Specialization of quadratic and symmetric bilinear forms

208 Pages·2010·1.71 MB·English

by  Manfred Knebusch (auth.)

Specialization of Quadratic and Symmetric
Bilinear Forms
Algebra and Applications
Volume11
ManagingEditor:
AlainVerschoren
UniversityofAntwerp,Belgium
SeriesEditors:
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EötvösLorándUniversity,Hungary
EricFriedlander
NorthwesternUniversity,USA
JohnGreenlees
SheffieldUniversity,UK
GerhardHiss
AachenUniversity,Germany
IekeMoerdijk
UtrechtUniversity,TheNetherlands
IdunReiten
NorwegianUniversityofScienceandTechnology,Norway
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HamburgUniversity,Germany
MinaTeicher
Bar-llanUniversity,Israel
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Manfred Knebusch
Specialization of Quadratic
and Symmetric Bilinear
Forms
Translated by Thomas Unger
123
Prof.Dr.ManfredKnebusch
DepartmentofMathematics
UniversityofRegensburg
Universita¨tsstr.31
93040Regensburg
Germany
[email protected]
Dr.ThomasUnger
SchoolofMathematicalSciences
UniversityCollegeDublin
Belfield
Dublin4
Ireland
[email protected]
ISBN978-1-84882-241-2 e-ISBN978-1-84882-242-9
DOI10.1007/978-1-84882-242-9
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Dedicatedtothememoryofmyteachers
EmilArtin1898–1962
HelBraun1914–1986
ErnstWitt1911–1991

