Torsors, reductive group schemes and extended affine Lie algebras

124 Pages·2013·1.1 MB·English

by  Philippe Gille

MEMOIRS
of the
American Mathematical Society
Volume 226 • Number 1063 (fourth of 5 numbers) • November 2013
Torsors, Reductive Group Schemes
and Extended Affine Lie Algebras
Philippe Gille
Arturo Pianzola
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 226 • Number 1063 (fourth of 5 numbers) • November 2013
Torsors, Reductive Group Schemes
and Extended Affine Lie Algebras
Philippe Gille
Arturo Pianzola
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Gille,Philippe,1968-author.
Torsors, reductive group schemes and extended affine Lie algebras / Philippe Gille, Arturo
Pianzola.
pagescm. –(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;number1063)
”November2013,volume226,number1063(fourthof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-8774-5(alk. paper)
1.Kac-Moodyalgebras. 2.Linearalgebraicgroups. 3.Geometry,Algebraic. I.Pianzola,
Arturo,1955-author. II.Title.
QA252.3.G55 2013
512’.482–dc23 2013025512
DOI:http://dx.doi.org/10.1090/S0065-9266-2013-00679-X
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Contents
Chapter 1. Introduction 1
Chapter 2. Generalities on the algebraic fundamental group, torsors, and
reductive group schemes 5
2.1. The fundamental group 5
2.2. Torsors 7
2.3. An example: Laurent polynomials in characteristic 0 8
2.4. Reductive group schemes: Irreducibility and isotropy 11
Chapter 3. Loop, finite and toral torsors 13
3.1. Loop torsors 13
3.2. Loop reductive groups 14
3.3. Loop torsors at a rational base point 15
3.4. Finite torsors 17
3.5. Toral torsors 18
Chapter 4. Semilinear considerations 21
4.1. Semilinear morphisms 21
4.2. Semilinear morphisms 22
4.3. Case of affine schemes 23
4.4. Group functors 24
4.5. Semilinear version of a theorem of Borel-Mostow 27
4.6. Existence of maximal tori in loop groups 29
4.7. Variations of a result of Sansuc 31
Chapter 5. Maximal tori of group schemes over the punctured line 35
5.1. Twin buildings 36
5.2. Proof of Theorem 5.1 37
Chapter 6. Internal characterization of loop torsors and applications 41
6.1. Internal characterization of loop torsors 41
6.2. Applications to (algebraic) Laurent series 45
Chapter 7. Isotropy of loop torsors 47
7.1. Fixed point statements 47
7.2. Case of flag varieties 49
7.3. Anisotropic loop torsors 52
Chapter 8. Acyclicity 57
8.1. The proof 57
8.2. Application: Witt-Tits decomposition 59
iii
iv CONTENTS
8.3. Classification of semisimple k–loop adjoint groups 59
8.4. Action of GL (Z) 61
n
Chapter 9. Small dimensions 71
9.1. The one-dimensional case 71
9.2. The two-dimensional case 71
Chapter 10. The case of orthogonal groups 83
Chapter 11. Groups of type G 85
2
Chapter12. Case ofgroups of typeF ,E andsimply connectedE innullity
4 8 7
3 87
Chapter 13. The case of PGL 91
d
13.1. Loop Azumaya algebras 91
13.2. The one-dimensional case 92
13.3. The geometric case 95
13.4. Loop algebras of inner type A 98
Chapter 14. Invariants attached to EALAs and multiloop algebras 99
Chapter 15. Appendix 1: Pseudo-parabolic subgroup schemes 101
15.1. The case of GLn,Z 102
15.2. The general case 104
Chapter 16. Appendix 2: Global automorphisms of G–torsors over the
projective line 107
Bibliography 109
Abstract
We give a detailed description of the torsors that correspond to multiloop
algebras. These algebras are twisted forms of simple Lie algebras extended over
Laurentpolynomialrings. TheyplayacrucialroleintheconstructionofExtended
Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody
Lie algebras). The torsor approach that we take draws heavily for the theory of
reductive group schemes developed by M. Demazure and A. Grothendieck. It also
allows us to find a bridge between multiloop algebras and the work of F. Bruhat
and J. Tits on reductive groups over complete local fields.
