SOME COMPLEMENTS TO THE LAZARD ISOMORPHISM
ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
Abstract. Lazard showed in his seminal work [L] that for rational coef-
ficients continuous group cohomology of p-adic Lie-groups is isomorphic
to Lie-algebra cohomology. We refine this result in two directions: firs*
*tly
we extend his isomorphism under certain conditions to integral coeffi-
cients and secondly, we show that for algebraic groups, his isomorphism
can be realized by differentiating locally analytic cochains.
Contents
1. Introduction 2
2. Review of some results by Lazard 3
2.1. Saturated groups 3
2.2. Serre's standard groups as examples 4
2.3. Valued rings, modules and the functor Sat 7
2.4. Group rings 8
2.5. Enveloping algebras 9
2.6. Group-like and Lie-algebra-like elements 9
2.7. Resolutions and Cohomology 11
3. An integral version of the Lazard isomorphism 12
3.1. Results 12
3.2. Some examples concerning the assumptions of Theorem 3.1.1 14
3.3. The case of standard groups and uniform pro-p-groups 15
3.4. Proof of Theorem 3.1.1 18
4. The Lazard isomorphism for algebraic group schemes 25
4.1. Group schemes 25
4.2. Analytic description of the Lazard morphism 26
4.3. The isomorphism over a general base 27
References 28
____________
Date: May 4, 2009.
1
2 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
1. Introduction
One of the main results of Lazard's magnum opus [L ] on p-adic Lie-groups
is a comparison isomorphism between continuous group cohomology, ana-
lytic group cohomology and Lie-algebra cohomology. This comparison iso-
morphism is an important tool in the cohomological study of Galois represen-
tations in arithmetic geometry. More recently, it also appeared in topology
and homotopy theory in connection with the formal groups associated to
cohomology theories and in particular with topological modular forms.
Lazard's comparison theorem holds for Qp-vector spaces and the isomor-
phism between continuous cohomology and Lie-algebra cohomology is ob-
tained from a difficult isomorphism between the saturated group ring and
the saturated universal enveloping algebra. For some applications (e.g. the
connection with the Bloch-Kato exponential map in [HK ]) it is important
to have a version for integral coefficients and a better understanding of the
map between the cohomology theories.
In this paper we extend and complement the comparison isomorphism in
two directions:
The first result is an integral version of the isomorphism assuming tech-
nical conditions (Theorem 3.1.1). For uniform pro-p-groups one gets a clean
result with only a mild condition on the module (Theorem 3.3.3). To our
knowledge and with the notable exception of Totaro's work [T ], this is the
first progress on a problem which Lazard wrote "reste `a faire" more than
forty years ago [L , Introduction 7,C)].
The second result concerns the definition of the isomorphism in the case
of smooth group schemes. Here one can define directly a map from an-
alytic group cohomology to Lie-algebra cohomology with constant coeffi-
cients by differentiating cochains (see Definition 4.2.1). We showed in [HK ]
(see Proposition 4.2.4) that the resulting map is Lazard's comparison iso-
morphism modulo the identification of continuous cohomology with analytic
cohomology. Serre mentioned to us that this was clear to him at the time
Lazard's paper was written, however, it was not included in the published
results. Unfortunately, we were so far not able to use this simple map to
obtain an independent proof of Lazard's comparison result.
The advantage of this description of the map is not only its simplicity
but also that it carries over to K-Lie-groups for finite extensions K=Qp. In
Theorem 4.3.1 we prove that this map is also an isomorphism in the case
of K-Lie-groups attached to smooth group schemes with connected generic
fiber over the integers of K. This generalizes results in [HK ] for GL n and
complements the result of Lazard, who only treats Qp-analytic groups.
The paper is organized as follows. In Section 2 we give a quick tour
through the notions from [L ] that we need. We hope that this section also
proves to be a useful overview of the central notions and results in [L ]. In
Section 3 we prove our integral refinement of Lazard's isomorphism. Finally
LAZARD ISOMORPHISM 3
Section 4 considers the isomorphism over a general base in the case of group
schemes.
Acknowledgments. We would like to thank B. Totaro for numerous in-
sightful remarks on a preliminary draft of this paper. The third author thanks
H.-W. Henn for discussions about Example 3.2.1.
2.Review of some results by Lazard
In this section we recall the basic notions about groups and group rings
we need to formulate our main results. As we proceed we illustrate the main
notions with the example of separated smooth group schemes. We hope that
this section helps to guide through the long and difficult paper by Lazard.
2.1. Saturated groups.
Definition 2.1.1. [L , II 1.1, II 1.2.10, III 2.1.2] A filtration on a group G
is a map
! : G ! R*+[ {1}
such that
1) For all x, y 2 G, !(xy-1 ) inf{!(x), !(y)}
2) For all x, y 2 G, !(x-1y-1 xy) !(x) + !(y).
G is called p-filtered if in addition for all x 2 G
!(xp) inf{!(x) + 1, p!(x)}.
G is called p-valued if ! satisfies
3) !(x) < 1 for x 6= e
4) !(x) > (p - 1)-1 for all x 2 G
5) !(xp) = !(x) + 1 for all x 2 G.
G is called p-divisible if it is p-valued and
6) For all x 2 G with !(x) > 1 + _1_p-1there exists y 2 G with yp = x.
Finally, a p-divisible group G is saturated, if
7) G is complete for the topology defined by the filtration.
Note that a filtration satisfying the conditions of a p-valuation is auto-
matically p-filtered.
Recall that a pro-p-group is the inverse limit of finite p-groups. This is
the case we are going to work with.
Proposition 2.1.2. [L , II 2.1.3] A p-filtered group is a pro-p-group if and
only if it is compact.
We denote by Fp[ffl] the polynomial ring with generator ffl in degree 1.
Definition and Lemma 2.1.3. [L , II 1.1., III 2.1.1, III 2.1.3] Let (G, !)
be filtered.
4 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
(1) For every 2 R*+
G := {x 2 G | !(x) }, G+ := {x 2 G | !(x) > } G
are normal subgroups.
(2) M
gr(G) := G =G+ ,
2R*+
gr(G) is a graded Lie-algebra over Fp, the Lie-bracket being induced
by the commutator in G.
(3) If (G, !) is p-valued, then gr(G) is even a graded Fp[ffl]-Lie-algebra,
the action of ffl being induced by x 7! xp (x 2 G , xp 2 G +1)
(4) In this case, gr(G) is free as a graded Fp[ffl]-module. The rank of G
is by definition the rank of the Fp[ffl]-module gr(G).
Example 2.1.4. [L , V 2.2.1] Let (G, !) be a complete p-valued group of
finite rank d.
If {xi}i=1,...,d G are representatives of an ordered basis of the Fp[ffl]-
Qd
module gr(G), then every y 2 G is uniquely an ordered product y = x~ii
i=1
with ~i2 Zp and
!(y) = inf(!(xi) + v(~i)),
i
where the valuation on Zp is normalized by v(p) = 1. (G, !) has rank d.
Definition 2.1.5. [L , V 2.2.1, V 2.2.7]
(1) In the situation of the example, the family {xi}i=1,...,dis called an
ordered basis of G.
(2) The p-valued group (G, !) is called equi-p-valued if it there exists an
ordered basis {xi} as above such that
!(xi) = !(xj) for all1 i, j d.
2.2. Serre's standard groups as examples. Let E be a finite extension
of Qp with ring of integers R. Let m be the maximal ideal of R. E is a
discretely valued field. We normalize its valuation v by v(p) = 1. Let e be
the ramification index of E=Qp.
Any formal group law F (X, Y ) in n variables over R defines a group
structure G on mn. These are the standard groups as defined by Serre.
