Notes on the A.B. Goncharov and Y. Manin article [GM02]
Contents 1 Remind on the category of motives DMgm (F )
1
2 Construction of the motive Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn 2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Some properties of Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 l-adic realisation 3.1 Construction of the l-adic realisation . . . . . . . . . . . . . . 3.2 Unramified motives . . . . . . . . . . . . . . . . . . . . . . . . 4 The motives Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn are defined over 4.1 The smooth proper base change theorem [Mil80] . . . . . . . 4.2 Proof of the themorem 4.1 . . . . . . . . . . . . . . . . . . . .
2 2 5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 8 10
Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 12 12
Remind on the category of motives DMgm (F )
1
Let Smcor(F ) be the category on (proper smooth) scheem over F with the correspondance as morphism. The f b “homological” category DMef gm (F ) is the localisation (makes 0) K (Smcor(F )) with respect to : 1. Homotopy : X × A → X 2. If X = U ∪ V then U ∩ V → U ⊕ V → X f DMef gm (F ) is the category of mixed effective motives over F ( let say a number field) with Q coefficients. f Definition 1.1. For X ∈ Smcor(F ) we write M (X) ∈ DMef gm (F ) for the complex 0 → X → 0 where X is in degree 0.
^ Definition 1.2. We write M (X) for the complex 0 → X → Spec(F ) → 0 with X in degree 0 Remark 1. We have the following : ^ • Z(1) = M (P1 )[−2] (the [−2] is because H2 (P1 ) = Z • M (P1 ) = Z(0) + Z(1) • We also have M (Gm ) = Z(0) + Z(1)[1] • Z(0) is the unit for ⊗ • ∀n ∈ N Z(n) = Z(1)⊗n f • for X ∈ DMef gm (F ), X(n) = X ⊗ Z(n) the twist operation
• there is also a shift operation which is the shifting of the complex C[n] 1I
wrote on my notes 0 → M (X) → M(Sp(F )) → 0
1
1
• Somewhere, here or after the localisation we will have to take the karoubisation of the category. f To be in DMgm (F ), we add new objects : (X, n) with X ∈ DMef gm (F ) and n ∈ Z and Hom((X, n); (Y, m)) = limN →+∞ Hom(X(N + n), Y (N + m))
Definition 1.3. for n ∈ N, Z(−n) = (M (Spec(F )), −n), and Z(−1) is the inverse of Z(1) for ⊗. Proposition 1.1. In DMgm (F ) there is an internal Hom written Hom an a duality2 f Theorem 1.2. The map DMef gm (F ) ,→ DMgm (F ) is fully faithful
Theorem 1.3. DMgm (F ) is a rigid tensor category We will work here with a cohomological version : / DMgm (F )
Smcor(F )
h:
X
/ H(X) = Hom(M (X); Z)
And so we will have to invert the signum of all the twist. Lemma 1.4 (blow up formula). Let Z ,→ X be a closed subscheme of codim d, and XZ the blow up of X on Z and E be the execptional divisor : / XZ E Z
/X
Then : M (E) =
d−1 M
M (Z)(i)[2i]
M (XZ ) = M (X)
and
d−1 M
M (Z)(i)[2i]3
i=1
i=1
and the dual version : H(E) =
d M
H(Z)(−i)[−2i]
and
H(XZ ) = H(X)
d M
H(Z)(−i)[−2i]
i=0
i=1
Construction of the motive Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn
2 2.1
Construction
Relative cohomology Let X be a smooth scheme over F and let B ⊂ X be a normal crossing divisor (NCD) in X we want to construct : ; H(X; B) “ = “ Hom(M (X/B); Z) | {z } ???
B =
n G
Bi . For I ⊂ {1, ..., n}, we define BI =
i=1
\
Bi .The inclusion maps leads to correspondance ϕI,j : BI →
i∈I
BI\{j} . So we can define the corespondance BI →
\
BI\{k} by
P (−1)k ϕI,k and we have a complex in Kb (Smcor(F ))
k
: · · · −→
M
BI −→ −k
2 How 3 To
n M
Bi −→
X
−1
0
i=1
I ⊂ {1, ..., n} |I| = k
degree
· · · −→
is the interna Hom define check
2
So there is a complex of object in DMgm (F ) : M
· · · −→
M (BI ) −→
n M
· · · −→
M (Bi ) −→ M (X)
i=1
I ⊂ {1, ..., n} |I| = k
Definition 2.1. This complex gives (using the associate total complex (or a sort of) and some work) an object in DMgm (F ). This object is named M (X/B). Remark 2. Taking the total complex (or a sort of) to get an object in DMgm (F ) is in some way what we do when we compute the relative homology (or the cohomoly) with spectral sequences. Example 2.1. For now the example just show that I didn’t understand what [ “some work” means and it is in contradiction with what I have just wrote. I copy my notes. If B = B1 B2 , we have a map in DMgm (F ), M (X/B) → M (B12 )[2] : ((B1 + B2 → X) →)
M (X/B) → M (B12 )[2]
M (X/B)
/ M (B12 )[2]
XO
/0 O
B1 +O B2
/0 O
B12
/ B12
Likewise we have : M (X) → (B1 + B2 → X) → B1 [1] + B2 [2] | {z } ??