ReceivedbytheeditorJuly3,2011,and,inrevisedform,February21,2012.
ArticleelectronicallypublishedonMay23,2013;S0065-9266(2013)00679-X.
2010 MathematicsSubjectClassification. Primary17B67,11E72,14L30,14E20.
Key words and phrases. Reductivegroupscheme,torsor,multiloopalgebra,extendedaffine
Liealgebras.
A.PianzolawishestothankNSERCandCONICETfortheircontinuoussupport.
Affiliations at time of publication: Philippe Gille, UMR 8553 du CNRS, Ecole Normale
Sup´erieure,45rued’Ulm,75005Paris,France,email: [email protected];andArturoPianzola,Depart-
ment of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada;
and Centro de Altos Estudios en Ciencia Exactas, Avenida de Mayo 866, (1084) Buenos Aires,
Argentina,email: [email protected].
(cid:2)c2013 American Mathematical Society
v

CHAPTER 1
Introduction
To our good friend Benedictus Margaux
Many interesting infinite dimensional Lie algebras can be thought as being
“finite dimensional” when viewed, not as algebras over the given base field, but
rather as algebras over their centroids. From this point of view, the algebras in
question look like “twisted forms” of simpler objects with which one is familiar.
The quintessential example of this type of behaviour is given by the affine Kac-
Moody Lie algebras. Indeed the algebras that we are most interested in, Extended
AffineLieAlgebras(orEALAsforshort),canroughlybethoughtofashighernullity
analogues of the affine Kac-Moody Lie algebras. Once the twisted form point of
view is taken the theory of reductive group schemes developed by Demazure and
Grothendieck [SGA3] arises naturally.
Two key concepts which are common to[GP2] and the present work are those
of a twisted form of an algebra, and of a multiloop algebra. At this point we briefly
recall what these objects are, not only for future reference, but also to help us
redact a more comprehensive Introduction.
***
Unless specific mention to the contrary throughout this paper k will denote
a field of characteristic 0, and k a fixed algebraic closure of k. We denote k–alg
the category of associative unital commutative k–algebras, and R object of k–alg.
Let n ≥ 0 and m > 0 be integers that we assume are fixed in our discussion.
Consider the Laurent polynomial rings R=R =k[t±1,...,t±1] and R(cid:3) =R =
n 1 n n,m
k[t±1 m1 ,...,t±nm1 ]. For convenience we also consider the direct limit R∞(cid:3) =−li→mRn,m
taken over m which in practice will allow us to “see” all the R at the same
n,m
time. The natural map R → R(cid:3) is not only faithfully flat but also ´etale. If k is
algebraically closed this extension is Galois and plays a crucial role in the study of
multiloop algebras. The explicit description of Gal(R(cid:3)/R) is given below.
LetAbeak–algebra. Weareingeneralinterestedinunderstandingforms(for
the fppf-topology) of the algebra A⊗ R, namely algebras L over R such that
k
(1.1) L⊗ S (cid:6)A⊗ S (cid:6)(A⊗ R)⊗ S
R k k R
for some faithfully flat and finitely presented extension S/R. The case which is of
most interest to us is when S can be taken to be a Galois extension R(cid:3) of R of
Laurent polynomial algebras described above.1
GivenaformLasaboveforwhich(1.1)holds,wesaythatListrivializedbyS.
TheR–isomorphismclassesofsuchalgebrascanbecomputedbymeansofcocycles,
1TheIsotrivialityTheoremof[GP1]and[GP3]showsthatthisassumptionissuperflousif
Aut(A)isaanalgebraick–groupwhoseconnectedcomponentisreductive,forexampleifAisa
finitedimensionalsimpleLiealgebra.
