Fields Institute Communications Volume 00, 0000

Line Bundles on Rigid Analytic Spaces

Urs T. Hartl

Abt. Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany

                             hartl@mathematik.uni-ulm.de

http://www.mathematik.uni-ulm.de/˜ hartl

Abstract. Let K be a complete discretely valued field and X_{K} a smooth proper rigid analytic space over K , which has a strictly semistable formal model. We construct a smooth rigid analytic group variety that represents the Picard functor of line bundles on X_{K} on the category of smooth rigid analytic spaces. Its identity component is an extension of a smooth proper rigid analytic group by an affine torus. The Néron-Severi group of X_{K} is finitely generated.

1 The Main Result

This is a report on joint work with W. Lütkebohmert [HL]. Our intention in this article is to explain the main result on the representability of the Picard functor, Theorem 1.5. We restrict ourselves at some points to indicate reasons rather than give detailed proofs and we leave out technicalities which are dispensable. The reader can find complete proofs of all our statements in [HL].

Rigid analytic spaces were introduced by J. Tate at a seminar at Harvard in 1961 following ideas of A. Grothendieck. His notes were later published in the Inventiones [Ta]. In the 1970’s M. Raynaud emphasized the relation between rigid analytic spaces and formal schemes [Ra]. Besides these two articles we refer the reader to [FRG] and [BGR] for an introduction to the subject.

Let K be a field of any characteristic which is assumed to be complete with respect to a non-archimedean discrete valuation of rank one and let R be its valuation ring. We denote by π a uniformizing parameter of R and by k the residue field of R . Moreover let X_{K} be a smooth rigid analytic space over K which is proper and connected and assume that there exists a K -rational point x : Sp K → X_{K} of X_{K} . We want to study line bundles on X_{K} . A line bundle L_{K} on X_{K} is

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

Supported by DFG grant LU 224/4-1.

a locally free sheaf of rank one on X_{K} , i.e. there exists an admissible covering {U_{K}^{i}} of X_{K} such that L_{K}|_{U_{K}^{i}} ≅ O_{U_{K}^{i}} . Since A. Grothendieck the classification of line bundles is approached by means of the relative Picard functor

\underline{Pic}_{X_{K}/K} : (Rigid analytic spaces) ⟶ (Sets) , V_{K} ⟼ \underline{Pic}_{X_{K}/K}(V_{K})

where

\underline{Pic}_{X_{K}/K}(V_{K}) := \{Isoclass (L_{K}, λ) : \begin{matrix}L_{K} line bundle on X_{K} ×_{K} V_{K} \\ λ : O_{V_{K}} ⟶^{∼} (x × id_{V_{K}})^{∗}L_{K} \end{matrix} \} .

See the diagram below. To a morphism f : V_{K} ⟶ W_{K} of rigid analytic spaces it assigns the morphism of sets

\underline{Pic}_{X_{K}/K}(f): \underline{Pic}_{X_{K}/K}(W_{K}) ⟶ \underline{Pic}_{X_{K}/K}(V_{K}) , (L_{K}, λ) ↦ ( id_{X_{K}} ×f)^{∗}(L_{K}, λ) .

So \underline{Pic}_{X_{K}/K} is a contravariant functor. It factors through the category of abelian groups. The isomorphism λ is called a rigidification. The purpose of the rigidification is twofold. Firstly it suppresses all line bundles coming from V_{K} and secondly it eliminates all automorphisms of L_{K} . Namely an isomorphism of two rigidified line bundles (L_{K}, λ) and (M_{K}, μ) on X_{K} ×_{K} V_{K} is by definition an isomorphism φ : L_{K} ⟶^{∼} M_{K} satisfying

μ = (x × id_{V_{K}})^{∗}φ ∘ λ : O_{V_{K}} ⟶^{∼} (x × id_{V_{K}})^{∗}M_{K} .

