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Foreword
Abilinearformdefinesanorthogonalgeometryonagivenlinearspaceorprojec-
tivemodule.Onceweomitdegenerateforms(roughlyspeaking,theseareforms
allowinganon-zerovectortobeorthogonaltotheentirespace),theremainingones
maybeclassifiedbytherelationofsimilarity(thetermisexplainedinSection1.1).
ThisleadsustothenotionoftheWittring,whichisanalgebraicstructureconsist-
ingofallthesimilarityclassesoffinitelygeneratedprojectivemodulesoveragiven
basering.Inasense,Wittringdescribesallthepossibleorthogonalgeometries
overthering/filedinquestion.Theleadingthemeofthisbookistostudymor-
phismsbetweenWittrings.Ingeometricterms,thiscanbeviewedasanalyzingto
whatextenttheorthogonalgeometriesdefinedoveroneringmaybetranscribed
toanotherring.Forexample,knowingthecriteriaforexistenceofanisomorphism
betweenWittrings,onemayverifywhethertworings/fieldsadmitthesameset
oforthogonalgeometries.IfthisisthecasewesaythatthetworingsareWitt
equivalent.Thisproblemhasbeenintensivelyresearchedinpreviousyears.The
completecriteriaareknownforfieldswithsmallsquareclassgroups(see[9])and
globalfields(seee.g.[44,53,52]).Theauthorofthisbookfoundcriteriafor
Wittequivalenceoffunctionfieldsandringsofregularfunctiononrealalgebraic
curves(summaryoftheseresultscanbefoundintheappendixtoChapter5).The
criterionofWittequivalenceofrealfunctionfieldshasbeenrecentlygeneralized
byN.Grenier-BoleyandD.Hoffmanntoarbitraryrealfieldswithu-invariantsnot
exceeding2.WepresenttheirresultinSection5.1.Inthesecondsectionofthe
lastchapterweapplytheideasusedearlierforringsofregularfunctiononreal
curvestoextendtheresultofGrenier-BoleyandHoffmannandobtainanecessary
condition(seeTheorem5.15)forWittequivalenceofrealholomorphyrings.This
generalizesourearlierresultobtainedin[27].
Chapter4copeswithasplittingoftheKnebusch-Milnorexactsequence.A
classicaltheoremduetoM.KnebuschandJ.Milnor(seeTheorem1.40)assert
thattheWittringWRofaDedekinddomainRinjectsintotheWittringofits
fieldoffractionsKandtheimageofthisembeddingcoincideswiththekernel
ofamapfromtheWittringofthefieldtotheco-productofWittringsofall
thelocalizationsofthebasering.Inanutshell,thissaysthatthestructureof