Define for (x1, . .,.xn) 2 mn
!(x1, . .,.xn) := inf{v(xi)}.
i
Then we have for ~ 0
G~ := {x 2 G : !(x) ~}.
Proposition 2.2.1. [S1, LG 4.23 Theorem 1 and Corollary and 4.25 Corol-
lary 2 of Theorem 2] G is a pro-p-group. For any ~ 0 the group G~ is a
normal subgroup of G. Moreover, the map ! defines a filtration on G in the
sense of definition 2.1.1,1)-3).
LAZARD ISOMORPHISM 5
One can show that a suitable open subgroup of the standard group G is
saturated.
Let ae denote the smallest integer which is larger than _e_p-1.
Lemma 2.2.2. Let E and G be as above. Then the subgroup (H, !),
ae oe
1
H := x 2 G : !(x) > _____ = Gae_
p - 1 e
is saturated and of finite rank. It is equi-p-valued if and only if e = 1.
Proof.Note first that according to [S1, LG 4.21, Corollary] the power series
fp, which defines the p-power map, is of the form
fp(X) = p(X + '(X)) + _(X)
with ord(') 2 and ord(_) p. It follows that
!(xp) inf{!(x) + 1, p!(x)}, x 2 G
because if x has coordinates xi, then xp has coordinates
fp,i(x1, . .,.xn) = pxi+ p'i(x) + _i(x)
and the valuations of the summands are bounded below by 1 + !(x), 1 +
2!(x) > 1 + !(x) and p!(x), respectively.
As !(x) > _1_p-1is equivalent to !(x) + 1 < p!(x), this implies on H
!(xp) !(x) + 1.
On the other hand, let xi be a coordinate of x with !(x) = v(xi). Then
!(xp) v(pxi+ pOEi(x) + _i(x)) = 1 + !(xi) = 1 + !(x)
and hence
!(xp) = !(x) + 1
for all x 2 H. This shows that (H, !) is p-valued. To see that H is saturated,
we note that by [S1, LG 4.26, Theorem 4], the p-power map induces an
isomorphism
~=
H~ -! H~+1
for all ~ 2 (__1_p-1, 1) \ v(m). As H is complete, this implies that the group
is saturated.
The valuation on R takes values in 1_eZ where e is the ramification index.
By definition H = Hae_e. We get
M
gr(H) ~= kn .
~21_eN,~ ae_e
As an Fp[ffl]-module gr(H) is freely generated by an Fp-basis of
M
kn
~21_eN,1+ae_e>~ ae_e
This is finite because [k : Fp] < 1.
6 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
If e = 1, then only a single ~ occurs in the sum, namely ae. If e > 1, then
1 + ae_e> ae+1_eand the sum has generators in more than one degree.
An important example of the above construction arises from separated
smooth group schemes G=R. The formal completion bGof G along its unit
section is a formal group over R and the associated standard group G is via
g 7! 1 + g isomorphic to
G ~=ker(G(R) ! G(k)).
Example 2.2.3. To have an even more concrete example, consider G = Gln
over R. Let ss be the uniformizer. Then
ae oe
1
ssaeR = x 2 R | v(x) > _____ .
p - 1
It follows from Lemma 2.2.2 that
H := 1 + ssaeMn(R) Gln(R)
is a saturated subgroup with respect to the filtration !(1+(xi,j)) = infi,j(v(x*
*i,j)).
As M
gr(H) ~= Mn(k)
~ ae_e,~21_eN
the rank of H is n2[R : Zp]. Note that this is not equi-p-valued for e > 1.
However, we can view Gln(R) as the group of Zp-valued points of the Weil-
restriction G0= ResR=ZpGln which is a separated smooth group scheme over
Zp [BLR , 7.6, Proposition 5]. This point of view yields a different valuation
on the corresponding standard group, which is
G0= bG0(Zp) = 1 + pMn(R) Gln(R).
By Lemma 2.2.2 (G0, !0) is saturated and equi-p-valued if p > 2.
As an explicit example, choose R = Zp[ss] with ss2 = p (hence e = 2) and
n = 1. Let p > 3 for simplicity. G = 1 + ssR has rank 2 with ordered basis
x1 = 1 + ss, x2 = 1 + ss2 with
1
!(x1) = __, !(x2) = 1
2
On the other hand, G0= 1+pR has also rank 2 with x01:= 1+p, x02:= 1+ss3
as an ordered basis. They satisfy
!0(x01) = !0(x02) = 1
G0is saturated and equi-p-valued. Compare this to
3
!(x01) = 1, !(x02) = __
2
under the inclusion G0 G.
LAZARD ISOMORPHISM 7
2.3. Valued rings, modules and the functor Sat.
Definition 2.3.1. [L , I 2.1.1. and I 2.2.1] A filtered ring is a ring togeth*
*er
with a map
v : ! R+ [ {1}
such that for ~, ~ 2
1) v(~ - ~) min(v(~), v(~))
2) v(~~) v(~) + v(~)
3) v(1) = 0.
Put
:= {~ 2 | v(~) }.
is called valued if in addition
2') v(~~) v(~) + v(~)
4) The topology defined on by the filtration is separated.
Definition 2.3.2. [L , I 2.1.3 and I 2.2.2] A filtered module M over a filtered
ring is an -module M together with a map
w : M ! R+ [ {1}
such that for x, y 2 M and ~ 2
1) w(x - y) min(w(x), w(y))
2) v(~x) v(~) + w(x)
Put
M := {x 2 M | w(x) }.
A filtered module over a valued ring is called valued if in addition
2') v(~x) v(~) + w(x)
3) The topology defined on M by the filtration M is separated.
Let be a commutative valued ring and A be an -algebra (e.g. a Lie-
algebra).
Definition 2.3.3. [L , I 2.2.4] An -algebra A over a commutative valued
ring is valued, if it is valued as a ring and (with the same valuation map)
valued as an -module.
The following definition is an important technical tool in Lazard's work.
Definition 2.3.4. [L , I 2.2.7] A valued module M over a commutative
valued ring is called divisible, if for all ~ 2 and x 2 M with v(~) w(x)
there exists y 2 M such that ~y = x. The module M is saturated if it is
divisible and complete.
Lazard shows in [L , I 2.2.10] that the completion of a divisible module is
saturated.
8 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
Definition 2.3.5. [L , I 2.2.11] The saturation SatM of a valued module M
over a commutative valued ring is the completion of
divM := {y 2 K M | w(y) 0} .
Here, K is the fraction field of and the valuation w on M is extended to
K M by w(~-1 m) := w(m) - v(~) (which is well-defined, see [L , I
2.2.8] ).
The saturation SatM satisfies the following universal property ([L , I.
2.2.11.]): For any morphism f : M ! N of M into a saturated -module N
there is a unique extension to a map SatM ! N.
2.4. Group rings. In this section we fix = Zp with the standard valua-
tion. All algebras are over Zp.
For any group G let Zp[G] be the group ring with coefficients in Zp.
Definition 2.4.1. [L , II 2.2.1] Let G be a pro-p-group. The completed group
ring Zp[[G]] is the projective limit
Zp[[G]] := lim-Zp[G=U],
where U runs through all open normal subgroups of G and every Zp[G=U]
carries the p-adic topology.
In [L ] this ring is denoted AlG.
Definition 2.4.2. [L , III 2.3.1.2] Let G be a p-filtered group. The induced
filtration w on Zp[G] is the lower bound for all filtrations (as Zp-algebra)
such that
w(x - 1) !(x) for allx 2 G .
Proposition 2.4.3. [L , III 2.3.3]Let G be p-valued. Then the induced fil-
tration w on Zp[G] is a valuation (as Zp-module). If G is compact (or
equivalently, pro-p), then Zp[[G]] is the completion of Zp[G] with respect to
the valuation topology.
