Prime divisors and noncommutative valuation theory

225 Pages·2012·1.44 MB·English

by  Hidetoshi Marubayashi, Freddy van Oystaeyen, Mathematiker

Lecture Notes in Mathematics 2059
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
(cid:127)
Hidetoshi Marubayashi Fred Van Oystaeyen
(cid:2)
Prime Divisors and
Noncommutative
Valuation Theory
123
HidetoshiMarubayashi FredVanOystaeyen
TokushimaBunriUniversity UniversityofAntwerp
FacultyofScienceandEngineering MathematicsandComputerScience
Shido,SanukiCity,Kagawa,Japan Antwerp,Belgium
ISBN978-3-642-31151-2 ISBN978-3-642-31152-9(eBook)
DOI10.1007/978-3-642-31152-9
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012945522
MathematicsSubjectClassification(2010):16W40,16W70,16S38,16H10,13J20,16T05
(cid:2)c Springer-VerlagBerlinHeidelberg2012
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Introduction
Theclassicaltheoryofvaluationringsinfieldshasnaturalapplicationsinalgebraic
geometryas well as in numbertheory,e.g. as local ringsof nonsingularpointsof
curvesor,respectively,asprimelocalizationsofringsofintegersinnumberfields.
IntheseapplicationstheringsareNoetherianandthevaluationsarediscrete,thatis
the valuegroupis the additivegroupof the integers,Z. The factthatnon-discrete
valuationringsarenotNoetherianmaybethereasonthatgeneral(cid:2)-valuationswith
totally ordered value group (cid:2), which need not be discrete, have not been used
to the same extent as discrete ones in the aforementioned branches of classical
mathematics.
However, function field extensions of arbitrary transcendence degree have
been considered; the transcendence degree is recorded as the dimension of some
correspondingalgebraicvariety,butitisalsothemaximalrankofavaluationring
in the function field. Algebraic geometry easily survives by using the theory of
local Noetherian rings, avoiding the non-Noetherian valuation rings dominating
them. The algebraic theory in this book deals with generalized (cid:2)-valuations,
often related to (cid:2)-filtrations, for totally ordered groups (cid:2), at places even totally
ordered semigroups appear. A source of problems and new ideas is the extension
theory for valuations from a subfield to the field. This theory is well developed
for number fields in connection with Galois theory, but is perhaps less used for
field extensions of finite transcendence degree probably because the (Noetherian)
geometric approach is more popular. We want to consider more general algebra
extensionsincludingskewfields,matrixrings,etc.
ThereforewedevelopinChap.1thegeneraltheorystartingfromanoncommuta-
tivebackground,expoundingaminimalcommutativebackgroundwhennecessary.
The primes or prime places introduced provide a very general approach to a
valuationtheoryforassociative ringsand algebras;this originatesin the theoryof
pseudoplacesasin[70,71]andtheworkofVanGeel[68,69].Thetheoryisrelated
to the considerationof filtrations and gradationsby totally orderedgroups;this is
thetopicofSect.1.3(andSect.1.8).Originallypseudoplacesweredevelopedwith
the aim to apply themto the constructionof genericcrossed productsandgeneric
skewfields, hence it is not really surprising that an interesting class of algebras
v
vi Introduction
inviting the development of a generalized valuation theory is the class of central
simple algebras, or more generally finitely dimensionalalgebras overfields. Thus
ourtheoryhasdirectlinkstothetheoryofordersandmaximalorders;thiswillbethe
subjectofChap.2.ThesecondpartofChap.1isdevotedtosomeparticularprimes:
Dubrovinrings.AfterthebasictheoryinSect.1.4andtheidealtheoryinSect.1.5
wefocusonDubrovinvaluationsoffinitedimensional(simple)algebrasinSect.1.6
andonGaussextensionsinSect.1.7.Toageneralizedvaluationringcorrespondsa
reductionoftheoriginalalgebraandtherelationbetweenpropertiesofthereduced
algebraandtheoriginaloneshouldbeclarified.Therearepropertiesofhomological
type or of (noncommutative) geometric type; in Appendix 1.9 we provide some
theoryrelatedtoAuslanderregularityofthealgebrasunderconsideration.
Chapter 2 deals with orders. It is not our aim to provide a complete theory of
ordersin(semisimple)Artinianalgebras,butweincludeashortintroductiontothis
importantrootofthetheoryrelatingtothefundamentalworkofAuslander,Reiner
andothers(see[57]).
ArithmeticalringswithsuitabledivisorialpropertiesandnoncommutativeKrull
rings are amongst the natural study objects here (Sects.2.1 and 2.2). Then we
