Topological K-theory of affinoid and dagger algebras

Let (A,|.|) be an ultrametric Banach ring, i.e. a ringA together with a map

| . | : A → R≥0 such that |x| = 0 iff x = 0, |x| = | − x|, |xy| ≤ |x||y|

and |x +y| ≤ max{|x|,|y|}, and such that A is complete for the metric (x, y) 7→ |y−x|. For example, any K- or R-affinoid algebra with a chosen norm is an ultrametric Banach ring. In [KV71] Karoubi and Villamayor define topological K-groups K_top^−∗(A) for arbitrary Banach rings and sketch a par-ticular approach for ultrametric Banach rings (using convergent power series instead of absolutely converging power series, see below), studied further by Adina Calvo [Cal85]. For unitary Banach rings it may be formulated as follows:

(1)Since the dagger space associated with an affineK-scheme is not affinoid, I am not quite sure what the “right” definition of topologicalK-theory of aK-scheme is.

130 CHAPTER 7. RELATIVEK-THEORY AND REGULATORS

Define

Ap :=Ahx₀, . . . , xpi/(X

i

xi−1), (7.1)

where

Ahx₀, . . . , x_pi={X

a_νx^ν ∈A[[x₀, . . . , x_p]]| |a_ν|−−−−→^|ν|→∞ 0}.

If φ : [p] → [q] is an increasing map, we define φ^∗ : A_q → A_p by x_i 7→

P

j∈[p]:φ(j)=ix_j. This is well defined, sinceAhx₀, . . . , x_piis complete (w.r.t. the Gauß norm) and the target elements are power bounded [BGR84, Proposition 1.4.3/1], and gives a simplicial ring A•.

Definition 7.1. — Thetopological K-groups of A are given by K_top⁻ⁱ(A) :=π_i(B•GL(A•)) =πi−1GL(A•), i≥1.

Remarks 7.2. — (i) It is clear from the definition, that the topological K-groups of an ultrametric Banach ring do not depend on the particular norm chosen. In particular, the topologicalK-groups of anR- orK-affinoid algebra are well defined.

(ii) For any simplicial groupG•, let ¯G_p:=Tp

i=1ker(∂_i)⊆G_p. Then ( ¯G_p, ∂₀)p≥0

is a chain complex of (non abelian) groups, whose homology groups are the homotopy groups of G•. Symmetrically, one can also use the chain complex (Tp−1

i=0 ker(∂i), ∂p) (see e.g. [May67, Proposition 17.4]).

(iii) The ring A• is additive contractible, i.e. the identity on A• is homotopic to the zero map by a homotopy which is compatible with the abelian group structure ofA•: The 1-simplex x0 ∈A1 corresponds to a mapfx0 : ∆[1]→A•

such that f_x0(0) = ∂₁(x₀) = 1, f_x0(1) = ∂₀(x₀) = 0, where 0 := δ¹ and 1 := δ⁰ ∈∆[1]₀ = Hom_∆([0],[1]) are the two vertices of ∆[1]⁽²⁾. The desired homotopy is then given by A• ×∆[1] →A•,(a, t) 7→ fx0(t)·a. In particular, the homotopy groupsπ∗(A•) vanish. More generally the simplicial ring ofr×r matrices Mat_r(A•) is additive contractible.

(2)For n ≥ 0 ∆[n] denotes the simplicial set Hom∆(. ,[n]) : ∆^op → Sets. Its geometric realisation is the standard simplex ∆ⁿ⊆Rⁿ⁺¹. For any simplicial setX•there is a natural isomorphism Hom(∆[n], X•) =Xn(Yoneda lemma).

7.1. TOPOLOGICALK-THEORY OF AFFINOID AND DAGGER ALGEBRAS 131

(iv) We also need the following refinement of the last remark. Equip Ahx₀, . . . , x_pi with the Gauß norm and A_p with the residue semi-norm and denote it byk.k. It is easy to see, that for any φ: [p]→[q], the induced homomorphism φ^∗ : Aq → Ap is contractive, i.e. kφ^∗(f)k ≤ kfk. Write A⁰_p :={f ∈Ap| kfk ≤1}. It follows, thatA⁰_• is a simplicial subring of A•. The semi-norm on Mat_r(A_p) is defined to be the maximum of the semi-norms of the entries. Write Mat_r(A_p)⁰⁰ := {g ∈ Mat_r(A_p)| kgk < 1}. Then Matr(A•)⁰⁰ is a simplicial subgroup of Matr(A•), which is moreover an A⁰_• -module.

