Universit¨at Regensburg Mathematik

Universit¨ at Regensburg Mathematik

Comparison of the Karoubi regulator and the p-adic Borel regulator

Georg Tamme

Preprint Nr. 21/2007

Universit¨ at Regensburg Mathematik

Comparison of the Karoubi regulator and the p-adic Borel regulator

Georg Tamme

Preprint Nr. 21/2007

Comparison of the Karoubi regulator and the p-adic Borel regulator

Georg Tamme 25th July 2007

Abstract

We give an explicit locally analytic cocycle which – composed with the Hurewicz map from algebraic K-theory to group homology – gives the p-adic Borel regulator defined by Annette Huber and Guido Kings for the K-theory of ap-adic field. Using this explicit cocycle we show that the p-adic Borel regulator equals Karoubi’s regulator up to a constant.

Contents

1 Karoubi’sp-adic regulator 2

1.1 Relative K-theory . . . 2 1.2 The regulator . . . 3 2 An explicit description of the p-adic Borel regulator 5 2.1 The construction of thep-adic Borel regulator . . . 5 2.2 The Lazard isomorphism . . . 5 2.3 An explicit cocycle . . . 6

3 Comparison of the two regulators 8

A Integration on the standard simplex 10

A.1 The ringAhx0, . . . , xni . . . 10 A.2 Integration of differential forms . . . 11 A.3 Dependence on parameters . . . 12

Introduction

LetK be a finite extension ofQp with valuation ring R. In [4] Annette Huber and Guido Kings introduce a regulatorb_p:K_2n−1(R)→Kand relate it via the Bloch-Kato exponential map to Soul´e’s regulator K_2n−1(K)→H¹(K,Qp(n)).

The definition ofbpparallels Borel’s construction of his regulator K_2n−1(C)→ C, only that the van Est isomorphism is replaced by the Lazard isomorphism

H_la²ⁿ⁻¹(GL_N(R), K)'H²ⁿ⁻¹(gl_N, K) between locally analytic group cohomol- ogy and Lie algebra cohomology. Huber and Kings describe explicitely an el- ement pn in the Lie algebra cohomology and by definition the p-adic Borel regulator b_p is the composition of the Hurewicz map from K-theory to group homology and the preimage of p_n under Lazard’s isomorphism.

There is another construction of a p-adic regulator from the relative K-theory of K to K due to Max Karoubi [5] which uses the Chern character ch^rel_2n−1 : K_2n−1^rel (K)→ HC_2n−2(K) from relative K-theory to topological cyclic homol- ogy and the periodicity operator HC_2n−2(K) → HC0(K) = K. There is a canonical isomorphismK_2n−1^rel (K)Q'K_2n−1(K)Q which allows us to compare both regulators. Karoubi’s regulator also factors through the Hurewicz map to the rational homology of a certain simplicial set and Nadia Hamida [3] described an explicit simplicial cocycle which gives this regulator.

In the present paper we show that Hamida’s cocycle induces a locally analytic cocycle for the group GL_N(R) and prove that this cocycle equals the one defining thep-adic Borel regulator up to a constant.

In the first section we shortly sketch Karoubi’s construction and recall the rel- evant results of Hamida. In the second section we recall the construction of the p-adic Borel regulator and construct a locally analytic cocycle similar to Hamida’s simplicial cocycle which gives thep-adic Borel regulator. In the third section we finally prove that both regulators agree up to a constant under the identificationsK_2n−1^rel (K)Q'K2n−1(K)Q 'K2n−1(R)Q. Since the explicit co- cycles involve the integration of a p-adic differential form over the standard simplex, we have included an appendix where the technical questions of inte- gration are discussed.

1 Karoubi’s p-adic regulator

1.1 Relative K-theory

Let A be an ultrametric Banach ring (cf. [1]), i.e. a ring A equipped with a ultrametric“quasi-norm” k.k :A →R+ verifying kak = 0 ⇐⇒ a= 0,kak = k −ak, kabk ≤ kakkbk, ka+bk ≤ max{kak,kbk}, for which it is complete.

Let Ahx₀, . . . , x_nidenote the ring of power series P

I∈Nⁿ⁺¹₀ a_Ix^I with a_I ∈A, x= (x0, . . . , xn) and kaIk|I|^r −−−−→^|I|→∞ 0 for everyr >∈N0. We call it the ring ofindefinitely integrable power series with coefficients inA.Ahx0, . . . , x_niis an ultrametric Fr´echet ring where the topology is given by the family of seminorms pr, r ∈ N0, pr(P

aIx^I) = sup_IkaIk · |I|^r (see the appendix for details). Let In ⊂Ahx0, . . . , xni be the principal ideal generated byx0+· · ·+xn−1 and defineA_n:=Ahx₀, . . . , x_ni/I_n.

A_∗ defines a simplicial ring with faces∂i and degeneraciessi induced by

∂i(xj) =







x_j ifj < i, 0 ifj=i, xj−1 ifj > i,

si(xj) =







x_j ifj < i, xi+xi+1 ifj=i, xj+1 ifj > i.

