Algebraic K-theory of Number Fields

149 Pages·2014·1.652 MB·English

by  Alexey Beshenov

rkK (O )
n F
Algebraic K-theory of Number Fields
0 1 2 3 4 5 6 7 8 9
r (cid:0)r (cid:1) r r (cid:0)r r r (cid:0)r
1 1 2 1 0 2 0 1 2 0 2 0 1 2
pSU{SOq Z{ Z{ Z Z
2 2 0 0 0 0
pSUq Z Z Z Z
0 0 0 0
Alexey Beshenov
Advised by Prof. Boas Erez
ALGANT Master Thesis
Università degli Studi di Milano / Université de Bordeaux
July 2014
ALGANT
Erasmus Mundus
2:3
ii
Preface
Никто не обнимет необъятного.
— Козьма Прутков
(One can’t embrace the unembraceable.
— Kozma Prutkov)
L
Oneofthecentraltopicsinnumbertheoryisthestudyof -functions. Probablythemostwell-known
Riemann zeta function
of these is the , which is defined by the series
ζ s n s 1 .
p q(cid:16) (cid:1) (cid:16) p s
n‚¥1 p„prime1(cid:1) (cid:1)
s C
This is convergent for Re 1, and it has analytic continuation to which is holomorphic, except for
s ¡ ζ s s
a simple pole at 1. We denote the analytic continuation also by . Its values at and 1 are
(cid:16) (cid:1)
functional equation
related by a
πs
ζ s π s s ζ s ,
1 cos 2 2 (cid:1) Γ
p (cid:1) q(cid:16) p q p q p q
2
(cid:1) (cid:9)
s gamma function n n
where Γ is the (which is Γ 1 ! for positive integers).
p q p q(cid:16)p (cid:1) q
ζ n n Z
One may ask what are the values of at . For instance, one special value is
p q P
ζ 1.
0
p q(cid:16)(cid:1)
2
n , , , ,... ζ n
If 3 5 7 9 are positive odd numbers, then the values are rather mysterious; the func-
(cid:16) p q n , , , ,...
tionalequationissupposedtorelatethemtothevaluesatnegativeevennumbers 2 4 6 8 ,
(cid:16)(cid:1) (cid:1) (cid:1) (cid:1)
but it just tells us that
ζ n n .
0 is a simple zero for 2 even
p(cid:1) q(cid:16) ¥
n , , , ,...
Less mysterious are the values at 2 4 6 8 They were discovered already by Euler about
(cid:16)
1749 (see [Ayo74] for a historical overview):
π
ζ 1 1 1 2,
2 1
p q(cid:16) (cid:0) 22 (cid:0) 32 (cid:0) 42 (cid:0)(cid:4)(cid:4)(cid:4)(cid:16) 6
π
ζ 1 1 1 4,
4 1
p q(cid:16) (cid:0) 24 (cid:0) 34 (cid:0) 44 (cid:0)(cid:4)(cid:4)(cid:4)(cid:16) 90
π
ζ 1 1 1 6 ,
6 1
p q(cid:16) (cid:0) 26 (cid:0) 36 (cid:0) 46 (cid:0)(cid:4)(cid:4)(cid:4)(cid:16) 945
π
ζ 1 1 1 8 ,
8 1
p q(cid:16) (cid:0) 28 (cid:0) 38 (cid:0) 48 (cid:0)(cid:4)(cid:4)(cid:4)(cid:16) 9450
.
.
.
iii
ζ ,ζ ,ζ ,ζ ,...
The pattern is more clear if we consider the corresponding values 1 3 5 7
These are some rational numbers. To explain them, introduce the Bep(cid:1)rnoqulpl(cid:1)i nqumpb(cid:1)erqs Bp(cid:1)n bqy a
generating function
T Tn T T T T T T
eT d(cid:16)ef Bn n (cid:16)1(cid:1) 1T(cid:0) 1 2 (cid:1) 1 4 (cid:0) 1 6 (cid:1) 1 8 (cid:0) 5 10 (cid:1) 691 12 (cid:0)(cid:4)(cid:4)(cid:4)
(cid:1)1 n‚¥0 ! 2 6 2! 30 4! 42 6! 30 8! 66 10! 2730 12!
