An Averaging Process for Unipotent Group Actions

0.21 MB·English

by  Amnon Yekutieli

AN AVERAGING PROCESS FOR UNIPOTENT GROUP
ACTIONS
6
0
0
AMNONYEKUTIELI
2

n
a Abstract. We present an averaging process for sections of a torsor under a
J unipotent group. Thisprocessallowsonetointegratelocal sections ofsucha
torsorintoaglobalsimplicialsection. Theresultsofthispaper haveapplica-
2
tionstodeformationquantizationofalgebraicvarieties.
1

]
T
R
0. Introduction
.
h Let K be a field of characteristic 0. For any natural number q let K[t ,...,t ]
t 0 q
a be the polynomial algebra,and let ∆q be the geometric q-dimensional simplex
K
m
∆q :=SpecK[t ,...,t ]/(t +···+t −1).
[ K 0 q 0 q

Our main result is the following theorem.
2
v Theorem 0.1. Let G be a unipotent algebraic group over K, let X be a K-scheme,
0
7 let Z be a G-torsor over X, and let Y be any X-scheme. Suppose f =(f0,...,fq)
1 is a sequence of X-morphisms fi :Y →Z. Then there is an X-morphism
5
wav (f):∆q ×Y →Z
0 G K
5
called the weightedaverage, such that the operation wav is symmetric, simplicial,
0 G
functorial in the data (G,X,Y,Z), and is the identity for q =0.
/
h
t “Symmetric” means that wav is equivariant for the action of the permutation
a G
m group of {0,...,q} on the sequence f and the scheme ∆qK. “Simplicial” says that
as q varies
:
v wav :Hom (Y,Z)q+1 →Hom (∆q ×Y,Z)
G X X K
i
X is a map of simplicial sets. “Functorial in the data (G,X,Y,Z)” means that e.g.
r given a morphism h:Z →Z′ of G-torsors over X, one has
a
h◦wav (f)=wav (h◦f),
G G
where h◦f is the sequence (h◦f ,...,h◦f ). Theorem 0.1 is repeated, in full
0 q
detail, as Theorem 1.11.
Observe that when we restrict the morphism wav (f) to each of the vertices
G
of ∆q we recover the original morphisms f ,...,f . This is due to the simplicial
K 0 q
propertyofwav . Thuswav (f)interpolatesbetweenf ,...,f . Thisisillustrated
G G 0 q
in Figure 1.
Date:12January2006.
Key words and phrases. Unipotentgroup,torsor,simplicialset.
Mathematics Subject Classification2000. Primary: 14L30; Secondary: 18G30, 20G15.
ThisworkwaspartiallysupportedbytheUS-IsraelBinationalScienceFoundation.
1
2 AMNONYEKUTIELI
Here is a naive corollary of Theorem 0.1, which can further explain the result.
Letus write G(K) for the groupofK-rationalpoints of G. By a weight sequencein
K we mean a sequence w =(w ,...,w ) of elements of K such that w =1.
0 q i
P
Corollary 0.2. Let G be a unipotent group over K. Suppose Z is a set with
G(K)-action which is transitive and has trivial stabilizers. Let z =(z ,...,z ) be a
0 q
sequence of points in Z and let w be a weight sequence in K. Then there is a point
wavG,w(z) ∈ Z called the weighted average. The operation wavG is symmetric,
functorial, simplicial, and is the identity for q =0.
The corollary is proved in Section 1. The idea is of course to take Y = X :=
SpecK in the theorem, and to note that w is a K-rationalpoint of ∆q.
K
Another consequence ofTheorem0.1 is a new proofof the fact that a unipotent
group in characteristic 0 is special. See Remark 1.12. (This observation is due to
Reichstein.)