Example 2.4.4. [L , V 2.2.1] Let G be p-valued, complete and of finite rank
d. Let {xi}i=1,...,d G be an ordered basis of G. Then Zp[[G]] admits
{zff| ff 2 Nd} Zp[[G]],
Yd
zff:= (xi- 1)ffi
i=1
as a topological Zp-basis satisfying
Xd
w(zff) = ffi!(xi).
i=1
The associated graded is UFp[ffl]gr(G), the universal enveloping algebra of the
Fp[ffl]-Lie-algebra gr(G).
LAZARD ISOMORPHISM 9
Remark 2.4.5. Note that if (G, !) is saturated and non-trivial, then Zp[[G]]
is never saturated. Indeed, since grG 6= 0 is a free Fp[ffl]-module, we have
gr G 6= 0 for arbitrarily large , in particular there exists g 2 G with
!(g) 1. Then x := g - 1 2 Zp[[G]] satisfies w(x) 1 = v(p), but x is not
divisible by p in Zp[[G]].
Lemma 2.4.6. The inclusion Zp[G] ! Zp[[G]] induces an isomorphism
SatZp[G] ~=SatZp[[G]].
Proof.By [L , I 2.2.2] the natural map Zp[G] ! Zp[[G]] is injective. It
extends to
SatZp[G] ! SatZp[[G]]
On the other hand, SatZp[G] is complete, hence there is a natural map
Zp[[G]] ! SatZp[G]. As the right hand side is saturated, it extends to
Sat Zp[[G]] ! SatZp[G] .
The two maps are inverse to each other.
2.5. Enveloping algebras. Let L be a valued Zp-Lie-algebra and UL its
enveloping algebra over Zp.
Definition 2.5.1. [L , IV 2.2.1] The canonical filtration
w : UL ! R+ [ {1}
is the lowest bound of all filtrations on UL turning it into a valued Zp-algebra
such that the canonical map L ! UL is a morphism of valued modules.
Lemma 2.5.2. [L , IV 2.2.5] UL equipped with the canonical filtration is a
valued Zp-algebra and the natural morphism
Ugr(L) ! gr(UL)
is an isomorphism.
2.6. Group-like and Lie-algebra-like elements. Everything in this sec-
tion applies to A = SatZp[[G]] where G is a p-valued pro-p-group. We fix
= Zp with its standard valuation.
Definition 2.6.1. [L , IV 1.3.1] Let A be a valued Zp-algebra with diagonal
: A ! Sat(A Zp A)
([L , IV 1.2.3]) and augmentation ffl. Then we define G, L, G* and L* by
(1) G = {x 2 A|ffl(x) = 1, (x) = x x}
(2) G*= {x 2 G|w(x) > (p - 1)-1}
(3) L = {x 2 A| (x) = x 1 + 1 x}
(4) L* = {x 2 L|w(x) > (p - 1)-1}
These subsets have the following structures.
10 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
Lemma 2.6.2. [L , IV 1.3.2.1 and 1.3.2.2] G and G* are monoids with respect
to the multiplication of A. If A is complete, G* is a group and L and L* are
Lie-algebras. Moreover, L = divL*.
For saturated Zp-algebras A we know much more:
Theorem 2.6.3. [L , IV 1.3.5] Let A be a saturated Zp-algebra with diagonal.
(1) The exponential maps G* to L* and the logarithm maps L* to G*.
They are inverse homeomorphisms.
(2) The Lie-algebra L is saturated. It is the saturation of L*.
(3) G* is a saturated group for the filtration !(x) = w(x - 1).
(4) The associated graded grL* and grG* are canonically isomorphic via
the logarithm map.
(5) L* and G* generate the same saturated associative subalgebra of A.
This has the following consequence for the universal enveloping algebra
UL of a valued Lie-algebra L.
Theorem 2.6.4. [L , IV 3.1.2 and IV 3.1.3] Let L be a valued Lie-algebra
over Zp and UL its universal enveloping algebra. Then
(1) SatUL = SatUSat L
(2) LSatUL = SatL
(3) GSatUL = G*SatUL
The next result concerns the saturation of the group ring Zp[G] (or equiv-
alently of Zp[[G]] by 2.4.6).
Theorem 2.6.5. [L , IV 3.2.5] Let G be a saturated group and A = SatZp[G]
then
G* = G.
Let UL be the universal enveloping algebra of L, then the canonical map
SatUL ! SatZp[G]
is an isomorphism.
We introduce new terminology.
Definition 2.6.6. Let G be a saturated group. We call
L*(G) = L* SatZp[G]
the integral Lazard Lie-algebra of G.
The last theorem then reads
SatUL*(G) ~=SatZp[G].
Example 2.6.7. Consider the saturated group H = 1 + ssaeMn(R) from
Example 2.2.3 and the algebra SatZp[H]. We claim that the Lie-algebra
LAZARD ISOMORPHISM 11
L* = L*(H) is ssaeMn(R) and L = Mn(R). To see this, note that by Theo-
rem 2.6.5 H = G* and that by Theorem 2.6.3 L* consists of the logarithms
of G*. By [L , III 1.1.4 and 1.1.5.].
X xn
(4) Log : 1 + ssaeMn(R)! ssaeMn(R) ; 1 + x 7! (-1)n+1 ___
n 1 n
X xn
(5) exp : ssaeMn(R)! 1 + ssaeMn(R) ; x 7! ___
n 0 n!
are both convergent and inverse to each other. By Theorem 2.6.3, L is the
saturation of L*, which is by Definition 2.3.5
L = {x 2 K R ssaeMn(R) | w(x) 0} = Mn(R).
Example 2.6.8. In general the Lazard Lie-algebra does not coincide with
the algebraic Lie-algebra. Let G be a separated smooth group scheme over
Zp and Lie(G) its Zp-Lie-algebra. If t1, . .,.tn are formal coordinates of G
around e, then _@_@t1, . .,._@_@tnare a Zp-basis of Lie(G).
Let G be the associated standard group over Zp as in Section 2.2. Let H
be the saturated subgroup of G, see Lemma 2.2.2. We have H = G = (pZp)n
if p 6= 2 and H = (4Z2)n if p = 2. Let x1, . .,.xn be the standard ordered
basis of H. We put
ffii= logxi2 SatZp[[H]].
By [L , IV 3.3.6] they form a Zp-basis of L*(H). As explained in [HK ]
Section 4.2 and 4.3 the ffii can be viewed as derivations on Zp[[t1, . .,.tn]],
the coordinate ring of ^G. Note, however, that the coordinate ~i in loc.cit.
takes values in all of Zp on G. Hence ~i= pti for p 6= 2. This implies
@
ffii= p___ |t=0
@ti
Hence under the identification of [HK , Proposition 4.3.1], we have
L*(H) = pLie(G).
For p = 2 the argument gives
L*(H) = 4Lie(G).
2.7. Resolutions and Cohomology.
Definition 2.7.1. [L , I 2.1.16, 2.1.17] Let A be a filtered Zp-algebra, M a
filtered A-module.
(1) A family of A-linearly independent elements (xi)i2I of M is called
filtered free if for every family (~i)i2I of elements of A, almost all
zero, _ !
X
w ~ixi = inf(w(xi) + v(~i)).
i2I i
M is called filtered free if it is generated by a filtered free family.
12 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
(2) Suppose A is complete. M is called complete free if it is the comple-
tion of the submodule generated by a filtered free family.
If A is complete and M is filtered free of finite rank, then M is also
complete free.
Definition 2.7.2. [L , V 1.1.3, 1.1.4, 1.1.7] Let A be a filtered augmented
Zp-algebra, M a filtered A-module.
