Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
8
Foreword
orthogonalgeometriesoverKisatleastasrichasthestructureovertheunderlying
Dedekinddomain.Surprisingly,notmuchisknownaboutthesplittingofthe
above-mentionedinjectionWRłWK.Theproblemwassolvedinthecaseof
algebraicintegersbyP.Shastriin[50]andforrealgeometricringsbytheauthor
in[26]and[28].Chapter4summarizestheresultsofthesetwopapers.Themain
resultofthischapter(seeTheorem4.3)assertsthat,ifRistheringofregular
functionsonasmoothrealcurve,thenWRisadirectsummandoftheWitt
ringofthefieldoffractionsofR(thefieldofrationalfunctionsonthiscurve).
Consequently,theKnebusch-Milnorexactsequenceslicesintoandispatchedby
twosplitexactsequences(c.f.Theorem4.17).Moreover,ifthecurveinquestion
issemi-algebraicallycompactandsemi-algebraicallyconnected,thentheWittring
oftheringofpolynomialfunctionisinturnadirectsummandofWRasshownin
Theorem4.19.
Ontheotherhand,itisnaturaltoreckonthatstartingfromaringwithacom-
plexWittring(i.e.witharichstructureoforthogonalgeometries)andappending
rootsof(quadratic)polynomialsonecansuccessivelykillelementsoftheWittring.
Therefore,itisexpectedthatthenaturalmorphismofWittringscorresponding
toanalgebraic(resp.:real,quadraticorintegral)closureofafield/ringshould
notbeinjective.Aclassicalexample:startfromthefieldQofrationals,letR
denoteitsrealclosureandR[√−1]bethealgebraicclosure.TheWittringofQ
hasquiteacomplexstructure(asadditivegroupitisadirectsumofinfinitely
manynontrivialterm,see[33,ChapterVI,Section4]),whileWRisisomorphicto
theringofintegersandWR[√−1]consistsofjusttwoelements.Thus,theinjec-
tionsQłRłR[√−1]correspondtothemapsWQ→WR→WR[√−1],both
havingstronglynontrivialkernels.InChapter3weconcentrateonananalogyof
thisphenomenoninthecaseoftheintegralclosureofaring.Forexample,weshow
(seeTheorem3.15)thatifPisseminormalbutnotquadraticallyclosedandRis
theintegralclosureofP,thenthenaturalmorphismWP→WRisnotinjective.
ThisproblemhasalsoanaturalinterpretationintermsofthePicardfunctor.This
connectionisstudiedinSection3.3.Weclosethischaptershowinghowtoapply
theseresultsinthecaseofcurvedesingularization.
TheWittfunctorofaringextensionisalsothesubjectofthesecondchapter.
Here,however,weconsideraquadraticextensionofalocalring.Wedevelop
ageneralizationofScharlau’stransferandproveananalogyofScharlau’snorm
principle.Thisallowsustoconstructexamplesofringextensionswhereboth
ringshavethesamefieldoffractionsbutthecorrespondingWittmorphismisnot
surjective(hencethereareclassesofformsoverthebiggerringnotcommingfrom
thesmallerone).
Thenewcontributionsinthisbookinclude:entireChapters2and3andSec-
tion5.2wherethemainnewresultsareTheorems:2.17,2.19,2.24,3.11,3.15,
3.24,5.15andProposition2.35.TheresultsofChapter4appearedearlierin