Theorem0.1wasdiscoveredinthecourseofresearchondeformationquantization
in algebraic geometry [Ye], in which we tried to apply ideas of Kontsevich [Ko] to
thealgebraiccontext. Hereisabriefoutline. SupposeX isasmoothn-dimensional
K-scheme. The coordinate bundle of X [GK, Ko] is an infinite dimensional bundle
Z → X which parameterizes formal coordinate systems on X. The bundle Z is a
torsor under an affine group scheme GLn,K ⋉G, where G is pro-unipotent. One
is interested in sections of the quotient bundle Z¯ := Z/GLn,K. If we are in the
differentiable setup (i.e. K = R and X is a C∞ manifold) then the fibers of Z¯ are
contractible (since they are isomorphic to G(R)), and therefore globalC∞ sections
exist. Howeverin the setupof algebraicgeometrysuchanargumentis invalid, and
we were forced to seek an alternative approach.
Our solution was to use simplicial sections of Z¯ (see Section 2, and especially
Corollary 2.7). This approach was inspired by work of Bott [Bo] on simplicial
connections (cf. also [HY]).
Acknowledgments. The author thanks David Kazhdan and Zinovy Reichstein
for useful conversations. Also thanks to the referee for reading the paper carefully
and suggesting a few improvements.
1. The Averaging Process
ThroughoutthepaperKdenotesafixedbasefieldofcharacteristic0. Allschemes
and all morphisms are over K.
We begin by recalling some standardfacts aboutthe combinatoricsof simplicial
objects. Let∆denotethecategorywithobjectstheorderedsets[q]:={0,1,...,q},
q ∈ N. The morphisms [p] → [q] are the order preserving functions, and we write
∆qp :=Hom∆([p],[q]). Thei-thco-facemap∂i :[p]→[p+1]istheinjectivefunction
that does not take the value i; and the i-th co-degeneracy map si : [p] → [p−1]
is the surjective function that takes the value i twice. All morphisms in ∆ are
compositions of various ∂i and si.
An element of ∆q may be thought of as a sequence i = (i ,...,i ) of integers
p 0 p
with 0 ≤ i ≤ ··· ≤ i ≤ q. Given i ∈ ∆m, j ∈ ∆p and α ∈ ∆q, we sometimes
0 p q m p
write α (i):=i◦α∈∆m and α∗(j):=α◦j ∈∆q .
∗ p m
Let C be some category. A cosimplicial object in C is a functor C :∆→C. We
shall usually refer to the cosimplicial object as C ={Cp}p∈N, and for any α∈∆qp
the corresponding morphism in C will be denoted by α∗ : Cp → Cq. A simplicial
AN AVERAGING PROCESS 3
object in C is a functor C : ∆op → C. The notation for a simplicial object will be
C ={Cp}p∈N and α∗ :Cq →Cp.
An importantexample is the cosimplicialscheme {∆pK}p∈N. The morphismsare
defined as follows. For any p we identify the orderedset [p] with the set of vertices
of ∆p. Given α ∈ ∆q the morphism α∗ : ∆p → ∆q is then the unique linear
K p K K
morphism extending α:[p]→[q].
LetGbeaunipotent(affinefinitetype)algebraicgroupoverK,with(nilpotent)
Lie algebra g. We write d(G) for the minimal number d such that there exists a
chainofclosednormalsubgroupsG=G ⊃G ···⊃G =1withG /G abelian
0 1 d k k+1
for all k∈{0,...,d−1}. The exponential mapexp :g→G is an isomorphismof
G
schemes, with inverse log ; see [Ho, Theorem VIII.1.1].
G
GivenaK-schemeX thereisaLiealgebrag×X (inthecategoryofX-schemes),
and a group-scheme G×X. There is also an induced exponential map
exp :=exp ×1 :g×X →G×X.
G×X G X
We will need the following result.
Lemma 1.1. Let G,G′ be two unipotent groups, with Lie algebras g,g′. Let X,X′
be schemes, let X →X′ be a morphism of schemes, and let φ :G×X →G′×X′
be a morphism of group-schemes over X′. Denote by dφ : g×X → g′ ×X′ the
induced Lie algebra morphism (the differential of φ). Then the diagram
g×X −e−x−pG−×−→X G×X
(1.2)
dφ φ
 
g′×yX′ −e−x−pG−′−×−X→′ G′×yX′
commutes.