Thus Aut(L_{K}, λ) = { id} . Equivalently we can describe the Picard functor as

\underline{Pic}_{X_{K}/K}(V_{K}) = Pic(X_{K} ×_{K} V_{K}) / (p × id_{V_{K}})^{∗} Pic(V_{K}) ,

with Pic(Y_{K}) being the group of line bundles on the rigid analytic space Y_{K} , i.e. the absolute Picard group of Y_{K} , and p : X_{K} ⟶ Sp K the structural morphism. Namely using the section x of p we associate to every line bundle L_{K} ∈ Pic(X_{K} ×_{K} V_{K}) the rigidified line bundle

(L_{K} ⊗ (p × id_{V_{K}})^{∗}(x × id_{V_{K}})^{∗}LK^{∨} , λ) ,

where λ : O_{V_{K}} ⟶^{∼} (x× id_{V_{K}})^{∗}L_{K} ⊗(x× id_{V_{K}})^{∗}LK^{∨} is the canonical isomorphism. This shows that the Picard functor \underline{Pic}_{X_{K}/K} is independent of the K -rational point x .

We now ask whether the Picard functor is representable. This means that there exists a rigid analytic space Pic and an isomorphism of functors

\underline{Pic}_{X_{K}/K} ≅ Hom_{K}( _{•} , Pic) .

Thus by the Yoneda-Lemma there is a line bundle P_{K} on X_{K} ×_{K} Pic together with a rigidification λ_{P} : O_{Pic} ⟶^{∼} (x × id_{Pic})^{∗}P_{K} such that for any rigid analytic space V_{K} and any rigidified line bundle L_{K} on X_{K} ×_{K} V_{K} there is a unique morphism f : V_{K} ⟶ Pic and an isomorphism of rigidified line bundles ( id_{X_{K}} ×f)^{∗}(P_{K}, λ_{P}) ≅ (L_{K}, λ) on X_{K} ×_{K} V_{K} .

[figure: X_{K}; Sp K; x; X_{K} ×_{K} V_{K}; V_{K}; x × id_{V_{K}}; L_{K}; X_{K} ×_{K} Pic; Pic; x × id_{Pic}; P_{K}; id_{X_{K}} ×f; ∃! f]

In this article we want to show the representability of the Picard functor on the category of smooth rigid analytic spaces under the additional assumption that X_{K} admits a strictly semi-stable formal model X over R . We define this notion analogous to A.J. de Jong [dJ, Definition 2.16].

Definition 1.1 Let X be an admissible formal R -scheme (cf. [FRG, I]) and let X_{0}^{σ} for σ = 1, …, s be the irreducible components of the special fiber X_{0} of X . For each nonempty M ⊆ N := {1, …, s} we define

X_{0}^{M} := ⋂_{σ∈M} X_{0}^{σ}

as the scheme-theoretic intersection. X is called strictly semi-stable over R if

(a) X_{0} is geometrically reduced,

(b) X_{0}^{σ} is a Cartier divisor on X for all σ ∈ N , i.e. X_{0}^{σ} is locally on X a principal divisor and

(c) X_{0}^{M} is smooth over k for all M ⊆ N and equidimensional of dimension dim X − #M .

Remark 1.2 The conditions imply that the rigid analytic fiber X_{rig} is smooth over K ; cf. Proposition 1.4.

Remark 1.3 Condition (b) implies that X is regular; cf. Proposition 1.4.

We can characterize strictly semi-stable formal R -schemes by the following proposition.

Proposition 1.4 Let X be an admissible formal R -scheme. The following are equivalent:

1.) X is strictly semi-stable.

2.) Every closed point x ∈ X_{0} of the special fiber admits an open neighborhood which for some r ∈ ℕ is formally smooth over the formal scheme

Spf R ⟨ξ_{i_{1}}, …, ξ_{i_{s}}⟩/(ξ_{i_{1}} · . . . · ξ_{i_{s}} − π) .

Proof (cf. [dJ, 2.16]) The sufficiency of condition 2.) is clear. We show its necessity. So let X be strictly semi-stable and x ∈ X_{0} a closed point of the special fiber. Say x lies on the irreducible components X_{0}^{i_{1}}, …, X_{0}^{i_{s}} and not on the other components. On a neighborhood U of x the Cartier divisor X_{0}^{σ} is principal, say generated by ξ_{σ} . By (a) and (b) these generators satisfy the relation ξ_{i_{1}} · . . . · ξ_{i_{s}} = u · π for a unit u on U . After replacing ξ_{i_{1}} by u · ξ_{i_{1}} , we may assume u = 1 and obtain a morphism

U ⟶ Spf R ⟨ξ_{i_{1}}, …, ξ_{i_{s}}⟩/(ξ_{i_{1}} · . . . · ξ_{i_{s}} − π) .

