De Rham cohomology

Let X be a smooth K-dagger space (cf. [GK99, p. 40] for the definition of smoothness) and Ω¹_X = Ω¹_X/Sp(K)the locally free sheaf of 1-forms onX[GK99, Lemma 5.3]. We denote its global sections simply by Ω¹(X). IfU = Sp(A)⊆X is affinoid, then Ω¹_X(U) = Ω¹_U(U) = Ω¹(A) is theuniversally finite differential module, i.e. d : A → Ω¹(A) is universal for K-derivations from A in finite A-modules [GK99, Lemma 5.1].

We define the sheaf of n-forms as Vn

OXΩ¹_X and get as usual the complex of sheaves Ω^∗_X. The de Rham cohomology of X is by definition

H_dR^∗ (X/K) :=H^∗(X,Ω^∗_X).

As usual, if X• is a smooth simplicial dagger space, the sheaves Ωⁿ_X

p on X_p, p≥0, together with the pullback maps form a sheaf on the simplicial dagger space X• and the de Rham cohomology of X• is by definition H_dR^∗ (X•/K) :=

H^∗(X•,Ω^∗_X•).

102 CHAPTER 5. CHERN-WEIL THEORY FOR SIMPLICIAL DAGGER SPACES

We need the analogue of Dupont’s theorem in the dagger context. The ana-logues of the standard simplices are the affinoid dagger spaces

∆^p := Sp Khx₀, . . . , x_pi^†

This map is well defined, since the elements P

j:φ(j)=ix_j have norm ≤ 1, hence are power bounded (cf. section 4.1). In particular, [p]7→ ∆^p defines a cosimplicial dagger space.

Definition 5.1. — A simplicialn-form on the simplicial dagger spaceX• is a family ofn-forms (ωp)p≥0, whereωp ∈Ωⁿ(∆^p×X_p) and for allp≥0, i= 0, . . . p

(δⁱ×1)^∗ω_p = (1×∂_i)^∗ωp−1 in Ωⁿ(∆^p−1×X_p).

The space of simplicialn-forms is denoted byDⁿ(X•). We get a commutative differential gradedK-algebra D^∗(X•) by applying the wedge product and the exterior differential component-wise.

Remarks 5.2. — (i) LetX and Y be two dagger spaces and consider their productX×Y with projectionsp₁ :X×Y →X, p₂ :X×Y →Y. Then there is a natural isomorphism

Ω¹_X×Y =p^∗₁Ω¹_X⊕p^∗₂Ω¹_Y.

In fact, the question being local, it suffices to consider the case, whereX and Y are affinoid, and there the result follows as in [BKKN67, 2.2.2.a)]. Hence we get a decomposition

Ωⁿ_X×Y = M

k+l=n

Ω^k,l_X×Y, where Ω^k,l_X×Y :=p^∗₁Ω^k_X ⊗_OX×Y p^∗₂Ω^l_Y.

5.1. DE RHAM COHOMOLOGY 103

Obviously, the differential Ωⁿ_X×Y −→^d Ωⁿ⁺¹_X×Y sends Ω^k,l_X×Y to Ω^k+1,l_X×Y ⊕Ω^k,l+1_X×Y and we denote the two components ofdbydX anddY respectively. Sincedd= 0, it follows, thatd_Xd_X = 0, d_Yd_Y = 0, d_Xd_Y =−d_Yd_X. In other words, (Ω^∗_X×Y, d) is the total complex associated with the double complex (Ω^∗,∗_X×Y, d_X, d_Y). This double complex is functorial in X and Y.

(ii) IfX is a dagger space, the complex of global sections Ω^∗(∆^p×X) is the total complex associated with the double complex (Ω^∗,∗(∆^p×X), d_∆, d_X). It follows, that if X• is a (strict) simplicial dagger space, then D^∗(X•) is the total complex associated with the double complex (D^∗,∗(X•), d∆, dX), where D^k,l(X•) consists of those forms ω = (ω_p)p≥0, such that ω_p ∈ Ω^k,l(∆^p×X_p) for all p≥0.

We denote by Fil^∗D^∗(X•) the filtration of D^∗(X•) with respect to the second index:

FilⁿD^∗(X•) = M

k+l=∗,l≥n

D^k,l(X•).

Our goal is to construct a filtered homotopy equivalence D^∗(X•) → Ω^∗(X•) given by integration along the standard simplices, similar to the classical case.

Here on the right hand side Ω^∗(X•) denotes the total complex of the cosim-plicial complex [p]7→Ω^∗(Xp) = Γ(Xp,Ω^∗_Xp).

