Grothendieck Rings and Motivic Integration

113 Pages·2009·0.785 MB·English

by  Karl Rökaeus

Grothendieck Rings and Motivic Integration

Grothendieck Rings and Motivic
Integration
Karl R(cid:246)kaeus
(cid:13)c KarlR(cid:246)kaeus,Stockholm2009
ISBN978-91-7155-834-3
PrintedinSwedenbyUS-AB,Stockholm2009
Distributor:StockholmUniversity,FacultyofScience,DepartmentofMathematics,10691Stockholm
Abstract
This thesis consists of three parts:
In Part I we study the Burnside ring of the (cid:28)nite group G. This ring
has a natural structure of a λ-ring, {λn}n∈N. However, a priori λn(S),
where S is a G-set, can only be computed recursively, by (cid:28)rst computing
λ1(S),...,λn−1(S). We establish an explicit formula, expressing λn(S)
as a linear combination of classes of G-sets. This formula is derived in
two ways: First we give a proof that uses the theory of representation
rings in an essential way. We then give an alternative, more intrinsic,
proof. This second proof is joint work with Serge Bouc.
In Part II we establish a formula for the classes of certain tori in the
Grothendieck ring of varieties K (Var ). More explicitly, K (Var ) has
0 k 0 k
a natural structure of a λ-ring, and we will see that if L∗ is the torus
of invertible elements in the n-dimensional separable k-algebra L then
[L∗] = Pn (−1)iλi(cid:0)[SpecL](cid:1)Ln−i, where L is the class of the a(cid:30)ne
i=0
line. This formula is suggested by the computation of the cohomology
of the torus. To prove it requires some rather explicit calculations in
K (Var ).Tobeabletomakethese,weintroduceahomomorphismfrom
0 k
the Burnside ring of the absolute Galois group of k, to K (Var ). In the
0 k
process we obtain some information about the structure of the subring
of K (Var ) generated by zero-dimensional varieties.
0 k
In Part III we give a version of geometric motivic integration that
specializes to p-adic integration via point counting. This has been done
before for stable sets, cf. [LS03]; we extend this to more general sets.
The main problem in doing this is that it requires to take limits, hence
the measure will have to take values in a completion of K (Var )[L−1].
0 k
The standard choice is to complete with respect to the dimension (cid:28)ltra-
tion.However,sincethepointcountinghomomorphismisnotcontinuous
with respect to this topology we have to use a stronger one. We thus be-
gin by de(cid:28)ning this stronger topology; we will then see that many of
the standard constructions of geometric motivic integration work also
in this setting. Using this theory, we are then able to give a geometric
explanation of the behavior of certain p-adic integrals, by computing the
corresponding motivic integrals.
5