1
2 1. INTRODUCTION
just as one does in Galois cohomology:
(cid:2) (cid:3)
(1.2) Isomorphism classes ofS/R–forms ofA⊗ R←→H1 S/R,Aut(A) .
k fppf
The right hand side is the part “trivialized by S” of the pointed set of non-
abelian cohomology on the flat site of Spec(R) with coefficients in the sheaf of
groups(cid:2) Aut(A). In(cid:3)the case when S is Galois over R we can inde(cid:2)ed identify
H1 S/R,Aut(A) with the “usual” Galois cohomology set H1 Gal(S/R),
fppf (cid:3)
Aut(A)(S) as in [Se].
Assume now that k is algebraically closed and fix a compatible set of primi-
tive m–th roots of unity ξ , namely such that ξe = ξ for all e > 0. We can
m me m
then identify Gal(R(cid:3)/R) with (Z/mZ)n where for each e = (e ,...,e ) ∈ Zn the
1 n
corresponding element e=(e1,··· ,en)∈Gal(R(cid:3)/R) acts on R(cid:3) via etim1 =ξmeitim1 .
The primary example of forms L of A⊗ R which are trivialized by a Galois
k
extension R(cid:3)/R as above are the multiloop algebras based on A. These are defined
asfollows. Considerann–tupleσσσ =(σ ,...,σ )ofcommutingelementsofAut (A)
1 n k
satisfying σm =1. For each n–tuple (i ,...,i )∈Zn we consider the simultaneous
i 1 n (cid:4)
eigenspace(cid:5)Ai1...in ={x∈A:σj(x)=ξmijx for all 1≤j ≤n}.ThenA= Ai1...in,
and A = A if we restrict the sum to those n–tuples (i ,...,i ) for which
i1...in 1 n
0≤i <m .
j j
The multiloop algebra corresponding to σσσ, commonly denoted by L(A,σσσ), is
defined by
L(A,σσσ)=(i1,...,⊕in)∈ZnAi1...in ⊗tim1 ...tnimn ⊂A⊗kR(cid:3) ⊂A⊗kR∞(cid:3)
NotethatL(A,σσσ),whichdoesnotdependonthechoiceofcommonperiodm,isnot
onlyak–algebra(ingeneralinfinite-dimensional), butalsonaturallyanR–algebra.
ItiswhenL(A,σσσ)isviewedasanR–algebrathatGaloiscohomologyandthetheory
of torsors enter into the picture. Indeed a rather simple calculation shows that
L(A,σσσ)⊗ R(cid:3) (cid:6)A⊗ R(cid:3) (cid:6)(A⊗ R)⊗ R(cid:3).
R k k R
Thus L(A,σσσ) corresponds to a torsor over Spec(R) under Aut(A).
When n = 1 multiloop algebras are called simply loop algebras. To illustrate
our methods, let us look at the case of (twisted) loop algebras as they appear in
the theory of affine Kac-Moody Lie algebras. Here n = 1, k = C and A = g is a
finite-dimensional simple Lie algebra. Any such L is naturally a Lie algebra over
R:=C[t±1] and L⊗RS (cid:6)g⊗CS (cid:6)(g⊗CR)⊗RS for some (unique) g, and some
finite´etaleextensionS/R.Inparticular,LisanS/R–formoftheR–algebrag⊗CR,
withrespect tothe´etale topology of Spec(R). Thus L corresponds toatorsor over
Spec(cid:2)(R) under(cid:3)Aut(g) whose isomorphism class is an element of the pointed set
H1´et R,Aut(g) . We may in fact take S to be R(cid:3) =C[t±m1 ].
Assume that A is a finite-dimensional. The crucial point in the classification
of forms of A⊗ R by cohomological methods is the exact sequence of pointed sets
k
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
(1.3) H1 R,Aut0(A) →H1 R,Aut(A) −ψ→H1 R,Out(A) ,
e´t e´t e´t
where Out(A) is the (finite constant) group of connected components of the alge-
braic k–group Aut(A).2
2StrictlyspeakingweshouldbeusingtheaffineR–groupschemeAut(A⊗kR)insteadofthe
algebraic k–group Aut(A). This harmless and useful abuse of notation will be used throughout
thepaper.