The classifying space BGL(A_∗) of the simplicial group GL(A_∗) is an H-space and by [6] 6.17 the natural map BGL(A)→BGL(A_∗) induces a homotopy fibre sequence

(GL(A_∗)/GL(A))⁺ −→^θ BGL(A)⁺→BGL(A_∗)

where (.)⁺ is Quillen’s +-construction andθis induced by the map GL(An)3 σ 7→ (σ(0)σ(1)⁻¹, . . . , σ(n−1)σ(n)⁻¹) ∈ BnGL(A) = GL(A)^×n. Hereσ(i) = (i)^∗σwhere (i) : [0]→[n] is the morphism in the simplicial category that sends 0∈[0] toi∈[n] (“the value ofσon thei^th vertex of the standard simplex”).

The topological and relativeK-groups ofAare by definition (cf. [5], [7]) K_n^top(A) :=πn(BGL(A∗)) and K_n^rel(A) :=πn (GL(A∗)/GL(A))⁺

.

In particular there are long exact sequences

· · · →K_n^rel(A)→Kn(A)→K_n^top(A)→K_n−1^rel (A)→. . . .

We are particularly interested in the case whereA=K is a finite extension of Q_p with residue fieldk. In this situation Adina Calvo ([1]) shows that one has exact sequences

0→Kn(k)→K_n^top(K)→K_n−1(k)→0.

By Quillen’s result on the K-groups of finite fields K_2i+1^top (K) = K_2i+2^top (K) = K_2i+1(k) is finite for i > 0. In particular the canonical map K_n^rel(K)_Q → Kn(K)Qis an isomorphism forn >2 (here we writeK_n^rel(K)QforK_n^rel(K)⊗ZQ etc.). Furthermore if R denotes the ring of interges in K there are canonical isomorphisms K_n^top(R)−^'→ K_n^top(k)' K_n(k). The localization sequence in al- gebraic K-theory yields isomorphismsKn(R)Q→Kn(K)Q forn >2. Thus we have canonical isomorphisms

K_n^rel(R)_Q ^' //

'

Kn(R)Q '

K_n^rel(K)_Q ^' //Kn(K)Q.

1.2 The regulator

We sketch the construction of Karoubi’sp-adic regulator as in [3]. More details may be found in [2].

Let K be a finite extension of Qp with ring of integers R and uniformiz- ing parameter π. Let

C∗^λtop(K), b

denote the complex defining the topo- logical cyclic homology (with ground ring Q) HC_∗^top(K) of K (see e.g. [6], [5]). Karoubi has constructed a relative Chern character ch^rel_2n−1:K_2n−1^rel (K)→ C_2n−2^λtop(K)

b

C_2n−1^λtop(K)

and Hamida proves that its image is contained in the subgroupHC_2n−2^top (K) ofC_2n−2^λtop(K)

b

C_2n−1^λtop(K)

([3] Def.-Prop. 2.1.2).

Definition. Thep-adic regulatorr_p is defined to be the composition rp:K_2n−1^rel (K) ^ch

rel

−−−−→2n−1 HC_2n−2^top (K)−→^S HC₀^top(K) =K, whereS is the (n−1)-fold iterate of Connes’ periodicity operator.

In order to compare the above regulator with the p-adic Borel regulator we need the explicit description of r_p given by Hamida which uses Goodwillie’s relative K-theory. We recall the relevant definitions and facts from [8] ch. 11.

For a ring A and a two-sided ideal I in A denote by K(A, I) the connected component of the basepoint of the homotopy fiber of BGL(A)⁺→BGL(A/I)⁺. For n ≥ 1, K_n(A, I) is defined to be π_n(K(A, I)). The space K(A, I) has a Volodin modelX(A, I) constructed as follows: For any orderingγof{1, . . . , n}

define T_n^γ(A, I) to be the subgroup {1 + (a_ij) ∈ GL_n(A) | a_ij ∈ I ifi

γ

6< j}

of GLn(A). ThenX(A, I) is the union of classifying spaces S

n,γBT_n^γ(A, I) in BGL(A). We considerX(A, I) also as a simplicial subset of BGL(A).

Proposition 1.1 ([8] Prop. 11.3.6, Cor. 11.3.8). There is a natural homo- topy equivalence X(A, I)⁺ ≈ K(A, I). In particularKn(A, I) =πn(X(A, I)⁺).

Moreover the direct sum of matrices induces an H-space structure onX(A, I)⁺ so that K_n(A, I)_Q is isomorphic to the primitive part PrimH_n(X(A, I),Q) of the rational homology ofX(A, I) via the Hurewicz homomorphism.

Now letA=RandI=πRthe maximal ideal ofR. Hamida proves the following Proposition 1.2 ([3] Thm. 1.3). There exists an isomorphism

Kn(R, πR)−^'→K_n^rel(R).

It is induced by the simplicial map ϕ:X(R, πR)→GL(R_∗)/GL(R)that sends (g1, . . . , gr)∈T^γ(R, πR)^×r⊂Xr(R, πR)toPr

i=0xigi+1· · ·gr∈GL(Rr).