Proof. ForthecaseX =X′ =SpecKthisiscontainedintheproofof[Ho,Theorem
VIII.1.2].
Inorderto handlethe generalcasewe firstrecallthe Campbell-Baker-Hausdorff
formula:
exp (γ )·exp (γ )=exp (F(γ ,γ )),
G 1 G 2 G 1 2
where
F(γ ,γ )=γ +γ + 1[γ ,γ ]+···
1 2 1 2 2 1 2
is a universal power series (see [Ho, Section XVI.2]). Hence if we define γ ∗γ :=
1 2
F(γ ,γ ),then(g,∗)becomesanalgebraicgroup(withunitelement0),andexp :
1 2 G
(g,∗)→(G,·) is a groupisomorphism. In this waywe mayeliminate G altogether,
and just look at the scheme g with its two structures: a Lie algebra and a group-
scheme. Note that nowg is its ownLie algebra,ascanbe seenfromthe 2-ndorder
term in the series F(γ ,γ ).
1 2
Consideramorphismφ:g×X →g′×X′ ofX′-schemes. Then φis amorphism
ofLiealgebrasoverX′ iffitisamorphismofgroup-schemes(forthemultiplications
∗). And moreoverdφ=φ. Therefore the diagram (1.2) is commutative. (cid:3)
From now on we shall write exp instead of exp , for the sake of brevity.
G G×X
Consider the following setup: X is a K-scheme, and Y,Z are two X-schemes.
Suppose Z is a torsor under the group scheme G×X. We denote the action of G
on Z by (g,z)7→g·z.
4 AMNONYEKUTIELI
Let f ,...,f : ∆q ×Y → Z be X-morphisms. We are going to define a new
0 q K
sequence of X-morphisms f′,...,f′ : ∆q ×Y → Z. Because Z is a torsor under
0 q K
G×X, for any i,j ∈{0,...,q} there exists a unique morphism g :∆q ×Y →G
i,j K
such that f (w,y) = g (w,y)·f (w,y) for all points w ∈ ∆q and y ∈ Y. Here
j i,j i K
w and y are scheme-theoretic points, i.e. w ∈ ∆qK(U) = HomK(U,∆qK) and y ∈
Y(U)=HomK(U,Y) for some K-scheme U. Define
q
(1.3) f′(w,y):=exp t (w)·log (g (w,y)) ·f (w,y).
i G(cid:16)X j G i,j (cid:17) i
j=0
In this formula we view t as a morphism t :∆q →A1, and we use the fact that
j j K K
g is a vector space (in the category of K-schemes).
For any set S let us write S∆q0 for the set of functions ∆q0 → S, which is the
same as the set of sequences (s ,...,s ) in S. As usual Hom (∆q ×Y,Z) is the
0 q X K
set of X-morphisms ∆q ×Y →Z. In this notation the sequences (f ,...,f ) and
K 0 q
(f0′,...,fq′) are elements of HomX(∆qK×Y,Z)∆q0. We denote by
wsymG :HomX(∆qK×Y,Z)∆q0 →HomX(∆qK×Y,Z)∆q0
the operation (f ,...,f )7→(f′,...,f′) given by the formula (1.3).
0 q 0 q
We will also need a similar operation
wG :HomX(Y,Z)∆q0 →HomX(∆qK×Y,Z)∆q0,
defined by
w (f ,...,f ):=(f′,...,f′)
G 0 q 0 q
with
q
(1.4) f′(w,y):=exp t (w)·log (g (y)) ·f (y).
i G(cid:16)X j G i,j (cid:17) i
j=0
It is clear that for q = 0 both operations w and wsym act as the identity,
G G
i.e. w (f ) = wsym (f ) = f for all f ∈ Hom (Y,Z). Both operations w and
G 0 G 0 0 0 X G
wsym are symmetric, namely they are equivariant for the simultaneous action
G
of the permutation group of {0,...,q} on ∆q and ∆q. Also if f = (f ,...,f ) ∈
0 K 0 q
HomX(∆qK×Y,Z)∆q0 isaconstantsequence,i.e.f0 =···=fq,thenwsymG(f)=f.