First we have to introduce some more notation: Let I := Sp(Khti^†). Then Ω¹(I) =Khti^†dt, Ωⁿ(I) = 0, if n >1. If X= Sp(A) is affinoid, thenI×X= Sp(Ahti^†), where Ahti^† := A⊗^†_K Khti^†. Explicitely, if A = Khxi^†/I, then Ahti^†=Khx, ti^†/(I).

Lemma 5.3. — There exists a unique A-linear map R1

0 . dt : Ahti^† → A, that sends t^k to _k+1¹ . If f ∈ Ahti^†, its formal derivative with respect to t,

∂f

∂t ∈Ahti^†, is well-defined and R1 0

∂f

∂tdt=f(1)−f(0).

This is the crucial point, where overconvergence and hence dagger spaces come into play.

Proof. — We first consider the caseA=W_n=Khx₁, . . . , x_ni^†. ThenAhti^†= W_n+1 = Khx₁, . . . , x_n, ti^†. If f ∈ W_n+1, there exists ρ > 1, such that f ∈

104 CHAPTER 5. CHERN-WEIL THEORY FOR SIMPLICIAL DAGGER SPACES Wn+1 is Wn-linear. The last formula of the assertion follows directly from the constructions.

In general,Amay be written as a quotientA=W_n/I. ThenAhti^†=W_n+1/I· W_n+1 and by linearity we have R1

0(I·W_n+1)dt⊆I. Hence, R1

0 . dt:W_n+1→ Wn induces the desired map Ahti^† → A. Similarly _∂t^∂ induces the morphism

∂

∂t :Ahti^†→Ahti^† and the last formula of the assertion follows from the case of the Washnitzer algebra treated before.

Remark 5.4. — For later reference we observe the following: Letρ >1 and f = P∞

Iterated application of the integration operator constructed in the lemma gives a K-linear morphismKhx₁, . . . , x_ni^†

(1)The valuation onQinduced by the valuation|.|onKis equivalent to thep-adic valuation

|.|p; hence there exists a constantc >0 such that|_k+1¹ |=|_k+1¹ |^cp≤(k+ 1)^c.

5.1. DE RHAM COHOMOLOGY 105

Lemma 5.5. — There is a natural OX-linear morphism K :p∗Ωⁿ_I×X →Ωⁿ⁻¹_X

satisfying dK+Kd=i^∗₁−i^∗₀.

Proof. — Let U = Sp(A) ⊆ X be open affinoid. Then p⁻¹(U) = I ×U = Sp(Ahti^†). We have

p∗Ωⁿ_I×X(U) = Ωⁿ(I×U) = Ω^0,n(I×U)⊕Ω^1,n−1(I×U)

=Ahti^†⊗_AΩⁿ(A)⊕Ahti^†dt⊗_AΩⁿ⁻¹(A).

We defineK(U) :p∗Ωⁿ_I×X(U)→Ωⁿ⁻¹(U) to be equal to the zero map on the first summand andf dt⊗ω7→(R1

0 f dt)·ω on the second summand.

If V = Sp(B) ⊆ U is an admissible open affinoid, given by a morphism of dagger algebras A → B, the maps K(U) and K(V) are clearly compatible with respect to the restriction map. Hence we get a well defined morphism of OX-modules p∗Ωⁿ_I×X →Ωⁿ⁻¹_X . This map is clearly natural in X.

Finally, we show that

dK+Kd_X = 0, Kd_I =i^∗₁−i^∗₀. (5.1) This in particular implies the last formula of the claim. Since (5.1) is local on X, we may assume that X = Sp(A) is affinoid. Choose a presentation A=Khx₁, . . . , xri^†/I, i.e. a closed immersionX ,→Sp(Khx₁, . . . , xri^†) =:B^r. By the naturality of K we have a commutative diagram

Ωⁿ(I×B^r)

K // // Ωⁿ(I×X)

K

Ωⁿ⁻¹(B^r) ^{// //} Ωⁿ⁻¹(X),

where the horizontal maps are surjections. Hence it suffices to prove the claim forX =B^r, where it can be checked by direct computation: Ann-form onI× B^ris a sum of forms of the typesg(x, t)dtdx_i1. . . dx_in−1 andf(x, t)dx_j1. . . dx_jn with g(x, t), f(x, t) ∈ Khx₁, . . . , x_r, ti^†. Let us check the first formula for

106 CHAPTER 5. CHERN-WEIL THEORY FOR SIMPLICIAL DAGGER SPACES

LetX• be a simplicial dagger space. For eachl≥0 we can consider the cosim-plicial group [p]7→Ω^l(Xp). The associated complex is denoted by (Ω^∗,l(X•), δ).