Remark 3. Using the cohomolical theory the object H(X; B) will be given by the complexe: H(X) −→
n M
H(Bi ) −→
i=1
degree Gysin maps :
0
M
· · · −→
H(BI ) −→
···
I ⊂ {1, ..., n} |I| = k
1
k
0
For p : Z → X proper of relative dimension d on X smooth, there is a cohomologycal map in DMgm (F ) p∗ : H(Z)(d)[2d] −→ H(X)
Coming from the homolical version : p∗ : M (X) −→ M (Z)(−d)[−2d] How it works (at least for Z smooth closed subscheme of X) • i : Z ,→ X the inclusion of Z smooth closed subcheme of codimension d of X smooth. • j : X \ Z = U ,→ X
;
M (U ) → M (X) := M (X/U ) | {z } in deg 0
Theorem 2.1. i : Z ,→ X, Z closed smooth subscheme of codimension d and U = X \ Z. There is a canonical map : M (X/U ) ' M (Z)(d)[2d]
3
Remark 4. We have a map ˜j : M (X) → M (X/U ) using4 :
0
/ M (X)
M (U )
/ M (X)
M (X) : ˜ j
M (X/U ) : Theorem 2.2. Let i : Z ,→ X like before, there is a canonical map : ˜ j
M (X)
/ M (X/(X \ Z))
∼
/ M (Z)(d)[−2d] 6
i∗
Using the cohomological version (Hom) we have : H(Z)(−d)[−2d] −→ H(X) and the Gysin distinguish triangle H(Z)(−d)[−2d] → H(X) → H(U ) → H(Z)(−d)[−2d + 1] Some ideas of how it could work : Some pictures and diagram ... X
X
Z•
Z×A1
• Z×1
Z•
A1
O
X ×A
Z×A1
O
1
99 99 9 999 99 99 99 99 99 A1 9
˜ Z
Y = BlZ×1 (X × A1 )
isomorphism 1 M (Y / (Y \ Z × A ) after lidem W W hRRR 6 W W+ l l localisation RRR lll l l v l ∼ l RR∼R RRR lll RRR lll l l l ∼ M (U/(U \ Z)) M (X/(X \ Z))
M (Z)(d)[2d] M (U )(d)[2d] / M C˘ Ui /(Ui \ Zi )
Need to ask fr some explanation about this thing a Proposition 2.3. If X is smooth, and projective, and A = Ai is a NCD in X we want to define a motive which correspond to X \ A. Using the standard notations : L L · · · / |I|=d H(AI ) / ··· / / H(X) i H(Ai )
∼
in
DMgm (F )
H(X \ A) Proof. Is the object H(X \ A) in DMgm (F ) defined as the total complex of the double complex ∗∗ or should we show with distinguish triangle and others thing that it works ? 4 To
tell the truth I don’t know where this morphism comes from
4
Notation. Let A and B be two normal crossing divisors in X smooth, projective. We say, using standard notation, \ that A + B is a NCD if and only if for all "stratum" AI of A AI B (∩BJ ) is a normal crossing divisor in B (BJ ), and likewise for BJ strata of B. Proposition 2.4. Let A, B be NCD in X smooth and projective of dimension n. Suppose that A + B is a NCD. We have a ”double complex” which defines an object in H(X \ A; B) ∈ DMgm (F ). Here AI,J = AI ∩ BJ : M
H(AI )(−n)[−2n]
|I|=n
M
H(AI,j )(−n)[−2n]
j,|I|=n
···
M
/
/
···
/ · · · /
|J|=p
|I|=n
H(AI,j )(−q)[−2q]
j,|I|=q
M
/ · · · /
/
···
/ · · · /
/ · · ·
H(AI,J )(−n)[−2n]
/ · · · /
M |J|=n
M
j,i
/ · · ·
/ · · · /
/ · · ·
|I|=q
/
H(AI,J )(−1)[−2]
|J|=p,i
/ · · ·
/ · · ·
/
M
H(AI,J )(−1)[−2]
|J|=n,i
9 > = H(X) > ;
L
j
9 > = H(Bj ) > ;
/ H(X \ A)
∼
/
∼
M
H(Bj \ A)
j
/ · · ·
M
H(AI,J )(−q)[−2q]
/
H(AI,j )(−1)[−2]
H(AI,J )(−q)[−2q]
|J|=p
/ · · ·
/
H(Ai )(−1)[−2]
/ · · ·
|I|=q
···
|J|=n
M
/
M i
/ · · ·
|I|=n
M
H(AI )(−q)[−2q]
|I|=q
/ · · ·
H(AI,J )(−n)[−2n]
M
9 > > > = H(BJ ) ∼ > |J|=p > > ; M
/
/ · · ·
/
···
M
H(BJ \ A)
|J|=p
···
9 > > M > = H(BJ ) ∼ / H(BJ \ A) > |J|=n |J|=n > > ; | {z } M
∼
H(X \ A; B) (2.1.1)