It is not a priori clear that Pr

i=0x_ig_i+1· · ·g_r is invertible in Mat(R_r). For a proof of a similar statement see lemma 2.3. Now the explicit description of the regulatorrp is the following

Proposition 1.3 ([3] Prop 2.1.3). The composition K2n−1(R, πR)−→^ϕ

' K_2n−1^rel (R)→K_2n−1^rel (K)−→^rp K is equal to the composition

K_2n−1(R, πR)→H_2n−1(X(R, πR))−→^φ K,

where the first arrow is the Hurewicz map andφis given by the simplicial cocycle that sends(g₁, . . . , g_2n−1)∈T^γ(R, πR)^×(2n−1) to

(−1)ⁿ(n−1)!

(2n−1)!(2n−2)!Tr Z

∆²ⁿ⁻¹

(dν·ν⁻¹)²ⁿ⁻¹∈K with ν=P2n−1

i=0 xigi+1· · ·g2n−1∈GL(R2n−1).

See the appendix for the definition of the integral.

2 An explicit description of the p-adic Borel reg- ulator

2.1 The construction of the p-adic Borel regulator

We recall the construction of thep-adic Borel regulator by Annette Huber and Guido Kings in [4]. LetK/Qp be a finite extension with ring of integersR and uniformizing parameterπ. Define UN(R) = 1 +πMatN(R)⊂GLN(R) and let gl_N denote theK-Lie algebra of GL_N. Forn≤N the mapV2n−1

gl_N →K, (X1, . . . , X2n−1)7→ ((n−1)!)²

(2n−1)!

X

σ∈S2n−1

sgn(σ)Tr(Xσ(1)· · ·X_σ(2n−1)), where S_2n−1 is the symmetric group, Tr ist the trace map gl_N →K and · is matrix multiplication, defines the primitive elementpn ∈H²ⁿ⁻¹(gl_N, K).

In [4] Huber and Kings prove the following version of Lazard’s theorem:

Theorem 2.1. There are natural isomorphisms

H_la²ⁿ⁻¹(GL_N(R), K)−^'→H_la²ⁿ⁻¹(U_N(R), K)−^'→H²ⁿ⁻¹(gl_N, K)

between the locally analytic group cohomology and the Lie algebra cohomology.

They also give an explicit description of this isomorphism which will be recalled in the next section.

Letb_p be the image of p_n under the composition H²ⁿ⁻¹(gl_N, K)←^'−H_la²ⁿ⁻¹(GLN(R), K) “forget la”

−−−−−−→H²ⁿ⁻¹(GLN(R), K).

Definition. Thep-adic Borel regulator K_2n−1(R)→K is defined to be the composition

K2n−1(R)−−−−−−→^Hurewicz H2n−1(GL(R),Q)'H2n−1(GLN(R),Q)−→^bp K.

Here one uses the fact that the canonical homomorphismH_2n−1(GLN(R),Q)→ H_2n−1(GL(R),Q) is an isomorphism ifN is big enough (depending onn).

2.2 The Lazard isomorphism

LetG= GLN(R) considered as a K-Lie group with unit element eand Lie al- gebragl_N. By [4], section 5, the Lazard isomorphismH_laⁿ(G, K)−^'→Hⁿ(gl_N, K) is induced by the map

Φ :O^la(G^×n)' O^la(G)^⊗nˆ →

n

^gl^∨_N, f1⊗ · · · ⊗fn 7→df1(e)∧ · · · ∧dfn(e), where O^la denotes the ring of locally analytic functions anddf(e) is the differ- ential off ate.

Now let exp be the exponential map of Gdefined on a neighbourhood of zero ingl_N. For a locally analytic functionf ∈ O^la(G^×n) we define ∆f ∈Vn

gl^∨_N by

∆f(X1, . . . , Xn) = X

σ∈Sn

sgn(σ) dⁿ dt1. . . dtn

f(exp(t1Xσ1), . . . ,exp(tnXσn)) _t=0

Iff is of the special formf =f₁⊗ · · · ⊗f_n one has d

dti

f(exp(t1Xσ1), . . . ,exp(tnXσn)) _t

i=0=

=f1(exp(t1Xσ1)). . . dfi(e)(Xσi). . . fn(exp(tnXσn)) and therefore

∆f(X₁, . . . , X_n) = X

σ∈Sn

sgn(σ)df₁(e)(X_σ1)· · ·df_n(e)(X_σn)

= df₁(e)∧ · · · ∧df_n(e)(X₁, . . . , X_n) = Φ(f)(X₁, . . . , X_n).

Since the functions of the form f1 ⊗ · · · ⊗fn are topological generators of O^la(G^×n) and Φ and ∆ are continuous for the Fr´echet topology on O^la(G^×n) we have proven

Proposition 2.2. The Lazard isomorphism H_laⁿ(GL_N(R), K)−^'→ Hⁿ(gl_N, K) is induced by ∆ :O^la(G^×n)→Vn

gl^∨_N.

2.3 An explicit cocycle

Recall the ringR_n=Rhx₀, . . . , x_ni/(Px_i−1) from section 1.1.