Since x₀ ∈ A⁰₁, the argument of the last remark shows, that Mat_r(A•)⁰⁰ is additive contractible, too.

The definition given above is not the one given by Karoubi–Villamayor and Calvo. Since the equivalence of both definitions is proved in the literature only in the case of discrete Banach rings, we give a proof here.

Proposition 7.3. — The topological K-groups defined above coincide with those defined by Karoubi-Villamayor and Calvo for i≥1.

Proof. — The agrument in the discrete case is due to Anderson [And73, Theo-rem 1.6]. First we recall Calvo’s definition. It is best, to work in the category of ultrametric Banach rings without unit. LetAbe such a ring. Then GLr(A) is by definition the group ofr×r-matrices invertible w.r.t. the formal group law (M, N)7→MN :=M N+M+N. IfAhas a unit, this is clearly equivalent to the usual definition via M 7→ M + 1. Denote by GL⁰_r(A) the subgroup of GLr(A) generated by the topologically nilpotent matrices. Define GL(A) and GL⁰(A) as the usual colimits. ThenK⁻¹(A) := GL(A)/GL⁰(A).

For any A as above, the path ringEA is the kernel ofp₀ :Ahti →A, t 7→ 0, i.e. EA = tAhti, the loop ring ΩA is the kernel of p1 : EA → A, “t 7→ 1”, i.e.P

ia_itⁱ7→P

ia_i. These are again ultrametric Banach rings and the higher Karoubi–VillamayorK-groups are defined by

K⁻ⁱ(A) :=K⁻¹(Ωⁱ⁻¹A).

132 CHAPTER 7. RELATIVEK-THEORY AND REGULATORS

A matrixM ∈GLr(A) is callednull-homotopic, if there existsMf∈GLr(EA), such thatp₁(M) =f M. Two matricesM, N are calledhomotopic, ifMN⁻¹is null-homotopic. Denote by GL⁰_r(A) = im (p₁ : GL_r(EA)→GL_r(A)) the group of null-homotopic matrices and let GL⁰(A) = colimrGL⁰_r(A). It follows as in [KV71, Appendice 3] that GL⁰(A) = GL⁰(A) (the arguments given there and in the cited references for unitary rings also work in the non-unitary context). In other words, p₁ : GL(EA) → GL(A) induces an isomorphism GL(EA)/GL(ΩA)∼= GL⁰(A).

Now let A be an unitary Banach ring and form the simplicial Banach ring A• as in (7.1). Applying the above isomorphism to the simplicial Banach ring ΩⁱA•, we get an isomorphism of simplicial groups

GL(EΩⁱA•)/GL(Ωⁱ⁺¹A•)∼= GL⁰(ΩⁱA•), i≥0. (7.2) We claim, that GL(EΩⁱA•) is contractible. In fact, the ring morphisms h_j : EΩⁱA_n =t(ΩⁱA)hti →EΩⁱA_n+1, j= 0, . . . , n, given by the degeneracy s_j on ΩⁱAn and by hj(t) = t(x0+· · ·+xj) define a simplicial homotopy between 0 and id_EΩⁱ_A• in the sense of [May67, §5]. Hence they induce a contracting homotopy of GL(EΩⁱA•). By (7.2) we get isomorphisms

πn(GL⁰(ΩⁱA•))−→^∼= πn−1(GL(Ωⁱ⁺¹A•)), i≥0, n≥1. (7.3) Next, by definition we have an isomorphism of simplicial groups

GL(ΩⁱA•)/GL⁰(ΩⁱA•) =K⁻¹(ΩⁱA•) (7.4) and we claim, that this last group is a constant simplicial group. Since An∼= Ahx₀, . . . , xn−1i it suffices to show, that for any Banach ring A the inclusion A ,→ Ahxi induces an isomorphism on K⁻¹. Since this inclusion is split by x 7→ 0, the induced map on K⁻¹ is an injection. Now consider h : Ahxi → Ahxihti=Ahx, ti, x7→tx. Then p0◦h(x) = 0, p1◦h(x) =x. Hence, anyM ∈ GL(Ahxi) is homotopic toM(0)∈GL(A)⊆GL(Ahxi), the null-homotopy for M M(0)⁻¹ being given byh(M)M(0)⁻¹ ∈GL(EAhxi). It follows, that K⁻¹(A)→K⁻¹(Ahxi) is also surjective.