ζ
Then the values of are related to these numbers as follows:
B
ζp(cid:1)nq(cid:16)(cid:1)nn(cid:0)1 for n¥1 odd.
1
(cid:0)
This is essentially the Euler’s calculation. In particular,
ζ 1 , ζ 1 , ζ 1 , ζ 1 , ζ 1 , ζ 691 , ...
1 3 5 7 9 11
p(cid:1) q(cid:16)(cid:1) p(cid:1) q(cid:16) p(cid:1) q(cid:16)(cid:1) p(cid:1) q(cid:16) p(cid:1) q(cid:16)(cid:1) p(cid:1) q(cid:16)
12 120 252 240 132 32760
ζ(s)
−9 −5 −1
s
−7 −3
− 1
12
We refer to [Neu99, Theorem VII.1.8] for a proof. Just to spice up this introduction, recall a proof
ζ
of 1 1 that one would suggest in the 18th century. If we formally differentiate the geometric
p(cid:1) q(cid:16)(cid:1)12
series formula
x x x 1 ,
1(cid:0) (cid:0) 2(cid:0) 3(cid:0)(cid:4)(cid:4)(cid:4)(cid:16) x
1
(cid:1)
then we get
x x x 1 .
1(cid:0)2 (cid:0)3 2(cid:0)4 3(cid:0)(cid:4)(cid:4)(cid:4)(cid:16) 1 x 2 (*)
p (cid:1) q
Now consider the sums (literally meaningless without the functional equation)
ζ
1 “ ” 1 2 3 4
p(cid:1) q (cid:16) (cid:0) (cid:0) (cid:0) (cid:0)(cid:4)(cid:4)(cid:4)
ζ
4 1 “ ” 4 8 12 16
p(cid:1) q (cid:16) (cid:0) (cid:0) (cid:0) (cid:0)(cid:4)(cid:4)(cid:4)
ζ ζ ζ
1 4 1 “ ” 3 1 “ ” 1 2 4 3 4 8
p(cid:1) q(cid:1) p(cid:1) q (cid:16) (cid:1) p(cid:1) q (cid:16) (cid:0)p (cid:1) q(cid:0) (cid:0)p (cid:1) q(cid:0)(cid:4)(cid:4)(cid:4)
1,
“ ” 1 2 3 4 “ ”
(cid:16) (cid:1) (cid:0) (cid:1) (cid:0)(cid:4)(cid:4)(cid:4) (cid:16)
4
x
where the last equality is thanks to the formula (*) with 1 (which may be considered wrong, but
(cid:16)(cid:1)
was used by Euler in his 1760 paper “De seriebus divergentibus”—cf. [BL76]). Therefore
ζ 1 .
1 “ ” 1 2 3 4 “ ”
p(cid:1) q (cid:16) (cid:0) (cid:0) (cid:0) (cid:0)(cid:4)(cid:4)(cid:4) (cid:16) (cid:1)
12
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The corresponding values at the positive even integers are
n B π n
ζ n p(cid:1)1q {2(cid:0)1 np2 q n .
p q(cid:16) n for ¥2 even
2 !
number field F
Now we want to generalize the situation and consider a , i.e. a finite algebraic
extension of the field of rational numbers Q. In F we have its ring of integers OF, which is a free
Z d F Q
-module of rank : .