Proof. Up to some work and distinguish triangle we have an objet in DMgm (F ) Proposition 2.5. Let k = (k1 , ..., kn ) be a sequence of positive integers with k1 > 2, n = k1 + ...kn . Let Ak be the divisor of singularities of the differential form on M0,n+3 associated to the multi-zetavalue ζ(k1 , ..., kn ). Let Bn be the bondary of the standard cell on M0,n . The motive Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn is weel defined. 5 Proof. The description of the bondary of M0,r (∀r) given in [DM69] or [BFLS99], shows that Ak + Bn is a NCD. Then the previous proposition with X = M0,n+3 , A = Ak and B = Bn gives that H(M0,n+3 \ Ak ; Bn \ Ak ∩ Bn ) is a well defined. Then we just take the Hn of the t-structure
2.2
Some properties of Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn
Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn is a Tate motive Proposition 2.6. Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn is in MT (Q) Proof. In order to show that Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn is Tate, it is enought to show H(M0,n+3 \ Ak ; Bn \ Ak ∩ Bn ) is in DMT (Q) = Db (MT ) The previous construction of H(M0,n+3 \ Ak ; Bn \ Ak ∩ Bn ) (2.1.1), shows that it is 5 Can we take the Hn (−) of the t-structure before knowing the motive is tate. Maybe the proposition should be "H(−) is well defined as an object of DMgm (F )"
5
built with extentions of the little pieces and so it is enough to proove that those former are Tate. As the (AI
\
BJ
are product of moduli spaces in genus 0, using the kunneth formula, we just have to show the H(M0,r ) are Tate. H(P1 ) = Q(0) + Q(1) is Tate. And then (P1 )r is Tate. Using the blow up formula 1.4 First proof We have the standard map M0,r → (P1 )r . We obtain M0,r blowing up some subspace Zi of (P1 )r . Those subspaces are isomorphics to (P1 )s for differents s 6 r − 1. Using the lemma, we have that M0,r is Tate. Second proof We can also use the description of the bondary of M0,r given by Keel in [Kee92] and use induction. Using the map M0,r → M0,r−1 × P1 , we see that from M0,r−1 × P1 we have to blow up some M0,s with s 6 r − 1 and some "diagonals" which are also Tate. M0,r
/ M0,r−1 × P1 JJJ 7 nnn JJ nnnσi % M0,r−1
The GrW n n 1. GrW M0,n+3 \ Ak ; Bn \ Ak ∩ Bn ' GrW 2n H 2n H (M0 , n + 3 \ Ak ) 2. GrW Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn ' GrW Hn (M0 , n + 3; Bn ) 0 0 n M0,n+3 \ Ak ; Bn \ Ak ∩ Bn are vector spaces 3. The GrW Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn and the GrW 0 2n H of dimension 1
Proposition 2.7.
Proof. Remark 5. n GrW 2n H
k GrW H (Z(a)[b] = 0 unless a = −l and b = −k 2l n M0,n+3 \ Ak ; Bn \ Ak ∩ Bn ' GrW 2n H (M0 , n + 3 \ Ak )
We have the (long) exact sequence : ` n−1 ´ (Bn \ Ak ) GrW 2n H
` n` ´´ / GrW M0,n+3 \ Ak ; Bn \ Ak ∩ Bn 2n H
` n ´ / GrW 2n H (M0 , n + 3 \ Ak )
n / GrW 2n (H (Bn \ Ak ))
We would like to show that H(Bn \ Ak ) as no GrW 2n We want to proove that all the GrW 2l (H(BJ \ A)) = 0 ifl > dim(BJ ) for all the J cf 2.1.1 We proove by induction on the dimension for all pair (A, B) of strata in M0,r sucht that A and B have no common component that GrW 2l (H(B \ A)) = 0 if l > dim(B). We assume first, it is true if A = ∅ and the GrW 2l (H(B \ A)) comes from the Gysin maps we have already seen. It is enough to consider one of them coming from : \ AI B ,→ B 6 with AI
\
B 7 is of codimesion d. This map gives : H(AI,BJ )(−d)[−2d] → H(BJ )8
We see that may be something is coming from H(AI,BJ )(−d) but as dim(AI ) + d = dim(BJ ), for m > dim(B) we have W W GrW 2m (H(B)) = 0. Gr2l (H(AI )) = 0 for all l > dim(AI ) and as m > dim AI + d, we have Gr2m (H(AI,B ))(−d) = 0 6A BJ ,→ B??? I 7 B ?? and in the next sentence B instead of B J J 8 H(Z)(−d)[−2d] → H(X) → H(U ) → H(Z)(−d)[−2d
T
+ 1] ??