Hence we get from (7.4), that

π_n(GL⁰(ΩⁱA•))∼=π_n(GL(ΩⁱA•)), i≥0, n≥1.

7.1. TOPOLOGICALK-THEORY OF AFFINOID AND DAGGER ALGEBRAS 133

Combining this with (7.3), we get

π_n(GL(A•))∼=πn−1(GL(ΩA•))∼=. . .∼=π₀(GL(ΩⁿA•)), n≥0.

Since the left hand side is K_top⁻ⁿ⁻¹(A) by definition, it suffices to show, that π0(GL(ΩⁿA•)) = K⁻ⁿ⁻¹(A) = K⁻¹(ΩⁿA). We have isomorphisms A1 ∼= Ahxi, x₀ 7→ x, x1 7→ 1−x, and A0 ∼= A, x0 7→ 1, under which ∂0, ∂1 are given by p₀:x7→0, p₁:x7→1 respectively. Hence by remark 7.2(ii)

π0(GL(ΩⁿA•)) = coker p1: ker p0: GL(ΩⁿAhxi)→GL(ΩⁿA)

→GL(ΩⁿA)

= coker(p₁: GL(EΩⁿA)→GL(ΩⁿA))

= GL(ΩⁿA)/GL⁰(ΩⁿA) =K⁻¹(ΩⁿA).

Now let R be a complete discrete valuation ring with maximal ideal (π), per-fect residue field R/(π) = k of characteristic p > 0 and field of fractions K of characteristic 0. We want to show, that the topological K-groups of affi-noid algebras may also be computed usingoverconvergent power series. More precisely: Define the R-dagger algebra R^†n :=Rhx₀, . . . , xni^†/(P

ixi−1) and similarly the K-dagger algebra Kn^†. As above, we get simplicial R- resp. K-dagger algebras R^†• andK•^†.

Definition 7.4. — LetA be anR-dagger algebra. We define the topological K-groups

K_top⁻ⁱ(A) :=πi(B•GL(A⊗^†_RR^†_•)) =πi−1(GL(A⊗^†_RR^†_•)), i≥1.

IfA is aK-dagger algebra, the definition is the same with R replaced by K.

Proposition 7.5. — Let Abe an R- orK-dagger algebra, and Aˆits comple-tion, an R- resp. K-affinoid algebra. Then

K_top⁻ⁿ(A)∼=K_top⁻ⁿ( ˆA), n≥1.

Remark 7.6. — In [Kar97] Karoubi states, that one can use “indefinitely in-tegrable power series” to define the topologicalK-theory of ultrametric Banach algebras and uses these for the construction of the relative Chern character.

The difference here is, that we do not use the full Banach algebra ˆA, but only the overconvergent partA of it.

134 CHAPTER 7. RELATIVEK-THEORY AND REGULATORS

Proof of the proposition. — The proof is the same forR- and K-dagger alge-bras and we restrict to the case ofR-dagger algebras. Choose a representation A=Rhyi^†/I and writeA_n:=A⊗^†_RRn^† =Rhy, x₀, . . . , x_ni^†/(I,P

ix_i−1).

The completion of A is given by ˆA = Rhyi/(I) and the ring ( ˆA)n appear-ing in the definition of the topological K-theory of A is given by ( ˆA)n = (Rhyi/(I))hx₀, . . . , x_ni/(P

ix_i−1) =Rhy, x₀, . . . , x_ni/(I,P

ix_i−1) =(A[_n).

Since π_n(GL(A•)) = lim−→^rπ_n(GL_r(A•)), it suffices to show, that for anyr ≥1 the natural map πn(GLr(A•))→πn(GLr( ˆA•)) is an isomorphism.

We begin with the surjectivity. A class in π_n(GL_r( ˆA•)) is represented by a g ∈GL_r( ˆA_n) such that ∂_ig= 1,i= 0, . . . , n. By lemma 7.7 below, there is a sequence of matricesgN ∈Matr(An) converging to g, where each gN satisfies

∂igN = 1,i= 0, . . . ,1. Since GLr( ˆAn)⊆Matr( ˆAn) is open,gN ∈GLr( ˆAn) for N large enough, and by lemma 7.8g_N ∈GL_r(A_n). We claim, that forN large enough, [g_N] = [g] inπ_n(GL_r( ˆA•)), thus showing surjectivity.