(1) A filtered acyclic resolution Xo is a chain complex of filtered A-
modules together with an augmentation ffl : Xo ! M such that for
all 2 R+ the morphism ffl : Xo ! M is a quasi-isomorphism.
(2) A split filtered resolution Xo of M is a morphism ffl : Xo ! M of chain
complexes of filtered A-modules together with filtered morphisms of
Zp-modules (sic, not A-linear!)
j : M ! X0, sn : Xn ! Xn+1
defining a homotopy between id and 0 on the extended complex
Xo ffl!M and such that s0j = 0. Note that a split filtered resolution
is a filtered acyclic resolution.
(3) Let A be complete. We call complete free acyclic resolution of M a
filtered acyclic resolution Xo by complete free modules.
(4) Let Xo be a complete free acyclic resolution of the trivial A-module
Zp, M a complete A-module with linear topology. We call
Hnc(A, M) = Hn (Hom c(Xo, M))
(with Hom c continuous A-linear maps) the n-th continuous cohomol-
ogy of A with coefficients in M.
(5) Let A be an augmented Zp-algebra, M an A-module. We call
Hn (A, M) = ExtnA(Zp, M)
the n-th cohomology of A with coefficients in M.
Let A be an augmented Zp-algebra. If Xo is a resolution of Zp by free
A-modules of finite rank, then Hom A (Xo, M) is a free resolution of M.
3. An integral version of the Lazard isomorphism
The purpose of this section is to establish that continuous group coho-
mology and Lie-algebra cohomology agree with integral coefficients, at least
under certain technical assumptions. This generalizes Lazard's result for
coefficients in Qp-vector spaces.
3.1. Results. We fix a saturated and compact group (G, !) of finite rank
d. In particular, G is a pro-p-group by Proposition 2.1.2. We assume
o (G, !) is equi-p-valued
o ! takes values in 1_eZ.
LAZARD ISOMORPHISM 13
Recall that the integral Lazard Lie-algebra
L*(G) = L*SatZp[[G]],
is a finite free Zp-Lie-algebra.
For technical reasons we fix a totally ramified extension Qp K of degree
e with ring of integers O K, uniformizer ss 2 O. The valuation on O is
normalized by v(p) = 1.
Let M be a linearly topologized complete Zp-module with a continuous,
Zp-linear action of G. Thus, M is a Zp[[G]]-module ([L , II 2.2.6]). We
assume that
o the Zp[[G]]-module structure on M extends to a SatZp[[G]]-module
structure.
M is canonically a L*(G)-module.
We are going to prove in Section 3.4:
Theorem 3.1.1. Let (G, !) and M be as above, then:
(1) There is an isomorphism of graded O-modules
OEG (M) : H*c(G, M) Zp O ' H*(L*(G), M) Zp O.
It is natural in M.
(2) If in addition M is a Qp-vector space, then this isomorphism agrees
with Lazard's in [L , V 2.4.9.]
(3) Let H be another group satisfying the assumptions of the Theorem
and f : G ! H a group homomorphism filtered for the chosen filtra-
tions. In addition assume that gr(H) is generated in degree 1_e. Then
the isomorphism is natural with respect to f.
(4) If gr(G) has generators in degree 1_e, then the isomorphism is com-
patible with cup-products as follows:
Assume that M0, M00satisfy the same assumptions as M does and
that
ff : M ^ZpM0 ! M00
is SatZp[[G]]-linear. Then the diagram
H*c(G, M) Zp H*c(G, M0) O ___________//H*c(G, M00) O
|OEG(M)|OEG(M0) |OEG(M00)|
fflffl| |fflffl
H*(L*(G), M) Zp H*(L*(G), M0) O _____//H*c(L*(G), M00) O
commutes. Here, the horizontal maps are the O-linear extensions of
the cup-product defined by ff.
Remark 3.1.2. i) If H*(L*(G), M) is a finitely generated Zp-module,
e.g. when M is of finite type, then this implies by the structure of
finitely generated modules over principal ideal domains the existence
of an isomorphism of graded Zp-modules
H*c(G, M) ' H*(L*(G), M).
14 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
However, it is not clear if this isomorphism is natural or compatible
with cup-products.
ii)According to [L , V 2.2.6.3 and 2.2.7.2] the mod p cohomology of an
equi-p-valued group G is simply an exterior algebra
H*c(G, Fp) = *Fp(H1c(G, Fp))
but the cohomology with torsion free coefficients is more interesting,
e.g. if G is not abelian then the Qp-Betti numbers of G are different
from the Fp-Betti numbers showing that H*c(G, Zp) contains non-
trivial torsion.
It is not obvious to see which groups satisfy the assumptions of The-
orem 3.1.1. We discuss in section 3.3 standard groups and uniform pro-p-
groups, which satisfy the assumptions of Theorem 3.1.1. The next section 3.2
discusses the assumptions with some examples.
3.2. Some examples concerning the assumptions of Theorem 3.1.1.
Here we illustrate the assumptions of Theorem 3.1.1 by a series of remarks
and examples.
The integral Lazard isomorphism may not hold for all topologically finitely
generated pro-p-groups without p-torsion. However, the assumptions of the
Theorem are too restrictive.
Example 3.2.1. Assume p 5 and let D=Qp be the quaternion-algebra,
O D its maximal order and 2 O a prime element. By Lemma 2.2.2
G := 1 + O O*
is p-saturated. From [R , Theorem 6.3.22] or [H , Proposition 7], we know
that
dim FpHic(G, Fp) = 1, 3, 4, 3, 1 (0 i 4).
In particular, H*c(G, Fp) 6= *H1c(G, Fp) and G does not admit an equi-p-
valuation by [L , V 2.2.6.3 and 2.2.7.2.] However, one can by direct arguments
establish an isomorphism
H*c(G, Fp) ' H*(L*(G), Fp)
of graded Fp-algebras; cf. Remark 3.4.10. The proof of [H , Proposition 7]
shows that the same result holds for coefficients in Zp.
Not even saturatedness is necessary.
Example 3.2.2. Let G = 1 + p2Zp for p 6= 2. This group is not saturated
for the obvious filtration, rather we have
Sat(G) = 1 + pZp.
Put
L*(G) L*(Sat(G))
the image of G under the logarithm map. We still get an isomorphism
H*(G, Zp) ! H*(L*(G), Zp)
LAZARD ISOMORPHISM 15
induced by log. It is compatible with the one for Sat(G).
Remark 3.2.3. (1) We are unaware of a group-theoretical characteri-
zation of those pro-p-groups satisfying the assumption of Theorem
3.1.1, but the remark on page 163 of [ST ] suggests that they are
closely related to uniform pro-p-groups.
(2) It is in general difficult to decide if a given Zp[[G]]-module structure
extends over SatZp[[G]], and we refer to [T , page 200] and especially
to the proof of [T , Corollary 9.3] for further discussion and useful
sufficient conditions.
(3) In Theorem 3.3.3 we have established a sufficient condition for both
problems, which have to be addressed here.
There are examples of groups which are saturated with respect to one
filtration but not with respect to another. It can also happen that the
group is saturated with respect to two filtrations but only equi-p-valued for
one of them.
Example 3.2.4. Let K=Qp be a finite extension with ramification index e.
Let O be its ring of integers with uniformizer ss. As discussed in Example
2.2.3 the group
1 + pMn(O)
carries two natural filtrations ! and !0. Recall that ae is the smallest integer
bigger than _e_p-1.
(1) If p = 5, e = 2, then ae = 1 and hence ssae6= 5. This implies that
1 + 5Mn(O) is saturated with respect to !0 but not with respect to
!.
(2) If p = 3, e = 2, then ae = 2 and hence ssae= 3. The group 1+3Mn(O)
is saturated with respect to ! and !0, but but only equi-3-valued with
respect to the second.