6
We still have to proove that GrW (we can proove it using 2l (H(B)) = 0 for l > dim(B).QIt is true if B = M0,r+3 P induction or the standard map M0,r+3 → (P1 )r ). In general B = M0,ri +3 with ri > 0 and ri = dim(B), and the result follows. So we have : n n M0,n+3 \ Ak ; Bn \ Ak ∩ Bn ' GrW GrW 2n H (M0 , n + 3 \ Ak ) 2n H
n =1 dim GrW 2n H (M0 , n + 3 \ Ak )
We are looking at the first line of 2.1.1 : −n M
H(AI )(−n)[−2n]
−1 / ···
/
|I|=n
M
/ ···
H(AI )(−q)[−2q]
|I|=q
/
M
H(Ai )(−1)[−2]
i
0 / H(X)
H(X \ A) O O L Z(−n)[−2n]
in the different degree in the H Hp+2q−q=p+q of weight 2n of weight 2(n−q)+2q = 2n?? M H(AI )(−n)[−2n] and is a one dimensional Qvector space Lemma 2.8. The only GrW 2n comes from this L
Z
2n−n=n
|I|=n
n Proof. To compute the GrW 2n H (M0 , n + 3 \ Ak ) we are in the category of Tate mixed Hodge structure (TMHS), and using a spectral sequence which degenerate in E2 9 (the d2 are zero): E1pq = Hq (deg p) ⇒ Hp+q (M0 , n + 3 \ Ak ) Then, writting it explicitly, we have : M E1pq = Hq+2p (AI )(−p)
the + 2p come from the shift [−2p]
|I|=p
So if we are looking the GrW 2n , with p < 0: M pq p+q q+2p (M0 , n + 3 \ Ak ) (AI ) ⇒ GrW GrW GrW 2n H 2n (E1 ) = 2n+2p H |I|=−p
The Hn ( − ) of the moduli space of curve are pures of weight n and so are the stratum of the boundary of M0,r q+2p (kunneth formula). We are looking at the Hn , so me must have p + q = n. The GrW H (A ) = 0 unless I 2n+2p 2n + 2p = q + 2p and with the condition p + q = n, we found that : n + p = 0 and q = 2n. We have a short exact sequence : ` ´ ` ´ L L ` n ´ / GrW / |I|=n H0 (AI ) / |I|=n−1 H2 (AI ) /0 0 2n H (M0 , n + 3 \ Ak ) The result follows from the fact that if D is a NCD, it has the same combinatoric as a sphere. : p+q / ⊕ H0 (AI ) / ⊕ H2 (AI ) /0 (−) 0 / GrW 2n H 0-skeleton •?? ? 9 degenerate
in E1 ??
7
1-skeleton +•
• •??+ ?
−
And so, n GrW 2n H (M0 , n + 3 \ Ak ) ' Q(−n)
But the isomorphism is non canonical. The form ωk which gives the divisor Ak , will provide a particular one. But we have to check that ωk is rational. R Let Ω be the "same" form on (P1 )n , we can see that Π Ω where Π is a certain product of circle around 0 or 1 10 is (2iπ)n and pulling back this domain on p : M0,n+3 with (M0,n+3 → (P1 )n , we have : Z Z (2iπ)n = Ω= ωk p∗ (Π)
Pi
we have used the following where A0 is the divisor of singularities of Ω and A0 = p−1 (A0 ) : H((P1 )n \ A0 ) ∗
H(M0,n+3 \ Ak )
j∗
p / H(M0,n+3 \ A0 )
3Ω p∗ (Ω) =
ΠO Π0 o
Π0
∗
j (ωk )
ωk
GrW Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn 0 : working the same way .
' GrW Hn (M0 , n + 3; Bn ) and it is a one dimensional vector space 0
Moreover the form ωk provides a canonical lifting for the framing ...
3 3.1
l-adic realisation Construction of the l-adic realisation
Some definitions and basic properties
Let X be a scheme.