consider Ore extensionsover Krullordersand noncommutativevaluationringsof
Ore extensions of simple Artinian rings in Sects.2.3 and 2.4. To end the chapter
we arrive at an arithmetical divisor theory, a noncommutative divisor calculus,
leading to a noncommutative Riemann–Roch theorem over a central curve. The
latter result fits in the geometry of rings with polynomial identities, cf. [76]; it
also fits in this book because it provides a good example of a concrete, almost
calculative, application of noncommutative valuation theory, even if here only
discretevaluationsappear.
InChap.3weleavethebiotopeoffinitedimensionalalgebrasandturnattention
to a recently popular class of infinite dimensional algebras containing quantum
groups,quantizedalgebrasanddeformations.Usuallyexamplesofsuchalgebrasare
givenby generatorsand relationsand thereforethey are equippedwith a standard
filtration. The reduction of such an algebra at a central valuation can be “good”
or “bad” depending on how the defining relations reduce modulo the maximal
idealofthe centralvaluationring.Similar phenomenaexistin algebraicgeometry
relatedtogoodreductionof(elliptic)curvesbutnowthereducing“equations”are
elementsoftheidealofrelationsinthefreealgebra.Thepropertyofgoodreduction
allows the extension of the central valuation to the (skewfield of fractions of the)
noncommutativealgebra;thisisthetopicofSect.3.1.Toestablishacasewherewe
cancompletelycalculateallvaluationswefocusontheWeylfieldinSect.3.2and
providesomedivisortheoryrelatedtoaverypeculiarsubringinSect.3.3.
Finally we consider finite dimensional Hopf algebras in the last section and
extend the valuation filtration of a central valuation to a Hopf filtration on the
Hopfalgebra.Thefiltration partof degreezerois a Hopf orderand we show how
certain Hopf orders, generalizing Larson orders in the case of finite group rings,
maybeexplicitlyconstructed.Theexplicitexamplesovernumberringsexhibitthe
importance of ramification properties of the value function corresponding to the
Hopforder.
Introduction vii
In our opinion noncommutative extensions of valuation theory deserve to be
furtherintegratedinnoncommutativealgebra(andnoncommutativegeometrytoo).
We do realize that this book does not provide a finished picture of possible
noncommutativeextensionsofvaluationtheory,butwehopetohaveopenedafew
doors for possible new developmentspointing at interesting directions for further
research.
(cid:127)
Contents
1 GeneralTheoryofPrimes................................................... 1
1.1 PrimePlacesofAssociativeRings.................................... 1
1.2 PrimesinAlgebras..................................................... 8
1.3 ValueFunctionsandPrimeFiltration................................. 19
1.4 DubrovinValuations................................................... 33
1.5 IdealTheoryofDubrovinValuationRings........................... 42
1.6 DubrovinValuationRingsofQwithFiniteDimension
OverItsCenter......................................................... 58
1.7 GaussExtensions...................................................... 70
1.8 FiltrationsbyTotallyOrderedGroups................................ 88
1.9 ReductionsofAlgebrasandFiltrations............................... 96
1.10 Appendix:GlobalDimensionandRegularityofReductions........ 101
2 MaximalOrdersandPrimes ............................................... 109
2.1 MaximalOrders........................................................ 109
2.2 KrullOrders............................................................ 115
2.3 OreExtensionsOverKrullOrders.................................... 132
2.4 Non-commutativeValuationRingsinK.XI(cid:3);ı/.................... 144
2.5 ArithmeticalPseudovaluationsandDivisors......................... 153
2.6 The Riemann–RochTheoremfor Central Simple
AlgebrasOverFunctionFieldsofCurves............................ 164
3 ExtensionsofValuationstoQuantizedAlgebras......................... 175
3.1 ExtensionofCentralValuations ...................................... 175
3.2 DiscreteValuationsontheWeylSkewfield .......................... 183
3.3 SomeDivisorTheoryforWeylFieldsOverFunctionFields........ 193
3.4 HopfValuationFiltration.............................................. 197
Bibliography...................................................................... 213
Index............................................................................... 217
ix