Construction

Let X• be a smooth separated simplicial K-scheme of finite type. We denote the associated simplicial dagger space byX•^†and byιthe canonical morphism ι:X•^†→X•. Definition 6.2. — Define therelative cohomology groups

H_rel^E,∗(X•, n) :=H^∗(X•,Cone(Ω^≥n

is an injective quasiisomorphism of complexes of abelian sheaves on X•^†

and eachI^k

X•^†

is injective (cf. Appendix A.2), and similarly for R(j∗ι∗p∗)Ω^∗

E•^†

.

6.1. CONSTRUCTION 119

Since p∗, the functor p⁻¹ being exact, maps injective sheaves to injectives, there exists a morphismI^∗

commute, and this morphism is unique up to homotopyunder Ω^∗

X•^†

and hence a map of the cones Cone(Ω^≥n

X•(logD•) → R(j∗ι∗)Ω^∗

), which is well defined up to homo-topy (cf. lemma A.2). We thus have a canonical morphism

p^∗:H_rel^∗ (X•, n)→H_rel^E,∗(X•, n).

This morphism fits in a long exact sequence

. . . // H_rel^E,i−1(X_•, n) //FilⁿH_dRⁱ (X_•/K) //H_dRⁱ (E_•^†/K) // H_rel^E,i(X_•, n) // . . .

(ii) As for the Hodge filtration of the de Rham cohomology one shows, that the definition of the relative cohomology groups does up to isomorphism not depend on the particular choice of the compactification X•. Since the family of all good compactifications is directed, one could take a colimit over all good compactifications to get a definition independent of choices.

(iii) By remark 5.15H_relⁱ (X•, n)∼=H_dRⁱ (X•/K)/FilⁿH_dRⁱ (X•/K) similar to the complex case.

If f :Y• → X• is a morphism of smooth simplicial K-schemes of finite type and E/X• as before, we can consider the pullback f^∗E and the associated

120CHAPTER 6. REFINED AND SECONDARY CLASSES FOR ALGEBRAIC BUNDLES

principal bundle f^∗E•^†. Whereas in the complex case we had functorial com-plexes defining the cone for the relative cohomologies, we have to be a little bit careful with functoriality here.

Lemma 6.4. — There are well defined pullback maps f^∗ : H_rel^E,∗(X•, n) → H_rel^f∗E,∗(Y•, n) and f^∗ :H_rel^∗ (X•, n)→H_rel^∗ (Y•, n)

Proof. — The proof is the same for both maps, and we restrict to the second one. Given good compactificationsY•,→Y•andX•,→X•, whose complement will as usual be denoted simply byD•, one can construct a good compactifi-cation Y• ,→ Y⁰_• together with maps f : Y⁰_• → X• and Y⁰_• → Y• fitting in a

Hence we may assume without loss of generality, that the given morphism f extends to a morphism f :Y• → X• of the compactifications. Thus we have a commutative diagram injective sheaves to injectives, the dotted arrow in the diagram

Ω^∗

exists and is unique up to homotopy under Ω^∗

X•^†

. Applying (j_X)∗(ι_X)∗ and composing with the natural maps Ω^≥n

X•(logD•) → (j_X)∗(ι_X)∗Ω^∗

X•^†

resp.

6.1. CONSTRUCTION 121 which is well defined up to homotopy (lemma A.2 again).

Choose an injective resolution Ω^≥n

exists and is unique up to homotopy under Ω^≥n

Y•(logD•). This induces a where the complex on the right hand side is well defined up to homotopy equivalence (cf. lemma A.3). Applyingf_∗, we get the natural map

Cone Composing this with (6.2) and noting that the last complex represents Rf_∗Cone relative cohomology groups. Similar one shows, that the map on cohomology does not depend on the choices of I^∗

X•^†

, I^∗

Y•^†

or J.

Remark 6.5. — That there is a unique way of definingf^∗ onH_rel^∗ (X•, n), is clear from remark 6.3(iii). Later on we have to use this lemma also in a slightly modified situation, where the conclusion of remark 6.3(iii) no longer holds.

Since lemma 2.11 applies equally in the dagger context, we have the analogue of proposition 2.10:

122CHAPTER 6. REFINED AND SECONDARY CLASSES FOR ALGEBRAIC BUNDLES

Proposition 6.6. — There exists a class Chf^rel_n (E)∈ H_rel^2n−1,E(X•, n), which is mapped to the n-th Chern character classChn(E) in FilⁿH_dR²ⁿ(X•/K), and which is functorial inX. Moreover, the assignment E 7→Chf^rel_n (E) is uniquely determined by these two properties.