Definition 3.1 (Etale site or Small etale site (cf. [Mil80])). The site Xet has as underlying category Et /X, whose objects are the etale morphisms U → X and whose arrows are the X-morphisms φ : U → V . The coverings are the surjective families of etale morphisms (Ui → U ) in Et /X. Definition 3.2 (Big etale site on X). The site XEt has as underlying category Sch/X . The coverings are the surjective families of etale X-morphisms (U i → U )i∈I . Definition 3.3 (Etale presheaf). Presheaf over Xet Definition 3.4 (Etale presheaf with transfert). Prasheaf over the site which has for underlying category EtCor /X whose objects are etale morphisms U → X and whose arrow are correspondances. The covering are the same as in Xet Definition 3.5 (Direct image). Let π : Y → X be a morphism of schemes, and let P be a presheaf on Yet . For U → X etale, define π∗ P(U ) = P(U ×X Y ). Since U ×X Y → Y is etale this definition makes sense. With the obvious restriction maps, π∗ P becomes a presheaf on Xet . 10 Iwrote
Π=
Q
S 1 but I think that in this case the integral is zero or divergent
8
Definition 3.6 (Inverse image). Let π : Y → X be a morphism of schemes. We shall define a left adjoint for the functor π∗ . Let P be a presheaf on Xet . For V → Y etale, define P ∗ (V ) = lim→ P(U ) where the direct limit is over the commutative diagrams : /U /X
V Y
with U → X etale. If P is a sheaf, we define π ∗ P to be the sheaf associate to the preshaef P 0 Diagramm path to construct the l-adic realisation [Ivo05] Let l be a prime number and S a scheme 11 . L will be in that paragraph the category of projective system over Z/l∗ , where Z/l∗ is the projective system of constant sheaf over S : Z/l∗ : Z/lZ ← Z/l2 Z ← · · · ← Z/lr Z ← · · · Z/lr Z is the etal constant sheaf over S We want to built a functor from the category SmCor/S to the category of complexe bounded above of L-object an then exentend it. First we need a functor from the category of scheme over S to C + (L) : / C b (SmCor/S ) / SmCor/S q m q q mmm vmmm xqqqF tot / + C + (L) / C b (C + (L)) C (L) Sm/S
/ Kb (SmCor/S )
/ Db (SmCor/S )
/ DMef f (F ) gm
/ Kb (L)
/ Db (L)
/ Db (L)
A key point is ti construct the map SmCor/S → C. A large part of the other maps are categorical machinery. In fact f b the DMef gm (F ) maps onto a subcategory of D (L) which objects ( the complex of projectives systems) are "isomorphic" to the complex of the projective limit of each system. Then we take in both side the H0 for the t-structure. F will be given by the Godement resolution. The constant sheaf with transfert Z/lr Z First we have to remark that the constant sheaf on S (or on X, a S scheme) Z/lr Z is a sheaf with transfert. If α ∈ Corr(X, Y ), α∗ will be the multiplication by the degree of α ∈ X × Y . Godement resolution of the sheaf with transfert Let X be an etale sheaf over S, we have a functorial map geometric i : Xet → Xdis = { X}. So we have two functors i∗ and i∗ . If F is an etale sheaf over X, we have in points of Q particular i∗ : F → x Fx . X ∗ We have a natural morphism η : id → i∗ i∗ and µ : i∗ i∗ → id. We write GN is = i∗ i . X ∗ X This functor GN is = i∗ i is a monade ([Ivo05][p.143]) and we can build a cosimplicial object B(GN is , F) were the X n+1 n-cosimplexe are given by GN is (F), the codegeneracy maps are : i
X X σin := GN is µGN is
The cofaces are :
i
n−i
X X δin := GN is ηGN is
X : GN is
n−i
n+1
n
X (F) → GN is (F)
n
X X : GN is (F) → GN is
n+1
(F)
i = 0, ...n − 1 i = 0, ...n
X This co-simplicial object leads to a complex C(GN is (F )). We want to proove that if F is a sheaf with transfert, then X C(GN (F)) will be a complex of sheafs with transfert. To proove that it is enought to proove that if F is a sheaf with is transfert, i∗ i∗ F is a sheaf with transfert.
Proposition 3.1. Let F be a Nisnevich sheaf with transfert over X then : Y Y X GN Fx = F(Xxh ) is (F)(X) = x
where
Xxh
11 may-be
is the henselisation of the local ring at the point x. Spec(K) where K is an algebraic closed field
9
x
Proposition 3.2. Let F be a Nisnevich sheaf with transfert over S. If to a S-scheme X we associate : X GN is (F)(X) = GN is (F)(X)
we have a presheaf. Moreover GN is (F) is a Nisnevich sheafs on S with transfert X Proof. The restriction of GN is (F) to the small Ninevich site of X is GN is (F), and so GN is (F) is a Nisnevich sheaf.