Lemma 2.3. Letg1, . . . , gn be elements ofUN(R). Thenν=Pn

i=0xigi+1· · ·gn

is invertible, i.e. lies inGLN(Rn).

Proof. Write g_i+1· · ·g_n = 1−h_i with h_i ∈ πMat_N(R). Then ν =Pn i=0x_i− Pn

i=0xihi = 1−Pn

i=0xihi =: 1−h. We show that P

k∈Nh^k converges in MatN(Rhx0, . . . , xni). Its image in MatN(Rn) will be an inverse ofν. Letpr, r∈ N0, be the family of seminorms defining the Fr´echet topology on Rhx0, . . . , xni and extendp_rin the obvious way to matrices. Sinceh^kis homogeneous of degree kwe have

pr(h^k)≤k^rmax

|I|=kkh0kⁱ⁰· · · khnkⁱⁿ≤k^rc^k

where c:= maxi=0,...,nkhik<1 andI runs through multiindices inNⁿ⁺¹₀ . But k^rc^k tends to zero as k tends to infinity and so P

k∈Nh^k indeed converges to an element of Mat_N(Rhx₀, . . . , x_ni).

Since thep-adic Borel regulatorbp∈H²ⁿ⁻¹(GL(R), K) is the image of a locally analytic cocycle andH_la²ⁿ⁻¹(GLN(R), K)'H_la²ⁿ⁻¹(UN(R), K),bpis determined by a locally analytic cocycleU_N(R)^×(2n−1)→K.

Theorem 2.4. The p-adic Borel regulator b_p is given by the locally analytic cocyclef :UN(R)^×(2n−1)→K,

f(g₁, . . . , g_2n−1) =−((n−1)!)² (2n−1)! Tr

Z

∆²ⁿ⁻¹

(dν·ν⁻¹)²ⁿ⁻¹ whereν =ν(g1, . . . , g_2n−1) =P2n−1

i=0 xigi+1· · ·g_2n−1∈GLN(R_2n−1).

Proof. The fact thatf is locally analytic is proven in the appendix (proposition A.8). For the proof thatf indeed defines a cocycle cf. [2], Prop. II 3.3.1.

We have to show that f is mapped to the primitive element p_n of section 2.1 under the Lazard isomorphism, i.e. ∆(f) =pn. Write∂iinstead of _dt^d

i. We have

∆(f)(X1, . . . , X_2n−1) =−((n−1)!)² (2n−1)!

X

σ∈S2n−1

sgn(σ)∂1. . . ∂_{2n−1 t}

1=···=0

Tr Z

∆²ⁿ⁻¹

(dν·ν⁻¹)²ⁿ⁻¹(exp(t₁X_σ1), . . . ,exp(t_2n−1X_σ(2n−1))).

By proposition A.7 we may interchange differentiation and integration. Let us first consider theσ= 1 summand. Write

ω :=

"_2n−1 X

i=0

dxiexp(ti+1Xi+1)· · ·exp(t_2n−1X_2n−1)

# ,

ω⁰ :=

"_2n−1 X

i=0

xiexp(ti+1Xi+1)· · ·exp(t_2n−1X_2n−1)

# .

Then

(dν·ν⁻¹)²ⁿ⁻¹(exp(t₁X₁), . . . ,exp(t_2n−1X_2n−1)) = (ω·ω⁰⁻¹)²ⁿ⁻¹ and

∂₁. . . ∂_2n−1(dν·ν⁻¹)²ⁿ⁻¹(exp(t₁X₁), . . . ,exp(t_2n−1X_2n−1)) _t=0=

=∂1. . . ∂2n−1(ω·ω⁰⁻¹)²ⁿ⁻¹ _t=0=

2n−1

X

i₁=1

· · ·

2n−1

X

i_2n−1=1

· · ·∂j ω·ω⁰⁻¹

· · · _t=0.

The last product in the sum is a product of 2n−1 factors (ω·ω⁰⁻¹) with ∂j

in front of the i^th_j factor (of course there may be several ∂’s in front of one factor). Note that ω

_t=0 = P2n−1

i=0 dxi = 0, so in the last sum all summands with (i₁, . . . , i_2n−1) not a permutation of (1, . . . ,2n−1) vanish. On the other hand using ω⁰

_t=0=P2n−1

i=0 xi= 1 we get

∂_j ω·ω⁰⁻¹ _t=0= (∂_jω)

_t=0·ω⁰⁻¹

_t=0+ω

_t=0·(∂_jω⁰⁻¹) _t=0=

= (∂jω) _t=0=

j−1

X

i=0

dxi·Xj.