Definition 6.7. — If X• is a smooth separated simplicial K-scheme and E/X• an algebraic GLr-bundle, the classChf^rel_n (E)∈H_rel^2n−1,E(X•, n) is called then-th refined Chern character class of E.

Assume, that the bundle E^† induced by E on X•^† admits a topological trivi-alization α, i.e. there exists a topological morphism α:X•^† E•GL^†_r,K, such thatp◦α=g^†, the classifying map ofE^†. Then αinduces a topological mor-phism α:X•^† E•^† right inverse top:E•^†→ X•^†. To define secondary classes as in section 2.3, we have to define a pullbackα^∗:H_rel^E,∗(X•, n)→H_rel^∗ (X•, n).

Again, this takes a little bit more work than in the complex analogue. For simplicity we restrict to the affine case (but see the remark below). This is enough for the construction of regulators.

Lemma 6.8. — In the above situation assume in addition that X• is affine.

Thenαinduces compatible left inversesα^∗ ofp^∗:H_dR^∗ (X•^†/K)→H_dR^∗ (E•^†/K) and of p^∗ :H_rel^∗ (X•, n)→H_rel^E,∗(X•, n).

Proof. — Using Theorem 5.6 α induces a section of p^∗ : Ω^∗(X•^†) → Ω^∗(E•^†), defined in the notation of the Theorem as the composition I◦α^∗◦E, which we also denote byα^∗. SinceEp^†= (Xp×GLr,K)^† (cf. remark 6.1) is the dagger space associated with an affineK-scheme, it is a Stein space and hence acyclic for the cohomology of coherent sheaves. Hence Ω^∗(E•^†) → RΓ(E•^†,Ω^∗

E•^†

) is a quasiisomorphism and on de Rham cohomology α^∗ is induced by the maps RΓ(E•^†,Ω^∗

E•^†)←^∼−Ω^∗(E•^†) ^α

∗

−→Ω^∗(X•^†)−→^∼ RΓ(X•^†,Ω^∗

X•^†).

On relative cohomology groups α^∗ is constructed as follows: Choose injective resolutions Ω^∗

6.1. CONSTRUCTION 123 Taking global sections we get the diagram

A^∗

where Γ^∗denotes the total complex associated with the obvious (strict) cosim-plicial complex, and A^∗ is defined by requiring that the left hand square is a quasi-pullback. In particular the left hand square commutes up to canonical homotopy, whereas the right hand square strictly commutes. Hence we get quasiisomorphisms Note, that the upper complex on the right hand side, hence also the complex on the left hand side, representsRΓ(X•,Cone(Ω^≥n pull-back on relative cohomology groups. This map is obviously compatible with the morphism α^∗ on de Rham cohomology constructed above.

Remark 6.9. — One can extend this to the case of separated smooth sim-plicial K-schemes of finite type as follows: Given such X• and an algebraic GL_r-bundle E/X•, topologically trivialized by α :X•^† E•GL^†_r,K as before, there exists astrict simplicial schemeU•→X• such that eachUp is a disjoint union of open affine subschemes ofXp, which coverXp. DefineX_•,•⁰ to be the

124CHAPTER 6. REFINED AND SECONDARY CLASSES FOR ALGEBRAIC BUNDLES

Cech nerve ofˇ U• → X•. Since X• is separated, X_•,•⁰ is affine, too. Moreover, the natural augmentation X_•,•⁰ → X• induces an isomorphism in cohomol-ogy.⁽¹⁾ Now the pullback E^0†_•,• ofE^†• toX_•,•⁰ is a bisimplicial Stein space and E^0†_•,• → E•^† induces an isomorphism in cohomology, too. By base change, α induces a topological morphism⁽²⁾ X_•,•⁰ → E^0†_•,•, which, using the extension of Dupont’s theorem 5.6 to the strict bisimplicial case, allows one to define the desired mapα^∗ on the de Rham and relative cohomology groups as in the lemma.

Definition 6.10. — LetX• be a smooth affine simplicialK-scheme of finite type, E/X• an algebraic GLr-bundle and α : X•^† E•GL^†_r,K a topological trivialization of the induced bundleE^†/X•^†. Then we define

Chf^rel_n (T, E, α) :=−α^∗Chf^rel_n (E)∈H_rel²ⁿ⁻¹(X•, n).