We use remark 7.2(iv). Choose N large enough, so that gNg⁻¹ − 1 ∈ Matr( ˆAn)⁰⁰. Since Matr( ˆA•)⁰⁰ is contractible, there existsh∈Matr( ˆAn+1)⁰⁰, such that ∂₀(h) = g_Ng⁻¹ −1, ∂_i(h) = 0, i > 0. Since khk < 1, 1 +h ∈ GL_r( ˆA_n+1). Moreover, ∂₀(1 +h) =g_Ng⁻¹ and ∂_i(1 +h) = 1 for i >0, hence [gNg⁻¹] = [1] and [gN] = [g] as claimed.

Next we prove the injectivity. Thus let g ∈ GL_r(A_n) with ∂_i(g) = 1, i = 0, . . . , n, and assume that there exists h ∈ GL_r( ˆA_n+1), such that ∂₀(h) = g,

∂i(h) = 1 ifi >0. As in remark 7.2(iii) π∗(Matr(A•)) = 0. Hence there exists a matrix eh ∈ Matr(An+1) such that ∂0(eh) = g, ∂i(eh) = 1, i = 1, . . . , n+ 1.

Now we can apply lemma 7.7 toh−ehto obtain a sequence of matrices h_N ∈ Mat_r(A_n+1) converging toh−ehand satisfying∂_i(h_N) = 0 fori= 0, . . . , n+ 1 and all N. Then hN +eh ∈ Matr(An+1) converges to h ∈ GLr( ˆAn+1), hence h_N+eh∈GLr(An+1) forN large enough, again by the openness of GL_R( ˆAn+1) and lemma 7.8. Moreover ∂₀(h_N +eh) = g, ∂_i(h_N +eh) = 1, i= 1, . . . , n+ 1, hence [g] = [1] in π_n(GL_r(A•)).

Lemma 7.7. — We use the notations of the above proof. Let g ∈Mat_r( ˆA_n) be such that ∂_ig = 0, i= 0, . . . , n. There exists a sequence of matrices g_N ∈

7.1. TOPOLOGICALK-THEORY OF AFFINOID AND DAGGER ALGEBRAS 135

Matr(An), N ≥0, which converges to g in Matr( ˆAn) and satisfies ∂igN = 0, for i= 0, . . . , n and all N.

Proof. — As in remark 7.2(iii) π∗(Mat_r( ˆA•)) = 0. Hence there exists h ∈ Matr( ˆAn+1), such that ∂0(h) =g,∂i(h) = 0, i >0.

We have an isomorphism ˆA_n = Rhy, x₁, . . . , x_ni/(I) given by x₀ 7→ 1 − P

i≥1x_i, x_i 7→x_i, i >0, and similar ˆA_n+1 =Rhy, x₁, . . . , x_n+1i/(I) . In terms of these isomorphisms ∂i, i > 0, is given by xj 7→ xj, if j < i, xi 7→ 0 and xj 7→xj−1, ifj > i.

Representh by a power series

˜h= X

ν∈Nⁿ⁺¹₀

a_ν(y)x^ν₁¹· · ·x^ν_n+1ⁿ⁺¹ ∈Mat_r(Rhy, x₁, . . . , x_n+1i), whereaν(y)∈Matr(Rhyi). Leti >0. Since

∂i˜h= X

ν∈Nⁿ⁺¹₀ :νi=0

aν(y)x^ν₁¹· · ·x^ν_i−1ⁱ⁻¹x^ν_iⁱ⁺¹· · ·x^ν_nⁿ⁺¹

represents ∂_ih = 0, a_ν(y) has entries in (I) ⊆Rhyi as soon as ν_i = 0 for one i, and we may assume without loss of generality, that a_ν(y) = 0 for those ν. Now let ˜hN ∈ Matr(R[y, x1, . . . , xn+1]) be the polynomial arising from ˜h by deleting all terms of total degree greater than N and denote by hN its image in Mat_r(A_n+1). By construction, we have∂_i(˜h_N) = 0, ifi > 0, and for N → ∞ the ˜h_N converge to ˜h. Hence also ∂_i(h_N) = 0 for i > 0 and theh_N converge to h in Matr( ˆAn+1). Now define gN := ∂0(hN) ∈ Matr(An). Then the gN converge to ∂0(h) = g, and ∂i(gN) = ∂i∂0(hN) = ∂0∂i+1(hN) = 0, i= 0, . . . , n.

Lemma 7.8. — Let g∈Matr(An) be a matrix, whose image in Matr( ˆAn) is invertible. Theng itself is invertible.