Theorem 5.6. — Let X• be a simplicial dagger space. For each l the two chain complexes(D^∗,l(X•), d∆)and(Ω^∗,l(X•), δ)are naturally chain homotopy each Xp is acyclic for coherent sheaves [Del74, (5.2.3)], e.g. affinoid or a Stein space, e.g. the dagger space associated with an affine K-scheme of finite type (cf. [GK99] Lemma 4.3 and p. 25 for the definition of a Stein space).

Proof of the Theorem. — We adapt Dupont’s proof of Theorem 1.2 [Dup76, Theorem 2.3].

5.1. DE RHAM COHOMOLOGY 107

For any j = 0, . . . , p consider the morphism gj : I ×∆^p → ∆^p given on dagger algebras byKhx₀, . . . , x_pi^†/(P

ix_i−1)→Khx₀, . . . , x_p, ti^†/(P

ix_i−1), x_i 7→δ_ij ·t+ (1−t)·x_i, whereδ_ij is the Kronecker delta. This is well-defined since the target elements are power bounded andP

i(δijt+ (1−t)xi) = 1 in Khx₀, . . . , xp, ti^†/(P

ixi−1). Thus gj is a homotopy between id∆^p and the constant mape_j : ∆^p →∆^p given byx_i7→δ_ij.

For any dagger space Y we can now define the homotopy operatorh_(j) to be the composition

h_(j): Ωⁿ(∆^p×Y) ^(gj×idY)

∗

−−−−−−→Ωⁿ(I×∆^p×Y)−→^K Ωⁿ⁻¹(∆^p×Y).

We have the analogue of [Dup76, Lemma 2.9]:

Lemma 5.8. — The operators h_(j),j = 0, . . . , p, satisfy h_(j)◦d_∆+d_∆◦h_(j)= (e_j×id_Y)^∗−id,

h_(j)◦dY +dY ◦h_(j)= 0 and for i= 0, . . . , p

(δⁱ×idY)^∗◦h_(j) =h_(j)◦(δⁱ×idY)^∗, i > j, (δⁱ×id_Y)^∗◦h_(j) =h_(j−1)◦(δⁱ×id_Y)^∗, i < j.

Proof. — Since everything follows by formal computation, we only check the first statement. Thus takeω ∈Ωⁿ(∆^p×Y). Then

h_(j)◦d_∆(ω) +d_∆◦h_(j)(ω) =

=K((g_j×id_Y)^∗d_∆ω) +d_∆K((g_j×id_Y)^∗ω)

=K(dI×∆(gj×idY)^∗ω)−K(d∆(gj×idY)^∗ω) cf. (5.1)

=K(d_I(g_j×id_Y)^∗ω)

= (i^∗₁−i^∗₀)(gj×idY)^∗ω by (5.1) again

= (e_j×id_Y)^∗ω−ω.

Here we used the naturality of the double complex of remark 5.2 (i) and applied the first formula of (5.1) with X = ∆^p×Y only for the ∆-component d_∆ of the differentiald∆×Y.

108 CHAPTER 5. CHERN-WEIL THEORY FOR SIMPLICIAL DAGGER SPACES

We define the integration mapI :D^k,l(X•)→Ω^l(X_k) as in the classical case:

I(ω) = (−1)^k(ek×idX_k)^∗(h(k−1)◦ · · · ◦h₍₀₎)(ωk). (5.6) Using the lemma and the compatibility condition of simplicial differential forms one checks (5.2).

Similarly, E is defined by the same formula as in the classical case: If ω ∈ Ω^l(X_k), the simplicial form E(ω) ∈ D^k,l(X•) is given on ∆^p ×X_p by 0, if is easy to see that E(ω) defines a simplicial form on X• and that E satisfies (5.3) and (5.4).

Also the homotopy operator s : D^k,l(X•) → D^k−1,l(X•) is defined by the same formula as in the complex situation, which again gives a well-defined differential form also in the dagger context. Thats(ω) really defines asimplicial differential form and thatssatisfies (5.5) follows again from the above lemma.

(Most of the computations are also carried out in [Dup78, proof of Theorem 2.16].)

Remark 5.9. — If Y is any dagger space, (5.6) defines an operator I: Ωⁿ(∆^k × Y) → Ω^n−k(Y), which we denote by R

∆^k. It may also be described as follows: Define a morphism ψ:I^k = Sp(Kht₁, . . . , t_ki^†)→ ∆^k = Sp(Khx₀, . . . , x_ki^†/(P

ix_i−1)) byx_i7→t₁· · ·t_i(1−t_i+1),i= 0, . . . , k, where we let tn+1 = 0.⁽²⁾ It follows directly from the definitions, thatR

∆^k is simply

Im Dokument The Relative Chern Character and Regulators (Seite 103-111)