By (c) the fiber X^{{i_{1},... ,i_{s}}} above the closed subscheme V(ξ_{i_{1}}, …, ξ_{i_{s}}) is smooth. This carries over to the formal completion along this closed subscheme. That in turn implies the formal smoothness of a whole neighborhood of x .

There is still the conjecture that every quasi-compact smooth rigid analytic space admits a strictly semi-stable formal model after a suitable base ring extension. If the residue characteristic is zero, one can apply the stable reduction theorem of Mumford; cf. [TE, p. 198] which provides the good model. So the result is valid without further conditions in this case.

In general a formal geometry analog of A.J. de Jong’s alterations result [dJ] is true. Namely, after a suitable finite separable field extension K̃ of K , there exists an étale surjective morphism X̃_{K} ⟶ X_{K} such that X̃_{K} admits a strictly semi-stable model X̃ over the valuation ring R̃ of K̃ . Indeed one can write X_{K} locally as a smooth curve fibration. In [L2, Theorems 5.2 and 5.3] it was shown that étale locally on X_{K} this curve fibration can be embedded into a smooth projective curve fibration which has a semi-stable formal model. Then we apply the methods of de Jong to obtain the desired strictly semi-stable model of X̃_{K} .

Now we are ready to state the main theorem.

Theorem 1.5 Let X_{K} be a smooth rigid analytic space over K which is proper and connected and assume that there exists a K -rational point x : Sp K → X_{K} of X_{K} . Further assume that X_{K} admits a strictly semi-stable formal model X over R .

Then there exists a unique (up to canonical isomorphism) smooth rigid analytic group variety Pic_{X_{K}/K} and a morphism of functors

Θ : Hom_{K}( _{•} , Pic_{X_{K}/K}) ⟶ \underline{Pic}_{X_{K}/K}

which is universal in the following sense: For any smooth rigid analytic space V_{K} , the map

Θ(V_{K}) : Hom_{K}(V_{K}, Pic_{X_{K}/K}) ⟶^{∼} \underline{Pic}_{X_{K}/K}(V_{K})

is bijective. In particular, there exists a line bundle P_{K} on X_{K} × Pic_{X_{K}/K} and an isomorphism λ_{P} : O_{Pic_{X_{K} /K}} ⟶^{∼} (x× id)^{∗}P_{K} such that, for any smooth rigid analytic space V_{K} and for any pair (L_{K}, λ) ∈ \underline{Pic}_{X_{K}/K}(V_{K}) , there is a unique morphism f : V_{K} ⟶ Pic_{X_{K}/K} and a unique isomorphism of rigidified line bundles (L_{K}, λ) ⟶^{∼} ( id × f)^{∗}(P_{K}, λ_{P}) .

The identity component Pic_{X_{K}/K}^{0} of Pic_{X_{K}/K} , i.e. the connected component containing the neutral element, is called the Picard variety of X_{K} and P_{K} is called the Poincaré bundle. After a base field extension the Picard variety is an extension of an abeloid variety by an affine torus; an abeloid variety is a connected smooth rigid analytic group variety with proper underlying space.

The Galois module NS_{X_{K}/K} := Pic_{X_{K}/K}(𝕂) / Pic_{X_{K}/K}^{0}(𝕂) where 𝕂 is the completion of an algebraic closure of K is called the Néron-Severi group of X_{K} . The Néron-Severi group of X_{K} is finitely generated.

Remark 1.6 (1) If X_{K} is the analytification of a proper algebraic variety, the rigid analytic Picard variety is the analytification of the classical algebraic Picard variety, due to the GAGA-principle; cf. [L1, Theorem 2.8].

(2) In the case of (1) the connected components of the Picard variety are proper as X_{K} is assumed to be smooth; cf. [FGA, n^{0}236, Theorem 2.1] or [BLR, Theorem 8.4/3].

(3) If X_{K} is a smooth connected proper rigid analytic group variety, the representability of \underline{Pic}_{X_{K}/K} was established by W. Lütkebohmert in [L2] resp. [BL]. In particular, he showed that Pic_{X_{K}/K}^{0} is smooth and proper and that the Néron-Severi group of X_{K} is finitely generated and torsion free.