Proof. — EquipRhy, xiwith the Gauß norm andA_n,Aˆ_n,Mat_r(A_n), etc., with the induced norms.

Let h ∈ GLr( ˆAn) be the inverse of g. Since An is dense in ˆAn, we may ap-proximate h by matricesh_N ∈Mat_r(A_n). Then h_N ·g −−−−→^N→∞ 1 in Mat_r(A_n), and, for N large enough, k1−h_Ngk ≤ |π| (recall that π is a uniformizer

136 CHAPTER 7. RELATIVEK-THEORY AND REGULATORS

for R). Then we can represent 1−hNg by a matrix of power series fN in Mat_r(Rhy, xi^†), with kf_Nk ≤ |π|. Then f_N = πf_N⁰ , where f_N⁰ is a matrix of power series with kf_N⁰ k ≤ 1. Since Rhy, xi^† is weakly complete (cf. section 4.1), the series P∞

k=0π^k(f_N⁰ )^k = P∞

k=0f_N^k converges in Matr(Rhy, xi^†) and defines an inverse of 1−fN. Hence its image eN in Matr(An) is an inverse of 1−(1−h_Ng) =h_Ng. Then e_N ·h_N is a left inverse ofg in Mat_r(A_n). By the same argument applied to 1−gh_N,g also possesses a right inverse, hence is invertible in Matr(An).

Proposition 7.9. — LetAbe anR-affinoid orR-dagger algebra and assume, that A/πAis regular. Then

K_top⁻ⁿ(A)∼=Kn(A/πA), n≥1.

Proof. — This follows from Calvo’s Proposition 2.1 [Cal85] and Gersten’s re-sult, that Karoubi–Villamayor theory for discrete noetherian regular rings coincides with Quillen’s K-theory [Ger73, Proposition 3.14]. In fact, similar methods as above show that πn−1(GL(A•)) = πn−1(GL((A/πA)•)), where (A/πA)n = (A/πA)[x0, . . . , xn]/(P

xi −1), and the right hand side is the Karoubi–VillamayorK-group K⁻ⁿ(A/πA).

7.2. Relative K-theory

Let R be as before. Let X = Spec(A) be an affine R-scheme of finite type.

Let ˆA denote theπ-adic completion of A, anR-affinoid algebra, andA^† ⊆Aˆ the weak completion of A, an R-dagger algebra. We define the topological K-groups of X to be

K_top⁻ⁱ(X) :=K_top⁻ⁱ( ˆA) =K_top⁻ⁱ(A^†) =π_i(B•GL(A^†⊗^†_RR^†_•)), i≥1.

Recall that Ki(X) =πi(|B•GL(A)|⁺). Sinceπ1(B•GL(A^†⊗^†_RR^†•)) =K_top⁻¹( ˆA) is abelian, the natural morphism|B_•GL(A)| → |B_•GL(A^†⊗^†_RR^†•)|factors up to homotopy uniquely through|B_•GL(A)| → |B_•GL(A)|⁺. We abbreviate the simplicial group GL(A^†⊗^†_RR^†•) byG•. As in the complex case (cf. section 3.2),

7.3. THE RELATIVE CHERN CHARACTER 137

we define the spaceFe and the simplicial setF by the pullback diagrams Fe

Let X = Spec(A) be a smooth affine R-scheme of finite type. Recall, that XˆK = Sp(A^†⊗_RK) denotes the generic fibre of the weak completion of X.

The relative cohomology groups H_rel^∗ (X/R, n) are defined as in the simplicial case (section 6.3).

We want to construct relative Chern character maps Ch^rel_n,i:K_i^rel(X)→H_rel²ⁿ⁻ⁱ⁻¹(X/R, n).

This is done as in the complex case: First of all, define the simplicial set Fr

by the pullback diagram Sp(A^†⊗_RK). The matrixg is determined by a morphism ofR-algebras

R[xij, y]/(det(xij)y−1)→A^†⊗^†_RR^†_p.

138 CHAPTER 7. RELATIVEK-THEORY AND REGULATORS

SinceA^†⊗^†_RR^†p is anR-dagger algebra as well, this morphism extends uniquely to a morphism

Rhx_ij, yi^†/(det(xij)y−1)→A^†⊗^†_RR^†_p

[MW68, Theorem 1.5], which in turn induces a morphism of dagger spaces Sp((A^†⊗^†_RR^†_p)⊗_RK)→Sp(Khx_ij, yi^†/(det(xij)y−1)),

that is,

Xˆ_K×∆^p →(GLc_r,R)_K.