Alltogether we obtain

∂₁. . . ∂_2n−1(dν·ν⁻¹)²ⁿ⁻¹(exp(t₁X₁), . . . ,exp(t_2n−1X_2n−1)) _t=0=

= X

τ∈S2n−1





τ(1)−1

X

i=0

dx_i·X_τ(1)



· · ·





τ(2n−1)−1

X

i=0

dx_i·X_τ(2n−1)





= X

τ∈S2n−1

X_τ(1)· · ·X_τ(2n−1)





τ(1)−1

X

i=0

dxi



· · ·





τ(2n−1)−1

X

i=0

dxi





= X

τ∈S2n−1

sgn(τ)X_τ(1)· · ·X_τ(2n−1)dx0dx1. . . dx_2n−2. It follows that

X

σ∈S2n−1

sgn(σ)∂1. . . ∂_2n−1

(dν·ν⁻¹)²ⁿ⁻¹(exp(t₁X_σ1), . . . ,exp(t_2n−1X_σ(2n−1))) _t=0=

= X

σ∈S2n−1

sgn(σ) X

τ∈S2n−1

sgn(τ)Xστ(1)· · ·X_στ(2n−1)dx0dx1. . . dx2n−2=

= (2n−1)! X

σ∈S2n−1

sgn(σ)X_σ(1)· · ·X_σ(2n−1)dx₀dx₁. . . dx_2n−2. Because

Z

∆²ⁿ⁻¹

dx0. . . dx_2n−2=− Z

∆²ⁿ⁻¹

dx_2n−1dx1. . . dx_2n−2

=− Z

∆²ⁿ⁻¹

dx1. . . dx_2n−1=− 1 (2n−1)!

(cf. the explicit formula in the proof of proposition A.4) we finally obtain

∆(f)(X1, . . . , X_2n−1) = ((n−1)!)² (2n−1)!

X

σ∈S2n−1

sgn(σ)Tr(Xσ1· · ·X_σ(2n−1)),

that is ∆(f) =pn.

3 Comparison of the two regulators

Theorem 3.1. Forn >1, the diagram K_2n−1^rel (K)Q

' //

r_p

''P

PP PP PP PP PP

PP K_2n−1(K)_Q oo ^' K_2n−1(R)_Q

(−1)n−1 (n−1)!(2n−2)!b_p

wwnnnnnnnnnnnnn

K is commutative.

Proof. We have a commutative diagram K_2n−1^rel (K)Q

' //K_2n−1(K)_Q

K2n−1(R, πR)Q

' //

'

K_2n−1^rel (R)Q '

OO

' //K2n−1(R)Q '

OO

'

PrimH_2n−1(X(R, πR),Q) ^β //PrimH_2n−1(BGL(R),Q) whereβ is induced by the composition

X(R, πR)−→^ϕ GL(R∗)/GL(R)−→^θ BGL(R)

(see section 1.1 for the definition ofθ and proposition 1.2 for that ofϕ) which is just the natural inclusionX(R, πR)⊂BGL(R) as one easily checks. We also denote byβ the induced map on homology. To prove the theorem it suffices to show that

H_2n−1(X(R, πR),Q) ^β //

r_p

''P

PP PP PP PP PP

PP H_2n−1(BGL(R),Q)

(−1)n−1 (n−1)!(2n−2)!bp

wwoooooooooooo

K commutes.

It follows from the long exact sequence of relative K-theory and the finiteness of K_i(R/πR) for i >0 that K_i(R, πR)_Q →K_i(R)_Q is an isomorphism for all i > 0. Thus we know that X(R, πR)⁺ → BGL(R)⁺ induces an isomorphism on the subspaces of primitive elements in rational homology and it follows from the theorem of Cartan-Milnor-Moore that it induces in fact an isomorphism H_∗(X(R, πR),Q)−^β→

' H_∗(BGL(R),Q).

Since for each N the subgroup UN(R) has finite index in GLN(R) it follows that H2n−1(BU(R),Q) → H2n−1(BGL(R),Q) is surjective where U(R) = lim−→U_N(R). NextBU(R) is actually contained inX(R, πR) and thus we have a commutative diagram

H_2n−1(BU(R),Q)

α

γ

**T

TT TT TT TT TT TT TT T

H2n−1(X(R, πR),Q) _'^β //H2n−1(BGL(R),Q)

withγ and hence alsoαsurjective. Now rp is given by the cocycleφof propo- sition 1.3 andbp◦γ is given by the cocyclef of theorem 2.4. From the explicit formulae forφandf it is clear thatφ◦α= (n−1)!(2n−2)!^(−1)n−1 f◦γwhich proves the theorem.

A Integration on the standard simplex

A.1 The ring Ahx

, . . . , x

i

LetAbe an ultrametric Banach ring. For simplicity writeAhxiforAhx0, . . . , x_ni (cf. section 1.1). Recall also the family of seminormsp_r,r∈N₀,p_r(Pa_Ix^I) = sup_IkaIk · |I|^r. We write alsok.kr forpr.

Proposition A.1. Ahxiis a sub-A-algebra of the algebra of formal power series with coefficients in A. Its underlying module is an ultrametric Fr´echet module.

Furthermore

kf·gk_r≤

r

X

s=0

r s

kfk_skgk_s−r. Proof. Letf =P

Ia_Ix^I andg=Pb_Ix^I be inAhxi. We want to show thatf · g=P

IcIx^IwithcI =P

K+L=IaKbLis also inAhxi. Fix a non negative integer rand letA:= sup_K(Pr

s=0 r s

|K|^skaKk),B:= sup_K(Pr s=0

r s

|K|^r−skbKk).