Lemma 1.5. Both operations w and wsym are simplicial. Namely, given α ∈
G G
∆q the diagram
p
HomX(∆qK×Y,Z)∆q0 −−w−sy−m−→G HomX(∆qK×Y,Z)∆q0
α∗ α∗
 
HomX(∆pKy×Y,Z)∆p0 −−w−sy−m−→G HomX(∆pKy×Y,Z)∆p0
is commutative, and likewise for w .
G
Proof. It suffices to consider α = ∂i or α = si. Since wsym is symmetric, we
G
may assume that α = ∂q : [q−1] → [q] or α = sq : [q+1] → [q]. Fix a sequence
f =(f0,...,fq)∈HomX(∆qK×Y,Z)∆q0. Letgi,j ∈HomK(∆qK×Y,G)besuchthat
f =g ·f , and let f′ =(f′,...,f′):=wsym (f).
j i,j i 0 q G
First let’s look at the case α = ∂q. Take w ∈ ∆q−1 and y ∈ Y, and let
K
v := α∗(w) ∈ ∆q. The coordinates of v are t (v) = t (w) for j ≤ q −1, and
K j j
AN AVERAGING PROCESS 5
t (v)=0. Then foranyi the i-thterm inthe sequenceα (f′), evaluatedat (w,y),
q ∗
equals
q
f′(v,y)=exp t (v)·log (g (v,y)) ·f (v,y)
i G(cid:16)X j G i,j (cid:17) i
j=0
(1.6)
q−1
=exp t (w)·log (g (v,y)) ·f (v,y).
G(cid:16)X j G i,j (cid:17) i
j=0
On the other hand, the i-th term of the sequence wsym (α (f)) is
G ∗
q−1
exp t (w)·log (α (g )(w,y)) ·α (f )(w,y)
G(cid:16)X j G ∗ i,j (cid:17) ∗ i
j=0
q−1
=exp t (w)·log (g (v,y)) ·f (v,y).
G(cid:16)X j G i,j (cid:17) i
j=0
So indeed α ◦wsym =wsym ◦α in this case.
∗ G G ∗
Next consider the case α = sq. Take w ∈ ∆q+1 and y ∈ Y, and let v :=
K
α∗(w) ∈ ∆q. The coordinates of v are t (v) = t (w) for j ≤ q−1, and t (v) =
K j j q
t (w)+t (w). For any i ≤ q the i-th term in the sequence α (f), evaluated at
q q+1 ∗
(w,y),isf′(v,y),whichwascalculatedin(1.6). The(q+1)-sttermisalsof′(v,y).
i q
On the other hand, for any i ≤ q the i-th term in the sequence wsym (α (f′)),
G ∗
evaluated at (w,y), is
q+1
z :=exp t (w)·log (α (g )(w,y)) ·α (f )(w,y).
i G(cid:16)X j G ∗ i,j (cid:17) ∗ i
j=0
But t (w) + t (w) = t (v), α (g )(w,y) = g (v,y) for j ≤ q, and
q q+1 q ∗ i,j i,j
α (g )(w,y)=g (v,y). Therefore
∗ i,q+1 i,q
q
z =exp t (v)·log (g (v,y)) ·f (v,y).
i G(cid:16)X j G i,j (cid:17) i
j=0
For i = q+1 one has z = z . We conclude that α ◦wsym = wsym ◦α in
q+1 q ∗ G G ∗
this case too.
The proof for w is the same. (cid:3)
G
Lemma 1.7. Both operations w and wsym are functorial in the data
G G
(G,X,Y,Z). Namely, given another such quadruple (G′,X′,Y′,Z′), a morphism
of schemes X → X′, a morphism of schemes e : Y′ → Y over X′, a morphism of
group-schemes φ:G×X →G′×X′ over X′, and a G×X -equivariant morphism
of schemes f :Z →Z′ over X′, the diagram
HomX(∆qK×Y,Z)∆q0 −−w−sy−m−→G HomX(∆qK×Y,Z)∆q0
(e,f) (e,f)
 
HomX′(∆qKy×Y′,Z′)∆q0 −w−s−y−m−G→′ HomX′(∆qKy×Y′,Z′)∆q0
is commutative, and likewise for w .