Contents
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
0.1.1 Overviewofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
0.1.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
0.2 Backgroundmaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
0.2.1 λ-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0.2.2 Representationrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0.2.3 TheGrothendieckringofvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
0.2.4 Burnsiderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Part I: The Burnside ring
1 The λ-structure on the Burnside ring . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.1 Introductionandstatementofthemaintheorem. . . . . . . . . . . . . . . . . . . . . . 21
1.2 ThemapfromtheBurnsideringtotherepresentationring . . . . . . . . . . . . . . . 23
1.3 TheSchursubringofB(Σn). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4 Theλ-operationsonB(Σn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Intrinsic proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 BackgroundMaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 ProofofTheorem1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Part II: The Grothendieck ring of varieties
3 The Grothendieck ring of varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 BackgroundMaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 ThesubringofArtinclassesinK0(Vark) . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Heuristicconsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 ComputationoftheclassofL∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6 Theclassofthetorusintermsoftheλ-operations . . . . . . . . . . . . . . . . . . . . 61
Part III: Motivic Integration
4 Motivic integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 ThecompletionofMk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 AppendixtoSection4.2:Resultsfrom[Eke09] . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 De(cid:28)nitionofthemotivicmeasure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Computing motivic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 BackgroundmaterialabouttheWittvectors . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Assumptionsandgeneralresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4 Changeofvariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.5 Theintegraloftheabsolutevalueofaproductofspeciallinearforms . . . . . . . . 98
5.6 Theintegraloftheabsolutevalueofaproductofarbitrarylinearforms. . . . . . . 104
5.7 Varyingtheprime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Part IV: Bibliography
6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
8
0.1 Introduction
This thesis is concerned with two main topics. Firstly, the investigation
of two Grothendieck rings: the Burnside ring, and the Grothendieck ring
of varieties. Secondly, the theory of motivic integration. These topics are
related by the fact that the motivic measure takes values in a certain
completed localization of the Grothendieck ring of varieties. Let us (cid:28)rst
say some words about that ring:
Let k be a (cid:28)eld and let Var be the category of k-varieties, by
k
which we mean the category of separated k-schemes of (cid:28)nite type.
The Grothendieck ring of k-varieties1, K (Var ), is by de(cid:28)nition the
0 k
free abelian group on the set of isomorphism classes of k-varieties [X],
modulo the relations [X] = [Z]+[X \Z] if Z ⊂ X is closed (the scissor
relations), and with a multiplication given by [X][Y] = [X × Y].
k
The class of the a(cid:30)ne line is of particular importance, and is given a
special symbol: L := [A1] ∈ K (Var ) (L is for Lefschetz). It follows
k 0 k
immediately from the de(cid:28)nition of the multiplication that [An] = Ln.
k
For an example of how the scissor relations work, note that since we
may choose a closed subscheme of Pn which is isomorphic to Pn−1, and
k k
has complement An, we have [Pn] = Ln+Ln−1+···+L+1 ∈ K (Var ).
k k 0 k
We want to use the theory of motivic integration in order to give
a geometric explanation of the behavior of certain p-adic integrals. As
an introduction to these ideas, let us illustrate how K0(VarFp) gives a
geometric way of counting solutions to polynomial equations modulo p.
Wedothiswithanexample:ForLaseparablek-algebra,de(cid:28)neL∗ tobe
the algebraic group of invertible elements of L, i.e., for every k-algebra
R,L∗(R) = (L⊗ R)×.IncaseL/k = F /F wehave|L∗(F )| = |F×| =
k p2 p p p2
p2 −1 and |L∗(F )| = |(F2 )×| = (p2 −1)2 = (p2)2 −2p2 +1. On the
p2 p2
other hand, computing in K0(VarFp), one may show that
[L∗] = L2−[SpecL]L+[SpecL]−1 ∈ K0(VarFp). (0.1)
Bydoingthis,wehavesimultaneouslycomputed|L∗(F )|foreverypower
q
q ofp.Becauseforeverysuchq wehaveapointcountinghomomorphism
Cq: K0(VarFp) → Z, induced by [X] 7→ |X(Fq)|. So by applying Cq to
[L∗] we obtain the number of F -points on L∗. In particular, using (0.1)
q
we again get |L∗(F )| = C ([L∗]) = p2 −1 and |L∗(F )| = C ([L∗]) =
p p p2 p2
(p2)2−2p2+1.
Beforewecontinue,letusmentionthatthisexampleisaspecialcaseof
a more general theorem. Namely, for any (cid:28)eld k, K (Var ) has a natural
0 k
structure of a λ-ring (a λ-ring is a ring together with a set of maps λn
behavinglikeexteriorpowers,seeSection0.2.1.),andusingthisstructure
1First introduced by Grothendieck, cf. the unpublished text Motifs, available at
www.grothendieckcircle.org
9
one may, for any separable n-dimensional k-algebra L, express [L∗] as
n
X
[L∗] = (−1)iλi([SpecL])Ln−i ∈ K (Var ). (0.2)
0 k
i=0
The validity of this formula will be the main result of Chapter 3. In
establishing it, we will be led to consider the structure of K (Var ) in
0 k
more detail, and this leads us to the subject of the Burnside ring.
TheBurnsideringofapro(cid:28)nitegroupG,denotedB(G),isconstructed
inmuchthesamewayasK (Var ),butinsteadofthecategoryofvarieties
0 k
one uses the category of (cid:28)nite, continuous G-sets. Let G be the absolute
Galois group of k. There is a natural map Art : B(G) → K (Var ), and
k 0 k
since the structure of the Burnside ring is much better known than that
of K (Var ), Art is useful for proving structure result about K (Var ).
0 k k 0 k
For an example of this, we may use Art to prove that Naumann’s con-
k
struction of zero divisors in K0(VarFp) actually works in K0(Vark) when
k is any (cid:28)eld which is not separably closed: Let L/k be a (cid:28)nite Ga-
lois extension of degree n. Then [SpecL]2 = [SpecL⊗ L] = [SpecLn] =
k
(cid:5)
(cid:0) (cid:1)
[∪ SpecL] = n[SpecL],hence[SpecL] [SpecL]−n = 0.Lookingatthe
n
preimage of [SpecL] under Art , it is easy to show that [SpecL] 6= 0,n;
k
consequently it is a zero divisor for any k (see Section 3.3).
Moreover, B(G) also has a natural structure of a λ-ring, and Art
k
commutes with the λ-operations. This allows us to move computations
of the λ-operations on K (Var ) to B(G). Actually, one of the proofs
0 k
of the validity of (0.2) given in Chapter 3 relies on a theorem about
Burnside rings (Theorem 1.1) which is proved in Chapter 1.
We now turn to motivic integration. Motivic integration was intro-
duced by Kontsevich in 1995, in order to strengthen Batyrev’s result
that birational Calabi-Yau manifolds have the same Betti numbers, to
also yield equality of Hodge numbers. It has since then rami(cid:28)ed in many
di(cid:27)erent directions; we are interested in utilizing it to give a geometric
wayofcomputingp-adicintegrals.Infact,motivicintegrationisinspired
by p-adic integration: one wants to de(cid:28)ne a measure on subsets of power
series rings of the type k[[t]], in much the same way as on subsets of
Z . However, since k[[t]] is not locally compact (whenever k is in(cid:28)nite)
p
it is not possible to do this in the classical way. Kontsevich’s method of
resolving this was to let the measure take values in a certain completion
of M := K (Var )[L−1]. This gives the original theory of geometric
k 0 k
motivic integration, developed in [DL99] and [DL02].
This idea may now be used also in the p-adic case: Let W be the ring
scheme of Witt vectors, constructed with respect to the prime p. (Recall
that W(F ), where q = pf, is the integers in the unrami(cid:28)ed degree f
q
extension of Q ; in particular W(F ) = Z .) Let F be an algebraic
p p p p
closure of F . W(F ) then contains all the W(F ), q = pf, as subrings.
p p q
10