Given ε >0 choose N >0 such that|K|^skaKk< _B^ε, |K|^r−skbKk < _A^ε for all s= 0, . . . , rand|K| ≥N. Then for every|I| ≥2N we have

|I|^rkcIk ≤ max

K+L=I((|K|+|L|)^rkaKkkbLk)

= max

K+L=I r

X

s=0

r s

|K|^ska_Kk|L|^r−skb_Lk

!

< ε.

Thusf·gin fact belongs toAhxi. The assertion onkf·gkrfollows immediately from the above computation.

It remains to show thatAhxiis complete. This is easy.

Remark. Let |f|s =Ps r=0

1

r!kfkr. Then it follows from the above proposition that (|.|s)s∈Nis a family of seminorms which defines the same topology onAhxi and satisfies|f ·g|s≤ |f|s|g|s.

Now letφ: [n]→[m] be a monotone map (a morphism in the simplicial category

∆). We want to defineφ^∗:Ahx0, . . . , xmi →Ahx0, . . . , xnibyxi7→P

φ(j)=ixj, i = 0, . . . m. We have to show thatφ^∗ is well defined and continuous. Slightly more generally we have:

Lemma A.2. Let g = (g0, . . . , gm) be a tuple of polynomials of degree 1 in Ahy0, . . . , yniwith integral coefficients. Then forf =P

IaIx^I ∈Ahx0, . . . , xmi the formal composition f◦g =P

Ia_Ig^I lies in Ahy0, . . . , y_niand f 7→f ◦g is continuous.

Proof. g^I is a polynomial of degree ≤ |I| with integral coefficients and thus kg^Ikr≤ k1k·|I|^r. SincekaIk|I|^rtends to zero whren|I|tends to infinity it follows that (P

|I|≤na_Ig^I)_n∈N is a Cauchy sequence. Since Ahy₀, . . . , y_ni is complete this Cauchy sequence converges and the limit isf ◦g.

Moreover kf ◦gkr = kP

IaIg^Ikr ≤ sup_IkaIg^Ikr ≤ sup_IkaIk · k1k · |I|^r = k1k · kfkr and thusf 7→f◦g is continuous.

Recall thatI_n ⊂Ahx₀, . . . , x_niis the principal ideal generated byx₀+· · ·+x_n−1 andAn :=Ahx0, . . . , xni/In.

Lemma A.3. The homomorphism η:Ahx0, . . . , xni →Aht1, . . . , tnithat sends x_i to t_i for i >0 and x₀ to 1−t₁− · · · −t_n induces an isomorphism A_n → Aht₁, . . . , t_ni.

Proof. We have the obvious continuous section ι: ti 7→xi, i > 0, so thatη is surjective. Assume f =P

IaIx^I is in the kernel ofη. Then f = f −ι(η(f)) =

= X

I

aI x^I−(1−x1− · · · −xn)ⁱ⁰xⁱ₁¹· · ·xⁱ_nⁿ

= X

I

aI·gI·(x0+· · ·+xn−1)

where thegI are polynomials with integral coefficients of total degree ≤ |I|. In particular kgIkr ≤ k1k · |I|^r and thus P

IaIg^I is an element of Ahx0, . . . , xni which satisfies (P

IaIg^I)·(x0+· · ·+xn−1) =f.

If f ∈ I_m then clearly φ^∗(f) ∈ I_n and thus there are induced continuous ho- momorphisms φ^∗ :A_m →A_n for every φ: [n]→[m] which make [n]7→A_n a simplicial Fr´echet ring.

A.2 Integration of differential forms

Fix a non archimedean field (K,|.|) of characteristic 0. We want to define the integral of a n-form with values in K over the standard simplex ∆ⁿ. Since integration produces denominators we assume that there are constants C >

0, s∈Nsuch that|¹_k| ≤Ck^sfor allk∈N.

We define Ω⁰(∆ⁿ) =Kn, Ω¹(∆ⁿ) = (Ln

i=0Kndxi)/(Pn

i=0dxi) and Ω^r(∆ⁿ) = Vr

K_nΩ¹(∆ⁿ) with the obvious differential d: Ωⁱ(∆ⁿ)→Ωⁱ⁺¹(∆ⁿ).

Since Ωⁿ(∆ⁿ) =K_ndx₁. . . dx_n everyn-form ωcan be written uniquely as ω= f dx₁. . . dx_n with f ∈ K_n. We also denote by f the image P

Ia_Ix^I of f in Khx1, . . . , xni. We want to define

Z

∆ⁿ

ω:=

Z

∆ⁿ

f dx1. . . dxn:=X

I

aI

Z

∆ⁿ

x^Idx1. . . dxn∈K,

where the integral on the right hand side is the usual integral of the n-form x^Idx1. . . dxn over the geometric standard simplex ∆ⁿ ⊂Rⁿ⁺¹ where the ori- entation of ∆ⁿ is given bydx1. . . dxn.

Proposition A.4. The above integral is well defined andω 7→ R

∆ⁿω gives a continuous homomorphismΩⁿ(∆ⁿ)→K.