G
Proof. This is due to the functoriality of the exponentialmap, see Lemma 1.1. (cid:3)
6 AMNONYEKUTIELI
Lemma 1.8. Assume G is abelian. Then for any (f ,...,f ) ∈ Hom (∆q ×
0 q X K
Y,Z)∆q0 the sequence wsymG(f0,...,fq) is constant.
Proof. In this case exp : g → G is an isomorphism of algebraic groups, where g is
viewed as an additive group. So we may assume that Z is a torsor under g×X.
Let (f′,...,f′) := wsym (f ,...,f ), and let γ : ∆q ×Y → g be morphisms
0 q G 0 q i,j K
such that f =γ +f . Take (w,y)∈∆q ×Y. Then
j i,j i K
q
f′(w,y)= t (w)·γ (w,y) +f (w,y)
i (cid:16)X j i,j (cid:17) i
j=0
for any i. Because γ =−γ =γ −γ , f =f +γ and q t (w) = 1, it
follows that f′(w,y)i=,j f′(w,jy,i). 0,j 0,i i 0 0,i Pj=0 j (cid:3)
i 0
Let’s write wsymd for the d-th iteration of the operation wsym .
G G
Lemma 1.9. For any f = (f0,...,fq) ∈ HomX(∆qK × Y,Z)∆q0 the sequence
wsymd(G)(f) is constant. For any d≥d(G) one has wsymd(f)=wsymd(G)(f).
G G G
Proof. For any k, the orbit of f ∈ Hom (∆q ×Y,Z) under the action of the
0 X K
group G (∆q × Y) will be denoted by G (∆q × Y) · f . Let (f′,...,f′) :=
k K k K 0 0 q
wsym (f ,...,f ). We will prove that if f ,...,f ∈ G (∆q × Y) · f then
G 0 q 1 q k K 0
f′,...,f′ ∈G (∆q ×Y)·f′. The assertions of the lemma will then follow.
1 q k+1 K 0
Let Y˜ = X˜ := ∆q ×Y and Z˜ := X˜ × Z. So Z˜ is a torsor under G×X˜,
K X
and f induces a morphism f˜ ∈ Hom (Y˜,Z˜). The morphism τ : G×X˜ → Z˜,
0 0 X˜
(g,x˜)7→g·f˜(x˜), is an isomorphism of X˜-schemes. Define W˜ := τ(G ×X˜)⊂ Z˜.
0 k
ThenW˜ isthe“geometricorbit”off˜ underG ×X˜;andinparticularW˜ isatorsor
0 k
under G ×X˜. By assumption f˜,...,f˜ ∈ Hom (Y˜,W˜). Define (f˜′,...,f˜′) :=
k 1 q X˜ 0 q
wsym (f˜,...,f˜). By Lemma 1.7 it suffices to prove that f˜′,...,f˜′ ∈G (Y˜)·
Gk 0 q 1 q k+1
f˜′.
0
DefineW¯ :=W˜/G . Thisisatorsorunderthegroupscheme(G /G )×X˜.
k+1 k k+1
Let f¯,...,f¯ ∈ Hom (Y˜,W¯) be the images of (f˜,...,f˜). Because the group
0 q X˜ 0 q
G /G is abelian, Lemma 1.8 says that wsym (f¯,...,f¯) is a constant
k k+1 Gk/Gk+1 0 q
sequence. AgainusingLemma1.7,weseethatinfactf˜′,...,f˜′ ∈G (Y˜)·f˜′. (cid:3)
1 q k+1 0
Given an X-scheme Y the collections Hom (∆q × Y,Z) and
(cid:8) X K (cid:9)q∈N
∆q
(cid:8)HomX(Y,Z) 0(cid:9)q∈N are simplicial sets. For q =0 there are equalities
(1.10) HomX(∆0K×Y,Z)=HomX(Y,Z)=HomX(Y,Z)∆00.