(cid:16)r s
F (cid:111)(cid:111) (cid:63)(cid:95)O
F
d d
Q(cid:111)(cid:111) (cid:63)(cid:95)Z
Dedekind zeta function F
By definition, the of is given by a series
ζFpsq(cid:16)LpSpecOF,sq(cid:16) pNaq(cid:1)s (cid:16) 1Np s,
a p 1(cid:1)p q(cid:1)
‚ „
where a runs through all nonzero ideals of OF, and p runs through all prime ideals of OF. By Na we
norm F Q
denote the of ideal. In particular, if , then this is the same as the Riemann zeta series
ζ s (cid:16)s C
as above. Again, this is convergent for Re 1, and has an analytic continuation to which is
p q s ¡
holomorphic, except for a simple pole at 1. The functional equation is
(cid:16)
ζF 1 s ∆F s(cid:1)1{2 cosπs r1(cid:0)r2 sinπs r2 2 2π (cid:1)sΓ s d ζF s ,
p (cid:1) q(cid:16)| | p q p q p q
2 2
(cid:1) (cid:9) (cid:1) (cid:9) (cid:0) (cid:8)
where
r real places F ª R
• 1 is the number of , i.e. embeddings (cid:209) .
r complex places F ª C
• 2 is the number of , i.e. conjugate pairs of embeddings (cid:209) .
ddef F Q r r F
• (cid:16) r : s(cid:16) 1(cid:0)2 2 is the degree of .
• ∆F is the discriminant of F.
F Q r r d
(If (cid:16) , then one has 1 (cid:16)1, 2 (cid:16)0, (cid:16)1, ∆F (cid:16)1.)
For basic facts about Dedekind zeta functions we refer to [Neu99, §VII.5].
ζ s s n n , , ,...
We again want to investigate the values F at points with 0 1 2 Looking at the
p q r (cid:16) (cid:1) (cid:16) totally real
functional equation,nwe notenthat these are zerosζ, unlenss 2 (cid:16)0 (when the number fierladtiiosnal ).
Inthelattercaseif 0or 1isodd, values F arenon-zero, actuallysome numbers.
ThefactthatζF n (cid:16)Qiskno¥wnas Siegel–Klingpe(cid:1)ntqheorem([Kli62]; cf. [Neu99,VII.9.9]). Thereare
p(cid:1) qP ζ n
certain ways to relate these values to some fundamental rational numbers, just as Euler related F
ζ p(cid:1) q
to Bernoulli numbers. For instance, a formula of Harder [Har71, §2.2] connects the values of F, for
F Euler–Poincaré characteristic
totally real to of arithmetic groups. In case of symplectic groups
Sp O
2np Fq the formula reads
χ Sp O 1 ζ i .
p 2np Fqq(cid:16) n d n Fp1(cid:1)2 q
2 p (cid:1) q i n
1⁄„⁄
χ Sp O i ζ n
Here p 2np Fqqnis a rational number. So by induction on , the last formula implies that Fp1(cid:1) q
are rational for even . We will not get into details and refer to [Ser71, §3.7] and [Bro74].
v
This may be seen as a manifestation of a general philosophical principle:
L
special values of -functions are captured by cohomological invariants.
zerosζ s s n
Inthistextwewillnotbetooambitiousandwewilllookatthe F at . Thismayseem
r r p q (cid:16)(cid:1)µ
trivial, but such zeros have multiplicities, depending on 1 and 2. Let us denote by n the multiplicity
s n µ
of zero at (if there is no zero, then n 0). The functional equation, together with the fact that
ζ s (cid:16)(cid:1) s (cid:16) s
F has no zeros for Re 1 and a simple pole at 1, shows readily
p q ¡ (cid:16)
r r , n ,
µ r1,(cid:0) 2(cid:1)1 n(cid:16)0 ,
n (cid:16)$ r2 r , n¥1 odd .