3.3. The case of standard groups and uniform pro-p-groups. We
discuss two examples, where the assumptions of Theorem 3.1.1 are satisfied.
First we consider standard groups and then uniform pro-p-groups.
Example 3.3.1. Let G=Zp be a separated smooth group scheme, G =
ker(G(Zp) ! G(Fp)) the associated standard group (see Section 2.2). Its
filtration takes values in Z. By Lemma 2.2.2, there is an open subgroup H
of G which is saturated and equi-p-valued. If p 6= 2, then H = G and the
generators have degree 1. If p = 2, then the generators have degree 2. H
satisfies the assumptions of the Theorem with e = 1.
Let M = Zp with the trivial operation of H. It also satisfies the as-
sumptions of the Theorem. Hence there is a natural isomorphism of graded
Zp-modules
H*c(H, Zp) ' H*(L*(H), Zp).
For p 6= 2 it is compatible with cup-products.
16 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
This example generalizes to a larger class of groups. Recall the notion
of a uniform or uniformly powerful pro-p-group from [DDMS , Definition 3.1
and Definition 4.1]:
Definition 3.3.2. A pro-p-group G is uniform if
i) G is topologically finitely generated._ ___
ii)For p 6= 2 (resp. p = 2), G=Gp (resp. G=G4 ) is abelian.
iii)Denoting G = G1 G2 . . .the lower p-series of G, we have
[Gi: Gi+1] = [G1 : G2] for all i 2.
To understand what is special about p = 2 here, note that the pro-2-group
Z*2= 1 + 2Z2 is not uniform but 1 + 4Z2 is.
The Lie-algebra g of a uniform pro-p-group G is constructed in [DDMS ,
x8.2] and coincides with the integral Lazard Lie algebra L*(G) by [DDMS ,
Lemma 8.14].
Theorem 3.3.3. Let p 6= 2 (resp. p = 2) be a prime, G a uniform pro-
p-group and M a finite free Zp-module with a continuous action of G such
that the resulting group homomorphism
% : G -! AutZp(M)
has image in 1 + pEnd Zp(M) (resp. in 1 + 4End Z2(M)). Then M is canon-
ically a module for the Lie-algebra g of G and there is an isomorphism of
graded Zp-modules
(6) H*c(G, M) ' H*(g, M)
which in case p 6= 2 is compatible with cup-products whenever these are
defined.
Remark 3.3.4. If G is an arbitrary Qp-analytic group acting continuously
on the finite free Zp-module M, then there are arbitrarily small open sub-
groups U G such that the action of U on M satisfies the assumptions of
Theorem 3.3.3.
Proof.We have the following two claims for p 6= 2 (resp. p = 2):
(1) G admits a valuation ! for which it is p-saturated of finite rank and
equi-p-valued with an ordered basis consisting of elements of filtra-
tion 1 (resp. filtration 2).
(2) The Zp-module M admit a valuation w for which it is saturated and
such that for all g 2 G, m 2 M : w((g - 1)m) w(m) + !(g).
Granting these claims, we see as in [T , pages 200-201] that the Zp[[G]]-
module structure of M extends over SatZp[[G]] and hence obtain (6) by
applying Theorem 3.1.1,(1) with O = Zp and remarking that g = L*((G, !)).
The proof of claim 1) is essentially given in [ST , Remark on page 163] but
we include details for convenience: The lower p-series
G = G1 G2 . . .
LAZARD ISOMORPHISM 17
[DDMS , Definition 4.1] consists of normal subgroups satisfying (Gn, Gm )
Gn+m and \n 1Gn = {e} [DDMS , Proposition 1.16] and hence
!(x) := sup{n 2 N | x 2 Gn}, x 2 G
defines a filtration of G by [L , II.1.1.2.4.]. Now [DDMS , Lemma 4.10] states
~=
that for all n, k 1 the pn-th power map of G is a homeomorphism Gk !
~=
Gk+n and induces bijections Gk=Gk+l ! Gk+n =Gk+n+l for all l 0.
In case p 6= 2, we get from this all the properties of ! in Definition 2.1.1
we need:
4) trivial since 1 > _1_p-1.
5) If x 2 G has filtration n = !(x) then [xp] 2 Gn+1=Gn+2 is non-
trivial, i.e. !(xp) = !(x) + 1.
6) If x 2 G satisfies !(x) > 1 + _1_p-1then !(x) 2, hence x 2 Gp.
As G is complete, we see that (G, !) is p-saturated, clearly of finite rank.
More precisely, from the above we get that grG is Fp[ffl]-free on gr1G and thus
G is equi-p-valued with an ordered basis consisting of elements of filtration
1. This settles claim 1) in case p 6= 2.
In case p = 2, ! satisfies all conditions in Definition 2.1.1, except 4), so (G*
*, !)
is in particular 2- filtered and grG has a structure of mixed Lie algebra over
F2 [L , II 1.2.5]. Note that the only 2 R+ with _1_p-1= 1 and gr G 6= 0
is gr1G. From Definition 3.3.2, ii) we have [gr1G, gr1G] = 0 which easily
implies that grG is abelian. Since ! has integer values, this means that
!([x, y]) !(x) + !(y) + 1 , x, y 2 G.
Using this, it is easy to see that !0 := ! + 1 is a filtration of G with the
properties stated in claim 1) in case p = 2.
As for claim 2), p being arbitrary now, we choose a Zp-basis {ei} M
and declare it to be a filtered basis with w(ei) = 0, i.e.
_ !
X
w ~iei = inf{v(~i)}, for ~i2 Zp
i i
Clearly, (M, w) is saturated. Assume p 6= 2.
We consider the continuous homomorphism of pro-p-groups
% : G -! 1 + pEnd Zp(M) =: G0
and claim that the lower-p-series of G0 is given by G0n= 1 + pnEnd Zp(M),
n 1. Since G0 is powerful, [DDMS , Lemma 2.4] gives G0n+1= (G0n), the
Frattini subgroup, for all n 1 and arguing inductively, it suffices to see
that (1 + pnEnd Zp(M)) = 1 + pn+1End Zp(M). Since the Frattini subgroup
is generated by p-th powers and commutators, we have " " and [DDMS ,
18 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
Proposition 1.16] then gives
(1 + pnEnd Zp(M))=(1 + pn+1End Zp(M))
= ((1 + pnEnd Zp(M)))=(1 + pn+1End Zp(M))
2
= ((Fp, +)n ) = 0.
Since % respects the lower p-series, we conclude that
%(Gn) 1 + pnEnd Zp(M) , n 1
which implies that w((g - 1)m) w(m) + !(g) for all g 2 G, m 2 M.
This settles claim 2) in case p 6= 2 and the argument in case p = 2 is an
obvious modification which we leave to the reader.
Finally, to see compatibility with cup products, assume p 6= 2 and M0, M00
satisfy the same assumptions as M does and
ff : M Zp M0 ! M00
is G-linear defining cup-products in H*c(G, -). Then both the source and
the target of ff are canonically SatZp[[G]]-modules as seen above and ff is
SatZp[[G]]-linear. Hence (6) is compatible with cup-products by Theorem
3.1.1,(4).
3.4. Proof of Theorem 3.1.1. We now describe the set-up for the rest of
the section.
We fix a saturated group (G, !) of finite rank d. Let
L*(G) = L*SatZp[[G]]
be its integral Lazard Lie-algebra. It is a finite free Zp-module.
We fix an ordered basis {x1, . .,.xd} G, and put !i:= !(xi). For every
0 k n let
Ik := {(i1, . .,.ik) | 1 i1 < . .<.ik n}
kP
and for I 2 Ik write |I| := !s. For I 2 I0 = ; we put by abuse of
s=1
notation |I| = 0.