12
13
We want to proove that GN is (F) is a sheaf with transfert. Let α be aY correspondance between X and Y . We Y want to define α∗ : GN is (F)(Y ) → GN is (F)(X),and as : GN is (F)(Y ) = Fy = F(Yyh ) and GN is (F)(X) = Y
Fx =
x∈X
Y
F(Xxh ).
So as F as transfert we have some α( x, y) :
y∈Y h F(Yy ) →
y∈Y h F(Xx ) if we
can associate to alpha a
x∈X
correspondance from Xxh → Yyh . To define those αx,y we look at localy the correspondance α. Over x we have finitely many yi such that (x, yi ) ∈ alpha, so restricting to the "neighbourhood" Xxh containing x and to the Yyhi we have well define correspondance αx,yi . If y is distinct from the yi , we define αx,y to be 0.
y3 • y2 •
Y
y1 • /o o/ • /o /o /o x
X
This construction hold for all x in X and we can define α∗ : ! X ∗ (sy )y∈Y 7→ (α (sy ))x x∈X = αx,y (sy ) y
3.2
x∈X
=
X
αx,y (sy )
y∈α−1 (x)
x∈X
Unramified motives
Definition 3.7. For Γ a subgroup of Ext1 (Q(0), Q(1)), MTΓ (F )is the subcategorie of MT(F ) such that M is in MTΓ (F ) , if and only if for all subquotients E of M which are an extension of Q(n) by Q(n + 1), the class of E in ∼
Ext1 (Q(0), Q(1) → Ext1 (Q(n), Q(n + 1)) is in Γ. Remark 6. This subcategory is stable by subquotient, tensor product and dual, it is a tannakienne subcategory of MT(F ) Definition 3.8. If F is a number field and v is a finite place of F , then let Ov be the localised ring at v, we have Ov∗ ⊗ ⊂ F ∗ ⊗ = Ext1 (Q(0), Q(1)). We say that a motive M ∈ MT(F ) in unramified at v if it is in MTOv∗ (F ). We say that a motive is unramified if it is unramified for all v. Proposition 3.3 ([GD03][p. 10, prop 1.8 ]). A motive M on F is unramified at v if and only if its l-adic realisation is unramfied at v for some prime l. X (Y ) for Y a X- Nisnevich scheme, I think we have G X (Y ) = don’t really understand what is GN N is is 13 May-be I dont need G N is but I may work with it 12 I
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Y ??? revoir f ∗ and f∗
Proof. Lets recall that the l-adic representation of Q(1) is given by Q ⊗ Zl (1) where Z(1) = proj lim µln (1). M is unramified at v if and only if the W−2n M/W−2(n+2) are unramified at v. An extension of Q(n) by Q(n + 1) is determined by its class in Ext1 (Q(n), Q(n + 1)). Lemma 3.4. An extension E of Q(n) by Q(n + 1) represented by x ∈ F ∗ ⊂ Ext1 (Q(n), Q(n + 1)) is unramified if and only if it l-adic realisation is unramified Proof. As Ext1 (Q(n), Q(n + 1)) ' Ext1 (Q(0), Q(1)) we just need to consider this last case. Let x be in F ∗ , we note K(x) the Kummer extension of Q(0) by Q(1) of class x. Its l-adic realisation Kl (x) for l different from the caracteristic of the residu field of Ov is given by the projective system · · · ← µln (x) × Z/ln Z ← · · · where µln (x) is the group of he ln roots of x. the action of Gal(Q/Q) is trivial on the second factor and the natural action on the roots of x. As l is different from the caracteristic of the residu field, the action of the inertia is trivial on Kl (x) if and only if neither x neither 1/x are in the ideal of v, so if and only if x is invertible in Ov So a motive M is unramified at v if and only if the l-adic realisation of the W2n M/W2(n+2) are unramified. In order to conclude we just need the next lemma. Lemma 3.5 ([GD03][lem. 1.8.1]). Let Kv be the fraction field of the henselisation of the local ring Ov and H a l-adic representation of Gal(Kv /Kv ). with an increasing filtration indexed by the even intregers such that the GrW 2n H is a direct sum of copies of Q(n). The representation H is unramified if and only if the representations W−2n /W−2(n+2) are unramified. Proof. Let Iv be the inertia groups. Its biggest pro-l quotient (the quotient is pro-l) is isomorphic to Zl (1) (it is the group corresponding to the ln roots of π a generator of the maximal ideal of the local ring). We proove by induction on r > 2 that the action of Iv(σ) on the W−2n /W−2(n+r) is trivial for all n. For r = 2 it is the hypothesis. Let n be an integer, and σ ∈ Iv(σ) . We write the action of σ on W−2n /W−2(n+r) 1 + v(σ). We will show that v(σ) factor through : W−2n /W−2(n+r)
/ / W−2n /W−2(n+1) = GrW vb(σ)/ GrW 2n 2n+r−1 = W−2n+r−1 /W−2(n+r)
/ W−2n /W−2(n+r) .