Proof. We have topological isomorphisms Ωⁿ(∆ⁿ) =Kndx1. . . dxn

−'→Kn

−'→ Khx1, . . . , xniand by construction the integralR

∆ⁿ factors through this isomor- phism. Thus we have to show that Khx₁, . . . , x_ni 3f 7→R

∆ⁿf dx₁. . . dx_n ∈K is well defined and continuous.

For simplicity we writeR

∆ⁿf forR

∆ⁿf dx₁. . . dx_n. ForI= (i1, . . . , in) one computes

Z

∆ⁿ

x^I =

n

Y

j=1





ij

X

l=0

ij

l

(−1)^l 1

n−j+ 1 +l+Pn k=j+1i_k



.

By our general assumption on Kwe have

1

n−j+ 1 +l+Pn−1 k=j+1i_k

≤C·



n−j+ 1 +l+

n

X

k=j+1

ik





s

≤C(n+|I|)^s,

thus

Z

∆ⁿ

x^I

≤Cⁿ(n+|I|)^sn≤C|I|˜ ^sn if|I| ≥1 with a constant ˜C depending only onC,nands.

Now, for f = P

IaIx^I ∈ Khx1, . . . , xni, |aI| · |I|^sn tends to zero when |I|

tends to infinity and thus R

∆ⁿf = P

IaI

R

∆ⁿx^I converges in A. Furthermore

|R

∆ⁿf| ≤sup_I|a_IR

∆ⁿx^I| ≤max{C˜·sup_I|a_I| · |I|^sn,|R

∆ⁿ1| · |a₀|)} ≤(const)· max{kfksn,kfk0}.It follows thatf 7→R

∆ⁿf is continuous.

Remark. In the definition of the integralR

∆ⁿωwe could take any representative off in Khx0, . . . , x_ni. The resulting value of the integral would be the same.

A.3 Dependence on parameters

For any K-Banach algebra A we denote by Fε(K^r, A) the Banach algebra of ε-convergent power series inrvariables, i.e. power seriesf =P

Ja_Jy^J,a_J∈A, y = (y₁, . . . , y_r),J ∈N^r₀ such that lim_|J|→∞ka_Jkε^|J| = 0, with norm kfk_ε = sup_JkaJkε^|J|.

Definition. Let M be a locally analytic r-dimensional K-manifold and f : M → Khx₀, . . . , x_ni a function. We say that f is locally analytic if for every u∈M there exists an ε >0 and a chartM ⊃V −→^ψ B_ε(0) ={|.| ≤ ε} ⊂K^r with ψ(u) = 0 such that f ◦ψ⁻¹ is given by P

Ia_Ix^I where the a_I are in Fε(K^r, K) and satisfykaIkε· |I|^t→0 as|I| → ∞for everyt∈N0.

Note that if the condition is satisfied foru∈M with chartψ:V →Bε(0) then it is also satisfied for allu⁰∈V with chartψ−ψ(u⁰).

It follows from the next Proposition that if the condition of the definition is satisfied by one chartψ :V →Bε(0) withu∈V andψ(u) = 0 then it is also satisfied by any other chart ψ⁰ :V⁰ →Bε⁰(0) withu∈V⁰ andψ⁰(u) = 0 after possibly shrinkingV⁰.

Remark. If we embedKhx0, . . . , xniin the Banach algebra F1(Kⁿ⁺¹, K) then f :M → F1(Kⁿ⁺¹, K) as above is locally analytic in the ordinary sense but not vice versa.

Proposition A.5. (a) Let f, g : M → Khx₀, . . . , x_ni be locally analytic.

Then alsof+g andf ·g are locally analytic.

(b) If ϕ : M⁰ → M is locally analytic then f ◦ϕ : M⁰ → Khx0, . . . , xni is locally analytic.

Proof. (a) is easy (cf. the proof of proposition A.1). (b) Letu⁰∈M⁰,u:=ϕ(u⁰) and choose a chart u∈V −→^ψ B_ε(0) withψ(v) = 0 such thatf ◦ψ⁻¹ is of the form P

IaIx^I with kaIkε· |I|^t → 0 as |I| → ∞ for every t ∈ N0. Choose a chart u⁰ ∈ V^{0 ψ}

0

−→ B_ε⁰(0) for M⁰ with ψ⁰(u⁰) = 0 such that ϕ(V⁰) ⊂ V. We may assume that the induced map ˜ϕ : Bε⁰(0) → Bε(0) is given by a power series. Since ˜ϕ(0) = 0 it follows that ˜ϕhas no constant term and therefore that kϕk˜ ε⁰⁰ ≤ ^ε_ε⁰⁰0kϕk˜ ε⁰ for everyε⁰⁰ ≤ε. Thus we may assume that kϕk˜ ε⁰ ≤ε. But thena_I◦ϕ˜is a well defined power series inF_ε⁰(K^r0, K) withka_I◦ϕk˜ _ε⁰ ≤ ka_Ik_ε. Nowf◦ϕ◦ψ⁰⁻¹=P

I(aI◦ϕ)x˜ ^I and the claim follows.

We call f :M →Kn locally analytic if it is locally analytic in the above sense under the identification K_n =Khx₁, . . . , x_ni. One can check that if g : M → Khx0, . . . , xniis locally analytic, so is the induced mapM →Kn.