(4) There are examples of (non-algebraic) proper smooth rigid analytic spaces where the Picard variety is not proper; cf. [Mu]. Let us briefly review the example. This is an analogue of the Hopf surface. It can be defined in the following way

X_{K} = (𝔸_{K}^{2} − {0})/Γ

where Γ = ⟨γ ⟩ is generated by a single element γ which acts on 𝔸_{K}^{2} − {0} by

γ(ξ_{1}, ξ_{2}) = (α_{1}ξ_{1} , α_{2}ξ_{2})

where α_{1}, α_{2} ∈ K^{∗} with 0 < |α_{1}| ≤ |α_{2}| < 1 . One can show that X_{K} is smooth and proper and that it admits a strictly semi-stable formal model. One computes that Pic_{X_{K}/K} = 𝔾_{m,K} .

Proof. The rest of this article is devoted to the proof of this theorem. For technical reasons we first demonstrate the existence of the identity component of the representing space.

We introduce the category

ℭ_{K} of pointed rigid analytic spaces (V_{K}, v_{K}) where V_{K} is smooth and connected over K and where v_{K} ∈ V_{K}(K) is a K-rational point. The morphisms in this category are the rigid analytic morphisms respecting the points.

Instead of the general Picard functor, we consider the slightly different functor

\underline{Pic}_{X_{K}/K}^{0} : ℭ_{K} ⟶ (Sets) , (V_{K}, v_{K}) ⟼ \underline{Pic}_{X_{K}/K}^{0}(V_{K}, v_{K})

of rigidified line bundles which are trivialized at the given point where

\underline{Pic}_{X_{K}/K}^{0}(V_{K}, v_{K}) = \{Isoclass (L_{K}, λ) : \begin{matrix}L_{K} line bundle on X_{K} ×_{K} V_{K} \\ λ : O_{V_{K}} ⟶^{∼} (x × id_{V_{K}})^{∗}L_{K} \\ ( id_{X_{K}} × v_{K})^{∗}L_{K} ≅ O_{X_{K}} trivial \end{matrix} \} .

These are in a sense the deformations of the trivial line bundle or phrased differently the line bundles which are “algebraically” equivalent to the trivial line bundle. Later on it turns out that the representing space of this functor is the identity component of the usual Picard functor.

We further show that the Néron-Severi group NS_{X_{K}/K} is finitely generated. Then we construct the rigid analytic Picard variety as the disjoint union

Pic_{X_{K}/K} := ∐_{[L_{K}] ∈ NS_{X_{K} /K}} L_{K} ⊗ Pic_{X_{K}/K}^{0} .

2 Typical Line Bundles

In order to show the representability of the functor \underline{Pic}_{X_{K}/K}^{0} , i.e. to construct the Picard variety we want to introduce two typical types of line bundles.

1.) The first type of line bundle arises from the combinatorial configuration of the irreducible components of the special fiber X_{0} . Let n ∈ H^{1}(X_{0}, ℤ) be a cohomology class. It can be represented by a cocycle n = (n_{ij}) with respect to an open covering {U^{i}} of X , i.e. n_{ij} ∈ ℤ(U^{i} ∩ U^{j}) . Then for a coordinate function ξ on the multiplicative group 𝔾_{m,K} we obtain elements ξ^{n_{ij}} ∈ O^{×}((U_{rig}^{i} ∩ U_{rig}^{j}) ×_{K} 𝔾_{m,K}) which clearly form a cocycle with respect to the covering {U_{rig}^{i}×_{K} 𝔾_{m,K}} of X_{K} ×_{K} 𝔾_{m,K} . We consider the line bundle on X_{K} ×_{K} 𝔾_{m,K} which is given by this cocycle and denote it by (ξ^{n}) . We call line bundles of this type multiplicative.

2.) To any formal line bundle L on X ×_{R} V we can consider the rigid analytic line bundle L_{rig} := L ⊗_{R} K on X_{K} ×_{K} V_{rig} .

These two types of line bundles indicate that Pic_{X_{K}/K}^{0} should have a formal part and a torus part. In a way all line bundles on X_{K} are built from these two types. More precisely the following is true.

