Finiteness of De Rham Cohomology in Rigid Analysis

For a large class of smooth dagger spaces—rigid spaces with overconvergent structure sheaf—we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of P. Berthelot’s rigid cohomology also in the nonsmooth case. We need a careful study of de Rham cohomology in situations of semistable reduction. Introduction Let R be a complete discrete valuation ring of mixed characteristic, let π ∈ R be a uniformizer, and let k = Frac(R), k̄ = R/(π). It is a simple observation that the de Rham cohomology H d R(X) of a positive-dimensional smooth affinoid k-rigid space X computed with respect to its (usual) structure sheaf is not finite-dimensional. The idea of instead using an overconvergent structure sheaf arises naturally from the paper of P. Monsky and G. Washnitzer [25]. The Monsky-Washnitzer cohomology of a smooth affine k̄-scheme Spec(A) is the de Rham cohomology of Æ ⊗R k, where Æ is a weakly complete formal lift of A. Monsky-Washnitzer cohomology has recently been shown to be finite-dimensional (independently by Berthelot [2] and, based on common work with G. Christol, by Z. Mebkhout [24]). The algebra Æ ⊗R k can be geometrically interpreted as a k-algebra of overconvergent functions on the rigid space Sp( à ⊗R k), where à is a lifting of A to a formally smooth π -adically complete R-algebra. In [11] we introduce a category of k-rigid spaces with overconvergent structure sheaf, which we call k-dagger spaces, and we study a functor X 7→ X ′ from this category to the category of k-rigid spaces which is not far from being an equivalence. For example, X and X ′ have the same underlying G-topological space and the same stalks of structure sheaves. Finiteness of Monsky-Washnitzer cohomology implies finiteness of de Rham cohomology for affinoid k-dagger spaces with good reduction; in the above notation, the algebra Æ gives rise to the affinoid k-dagger space X with 0(X,OX ) = Æ ⊗R k. Our main result generalizes this statement as follows. DUKE MATHEMATICAL JOURNAL Vol. 113, No. 1, c © 2002 Received 4 July 2000. Revision received 15 May 2001. 2000 Mathematics Subject Classification. Primary 14F30, 14G22. 57 58 ELMAR GROSSE-KLÖNNE THEOREM A (Corollary 3.5 plus Theorem 3.6) Let X be a quasi-compact smooth k-dagger space, let U ⊂ X be a quasi-compact open subset, and let Z → X be a closed immersion. Then T = X − (U ∪ Z) has finite-dimensional de Rham cohomology H d R(T ). By [11, Theorem 3.2], this implies finiteness of de Rham cohomology also for certain smooth k-rigid spaces Y , for example, if Y admits a closed immersion i into a polydisk without boundary (at least if i extends to a closed immersion with bigger radius) or if Y is the complement of a quasi-compact open subspace in a smooth proper k-rigid space. But our main corollary is of course the following. COROLLARY B (Corollary 3.8) For a k̄-scheme X of finite type, the k-vector spaces H rig(X/k) (see [2]) are finitedimensional. We do not re-prove finiteness of Monsky-Washnitzer cohomology; rather, we reduce our Theorem A to it. A big part of this paper is devoted to the study of de Rham cohomology in situations of semistable reduction. We need and prove the following theorem. THEOREM C (Theorem 2.3) Let X be a strictly semistable formal R-scheme, and let Xk̄ = ⋃ i∈I Yi be the decomposition of the closed fibre into irreducible components. For K ⊂ I , set YK = ⋂ i∈K Yi . Let X † be a k-dagger space such that its associated rigid space is identified with Xk . For a subscheme Y ⊂ Xk̄ , let ]Y [ † X be the open dagger subspace of X † corresponding to the open rigid subspace ]Y [X of Xk . Then for any ∅ 6= J ⊂ I , the canonical map H d R(]YJ [ † X )→ H ∗ d R ( ]YJ − (YJ ∩ ( ⋃ i∈I−J Yi ))[ † X ) is bijective. Another important tool is A. de Jong’s theorem on alterations by strictly semistable pairs, in its strongest sense. We proceed as follows. After recalling some facts on dagger spaces in Section 0, we formulate in Section 1 some basic concepts about D-modules on rigid and dagger spaces. This follows the complex analytic case (see, e.g., [23]). Instead of reproducing well-known arguments, we focus only on what is specific to the nonarchimedean case. Then we construct a long exact sequence for de Rham cohomology with supports in blowing-up situations. As in [18] (for algebraic k-schemes), this results from the existence of certain trace maps for proper morphisms; we define such trace maps based on constructions from [8], [3], and [26]. Finally, we prove the important technical fact that the de Rham cohomology H d R(X) of a smooth dagger FINITENESS OF DE RHAM COHOMOLOGY IN RIGID ANALYSIS 59 space X depends only on its associated rigid space X ; hence knowledge of X ′ (e.g., a decomposition into a fibre product) gives information about H d R(X). In Section 2, we begin to look at R-models of the associated rigid spaces; more specifically, we consider the case of semistable reduction. Its main result is Theorem C. It enables us to reduce Theorem A, in the case where U = ∅ and X has semistable reduction, to the finiteness of Monsky-Washnitzer cohomology. In Section 3, we first prove Theorem A in the case when U = ∅ = Z . After reduction to the case where X is affinoid and defined by polynomials, we apply de Jong’s theorem to an R-model of a projective compactification of X to reduce to the finiteness result of Section 2. The case of general Z is handled by a resolution of singularities (see [4]). Then we treat the case of general U by another application of de Jong’s theorem. The formal appearence of these last arguments bears some resemblance to the finiteness proofs in [2] and [18]. But there are also distinctive features, in the simultaneous control of special and generic fibre, and in particular in our second application of de Jong’s theorem. We apply it to a certain closed immersion of R-schemes X̄ k̄ ∪ Ȳ → X̄ , where the space X − U we are interested in is realized in (the tube ]Ȳ [ of) the compactifying divisor—not in its open complement. 