Preface
AMathematicianSaidWho
CanQuoteMeaTheoremthat’sTrue?
FortheonesthatIKnow
AreSimplynotSo,
WhentheCharacteristicisTwo!
This pretty limerick first came to my ears in May 1998 during a talk by T.Y. Lam
on field invariants from the theory of quadratic forms.1 It is—poetic exaggeration
allowed—asuitablemottoforthismonograph.
What is it about? At the beginning of the seventies I drew up a specialization
theoryofquadraticandsymmetricbilinearformsoverfields[32].Letλ: K → L∪∞
beaplace.Thenonecanassignaformλ∗(ϕ)toaformϕoverKinameaningfulway
ifϕhas “good reduction” withrespecttoλ(see§1.1).The basicidea istosimply
applytheplaceλtothecoefficientsofϕ,whichmustthereforebeinthevaluation
ringofλ.
The specialization theory of that time was satisfactory as long as the field L,
and therefore also K, had characteristic (cid:2) 2. It served me in the first place as the
foundation for a theory of generic splitting of quadratic forms [33], [34]. After a
verymodestbeginning,thistheoryisnowinfullbloom.Itbecameimportantforthe
understandingofquadraticformsoverfields,ascanbeseenfromthebook[26]of
Izhboldin–Kahn–Karpenko–Vishikforinstance.Oneshouldnotethatthereexistsa
theoryof(partial)genericsplittingofcentralsimplealgebrasandreductivealgebraic
groups, parallel to the theory of generic splitting of quadratic forms (see [29] and
theliteraturecitedthere).
InthisbookIwouldliketopresentaspecializationtheoryofquadraticandsym-
metricbilinearformswithrespecttoaplaceλ: K → L∪∞,withouttheassumption
that charL (cid:2) 2. This is where complications arise. We have to make a distinction
1“Somereflectionsonquadraticinvariantsoffields”,3May1998inNotreDame(Indiana)onthe
occasionofO.T.O’Meara’s70thbirthday.
vii
viii Preface
between bilinear and quadratic forms and study them both over fields and valua-
tion rings. From the viewpoint of reductive algebraic groups, the so-called regular
quadratic forms (see below) are the natural objects. But, even if we are interested
only in such forms, we have to know a bit about specialization of nondegenerate
symmetricbilinearforms,sincetheyoccuras“multipliers”ofquadraticforms:ifϕ
issuchabilinearformandψisaregularquadraticform,thenwecanformatensor
productϕ⊗ψ,see§1.5.Thisisaquadraticform,whichisagainregularwhenψhas
evendimension(dimψ =numberofvariablesoccurringinψ).However—andhere
alreadywerunintotrouble—whendimψisodd,ϕ⊗ψisnotnecessarilyregular.
EvenifweonlywanttounderstandquadraticformsoverafieldKofcharacteris-
ticzero,itmightbenecessarytolookatspecializationswithrespecttoplacesfrom
K tofieldsofcharacteristic2,especiallyinarithmeticinvestigations.WhenK itself
hascharacteristic2,anoftenmorecomplicatedsituationmayoccur,forwhichwe
arenotpreparedbytheavailableliterature.Certainlyfieldsofcharacteristic2were
alreadyallowedinmyworkonspecializationsin1973[32],butfromtoday’spoint
of view satisfactory results were only obtained for symmetric bilinear forms. For
quadraticformstherearegapingholes.Wehavetostudyquadraticformsoveraval-
uationringinwhich2isnotaunit.EventhebeautifulandextensivebookofRicardo
Baeza[6]doesn’tgiveusenoughforthetheoryofspecializations,althoughBaeza
even allows semilocal rings instead of valuation rings. He studies only quadratic
formswhoseassociatedbilinearformsarenondegenerate.Thisforcesthoseforms
tohaveevendimension.
Letmenowdiscussthecontentsofthisbook.Afteranintroductiontotheprob-
lemin§1.1,whichcanbeunderstoodwithoutanypreviousknowledgeofquadratic
and bilinear forms, the specialization theory of symmetric bilinear forms is pre-
sentedin§1.2–§1.3.Therearegood,generallyaccessiblesourcesavailableforthe
foundations of the algebraic theory of symmetric bilinear forms. Therefore many
resultsarepresentedwithoutaproof,butwithareferencetotheliteratureinstead.
Asanimportantapplication,theoutlinesofthetheoryofgenericsplittinginchar-
acteristic(cid:2)2aresketchedin§1.4,nearlywithoutproofs.
From §1.5 onwards we address the theory of quadratic forms. In characteris-
tic 2 fewer results can be found in the literature for such forms than for bilinear
forms, even at the basic level. Therefore we present most of the proofs. We also
concern ourselves with the so-called “weak specialization” (see §1.1) and get into
areaswhichmayseemstrangeeventospecialistsinthetheoryofquadraticforms.
InparticularwehavetorequireaquadraticformoverKtobe“obedient”inorderto
weaklyspecializeitwithrespecttoaplaceλ: K → L∪∞(see§1.7).Ihavenever
encounteredsuchathinganywhereintheliterature.
AttheendofChapter1wereachalevelinthespecializationtheoryofquadratic
formsthatfacilitatesagenericsplittingtheory,usefulformanyapplications.Inthe
firsttwosections(§2.1,§2.2)ofChapter2weproducesuchagenericsplittingtheory
intwoversions,bothofwhichdeserveinterestintheirownright.
Wecallaquadraticformϕoverafieldk nondegeneratewhenitsquasilinearpart
(cf.Arf[3]),whichwedenotebyQL(ϕ),isanisotropic.Wefurthercall—deviating
Preface ix
from Arf [3]—ϕ regular when QL(ϕ) is at most one-dimensional and strictly reg-
ular when QL(ϕ) = 0 (cf. §1.6, Definition 1.59). When k has characteristic (cid:2) 2,
every nondegenerate form is strictly regular, but in characteristic 2 the quasilinear
part causes complications. For in this case ϕ can become degenerate under a field
extensionL⊃k.Onlyintheregularcaseisthisimpossible.
In §2.1 we study the splitting behaviour of a regular quadratic form ϕ over k
underfieldextensions,whilein§2.2anynondegenerateformϕ,butonlyseparable
extensionsofk areallowed.Thetheoryof§2.1incorporatesthetheoryof§1.4,so
themissingproofsof§1.4aresubsequentlyfilledin.
Untilthe end of §2.2 our specialization theory is based on an obvious “canoni-
cal”conceptofgoodreductionofaformϕoverafield K (quadraticorsymmetric
bilinear) to a valuation ring o of K, similar to what is known under this name in
otherareasofmathematics(e.g.abelianvarieties).Thereisnothingwrongwiththis
theory;however,formanyapplicationsitistoolimited.
This is particularly clear when studying specializations with respect to a place
λ : K → L∪∞ with charK = 0, charL = 2. If ϕ is a nondegenerate quadratic
form over K with good reduction with respect to λ, then the specialization λ∗(ϕ)
is automatically strictly regular. However, we would like to have a more general
specialization concept, in which forms with quasilinear part (cid:2) 0 can arise over L.
Conversely, if the place λ is surjective, i.e. λ(K) = L∪∞, we would like to “lift”
everynondegeneratequadraticformψoverLwithrespecttoλtoaformϕoverK,
i.e.tofindaformϕoverK whichspecializestoψwithrespecttoλ.Thenwecould
usethetheoryofformsoverK tomakestatementsaboutψ.
Wepresentsuchageneralspecializationtheoryin§2.3.Itisbasedontheconcept
of“fairreduction”,whichislessorthodoxthangoodreduction,butwhichneverthe-
lesspossessesquitesatisfyingproperties.
Next,in§2.4,wepresentatheoryofgenericsplitting,whichunitesthetheories
of §1.4, §2.1 and §2.2 under one roof and which incorporates fair reduction. This
theory is deepened in §2.5 and §2.6 through the study of generic splitting towers,
andthuswereachtheendofChapter2.
Chapter3(§3.1–§3.13)isalongchapterinwhichwepresentapanoramaofre-
sultsaboutquadraticformsoverfieldsforwhichspecializationandgenericsplitting
of forms play an important role. This only scratches the surface of applications of
the specialization theory of Chapters 1 and 2. Certainly many more results can be
unearthed.
WereturntothefoundationsofspecializationtheoryinthefinalshortChapter4
(§4.1–§4.5). Quadratic and bilinear forms over a field can be specialized with re-
specttoamoregeneral“quadraticplace”Λ : K → L∪∞(definedin§4.1)instead
ofausualplaceλ : K → L∪∞.Thisrepresentsaconsiderablebroadeningofthe
specializationtheoryofChapters1and2.Ofcoursewerequireagain“obedience”
from a quadratic form q over K in order for its specialization Λ∗(q) to reasonably
exist. It then turns out that the generic splitting behaviour of Λ∗(q) is governed by
thesplittingbehaviourofqandΛ,insofarasgoodorfairreductionispresentina
weaksense,aselucidatedforordinaryplacesinChapter2.