& 1(cid:0) 2 ¥2 even
F Q i r r
Here is an example of zeta function%for (cid:16) p q. In this case 1 (cid:16)0 and 2 (cid:16)1, hence all negative
integers are simple zeros:
ζQ(i)(s)
−6 −5 −3 −1
s
−4 −2
F Q α α X X r r
ζIf we take (cid:16) p q where is a root of polynomial 3(cid:0) (cid:0)1, then 1 (cid:16) 2 (cid:16)1, and simple zeros
of Q α alternate with zeros of multiplicity two:
p q
ζQ(α)(s)
−3 −1
s
−4 −2
cohomological account
We are going to see some of these multiplicities of zeros!
F ideal class group F
Recall that for a number field one can define its Cl [Neu99, I.3]. This was
p q
studied already by Gauss, Kummer, Dedekind, and other 19th century mathematicians. It is some
O
abelian group which vanishes if and only if F is a principal ideal domain. Moreover,
F .
Cl is finite
p q
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group of units O
Another basic invariant is the F(cid:2)—the multiplicative group of invertible elements in
O O r r
F. A remarkable theoµrem of Dirichlet tells that F(cid:2) is finFitely generated, it has rank exactly 1(cid:0) 2(cid:1)1,
and its torsion part is F, the group of roots of unity in :
OF(cid:2) Zr1(cid:0)r2(cid:1)1 µF.
(cid:21) ‘
F O
We will review briefly Cl and F(cid:2) in chapter 1.
p q
R
Now the main objects of our study come into play. For any ring (and actually any scheme, if you
algebraic K-groups
like) one can define a whole series of intricate algebraic invariants, named :
K R ,K R ,K R ,K R ,K R ,...
0p q 1p q 2p q 3p q 4p q
These are some abelian groups. The first invariants in this list were introduced in the 50s and 60s by
K K K
Grothendieck ( 0); Hyman Bass, Stephen Schanuel ( 1); and John Milnor ( 2). A brief review that fits
K R i
our needs constitutes chapter 1. The general definition of i for 2 (both pretty technical and
p q ¥
conceptual) is due to Quillen and it is the subject of chapter 2 and also appendix Q.
R O
The only ring that interests us is F, and in this case
(cid:16)
K O F Z K O O .
0p Fq(cid:21)Clp q‘ and 1p Fq(cid:21) F(cid:2)
K
So Gauss, Dirichlet, Kummer, and Dedekind were all actually studying algebraic -theory of number
K O F Z K
fields! We note that the isomorphism 0p Fq(cid:21)Clp q‘ is prettyKobOvious (Osee § 1.1) since 0 is really
a kind of generalization of the class group. On the other hand, 1p Fq (cid:21) F(cid:2) is a nontrivial theorem
due to Bass, Milnor, and Serre (see § 1.2).
K K O ,K O ,K O ,... O
As for the higher -groups 2p Fq F3p Fq O4p Fq for F, one can think of them asKof some
analogues of the two basic invariants Cl and F(cid:2). The first important result about higher -groups
O p qK O
of F, due to Quillen [Qui73a], is that all n F are finitely generated abelian groups. Next it is
p Kq O
natKuraOl to asrk abrout their ranks. Of course rk 0p Fq (cid:16) 1 (by finiteness of the class group) and
rk 1p Fq(cid:16)K1(cid:0)O 2(cid:1)1(byDirichlet). TheoKtherOranksaremuchhardertoget. ItisaresultofGarland
[Gar71] that 2p Fq is a finite group, i.e. rk 2p Fq(cid:16)K0. TOhis was generalized by Armand Borerl [Bor7r4]
whose intricate calculation tells that the ranks of rk np Fq are periodic, depending only on 1 and 2.
Putting together the results of Dirichlet, Garland, and Borel, we have
, n ,
1 0
r r , n(cid:16) ,
K O $ 1, (cid:0) 2(cid:1)1 n(cid:16)1i, i
rk np Fq(cid:16)’’’’& r0r1,(cid:0)r2, nn(cid:16)(cid:16)42ii(cid:0)1¡,, 0ii¡0,.