0. Dagger spaces Let k be a field of characteristic zero, complete with respect to a nonarchimedean valuation |.|, with algebraic closure ka , and let 0 = |k a | = |k ∗ | ⊗Q. We gather some facts from [11]. For ρ ∈ 0, the k-affinoid algebra Tn(ρ) consists of all series ∑ aνX ∈ k[[X1, . . . , Xn]] such that |aν |ρ tends to zero if |ν| → ∞. The algebra Wn is defined to be Wn = ⋃ ρ>1 ρ∈0∗ Tn(ρ). A k-dagger algebra A is a quotient of some Wn; a surjection Wn → A endows it with a norm that is the quotient seminorm of the Gauss norm on Wn . All k-algebra morphisms between k-dagger algebras are continuous with respect to these norms, and the completion of a k-dagger algebra A is a k-affinoid algebra A in the sense of [6]. There is a tensor product ⊗k in the category of k-dagger algebras. As for k-affinoid algebras, one has for the set Sp(A) of maximal ideals of A the notions of rational and affinoid subdomains, and for these the analogue of Tate’s acyclicity theorem (see [6, Theorem 8.2.1]) holds. The natural map Sp(A)→ Sp(A) of sets is bijective, and via this map the affinoid subdomains of Sp(A) form a basis for the strong G-topology on Sp(A) from [6]. Imposing this G-topology on Sp(A), one gets a locally G-ringed space, an affinoid k-dagger space. (Global) k-dagger spaces are built from affinoid ones precisely as in [6]. The fundamental concepts and properties from [6] translate to k-dagger spaces. The notation Wn is taken from [16]. There the author assigns only the name of Washnitzer to this algebra. However, the referee pointed out that the name Monsky-Washnitzer algebra is the usual one. 60 ELMAR GROSSE-KLÖNNE There is a faithful functor from the category of k-dagger spaces to the category of k-rigid spaces, assigning to a k-dagger space X a k-rigid space X ′ (to which we refer as the associated rigid space; however, we use the notation (?) not only for this functor). X and X ′ have the same underlying G-topological space and the same stalks of structure sheaf. A smooth k-rigid space Y admits an admissible open affinoid covering Y = ⋃ Vi such that Vi = U ′ i for uniquely determined (up to noncanonical isomorphisms) affinoid k-dagger spaces Ui . Furthermore, this functor induces an equivalence between the respective subcategories formed by partially proper spaces as defined below. In particular, there is an analytification functor from k-schemes of finite type to k-dagger spaces. For a smooth partially proper k-dagger space X with associated k-rigid space X , the canonical map H d R(X) → H ∗ d R(X ) between the de Rham cohomology groups is an isomorphism. This follows from applying [11, Theorem 3.2] to the morphism between the respective Hodge – de Rham spectral sequences. By a dagger space not specified otherwise, we mean a k-dagger space; we use similarly the terms dagger algebras, rigid spaces, and so on. In the sequel, all dagger spaces and rigid spaces are assumed to be quasiseparated. We denote by D = {x ∈ k; |x | ≤ 1} (resp., D0 = {x ∈ k; |x | < 1}) the unit disk with (resp., without) boundary, with its canonical structure of k-dagger or k-rigid space, depending on the context. For ∈ 0, the ring of global functions on the polydisk {x ∈ kn; all |xi | ≤ }, endowed with its canonical structure of kdagger space, is denoted by k〈 −1 · X1, . . . , −1 · Xn〉. The dimension dim(X) of a dagger space X is the maximum of all dim(OX,x ) for x ∈ X . We say that X is pure-dimensional if dim(X) = dim(OX,x ) for all x ∈ X . A morphism f : X → Y of rigid or dagger spaces is called partially proper (cf. [19, p. 59]) if f is separated, if there is an admissible open affinoid covering Y = ⋃ i Yi , and if for all i there are admissible open affinoid coverings f (Yi ) = ⋃ j∈Ji X i j = ⋃ j∈Ji X ′ i j with X i j ⊂⊂Yi X ′ i j for every j ∈ Ji (where ⊂⊂Yi is defined as in [6]). LEMMA 1 Let Z → X be a closed immersion into an affinoid smooth dagger space. There is a proper surjective morphism g : X̃ → X with X̃ smooth, g−1(Z) a divisor with normal crossings on X̃ , and g−1(X − Z)→ (X − Z) an isomorphism. Proof Write X = Sp(Wn/I ), Z = Sp(Wn/J ) with ideals I ⊂ J ⊂ Wn . Since these ideals are finitely generated, there are a ρ > 1 and ideals Iρ ⊂ Jρ ⊂ Tn(ρ) such that I = Iρ ·Wn and J = Jρ ·Wn , and such that the rigid space Xρ = Sp(Tn(ρ)/Iρ) is smooth. FINITENESS OF DE RHAM COHOMOLOGY IN RIGID ANALYSIS 61 Apply [4, Theorem 1.10] to the closed immersion Zρ = Sp(Tn(ρ)/Jρ) → Xρ to get a morphism of rigid spaces X̃ρ → Xρ with the desired properties. Its restriction to the partially proper open subspace ⋃ ρ<ρ Sp(Tn(ρ )/(Iρ)) ⊂ Xρ is a morphism of partially proper spaces (compositions of partially proper morphisms are partially proper; see [19]) and hence by [11, Theorem 2.27] is equivalent to a morphism of dagger spaces. The restriction of the latter to X does the job.