Proposition A.6. Letf :M →Knbe locally analytic. ThenM 3u7→ϕ(u) :=

R

∆ⁿf(u)dx₁. . . dx_n∈K is locally analytic.

Proof. Fix a chartM ⊃V −→^ψ B_ε(0) such thatf◦ψ⁻¹=P

Ia_Ix^I withka_Ik_ε·

|I|^s → 0 as |I| → ∞ for every s ∈ N0 as in the definition. For I fixed the functionBε(0)3v7→R

∆ⁿaI(v)x^Idx1. . . dxn=aI(v)R

∆ⁿx^Idx1. . . dxn is given by the power series a_I ·(R

∆ⁿx^Idx₁. . . dx_n) ∈ Fε(K^r, K). For|I| ≥1 we have kaI·(R

∆ⁿx^Idx1. . . dxn)kε≤ kaIkε· |(R

∆ⁿx^Idx1. . . dxn)| ≤ kaIkε·C|I|˜ ^sn with C˜as in the proof of proposition A.4. SincekaIkε|I|^sn→0 as|I| → ∞it follows thatP

IaI·(R

∆ⁿx^Idx1. . . dxn) converges inFε(K^r, K). The claim follows since obviouslyϕ◦ψ⁻¹=P

Ia_I ·(R

∆ⁿx^Idx₁. . . dx_n).

We will also writeR

∆ⁿf dx1. . . dxn for the functionϕin the above proposition.

Proposition A.7. Assume thatM =Bε(0) ⊂K^r and f :M →Kn is given by P

Ia_Ix^I with ka_Ik_ε|I|^t→0 as|I| → ∞.

(i) ∂if :=P

I(∂iaI)x^I is well defined and locally analytic.

(ii) R

∆ⁿ(∂if)dx1. . . dxn=∂i

R

∆ⁿf dx1. . . dxn.

(iii) Ifg:M →K_n is of the same type then∂_i(f g) = (∂_if)g+f(∂_ig).

Proof. One easily sees that∂i:Fε(K^r, K)→ Fε(K^r, K) is well defined and con- tinuous with k∂iakε≤ ε⁻¹kakε. Thus (i) follows. Then R

∆ⁿ(∂if)dx1. . . dxn = P

I(∂iaI)R

∆ⁿx^Idx1. . . dxn=∂i P

IaI

R

∆ⁿx^Idx1. . . dxn

=∂i

R

∆ⁿf dx1. . . dxn

by definition of the integral and the continuouity of ∂_i. The last assertion is clear.

More generally, if P is a free K_n-module of finite rank we say that a function f : M → P is locallay analytic if all comoponent functions with repsect to a given basis ofP are locally analytic. Then analogues of the above propositions hold. In particular we are interested in the case whereP = Mat_N(K_n) or P = Mat_N(K_n)⊗_KnΩ^r(∆ⁿ).

Now we can prove that the cocycle f in theorem 2.4 is in fact locally analytic.

Proposition A.8. The functionUN(R)^×(2n−1)→K, (g1, . . . , g_2n−1)7→Tr

Z

∆²ⁿ⁻¹

(dν·ν⁻¹)²ⁿ⁻¹ where ν =ν(g₁, . . . , g_2n−1) =P2n−1

i=0 x_ig_i+1· · ·g_2n−1 ∈GL_N(R_2n−1) is locally analytic.

Proof. It suffices to show that ν⁻¹ : (g1, . . . , g_2n−1)7→ν(g1, . . . , g_2n−1)⁻¹ and dν : (g1, . . . , g2n−1)7→d(ν(g1, . . . , g2n−1)) are locally analytic, where the above functions are considered as functions on U_N(R)^×(2n−1) with values in N×N- matrices with coefficients in Ω⁰(∆²ⁿ⁻¹) =K_2n−1 resp. Ω¹(∆²ⁿ⁻¹).

This is clear fordν. Setε:=|π|and consider the global chartψ:UN(R)^×(2n−1)→ πMat_N(R) = B_ε(0) ⊂ K^N×N whose inverse is given by (M₁, . . . , M_2n−1) 7→

(1 +M₁, . . . ,1 +M_2n−1). Thenν⁻¹◦ψ⁻¹is given byP∞

k=0(P2n−1

i=0 x_ih_i)^k where hi : πMatN(R) → MatN(K) is the function (M1, . . . , M_2n−1) 7→ 1−(1 + Mi+1)· · ·(1 +M2n−1) (cf. the proof of lemma 2.3). Since hi has no constant term and only integral coefficients we have khikε ≤ε. The coefficient ofx^I in the above expansion of ν⁻¹◦ψ⁻¹ is of the form h^I + permutations and thus k(coefficient ofx^I)kε≤ kh0kⁱ_ε⁰· · · kh_2n−1kⁱε²ⁿ⁻¹≤ε^|I|. Sinceε <1 it follows that k(coefficient ofx^I)kε· |I|^ttends to zero as|I|tends to infinity for everyt∈N0

and thus that ν⁻¹ is locally analytic.

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