We assume that there exists an integer e 1 such that !(G) 1_eZ and
fix a totally ramified extension Qp K of degree e with ring of integers
O K, uniformizer ss 2 O. The valuation on O is normalized by v(p) = 1.
The artificial introduction of O is a trick invented by Totaro in [T ]. In
this section all valued modules and algebras are over O. In particular, the
saturation functor is taken in the category of valued O-modules.
The inclusion Zp O induces
Fp[ffl] = grZp grO = Fp[fflK ]
where ffl (resp. fflK ) is the leading term of p 2 Zp (resp. ss 2 O). We have
ffleK2 F*p. ffl, in particular the degree of fflK is 1_e.
LAZARD ISOMORPHISM 19
If M is a valued O-module, gr(M) is canonically a Fp[fflK ]-module. As
pointed out by Totaro ([T , p. 201]) it follows directly from the definitions
that
1
gr(Sat(M)) = grM Fp[fflKF]p[fflKd]egree.0
Let
A := O[[G]] := lim- O[G=U],
U G open normal
and
B := UO (L*(G) Zp O)^ = UZp(L*(G))^ Zp O,
the completion of the universal enveloping algebra with respect to its canon-
ical filtration. (This filtration is easily seen (using Poincar'e-Birkhoff-Witt)
to be the p-adic filtration, hence the claimed equality because O is finite
free as a Zp-module.) We finally introduce using Theorem 2.6.5
C := SatA ~=SatB.
Lemma 3.4.1. gr(A) = gr(B) inside gr(C).
Proof.We have
grA = gr(Zp[[G]] ZpO) = UFp[ffl](grG) Fp[ffl]Fp[fflK ] = UFp[fflK(]grG Fp[ffl]F*
*p[fflK ]),
whereas
grB = gr(UZp(L*(G)) Zp O) = UFp[fflK(]grL*(G) Fp[ffl]Fp[fflK ]).
Since grL*(G) = grG by Theorem 2.6.5 and Theorem 2.6.3, (4) the claim
follows.
Remark 3.4.2. Moreover, Totaro shows in [T , pages 201-202] that for
t := (grG Fp[ffl]Fp[fflK1])degree 0,
a finite graded free Fp[fflK ]-Lie-algebra with generators in degree zero, we
have grC = UFp[fflK(]t).
Lemma 3.4.3. (1) Let X be a filtered free A-module with A-basis e1, . .,.e*
*r.
Then SatX is a filtered free C-module on generators
e0i= ss-ew(ei)ei, i = 1, . .,.r
(2) Let Y be a filtered free B-module with B-basis f1, . .,.fs. Then SatY
is a filtered free C-module on generators
f0j= ss-ew(fj)fj, j = 1, . .,.s
Proof.It suffices to consider the case of the algebra A. The argument for B
is the same. Without loss of generality r = 1. By construction (and because
20 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
X is torsion-free), there are embeddings
divX8PP8
rrrr PPP
rrrr PPPP
rrr PP''P
X KKK K77OoX
KKK oooo
KKK oooo
KK%% oo
(divA) A X
By assumption, any element x of K X can be written in the form
x = ssvae1 = ssv+ew(e1)ae01, 2 Z, a 2 A
It is in divX if and only if
v
w(x) = __+ w(a) + w(e1) 0
e
This is equivalent to ssv+ew(e1)a 2 divA and hence x 2 div(A)e01. Hence
divX = (divA) A X.
Finally, apply the completion functor to the equality.
Remark 3.4.4. This is the step where we make use of the coefficient ex-
tension to O.
Both A and B are canonically subrings of C, and our first aim is to
compare the cohomology of the (abstract) rings A and B with that of C.
Both A and B are augmented O-algebras, hence we have an A- (resp. B-)
module structure on O which we will refer to as trivial.
Proposition 3.4.5. (1) The trivial A-module O admits a resolution Xo
such that Xk is filtered free of rank dkover A on generators {eI | I 2
Ik} of filtration w(eI) = |I|.
(2) The trivial B-module O admits a resolution Yo such that Yk is fil-
tered free of rank dk over B on generators {fI | I 2 Ik} of filtration
w(fI) = |I|.
(3) Furthermore, Xo and Yo can be chosen such that grXo = grYo as
complexes of grA = grB-modules.
Proof. (1) The base extension from Zp to O of the quasi-minimal com-
plex of G has the desired properties [L , V 2.2.2.]. To see that the
generators have the indicated filtration, remember that the quasi-_
minimal complex is obtained by lifting the standard_complex X o
of the Fp[ffl]-Lie-algebra gr(G) which has X k = kFp[ffl](gr(G)) finite
graded free on
{xi1G+!1^ . .^.xikG+!k}.
(2) The Lie-algebra L*(G) is Zp-free on generators Log(xi) of filtration
!i. Hence the standard complex of L*(G) Zp O is as desired.
LAZARD ISOMORPHISM 21
(3) The equality grXo = grYo follows by construction from gr(G) ~=
grL*(G).
Example 3.4.6. If G is equi-p-valued, i.e, !i= !j for all i, j, then Xo and
Yo are minimal in the sense of [L , V 2.2.5], i.e., Xo Fp and Yo Fp have
zero differentials.
For the following, we fix complexes Xo and Yo satisfying the conclusion
of Proposition 3.4.5. Note that C is an augmented O-algebra with augmen-
tation extending both the one of A and the one of B.
Lemma 3.4.7. Both SatXo and SatYo are finite filtered resolutions of the
trivial C-module O, the modules SatXk (resp. SatYk) being filtered free on
generators {ss-e|I|eI | I 2 Ik} (resp. {ss-e|I|fI | I 2 Ik}) of filtration zero
over C.
Proof.Clearly, SatXo and SatYo are canonically complexes of C-modules.
Since both Xo and Yo admit the structure of a split resolution, and this
structure is preserved by the additive functor Sat, both SatXo and SatYo
are resolutions of SatO = O.
The statement on generators follows directly from Lemma 3.4.3.
For 0 k n and I 2 Ik denote by e0I2 SatXk (resp. f0I2 SatYk) the
C-generators found above, i.e. e0I:= ss-e|I|eI, f0I:= ss-e|I|fI.
We see that the canonical morphisms of complexes over C
C A Xo ,! SatXo
and
C B Yo ,! SatYo
are injective.
We pause to remark that, evidently, the above injections are isomorphisms
rationally, a key input in Lazard's comparison isomorphism for rational co-
efficients. Similarly, an integral version of this comparison isomorphism is
essentially equivalent to C A Xo being isomorphic to C B Yo and we pro-
ceed to prove this in a special case as follows.
Proposition 3.4.8. There exists an isomorphism
OE : SatXo ! SatYo
of filtered complexes over C such that grOE = id and H0(OE) is the identity
of O. Any two such OE are chain homotopic where the homotopy h can be
chosen such that gr(h) = 0.
Proof.In order to construct the isomorphism it suffices using [L , V 2.1.5]
(applicable by Proposition 3.4.7) to canonically identify the complexes grSatXo
and grSatYo of grC-modules. Recall from Proposition 3.4.5 that
grXo = grYo
22 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
This implies
grSatXo = (grXo Fp[fflKF]p[fflK1])degree 0
= (grYo Fp[fflKF]p[fflK1])degree 0= grSatYo.
Now, H0(OE) is an O-linear automorphism of O, hence given by multipli-
cation with a unit ff 2 O*. Using that its associated graded is the identity,
one easily obtains ff = 1, as claimed.