Theorem1.11. LetGbeaunipotentalgebraic groupoverK, letX beaK-scheme,
and let Z →X be a G-torsor over X. For any X-scheme Y and natural number q
there is a function
wavG :HomX(Y,Z)∆q0 →HomX(∆qK×Y,Z)
called the weighted average. The function wav enjoys the following properties.
G
(1) Symmetric: wav is equivariant for the action of the permutation group of
G
{0,...,q} on ∆q and on ∆q.
0 K
(2) Simplicial: wav is a map of simplicial sets
G
(cid:8)HomX(Y,Z)∆q0(cid:9)q∈N →(cid:8)HomX(∆qK×Y,Z)(cid:9)q∈N.
AN AVERAGING PROCESS 7
(3) Functorial: given another such quadruple (G′,X′,Y′,Z′), a morphism of
schemes X → X′, a morphism of X′-group-schemes G×X → G′×X′, a
G×X -equivariant morphism of X′-schemes f :Z →Z′ and a morphism
of X′-schemes e:Y′ →Y, the diagram
HomX(Y,Z)∆q0 −−w−av−G→ HomX(∆qK×Y,Z)
(e,f) (e,f)
 
HomX′(yY′,Z′)∆q0 −w−s−y−m−G→′ HomX′(∆yqK×Y′,Z′)
is commutative.
(4) If q =0 then wav is the identity map of Hom (Y,Z).
G X
Proof. Givenasequencef =(f0,...,fq)∈HomX(Y,Z)∆q0 definewavG(f):=f′ ∈
Hom (∆q ×Y,Z) to be the morphism such that
X K
(wsymd(G)◦w )(f ,...,f )=(f′,...,f′);
G G 0 q
see Lemma 1.9. Properties (1)-(4) follow from the corresponding properties of w
G
and wsym . (cid:3)
G
Proof of Corollary 0.2. Take X =Y :=SpecK in Theorem 1.11, and consider the
G-torsorZ :=G. Choose any base point z ∈Z; this defines an isomorphismof left
G(K)-sets Z(K) ∼= Z. The weight sequence w can be considered as a K-rational
point of ∆q, and we define
K
wavG,w(z):=wavG(z)(w)∈Z.
If we were to choose another base point z′ ∈ Z this would amount to applying
an automorphism of the torsor Z, namely right multiplication by some element of
G(K). Due to the functoriality of wavG the point wavG,w(z) will be unchanged.
The properties of this set-theoretical averagingprocess are now immediate con-
sequences of the corresponding properties of the geometric average. (cid:3)
Remark 1.12. Z. Reichstein observed that our averaging process provides a new
proof (in characteristic 0) of the fact that a unipotent group G is special, namely
any G-torsor Z over K has a K-rationalpoint. Let us explain the idea.
Let z ∈ Z be some closed point. Choose a finite Galois extension L of K
0
containing the residue field k(z ). Let Γ be the Galois group of L over K, which
0
acts on the set Z(L). Let z ,...,z ∈ Z(L) be the Γ-conjugates of z . The group
0 q 0
Γ acts on the sequence z := (z ,...,z ) by permutations. Thus the simultaneous
0 q
action of Γ on
∆q ∆q
Z(L) 0 =HomSpecK(SpecL,Z) 0
fixes z.
We know that the operator wav is symmetric. And functoriality says that
G
the action of the Galois group on SpecL is also respected. Since z is fixed by
the simultaneous action of Γ, so is wav (z). Take the uniform weight sequence
G
w := ( 1 ,..., 1 ) and define z′ := wav (z)(w) ∈ Z(L). Because w is fixed by
q+1 q+1 G
the permutation group we conclude that z′ is Γ-invariant, and hence z′ ∈Z(K).
Remark1.13. Theorem1.11hasaratherobviousparallelindifferentialgeometry.
Indeed, a simply connected nilpotent Lie group is the same as the group G(R) of
rational points of a unipotent algebraic group G over R.
8 AMNONYEKUTIELI
2. Simplicial Sections
In this section we show how the averaging process is used to obtain simplicial
sections of certain bundles.