The incresing filtration is: W−2(n+r) ⊂ W−2(n+r−1) ⊂ · · · ⊂ W−2(n+1) ⊂ W−2n . GrW 2n+r−1
⊂ W−2n /W−2(n+r) and W−2(n+r) ⊂ W−2(n+1) Let x be in W−2n , by induction, we know So we have that that v(σ)(x) = 1 − σ(x) ∈ W−2(n+r−1) (the action is trivial on W−2n /W−2(n+r−1) ). Let y be in W−2(n+1) , we want to show that the image of v(σ)(x + y) = v(σ)(x) + v(σ)(y) in W−2(n+r−1) /W−2(n+r) is equal to the action of 1 − σ on W−2n /W−2(n+r) apply to x. Using the induction for n + 1 we have that v(σ)(y) ∈ W−2(n+r) and v(σ) factor v b(σ)
W 0 through GrW 2n −→ Gr2n+r−1 . Using twice the induction, we see that v(σ)v(σ )(x) ∈ W−2(n+r) in x ∈ W−2n then 0 0 vb(σσ ) = vb(σ) + vb(σ ) ((1 − ab) = (1 − a) + (1 − b) − (1 − a)(1 − b)). Then the application σ 7→ vb(σ) factor through Zl (1) n W 14 . But v(σ) can be written n.tl (σ) where tl (σ) is in Zl (1) and n : GrW 2n (1) −→ Gr2n+r−1 is Gal(Kv /Kv ) equivariant. Then as r > 2 : n ∈ HomGal(Kv /Kv ) (Ql (n + 1), Ql (n + r − 1)) = 0.
Then vb(σ) = 0 and the action of Iv on W−2n /W−2(n+r) is trivial.
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The motives Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn are defined over Z The goal of this section is to proove :
Theorem 4.1. The motives Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn are unramified over p for all prime p 14 why
???
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4.1
The smooth proper base change theorem [Mil80]
We are working with the etal cohomology. Theorem 4.2 (Proper-smooth base change). Let X and S, be variety on an algebraic closed field k. If π : X → S is proper and smooth and F is locally constant with finite stalks, then Rr π? F is locally constant with finite stalks, provided the torsion in F is prime to the characteristic of k. Theorem 4.3. Let π : X → S be proper and smooth and F be a locally constant sheaf on X. Assume that F have finite stalks with torsion prime to the caracteristic of k. For any pair of geometric points of S s¯0 and s¯1 with s¯0 a specialisation of s¯1 , the cospecialisation map : ∼
Hr (Xs¯0 ; F) −→ Hr (Xs¯1 ; F) is an isomorphism. Here Xs = X ×S s is the fiber of X over s. We will apply the former theorem in the following situation : for π : X → Spec(Zp = S) a smooth proper scheme defined over Zp . We will write η the generic point of S and µ its special point. We will write X η = XQp and X µ = XFp for the fiber of X over η and µ. As the moduli spaces of curve and the stratum of their boundries are smooth over Z, we have : Corollary 4.4.
• for all k we have an ismorphism : η
∼
µ
Hr (M0,k , Ql ) −→ Hr (M0,k , Ql ) • for all Y , strata of ∂M0,k , we have an ismorphism : ∼
Hr (Y η , Ql ) −→ Hr (Y µ , Ql )
4.2
Proof of the themorem 4.1
We would like to use the same trick we have use to proove that the Hn M0,n+3 \ Ak ; Bn \ Ak ∩ Bn are Tate and use the diagram 2.1.1. However we can’t from this diagram control the extension which arise in the final motive. As the definition of begin unramified is a condition on the W−2n /W−2(n+2) , it is hopeless to proove directly that those motives are unramified. We will use the l-adic realistion and the theorem 3.3 Definition 4.1. Let D be a normal crossing divisor in a regular (and smooth) scheme X over Zp . Assume that the pair (D; X) is proper over Zp . We say that reduction modulo p does not change the combinatorics of (D; X) if X and every stratum of D are smooth over Zp , and the reduction map from the strata of D to ones at the special fiber is a bijection. If we take for X, M0,n+3 and for D, the union of Ak ∪ Bn . We have that the reduction modulo p does’nt change the combinatoric of (X, D) because M0,n+3 is defined over Z. Let have a look to how from (X, D) define over Z we will go in one side to (XQ , DQ ) and on the other side to (XQp , DQp ) and (XFp , DFp ). We will write k instead of Spec(k) (XFp , DFp ) (4.2.1) (X un , DZun )o p 3 Zp g g g g | g g g ggg || g g g | g g un || ggggg j5 Zp kXXXXXXXXX || j ggggg g j g | j g X g XXXX || jjj gggg ~| XXXXXX jjjj ggggg j X j j / (XQp , DQp ) jj / (XZp , DZp ) (XQp , DQp ) (X , D F Fp ) Fp p j j j n o j } o n o j o j n o } o n o }} wooo wooo jjjjj vnnn }} / (XQ , DQ ) jj / (XZ , DZ ) } (XQ , DQ ) jj }} jjjj }} j j } j jjjj ~} / Zp vo / Qp Fp Qp m n n n n mm nnn nnn mmm wnnnnn wnnnnn vmmmmm /Q /Z Q