We turn to the construction of the homotopy. Let OE, OE0be isomorphisms
as above. Let e0I2 Sat(X0) be a basis element. We need to define h0(eI) 2
Sat(Y0) such that
dh0(e0I) = (OE - OE0)(e0I) =: yI
By assumption gr(OE - OE0) = 0, and hence yI 2 Sat(Y0)1_e. As OE and OE0 are
isomorphisms of resolution of O, we have ffl(yI) = 0. Recall that SatXo and
SatYo are filtered resolutions. Hence yI has a preimage "yI2 Sat(Y1)1_e. Put
h0(e0I) = "yI
By C-linearity, this defines h0. It satisfies gr(h0) = 0. As usual, the same
argument can be used inductively to define hi for all i 0.
Proposition 3.4.9. If, in the situation of Proposition 3.4.8, (G, !) is as-
sumed to be equi-p-valued, then OE restricts to an isomorphism
_ : C A Xo ! C B Yo
of complexes over C. If moreover gr(G) is generated in degree 1_e, then any
two such isomorphisms are homotopic.
Proof.We have the solid diagram of complexes over C
'"1
C A_Xo O____//SatXo
___ |
________ ' OE|
fflffl___O'"2fflffl|
C B Yo _____//_SatYo
Since the horizontal maps are injective, OE factors as a chain-map if for every
0 k n we have
(*) OEk(C A Xk) C B Yk.
If _ exists, it is necessarily an isomorphism by completeness and the fact
that its associated graded is the identity. Alternatively observe that the
following argument applies likewise to OE-1 to produce an inverse of _.
To see what (*) means, fix 0 k n and remember the C-generators
eI 2 C A Xk, e0I2 SatXk, fI 2 C B Yk and f0I2 SatYk (I 2 Ik) satisfying
'1(eI) = sse|I|e0Iand '2(fI) = sse|I|f0I. Expanding
X
OEk(e0I) = cI,Jf0J, cI,J2 C
J2Ik
LAZARD ISOMORPHISM 23
we see, using that C is saturated, that (*) for our fixed k is equivalent to
(**) 8I, J 2 Ik : w(cI,J) |J| - |I|,
w denoting the filtration of C. If (G, !) is equi-p-valued, all the differences
on the right-hand-side of (**) are zero, so that (**) is trivially true.
By Proposition 3.4.8 any two such OE are chain homotopic via a homotopy
h : SatXo ! SatYo such that gr(h) = 0. It remains to check that it restricts
to a homotopy h : C A Xo ! C B Yo. We use the same generators
as before. The additional assumption that gr(G) is generated in degree 1_e
implies |I| = k_efor I 2 Ik.
Consider e0Ifor I 2 Ik. Then hk(e0I) 2 SatYk+1 and expands as
X
hk(e0I) = dI,Jf0J, dI,J2 C
J2Ik+1
gr(h) = 0, hence ss|dI,Jfor all I, J. As e0I= ss-k eI and f0J= ss-(k+1)fJ this
implies X
hk(eI) = dI,Jss-1 fJ .
J2Ik+1
with dI,Jss-1 2 C as required.
Remark 3.4.10. It seems difficult to directly relate the complexes C A
Xo and C B Yo using the filtration techniques successfully employed for
example in [ST ] and [T ], essentially because these complexes do not satisfy
any reasonable exactness properties.
In fact, we have H*(C A Xo) = TorA*(C, O) and H*(C B Yo) = TorB*(C, O)
and one can check that, unless G = {e}, the algebra C is not flat over neither
A nor B.
We have examples of saturated but not equi-p-valued groups and an iso-
morphism OE as above which does not restrict as in Proposition 3.4.9, but in
all these examples it was possible by inspection to modify OE suitably.
It thus remains a tantalizing open problem to decide whether the assump-
tion "equi-p-valued" is superfluous in Proposition 3.4.9. Of course, a positive
answer would greatly extend the range of applicability of our integral Lazard
comparison isomorphism.
Proof of Theorem 3.1.1.There is a filtration ! of G such that (G, !) is p-
saturated, equi-p-valued of finite rank and !(G) 1_eZ for some integer e 1,
hence we are in the situation studied in this subsection and in particular
recall O, A, B, C, Xo and Yo from above. The continuous group cohomology
H*c(G, M) is defined using continuous cochains and the Bar-differential as
in [L , V 2.3.1.]. By [L , V 1.2.6 and 2.2.3.1] we have
H*c(G, M) ' H*c(Zp[[G]], M) ' Ext*Zp[[G]](Zp, M)
and analogously
H*c(G, M Zp O) ' Ext*A(O, M Zp O)
24 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
by the flatness of O over Zp. Introduce N := M Zp O. Since Xo is a finite
free resolution of O over A, we obtain
Ext*A(O, N) = H*Hom A(Xo, N) = H*Hom C(C A Xo, N)
using that the A-module structure on N extends to a C-module structure,
and then
Prop.3.4.9* *
... ' H Hom C(C B Yo, N) = H Hom B(Yo, N)
' H*(L*(G) Zp O, N) ' H*(L*(G), M) Zp O,
the last but one isomorphism by [T , Lemma 9.2]. Summing up we have
(7) H*c(G, M) Zp O ' H*(L*(G), M) Zp O
of O-modules.
We now turn to functoriality. Let f : G ! H be filtered group homo-
morphism. We use A(G), C(G), Xo(G), Yo(G) (resp. A(H), C(H), Xo(G),
Yo(G)) for the rings A, C and complexes Xo, Yo corresponding to the group
G (resp. H). The group homomorphism induces a commutative diagram
gr(f)
gr(G) _________//gr(H)
~=|| |~=|
fflffl| fflffl|
gr(L*(H)) _____//gr(L*(H))
As in Proposition 3.4.8 it lifts to a diagram of filtered complexes of SatA(G)-
modules
SatXo(G) _____//SatXo(H)
|~=| |~=|
fflffl| fflffl|
SatYo(G) _____//_SatYo(H)
which commutes up to homotopy and such that taking gradeds gives back
the previous diagram, and such that taking gradeds of the homotopy is 0.
As in Proposition 3.4.9 it restricts to a diagram of filtered complexes of
C(G)-modules
C(G) Xo(G) _____//C(G) Xo(H)
~=|| |~=|
fflffl| fflffl|
C(G) Yo(G) _____//_C(G) Yo(H)
which commutes up to homotopy.
Compatibility with cup-products is the case : G ! G x G. Note that
the generators of G x G are of the form (x, 1) and (1, y) for generators x, y
of G. Their filtration is the same as that of x, y.
LAZARD ISOMORPHISM 25
4. The Lazard isomorphism for algebraic group schemes
In this section we give, in the case of p-adic Lie-groups arising from alge-
braic groups, a direct description of a map from analytic group cohomology
to Lie-algebra cohomology by differentiating the cochains. In 4.2.4 we show
that this coincides with Lazard's isomorphism.
4.1. Group schemes. Let p be a prime number, K be a finite extension of
Qp, let R be its ring of integers with prime element ss. Throughout G will
be a separated smooth group scheme over R and g its R-Lie-algebra in the
following sense:
g = Lie(G) = DerR (OG,e, R) .
Then gK := g R K = Lie(GK ) is its Lie-algebra as a K-manifold.
Note that this category is stable under base change and Weil restriction
for finite flat ring extensions R ! S. If A ! B is a ring extension with B
finite and locally free over A and X an A-scheme, we write XB = X xA
SpecB. If Y is a B-scheme, we write ResB=A X for the Weil restriction, i.e.,
ResB=A X(T ) = X(TB ) for all A-schemes T . See [BLR , x7.6] for properties
of the Weil restriction. In particular, if G is a group scheme over a discrete
valuation ring R, then G is quasi-projective by [BLR , x6.4, Theorem 1]. This
suffices to guarantee that ResS=R(G) exists for finite extensions S=R.