Proposition 2.1 Let V be smooth over R⟨ζ_{1}, …, ζ_{t+1}⟩/(ζ_{1} · . . . · ζ_{t+1} − π) . Let v ∈ V (R) be a point above {ζ_{1} = … = ζ_{t} = 1} . Consider a rigid analytic line bundle L_{K} on X_{K} ×_{K} V_{rig} which is trivialized along v .

Then L_{K} is a tensor product L_{K} ≅ M_{K} ⊗ N_{rig} of two line bundles where

(1) M_{K} ≅ (ζ_{1}^{n_{1}}) ⊗ … ⊗ (ζ_{t}^{n_{t}}) is the multiplicative line bundle on X_{K} ×_{K} V_{rig} associated to suitable elements n_{1}, …, n_{t} ∈ H^{1}(X_{0}, ℤ) .

(2) N is a formal line bundle on X ×_{R} V which is trivialized along v .

The decomposition L_{K} ≅ M_{K} ⊗ N_{rig} is unique. In particular, the Picard group of line bundles which are trivial along v decomposes into

Pic^{0}(X_{K} ×_{K} V_{rig}) = H^{1}(X_{0}, ℤ)^{t} ⊕ Pic^{0}(X ×_{R} V) .

Proof In order to prove the proposition we try to extend the line bundle L_{K} to a formal line bundle on X ×_{R} V . If X ×_{R} V were regular, this would be possible. However that is not the case in general. On the other hand X possesses a covering by open sets U which are smooth over Spf R⟨ξ_{1}, …, ξ_{s}⟩/(ξ_{1} ·. . .·ξ_{s} −π) . We can desingularize the product U ×_{R} V by successive blowing-ups. Namely we blow up

Spf R⟨ξ_{1}, …, ξ_{s}, ζ_{1}, …, ζ_{t+1}⟩/(ξ_{1} · . . . · ξ_{s} − π , ζ_{1} · . . . · ζ_{t+1} − π)

in the open ideal (ξ_{1}, ζ_{1}) and obtain two charts of the blowing-up Y′ :

• ξ_{1} = ζ_{1} · ξ′_{1} : The relations are equivalent to

\begin{aligned}ξ′_{1} · ξ_{2} · . . . · ξ_{s} − ζ_{2} · . . . · ζ_{t+1} & = & 0 and \\ ξ_{1} · . . . · ξ_{s} − π & = & 0 . \end{aligned}

Thus the number t was decreased by 1 while s stayed constant.

• ζ_{1} = ξ_{1} · ζ′_{1} : The relations are equivalent to

\begin{aligned}ξ_{2} · . . . · ξ_{s} − ζ′_{1} · ζ_{2} · . . . · ζ_{t+1} & = & 0 and \\ ξ_{1} · . . . · ξ_{s} − π & = & 0 . \end{aligned}

Thus the number s was decreased by 1 while t stayed constant.

By repeating this step we obtain a desingularization U ×_{R} V ⟵ … ⟵ Y^{n} . Since Y^{n} is regular with Y_{rig}^{n} = (U ×_{R} V)_{rig} the line bundle L_{K} extends to a formal line bundle L^{(n)} on Y^{n} . Further the above desingularization allows us to control the centers of the blowing-ups. A careful analysis shows that the line bundle L^{(n)} descends step by step to a line bundle L on U ×_{R} V which extends the rigid analytic line bundle L_{K} . We can further choose L to be trivialized along v .

Now we would like to glue the local extensions L and L′ of our rigid analytic line bundle L_{K} above the intersection (U ∩ U′) ×_{R} V =: W . On the rigid analytic fiber W_{rig} we have an isomorphism

ψ_{K} : L_{rig} ⟶^{∼} L′_{rig}.

In order to extend it to an isomorphism of the formal line bundles we have to study the invertible functions on W_{rig} . These obviously include all formal units O_{W} (W)^{×} as well as ξ_{1}, …, ξ_{s} and ζ_{1}, …, ζ_{t+1} . If W is connected it turns out that all invertible functions are built up from these

O_{W_{rig}}(W_{rig})^{×} = O_{W} (W)^{×} ⊕ ((ξ_{1}^{ℤ} ⊕ … ⊕ ξ_{s}^{ℤ}) + (ζ_{1}^{ℤ} ⊕ … ⊕ ζ_{t+1}^{ℤ})).