PHILIPPEGILLEandARTUROPIANZOLA 3
Grot(cid:2)hendieck’s (cid:3)theory of the algebraic fundamental group allows us to iden-
tify H1 R,Out(A) with the set of conjugacy classes of n–tuples of commuting
e´t
elements of the corresponding finite (abstract) group Out(A) (again under the as-
sumptionthatk isalgebraicallyclosed). Thisisanimportantcohomologicalinvari-
ant attached to any twisted form of A⊗ R. We point out that the cohomological
k
information is always about the twisted forms viewed as algebras over R (and not
k). In practice, as the affine Kac-Moody case illustrates, one is interested in un-
derstanding these algebras as objects over k (and not R). A technical tool (the
centroid trick) developed and used in [ABP2] and [GP2] allows us to compare k
vs R information.
We begin by looking at the nullity n=1 case. The map ψ of (1.3) is injective
[P1]. This fundamental fact follows from a general result about the vanishing of
H1 for reductive group schemes over certain Dedekind rings which includes k[t±1].
This result can be thought of as an analogue of “Serre Conjecture I” for some
very special rings of dimension 1. It follows from what has been said that we
can attach a conjugacy class of the finite group Out(A) that characterizes L up
to R–isomorphism. In particular, if Aut(A) is connected, then all forms (and
consequently, all twisted loop algebras) of A are trivial, i.e. isomorphic to A⊗ R
k
as R–algebras. This yields the classification of the affine Kac-Moody Lie algebras
bypurelycohomologicalmethods. Onecaninfactdefinetheaffinealgebrasbysuch
methods (which is a completely different approach than the classical definition by
generators and relations).
Surprisingly enough the analogue of “Serre Conjecture II” for k[t±1,t±1] fails,
1 2
asexplainedin[GP2]. Thesingle familyofcounterexamplesknownarethetheso-
calledMargauxalgebras. Theclassificationofformsinnullity2caseisinfactquite
interestingandchallenging. Unlikethenullityonecasethereareformswhicharenot
multiloop algebras (the Margaux algebra is one such example). The classification
in nullity 2 by cohomological methods, both over R and over k, will be given
in §9 as an application of one of our main results (the Acyclicity Theorem). This
classification(overkbutnotoverR)canalsobeattainedentirelybyEALAmethods
[ABP3]. The twoapproaches complement each other and are the culmination of a
projectstartedadecadeago. WealsoprovideclassificationresultsforloopAzumaya
algebras in §13.
Questions related to the classification and characterization of EALAs in ar-
bitrary nullity are at the heart of our work. In this situation A = g is a finite
dimensional simple Lie algebraover k. The twistedforms relevantto EALAtheory
are always multiloop algebras based on g [ABF(cid:2)P]. It is th(cid:3)erefore desirable to try
to characterize and understand the part of H1 R,Aut(g) corresponding to mul-
e´t
tiloop algebras. We address this problem by introducing the concept of loop and
toral torsors (with k not necessarily algebraically closed). These concepts are key
ideaswithinourwork. ItiseasytoshowusingatheoremofBorelandMostowthat
a multiloop algebra based on g, viewed as a Lie algebra over R , always admits a
n
Cartan subalgebra (in the sense of [SGA3]). We establish that the converse also
holds.
Central to our work is the study of the canonical map
(cid:2) (cid:3) (cid:2) (cid:3)
(1.4) H1 R ,Aut(g) →H1 F ,Aut(g)
e´t n e´t n
where F stands for the iterated Laurent series field k((t ))...((t )). The Acyclic-
n 1 n
ity Theorem proved in §8 shows that the restriction of the canonical map (1.4)