Suppose H and G are affine group schemes over K, and H acts on G by auto-
morphisms. Namely there is a morphism of schemes H ×G → G which for every
K-scheme Y induces a group homomorphism H(Y) → Aut (G(Y)). Then
Groups
H ×G has a structure of a group scheme, and we denote this group by H ⋉G; it
is a geometric semi-direct product.
Recall that an affine group scheme G is called pro-unipotent if it is isomorphic
to an inverse limit lim G of an inverse system {G } of (finite type affine)
←i i i i≥0
unipotent groups. One may assume that each of the morphisms G → G → G
i i−1
is surjective. Thus G ∼=G/N where N is a normal closed subgroup of G.
i i i
We will be concerned with the following geometric situation.
Scenario 2.1. Let H ⋉G be an affine group scheme over K. Assume G is pro-
unipotent, and moreover there exists a sequence {N } of H-invariant closed
i i≥0
normal subgroups of G such that G ∼= lim←iG/Ni and each G/Ni is unipotent.
Let π : Z →X be an H ⋉G -torsor over X which is locally trivial for the Zariski
topology of X. Define Z¯ :=Z/H and let π¯ :Z¯ →X be the projection.
Theorem 2.2. Assume Scenario 2.1. Suppose U ⊂ X is an open set and
σ ,...,σ :U →Z¯ are sections of π¯. Then there exists a morphism
0 q
σ :∆q ×U →Z¯
K
such that the diagram
∆q ×U −−−σ−→ Z¯
K
p2 π¯
 
y y
U −−−−→ X
is commutative. The morphism σ depends functorially on U and simplicially on
the sequence (σ ,...,σ ). If q =0 then σ =σ .
0 q 0
Proof. We might as well assume that U = X. Consider the quotient Z/G. Since
G is normal in H ⋉G it follows that Z/G is a torsor under H ×X. Let’s write
π :Z →Z¯ =Z/H and π :Z →Z/G for the projections.
H G
PickanopensetV ⊂X whichtrivializesπ :Z →X. Let’swriteZ| :=π−1(V).
V
Becauseπ | :Z| →Z¯| isatrivialtorsorunderH×Z¯| ,wecanliftthesections
H V V V V
σ ,...,σ to sections σ˜ ,...,σ˜ : V → Z such that π ◦σ˜ = σ . Furthermore,
0 q 0 q H j j
since π :Z →Z/G is H-equivariant and Z/G is a torsor under H×X, it follows
G
thatwe canchooseσ˜ ,...,σ˜ suchthat π ◦σ˜ =τ forsome sectionτ :V →Z/G.
0 q G j
Let F ⊂ Z| be the fiber over τ, i.e. F := V × Z via the morphisms π :
V Z/G G
Z → Z/G and τ : V → Z/G. Then F is a torsor under G×V, and σ˜ ,...,σ˜ ∈
0 q
AN AVERAGING PROCESS 9
Hom (V,F). See diagram below.
X
Z oo ⊃ F
zzuπuuHuuuuuu CCCCCπCGC!! OO(cid:31)(cid:31)(cid:31)
Z¯ =Z/H π Z/G (cid:31)
ZZ IIIIIπ¯IIII$$ (cid:15)(cid:15) }}{{{{{{{ XX222 σ˜i (cid:31)(cid:31)(cid:31)
2
Xσi hhRRRRRRRR⊃RRRτRR2RR22 (cid:31)(cid:31)(cid:31)
V
ForanyidefineF :=F/N ,whichisatorsorunder(G/N )×V. Letα :F →F
i i i i i
be the projection, so α ◦σ˜ ∈Hom (V,F ). By Theorem 1.11 we get an average
i j X i
(2.3) ρ :=wav (α ◦σ˜ ,...,α ◦σ˜ ):∆q ×V →F .
i G/Ni i 0 i q K i
The functoriality of wav says that the ρ form an inverse system, and we let
i
(2.4) ρ:=limρ :∆q ×V →F
i K
←i
and
(2.5) σ :=π ◦ρ:∆q ×V →Z¯.