Hence the above diagram (7.5) gives rise to a morphism of simplicialR-schemes gr:X⊗Fr→B•GLr,R,

together with a commutative diagram

E•(GLc_r,R)_K

p

XˆK⊗Fr

(ˆgr)K //

6v 6v 6vα6vr6v 6v 6v 6v 66

B•(GLc_r,R)_K

of topological morphisms of dagger spaces. Thus we are exactly in the situation of section 6.3 and have relative Chern character classes

Chf^rel_n (T_r, E_r, α_r/R)∈H_rel²ⁿ⁻¹(X⊗Fr/R, n),⁽³⁾

where we denote by Tr the trivial GLr-bundle and by Er the algebraic GLr -bundle classified by g_r. Similar as for the complex analogue one shows, that these classes are compatible for different r:

Lemma 7.11. — The classChf^rel_n (T_r+1, E_r+1, α_r+1/R) is mapped to the class Chf^rel_n (T_r, E_r, α_r/R) by the natural mapH_rel²ⁿ⁻¹(X⊗Fr+1/R, n)→H_rel²ⁿ⁻¹(X⊗ Fr/R, n) induced by the inclusion j: GL_r ,→GL_r+1 in the upper left corner.

The other input we need for the definition of Chern character maps is

(3)Here we tacitly extended the definition of relative cohomology to the case of simplicial schemes of the formX⊗S, which obviously does make sense.

7.3. THE RELATIVE CHERN CHARACTER 139

Lemma 7.12. — Let X be a smooth separatedR-scheme of finite type andS a simplicial set. Then we have natural isomorphisms

H_rel^k (X⊗S/R, n)∼= M

p+q=k

Hom(Hp(S), H_rel^q (X/R, n)).

Proof. — Choose a good compactification j : X_K ,→ X_K with complement D. This induces a good compactification j:XK⊗S ,→XK⊗S. Denote the natural morphisms ˆX_K →X_Kand ˆX_K⊗S→X_K⊗Sbyι. Choose a complex I^∗of injective sheaves onX_K representing Cone(Ω^≥n

XK(logD)→Rj∗Rι∗Ω^∗_ˆ

XK).

Thus Γ(X_K, I^∗) is a complex computing H_rel^∗ (X/R, n).

In an obvious way, I^∗ induces a complex of sheaves I^∗⊗S on the simplicial schemeXK⊗S, which represents Cone(Ω^≥n

XK⊗S(logD⊗S)→Rj∗Rι∗Ω^∗_ˆ

XK⊗S), sinceRj∗Rι∗Ω^∗_ˆ

XK⊗Scan be computed “degree-wise” [Del74, (5.2.5)]. Moreover, eachI^q⊗S_pis an injective sheaf onX_K⊗S_p, and hence the relative cohomology H_rel^∗ (X⊗S/R, n) is just the cohomology of the total complex associated with the cosimplicial complex [p]7→Γ(X_K⊗S_p, I^∗⊗S_p) =Q

σ∈SpΓ(X_K, I^∗). Now the claim follows as in lemma 3.5.

Putting everything together, we can now define:

Definition. — LetX = Spec(A) be a smooth affineR-scheme of finite type.

The relative Chern character

Ch^rel_n,i:K_i^rel(X)→H_rel²ⁿ⁻ⁱ⁻¹(X/R, n) is given by the composition

K_i^rel(X) =πi(Fe)−^Hur.−−→Hi(F ,e Z)∼=Hi(F,Z) = lim−→^rH_i(Fr,Z) ^lim−→r

Chf^rel_n (Tr,Er,αr/R)

−−−−−−−−−−−−−−→H_rel²ⁿ⁻ⁱ⁻¹(X/R, n).

Remark 7.13. — The construction of the relative Chern character for ul-trametric Banach algebras is also due to Karoubi [Kar83, Kar97]. Instead of overconvergent power series he uses indefinitely integrable power series and in contrast to our construction his relative Chern character takes values in the cohomology of the truncated de Rham complex of the rigid space Sp( ˆA_K).

140 CHAPTER 7. RELATIVEK-THEORY AND REGULATORS

That is, he does not take logarithmic singularities or overconvergence into account.

7.4. The case X = Spec(R): Comparison with the p-adic Borel

Im Dokument The Relative Chern Character and Regulators (Seite 131-142)