IfOwe recall the Dirichlet’s theorem p’’’’%roof2[Neu99, §I.7],rf(cid:16)or4Kr(cid:1)1pO1Fq(cid:21)¡O0F(cid:2) it is not very difficult to see
tKhat F(cid:2) is finitely generated, but getting the exact raKnkO1 (cid:0) 2 (cid:1)1 requires more work. For higher
-groups this is similar: it is a very nice result that n F are finitely generated, but calculating the
p q
ranks is much harder. A detailed exposition of this is the main point of this mémoire.
F
As we promised, this is related to the zeta function of ; we note that these ranks are exactly the
ζ n
multiplicities of zeros F :
p(cid:1) q
n
: 0 1 2 3 4 5 6 7 8 9
K O r r r r r r r r (cid:4)(cid:4)(cid:4)
rk np Fq: 1 1(cid:0) 2µ(cid:1)1 0 2µ 0 1(cid:0)µ2 0 2µ 0 1(cid:0)µ2 (cid:4)(cid:4)(cid:4)
(cid:16) 0 (cid:16) 1 (cid:16) 2 (cid:16) 3 (cid:16) 4
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Bottperiodicity
Tointroducemoreintriguingnumerology,werecallthat givesushomotopygroups
of the infinite orthogonal group O R def limOn R (cf. [Bot70]). They are periodic with period eight:
p q (cid:16) p q
(cid:221)(cid:209)
n
: 0 1 2 3 4 5 6 7
π O R Z Z Z Z
n : 2 2 0 0 0 0
p p qq { {
π O R Q
If we are interested only in rational homotopy, then n is periodic with period four. The
K O p p qqb
same period in -groups of F has the same nature. This will pop up during the calculation (§ 4.6).
Often one is interested in the ring of S-integers OF,S for S a finite set of primes in OF. In this case
K
-groups have the same rank, and they are finitely generated as well:
K O ,
rk 0p F,Sq(cid:16)1
K O O S r r ,
rk 1p F,Sq(cid:16)rk F(cid:2),S (cid:16)| |(cid:0) 1(cid:0) 2(cid:1)1
K O K O . n
rk n F,S rk n F 2
p q(cid:16) p q p ¥ q
—this is an easy consequence of the so-called “localization exact sequence”, as will be explained in
corollary 2.5.7. It was also established by Borel in [Bor81] using different arguments.
F
Similarly, if we take the algebraic number field itself, then
K F Z,
0p q(cid:21)
K F F ,
(cid:2)
1p q(cid:21)
K F Q K O Q. n
n Z n F Z 2
p qb (cid:21) p qb p ¥ q
K F Q K O Q
In this case, however, the groups are not finitely generated: while n Z n F Z , there
K F K Q p qb (cid:21) p qb
may be infinite torsion in np q. E.g. this is obvious already for 1p q, and the infinite torsion
K Q Z Z Z Z Z Z Z Z Z
2p q(cid:21) {2‘p {3 q(cid:2)‘p {5 q(cid:2)‘p {7 q(cid:2)‘p {11 q(cid:2)‘(cid:4)(cid:4)(cid:4)
has interesting arithmetic meaning, cf. [Mil71, §11] and [BT73].
K O F
Thetorsionin -groupsof F or isveryimportantforarithmetic,butitwillnotbedealthere. We
refer to surveys [Wei05], [Kah05], and [Gon05] for the general picture. The rest of this text examines
K O
just ranks of n F . Here is a brief outline of the text.
p q
K R K R K R
• Chapter 1 introduces the groups 0p q, 1p q, and 2p q.
K plus-construction
• Chapter 2 defines higher -groups of rings via the so-called . We also collect
some facts from Quillen’s papers [Qui73b] and [Qui73a].
• Chapter 3 reviews some rational homotopy theory and shows that in order to calculate ranks of
K O H SL O ,R
n F , it is enough to know the cohomology ring (cid:13) F .
p q p p q q
K O
• Chapter 4 finally gets the ranks of n F , assuming certain difficult and technical result about
p q
stable cohomology of arithmetic groups.