H K
We claim that the morphism σ does not depend on the choice of the section
τ : V → Z/G. Suppose τ′ : V → Z/G is another such section. Let F′ be the
fiber over τ′, and let ρ′ : ∆q × V → F′ be the corresponding morphism as in
K
(2.4). Now τ′ = h·τ for some morphism h : V → H. Then F′ = h·F, and
h : F → F′ is a G × V -equivariant morphism of torsors, with respect to the
group-scheme automorphism Ad(h) : G × V → G × V. The new lift of σ is
j
σ˜′ := h·σ˜ : V → F′. Define F′ := F′/N , and let ρ′ : ∆q ×V → F′ be the
j j i i i K i
morphism as in (2.3). Since N ×V = Ad(h)(N ×V), we get a group-scheme
i i
automorphism Ad(h):(G/N )×V →(G/N )×V, and a (G/N )×V -equivariant
i i i
morphismoftorsorsh:F →h·F′. Byfunctorialityofwav(property3inTheorem
i i
1.11) it follows that ρ′ =h·ρ . Therefore ρ′ =h·ρ, and π ◦ρ′ =π ◦ρ=σ.
i i H H
Property 2 in Theorem 1.11 implies that σ depends simplicially on (σ ,...,σ ).
0 q
FinallytakeanopencoveringX = V suchthateachV trivializesπ :Z →X,
j j
and let σ : ∆q × V → Z¯| be thSe morphism constructed in (2.5). Since no
j K j Vj
choices were made we have σ | = σ | for any two indices. Therefore
j Vj∩Vk k Vj∩Vk
these sections can be glued to a morphism σ : ∆q ×X → Z¯. The functorial and
K
simplicial properties of σ are clear from its construction. (cid:3)
Let X be a K-scheme, and let X = m U be an open covering, with in-
Si=0 (i)
clusions g : U → X. We denote this covering by U. For any multi-index
(i) (i)
i = (i0,...,iq) ∈ ∆mq we write Ui := Tqj=0U(ij), and we define the scheme
Uq := `i∈∆mq Ui. Given α ∈ ∆qp and i ∈ ∆mq there is an inclusion of open sets
α∗ : Ui → Uα∗(i). These patch to a morphism of schemes α∗ : Uq → Up, making
{Uq}q∈N into a simplicial scheme. The inclusions g(i) :U(i) →X induce inclusions
gi : Ui → X and morphisms gq : Uq → X; and one has the relations gp◦α∗ = gq
for any α∈∆q.
p
10 AMNONYEKUTIELI
Definition 2.6. Letπ :Z →X be amorphismofK-schemes. Asimplicial section
of π based on the covering U is a sequence of morphisms
σ ={σq :∆qK×Uq →Z}q∈N
satisfying the following conditions.
(i) For any q the diagram
∆q ×U −−−σq−→ Z
K q
p2 π
 
Uy −−−gq−→ Xy
q
is commutative.
(ii) For any α∈∆q the diagram
p
∆p ×U
K p
1qq×qαq∗qqqq88 GGGGσGpGGG##
∆pK×Uq ;;Z
αMM∗M×M1MMMM&& wwwwσwqwww
∆q ×U
K q
is commutative.
Corollary 2.7. Assume Scenario 2.1. Let U = {U }m be an open covering of
(i) i=0
X. Suppose that for any i ∈ {0,...,m} we are given some section σ : U → Z¯
(i) (i)
of π¯. Then there exists a simplicial section
σ ={σq :∆qK×Uq →Z¯}q∈N
based on U, such that σ | =σ for all i∈{0,...,m}.
0 U(i) (i)
Proof. For any multi-index i = (i0,...,iq) we have sections σ(i0),...,σ(iq) : Ui →
Z¯. Let σi : ∆qK×Ui → Z¯ be the morphism provided by Theorem 2.2. For fixed
q these patch to a morphism σ : ∆q ×U → Z¯. The functorial and simplicial
q K q
properties in Theorem 2.2 imply that this is a simplicial section. (cid:3)
This result (with H trivial) is illustrated in Figure 1.
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