The later sum is not direct. There is one relation ξ_{1} · . . . · ξ_{s} = π = ζ_{1} · . . . · ζ_{t+1} .

Since L and L′ are both trivialized along v we see that there is no contribution of the ξ ’s in ψ_{K} , i.e.

ψ_{K} = ζ_{1}^{n_{1}} · . . . · ζ_{t}^{n_{t}} · ψ

Since the collection of the ψ_{K} satisfies a cocycle condition the same is true for n_{1} up to n_{t} and for ψ . So they form cocycles n_{1}, …, n_{t} ∈ H^{1}(X_{0}, ℤ) . The cocycle (ζ_{1}^{n_{1}} · . . . · ζ_{t}^{n_{t}}) with respect to the covering consisting of the U ×_{R} V ’s defines the multiplicative line bundle M_{K} := (ζ_{1}^{n_{1}})⊗…⊗(ζ_{t}^{n_{t}}) on X_{K} ×_{K} V_{rig} . In particular M_{K}|_{(U×_{R}V)_{rig}} is trivial. The collection of the isomorphisms ψ : L ⟶^{∼} L′ forms a cocycle which allows us to glue the local extensions L to a formal line bundle N on X ×_{R} V . This yields the decomposition

L_{K} ≅ M_{K} ⊗ N_{rig} .

we are looking for. It is unique since the decomposition of the isomorphism was unique.

3 The Construction of the Identity Component

We want to illustrate the construction of the Picard variety by discussing the example of the Tate curve. The Tate curve is the quotient X_{K} = 𝔾_{m,K}/q^{ℤ} for some q ∈ K^{×} with |q| < 1 . It is an algebraic curve, connected proper and smooth over K and admits a strictly semi-stable formal model X over R . The special fiber X_{0} of this model is a closed chain of projective lines. Since X_{K} is an elliptic curve, \underline{Pic}_{X_{K}/K}^{0} is represented by the Jacobi variety J_{K} of X_{K} . The Jacobi variety is isomorphic to X_{K} and therefore also possesses a strictly semi-stable model J . Let \overline{J} be the identity component of the open part of J which is formally smooth over R . Then

\overline{J} = \overline{𝔾}_{m,R} := lim_{⟶} 𝔾_{m,R_{n}}

where we denote the n -th infinitesimal neighborhood by R_{n} := R/(π^{n+1}) and X_{n} := X⊗_{R}R_{n} . The multiplicative group 𝔾_{m,R_{n}} represents the functor \underline{Pic}_{X_{n}/R_{n}}^{0} of line bundles which have degree 0 on each irreducible component of X_{0} . This is due to the fact that every such line bundle on X_{0} is trivial on each of the projective lines contained in X_{0} . Namely choose local sections of the line bundle on each of these projective lines X_{0}^{σ} , i.e. meromorphic functions f_{σ} , which neither have poles nor zeroes in the double points. Starting on one component X_{0}^{1} we can multiply the meromorphic function f_{2} on the next component X_{0}^{2} by a constant a so that the two functions f_{1} and af_{2} agree at the double point X_{0}^{1} ∩ X_{0}^{2} . Repeating this, we go round the circle once. The ratio f_{1}/f_{s} at the last double point where the chain of projective lines closes is the gluing data that determines this line bundle. It is an element of 𝔾_{m,k} .

The group \overline{J} represents the formally smooth deformations of the trivial line bundle on X . Its rigid analytic fiber is

\overline{J}_{rig} = \overline{𝔾}_{m,K} := Sp K⟨ζ, ζ^{−1}⟩ = \{z ∈ K^{alg} : |z| = 1\}/ Gal .

It can be embedded into the multiplicative group

\overline{J}_{rig} = \overline{𝔾}_{m,K} ⊂⟶ 𝔾_{m,K} =: Ĵ_{K} .

We interpret Ĵ_{K} in the following way. We have H^{1}(X_{K}, ℤ) = H^{1}(X_{0}, ℤ) = ℤ · b . This induces the multiplicative line bundle (ζ^{b}) on X_{K} ×_{K} 𝔾_{m,K} which gives rise to a morphism 𝔾_{m,K} ⟶ J_{K} . The group Ĵ_{K} contains a lattice

Λ := {ζ ∈ Ĵ_{K} : (ζ^{b}) is trivial } .

