoviteoremydvol'stvennosti.English]
Homology ofanalyticsheavesand dualitytheorems / V, D. Golovin; translated
fromRussianbyNormanStein.
p. cm.——(ContemporarySovietmathematics)
Translationof:Gomologiianaiiticheskikhpuchkoviteoremydvoistvennosti.
Bibliography:p.
includesindex.
ISBN0306]1024-5
1.Analyticsheaves.2.Homologytheory.3.Dualitytheory(Mathematics)i.Title.
ii.Series.
QA612.36.06513 1989 89-458
514’.224—dc19 CIP
Thistranslationispublishedunderanagreementwiththe
CopyrightAgencyoftheUSSR(VAAP)
©1989ConsultantsBureau,NewYork
ADivisionofPlenumPublishingCorporation
233SpringStreet, NewYork, N.Y. 10013
Allrightsreserved
Nopartofthisbookmaybereproduced,storedinaretrievalsystem, ortransmitted
inanyformorbyanymeans,electronic, mechanical,photocopying, microfilming,
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PREFACE
Thehomologyofanalytic sheavesisanaturalapparatusinthetheory
ofduality on complex spaces. The corresponding apparatus in algebraic
geometry was developedbyGrothendieckin thefifties. Incomplex ana-
lytic geometry the apparatusofhomologywasmissinguntilrecently, and
inits steadthehypercohomologyofcomplexsheaves(thehyper-Extfunc-
tors) and the Aleksandrov—Cech homology with coefficients in co-
presheaves were used. The homology of analytic sheaves, sheaves of
germsofhomologyandhomology groupsofanalytic sheaves,wereintro-
ducedandstudiedinthemid-seventiesinanumberofpapersbytheauthor.
Themaingoalofthisbookis to givea systematicanddetailedaccountof
the homology theory ofanalytic sheaves and some ofits applications to
duality theoryon complex spaces and tothe theoryofhyperfunctions. In
ordertoreadthisbookonemustbeacquaintedwith thefoundationsofho-
mological algebra and the theory oftopological vector spaces. Only the
mostelementaryconcepts andresultsfromthetheoryoffunctionsofsev-
eralcomplexvariablesareassumedtobeknown. Theinformation needed
aboutsheavesandcomplex spacesisrecountedbriefly atthebeginningof
thefirstchapter.
V. D. Golovin

CONTENTS
Chapter 1. ANALYTICSHEAVES .................................... 1
1. PreliminaryInformation .................................... 1
2. InjectivityTest................................................ 16
3. LocalDuality ................................................ 24
4. InjectiveandGlobalDimension ........................... 36
5. PropertiesofFineSheaves ................................. 46
Chapter2. HOMOLOGYTHEORY.................................... 63
1. SheavesofGermsofHomology........................... 63
2. HomologyGroups .......................................... 70
3. ConnectionwiththeFunctorsExt ........................ 76
4. PoincareDuality ............................................. 80
5. DualizingComplexes ....................................... 83
Chapter3. DUALITYTHEOREMS .................................... 87
1. HomologyofCoverings .................................... 87
2. Cohomologywith Arbitrary Supports..................... 91
3. Cohomologywith CompactSupports..................... 94
4. HolomorphicallyCompleteSpaces ........................ 98
5. Leray Spectra] Sequence .................................... 103
Chapter4. LOCALHOMOLOGY .................................... 109
l. HomologyofaClosedSet ................................. 109
2. LocalHomology ............................................. 111
3. DualityTheorems .......................................... 114
vii
viii CONTENTS
4. InductiveandProjectiveLimits ........................... 127
5. TestsforSeparatedness .................................... 135
Chapter5. APPLICATIONS .......................................... 143
1. DimensionofSupport....................................... 143
2. HomologicalCodimension ................................. 147
3. TheResultsofAndreottieGrauert ........................ 149
4. LefschetzTheorem .......................................... 151
5. TheoryofHyperfunctions ................................. 154
NOTES ..................................................................... 165
REFERENCES ............................................................ 199