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Proposition 4.5. With the previous conventions, assume the reduction modulo p does not change the combinatoric of (D, X) . Let A, B be two divisor such that A ∪ B = D and A and B don’t have any common irreductible maximal component. Then for l a prime number different from pwe have an isomorphism ∼
Hnet (X η \ Aη ; (B \ B ∩ A)η ; Ql ) −→ Hnet (X µ \ Aµ ; (B \ B ∩ A)µ ; Ql ). In particular Hnet (X \ A; B \ B ∩ A; Ql ) is unramified at p. First let see what means the last part of the proposition. The space X and the two part of D are define over Z. Let write Y := X \ A and B := B \ B ∩ A. We have the following diagram : Hnet ((Y, B)Q , Ql ) o
Gal(Q/Q) o
Hnet ((Y, B)Qp , Ql ) o ?_
Hnet ((Y, B)Zp , Ql )
/ Hnet ((Y, B)F , Ql ) p
Dp = Gal(Qp /Qp )
/ Gal(Fp /Fp )
So the isomorphism between Hnet ((Y, B)Qp , Ql ) and Hnet ((Y, B)Fp , Ql ) says that the action of Gal(Fp /Fp ) can be lift. Proof. From corolary 4.4 we have an isomorphisme on each strata Y : ∼
Hr (Y η , Ql ) −→ Hr (Y µ , Ql ) To compute Hn (Y ; B, Ql ) we use its standard simplicial resolutionS• and the associate spectral sequence which E1 terms are : H2n (Y, Ql ) / · · · / ⊕|I|=k H2n (BI \ A, Ql ) / · · · ···
/ ···
/ ···
/ ···
Hl (Y, Ql )
/ ···
/ ⊕|I|=k Hl (BI \ A, Ql )
/ ···
···
/ ···
/ ···
/ ···
.We have maps of spectral sequences : E1 (YQp , BQp , Ql ) −→ E1 (YZun , BZun , Ql ) ←− E1 (YFp , BFp , Ql ) As the reduction modulo p does not change the combinatoric the reduced modulo p simplicial object S•µ is computing the Hn (X µ , B µ , Ql ). Then if we have the isomorphisme of the E1 terms, the morphisms of spectral sequences became isomorphisms and the result is prooved. We want to proove that : Lemma 4.6. Let X be a proper scheme over Zp and B is a proper normal crossing divisor in X. Suppose that reduction modulo p does not change the combinatorics of B. Then there is an isomorphism Hn (X η \ B η ; Ql ) ' Hn (X µ \ B µ ; Ql ) Proof. By duality it is enought to do it for Hn (X, B; Ql ). We use the same methode as above, the spectral sequence associated to simplicial resolution S• of (X, B). We again have the morphisme of spectral sequence and as the reduction mu modulo p doesn’t change the combinatoric of B, the reduction Sbullet compute the righthand side.The proper-smooth base change theorem give and isomorphisme on the E1 terms and we have an isomorphisme of spectral sequence. The lemma is prooved and so the proposition.
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References [BFLS99] X. Buf, J. Fehrenbach, P. Lochak, and L. Schneps, Espace de modules de courbes, groupes modulaires et théorie des champs, Panorama et Synthèse, no. 7, SMF, 1999. [DM69]
Pierre Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Pub. Math. Institut des Hautes Etudes Scientifiques (1969), no. 36.
[GD03]
A. B. Goncharov and P. Deligne, Groupe fondamentaux motivique de tate mixte, www.arxiv.org/abs/math.NT/0302267, Fevrier 2003.
e-print,
[GM02]
A. B. Goncharov and Yu. I Manin, Multiple ζ-motives and moduli spaces M0,n , www.arxiv.org/abs/math.AG/0204102, April 2002.
e-print,
[Ivo05]
F. Ivorra, Réalisation l-adic des motifs mixtes, Ph.D. thesis, Université Paris 6, 2005.
[Kee92]
Sean Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574.
[Mil80]
J.S. Milne, Étale cohomology, Princeton University Press, 1980.
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