The following bit of algebraic geometry will be needed in the proofs.
Lemma 4.1.1. Let L=K be finite extension, S the ring of integers of L.
Consider a separated smooth group scheme G over S. Then G is a direct
factor of ResS=R(G)S.
Proof.Let X be an S-scheme. For all S-schemes T we describe T -valued
points of ResS=R(X)S:
Mor S(T, ResS=R(X)S) = Mor R(T, ResS=R(X)) = Mor S(T xR SpecS, X)
= Mor S(T xS Spec(S R S), X)
The natural map ' : S ! S R S which maps s 7! s 1 induces the
transformation of functors
Mor S(T xSSpec S, X) '-!MorS (T xSSpec (S R S), X) = Mor S(T, ResS=R(X)S)
and hence a morphism
' : X ! ResS=R(X)S
This is nothing but the adjunction morphism.
The multiplication ~S : S R S ! S is a section of '. This again induces
a transformation of functors
Mor S(T, ResS=R(X)S) = Mor S(T xS Spec(S R S), X) ~S--!MorS(T xS S, X)
and hence a morphism
~S : ResS=R(X)S ! X .
26 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
(Put T = ResS=R(X)S and the identity on the left.) By construction ~S is a
section of '. Both are natural in X, hence G is a direct factor of ResS=R(G)S
as group schemes.
Remark 4.1.2. If L=K is Galois of degree d, then ResL=K (G)L ~=Gd. This
carries over to the integral case if the extension is unramified. The assertion
becomes false for ramified covers. Note, however, that the weaker statement
of the lemma remains true.
4.2. Analytic description of the Lazard morphism. Let G be a smooth
connected group scheme over R with Lie-algebra g. Let G G(R) be an
open sub-Lie-group.
We denote by Ola(G) (locally) analytic functions on G, i.e. those that
can be locally written as a converging power series with coefficients in K.
We denote by Hila(G, K) (locally) analytic group cohomology, i.e., cohomol-
ogy of the bar complex Ola(Gn)n 0 with the usual differential. We denote
Hi(g, K) Lie-algebra cohomology, i.e. cohomology of the complex *(g_K)
with differential induced by the dual of the Lie-bracket.
Definition 4.2.1. The Lazard morphism is the map
: Hila(G, K) ! Hi(g, K)
induced by the morphism of complexes
Ola(Gn) ! (gnK)_ ! ng_K
f 7! dfe
Remark 4.2.2. It is not completely obvious that is a morphism of com-
plexes. See [HK , Section 4.6. and Section 4.7.].
Remark 4.2.3. is compatible with the multiplicative structure.
Recall from Lemma 2.2.2 that in the case K = Qp, the kernel G of
G(Zp) ! G(Fp) is filtered and has a subgroup G of finite index which is
saturated and equi-p-valued. Indeed for p 6= 2, we have G = G.
Let L* = L*(G) be its integral Lazard Lie-algebra (see Definition 2.6.6).
As reviewed in Example 2.6.8 there is a natural isomorphism
g Qp ~=L* Qp .
Proposition 4.2.4. [HK , Theorem 4.7.1] For K = Qp and G saturated, the
Lazard morphism (see Definition 4.2.1) agrees under the identification of
Lie-algebras in Example 2.6.8 with the isomorphism defined by Lazard ([L ,
V 2.4.9, V 2.4.10])
In particular, is an isomorphism in this case.
Remark 4.2.5. This is a case where our integral version of the result (The-
orem 3.1.1) can be applied. As shown there this is again the same isomor-
phism.
LAZARD ISOMORPHISM 27
4.3. The isomorphism over a general base.
Theorem 4.3.1. Let G be a smooth group scheme over R with connected
generic fiber and G G(R) an open subgroup. Then the Lazard morphism
(see Definition 4.2.1) is an isomorphism.
The proof will take the rest of this note.
Remark 4.3.2. Let us sketch the argument. We are first going to show
injectivity. For this we can restrict to smaller and smaller subgroups G and
even to their limit. In the limit, the statement follows by base change from
Lazard's result for R = Zp. We then show surjectivity. Finite dimensionality
of Lie-algebra cohomology implies that the morphism is surjective for small
enough G. Algebraicity then implies surjectivity also for the maximal G.
By construction, the Lazard morphism depends only on an infinitesimal
neighborhood of e in G. Hence it factors through the Lazard morphism for
all open sub-Lie-groups of G and even through its limit
1 : lim-!Hila(G0, K) ! Hi(g, K)
G0 G
Lemma 4.3.3. The limit morphism 1 is an isomorphism.
Proof.For K = Qp this holds by Proposition 4.2.4 and the work of Lazard
[L , V 2.4.9 and V 2.4.10].
The system of open sub-Lie-groups of G is filtered, hence
lim-!Hila(G0, K) = Hi(Ola(Go)e)
G0 G
where Ola(Gn)e is the ring of germs of locally analytic functions in e. Note
that G(Zp) also carries the structure of a rigid analytic variety and germs of
locally analytic functions are nothing but germs or rigid analytic functions.
Hence they can be identified with a limit of Tate algebras.
First suppose that G = HK for a smooth group scheme H over Qp. Then
Ola(Gn)e ~=Ola(H(Zp)n) K
because Tate algebras are well-behaved under base change (see [BGR , Chap-
ter 6.1, Corollary 8]). Moreover, 1 is compatible with base change. As it
is an isomorphism for H it is also an isomorphism for G.
Now consider general G. By Lemma 4.1.1 G is direct factor of some
group of the form HK with H a group over Zp. Indeed, H = ResR=Zp(G).
By naturality, 1,G is a direct factor of 1,HR and hence by the special
case an isomorphism.
Corollary 4.3.4. is injective.
Proof.G(R) is compact, hence all open sub-Lie-groups are of finite index.
If G0 G is an open normal subgroup, we have
0
Hila(G, K) ~=Hila(G0, K)G=G
28 ANNETTE HUBER, GUIDO KINGS AND NIKO NAUMANN
Hence the restriction maps
Hila(G, K) ! Hila(G0, K)
are injective. As the system of open normal subgroups is filtered, this also
implies that
Hila(G, K) ! lim-!Hila(G0, K)
G0
is injective. The injectivity of follows from the injectivity of 1 .
Lemma 4.3.5. Let G G(R) be an open subgroup. Then there is an open
subgroup H G such that the Lazard morphism for H is bijective.
Remark 4.3.6. Note that this is precisely what Lazard proves over Qp with
H the saturated subgroup of G = G(Zp).
Proof.As 1 is bijective and injective, it suffices to show that there is H
such that the restriction map
Hila(H, K) ! lim-!Hila(G0, K)
G0
is surjective. Let ff be a cocycle with class [ff] 2 lim-!G0Hila(G0, K). By
definition it is represented by a cochain on some G0. It is a cocycle (possibly
on some smaller G0). Hence [ff] is in the image of the restriction map for G0.
Lie-algebra cohomology is finite dimensional by definition, hence this is
also true for lim-!G0Hila(G0, K). By intersecting the G0 for a basis we get the
group H we wanted to construct.
Proof of Theorem 4.3.1.Injectivity has already been proved in Corollary
4.3.4. We use an argument of Casselman-Wigner [CW , x3] to conclude.
The operation of GK on Hi(g, K) is algebraic. Hence the stabilizer SK is a
closed subgroup of GK . On the other hand it contains some open subgroup
of G(R). This implies that SK = GK because GK is connected. Hence
G G(R) operates trivially and thus
: Hila(G, K) ! Hi(g, K)
is surjective.
Remark 4.3.7. The argument also works for cohomology with coefficients
in a finite dimensional algebraic representation of the group.
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