The rest is devoted to certain steps in the direction of that “technical result”. One who is interested
K O
only in the general strategy of computing rk n F may content themselves with chapters 1–4.
p q
Matsushima’sconstant
• Chapter5examinesatheoremofMatsushimathatinvolvestheso-called
m G R
that is very important for stable cohomology.
p p qq
• Chapter 6 proves certain variation of Matsushima’s result, due to Garland.
viii
I tried to make the exposition as much coherent and self-contained as possible. I did my best to
give motivation and explain used facts, reviewing the proofs—when they are instructive and not too
technical—or providing the references. Certain constructions are both very interesting and hard to
take on hearsay, so I included a long discussion of them. The tools that one would consider standard
are included in the appendices. They serve to fix definitions and notation, and summarize some basic
Q
facts to be used in the main text. The additional appendix Q outlines Quillen’s -construction, which
is not crucial for the main text, although at some point we should assume results that are normally
proved using that.
Some notation
Let us fix some notation for all the subsequent chapters:
F
• is a number field.
O F
• F is the ring of integers in .
µ F
• F denotes the group of roots of unity in .
r
• 1 is the number of real places.
r
• 2 is the number of complex places.
ddef F Q r r F
• (cid:16) r : s(cid:16) 1(cid:0)2 2 is the degree of .
F
• ∆F is the discriminant of .
G,H,K
Letters like will often denote Lie groups, and the corresponding Lie algebras are written in
g,h,k
the Fraktur script like .
(cid:4)
As usual, the end of a proof is denoted by a tombstone sign ; when there is no proof, I mark it
(cid:78)
with (unless it is something really well-known). End of an example is marked with .
(cid:47)
References
The primary sources that I used writing this text worth a separate mention: the original Borel’s article
is [Bor74], and there are also some surveys written by Borel himself, notably [Bor06], [Bor95], and a
monograph [BW00] by Borel and Wallach.
K
I hope this text will be useful for someone who wants to learn about algebraic -theory of number
fields.
A note about this version
K O
Myintentionwastocoverallthedetailsandpreliminariesneededtocalculaterk n F . Atsomepoint
thpe tqechnical result
the text became quite long, so I took decision to explain only first steps towards
(theorem 4.7.2), to avoid making all fifty pages longer. Understanding nuts and bolts of Borel’s proofs
is a starting point of my future PhD project suggested by Boas Erez, so I will soon post online a more
detailed and lengthy version of these notes (it more resembles a book than a mémoire!).
[email protected]
Please send all your comments to .
ix
Acknowledgments
I thank Yuri Bilu and Giuseppe Molteni who taught me arithmetic, and Boas Erez who further taught
me its cohomological aspects. It was Prof. Erez who suggested the topic, which turned out to be very
exciting, and I have learned lots of things while working on this mémoire. Nicola Mazzari and Dajano
Tossici were very kind to read a preliminary version of the text and give their valuable advices.
Finally, I thank all my ALGANT professors and classmates. The program has given me and many
other students a priceless opportunity to concentrate on the most interesting thing ever: exploring
mathematics.
Voglio ringraziare i miei compagni di corso italiani: Andrea Gagna, Pietro Gatti, Roberto Gualdi,
Dino Destefano, Mauro Mantegazza ed Alessandro Pezzoni. Un paio d’anni fa quando arrivai a Milano,
non immaginavo che io avrei potuto parlare e pensare di matematica in italiano in così poco tempo.
Ringrazio soprattutto Francesco e tutta la sua grande famiglia salentina; Majd, Maurizio, Anna e tutti
gli altri amici che non essendo matematici, forse non apprezzerebbero questa tesi, ma mi insegnarono
lo stesso molte cose importanti.
TambiénagradezcoparticularmenteaJoséIbrahimVillanuevaGutiérrez, nosólouncolega, sinoun
buen amigo y una gran persona.
— Ale
x