One can show that the quotient of Ĵ_{K} by Λ is J_{K} ; cf. [BL, Theorem 1.2].

In the general case we construct the Picard variety in a similar manner. We consider the infinitesimal neighborhoods R_{n} := R/(π^{n+1}) and X_{n} := X ⊗_{R} R_{n} . By assumption the scheme X_{n} is proper and flat over R_{n} with geometrically reduced special fiber. The point x ∈ X(R) induces a point x_{n} ∈ X_{n}(R_{n}) for all n ∈ ℕ . Due to the classical result of M. Artin [Ar 1, Theorem 7.3] the functor \underline{Pic}_{X_{n}/R_{n}} is representable by an algebraic space locally of finite type over R_{n} . This is a group scheme over R_{n} since an algebraic group space over an Artinian base is a scheme; cf. [Ar 2, Theorem 3.5]. It is locally of finite type over R_{n} . Let P′_{n} be its identity component

P′_{n} := Pic_{X_{n}/R_{n}}^{0} .

It is a group scheme of finite type over R_{n} ; cf. [SGA 3, I, Exposé VI_{A}, Proposition 2.4]. Furthermore we have the Poincaré bundle P′_{n} on X_{n} ×_{R_{n}} P′_{n} . We set

P′ := lim_{⟶} P′_{n} .

The limit is defined via the projections P′_{n+1} → P′_{n} ; notice that the identity X_{n+1} ⊗_{R_{n+1}} R_{n} = X_{n} implies P′_{n+1} ⊗_{R_{n+1}} R_{n} = P′_{n} . The formal scheme P′ is of topological finite type over R (cf. [FRG, I]). The system of the Poincaré bundles P′_{n} is compatible and gives rise to the Poincaré bundle P′ := lim_{⟵} P′_{n} on X ×_{R} P′ .

We investigate the structure of P′ . Its special fiber P′_{0} does not contain an additive group 𝔾_{a,\overline{k}} where \overline{k} is an algebraic closure of k . The reason for this lies in the semi-stability of X . Namely every line bundle on X_{0} ×_{k} 𝔾_{a,\overline{k}} which is trivialized along 0 is already globally trivial. In particular the torsion points of order prime to char k are dense in P′_{0} . However P′ may not be smooth, in fact it will in general not even be flat over R . By dividing out its π -torsion and its nilpotent structure we obtain an admissible formal R -scheme \overline{P′} . Its rigid analytic fiber is smooth but maybe not connected. The smoothness comes from the fact that the mentioned torsion points of order prime to char k lift to \overline{P′} . We now consider the identity component \overline{P}_{K} of \overline{P′}_{rig} and let \overline{P} be the formal Néron model of \overline{P}_{K} ; cf. [BS, Theorem 1.2]. It is a smooth formal group scheme over R and can be obtained as the identity component of the group smoothening of \overline{P′} ; cf. [BLR, Theorem 7.1/5]. We have the following diagram

\overline{P}_{K} ⊂⟶ \overline{P}
⊂⟶  ⟶
\overline{P′}_{rig} ⊂⟶ \overline{P′} ⊂⟶ P′

The vertical arrow on the left is a finite morphism, so also the vertical arrow on the right is finite; cf. [BGR, Theorem 6.3.5/1]. By pulling back the line bundle P′ we obtain the Poincaré bundle \overline{P} on X ×_{R} \overline{P} . The group \overline{P} represents the formally smooth deformations of the trivial line bundle on X , because every morphism from a smooth connected admissible formal R -scheme into P′ factors through \overline{P} due to the universal property of the Néron model.

Since P′_{0} contains no additive group and \overline{P} ⟶ P′ is finite, the unipotent part of the special fiber \overline{P}_{0} is trivial [SGA 3, II, Exposé XVII, Proposition 4.1.1]. This implies that \overline{P} is semi-abelian, i.e.

1 ⟶ \overline{T} ⟶ \overline{P} ⟶ B ⟶ 1

the formal group scheme \overline{P} is an extension of a formal abelian scheme B by a formal torus \overline{T} ≅ \overline{𝔾}_{m,R}^{r} = lim_{⟶} 𝔾_{m,R_{n}}^{r} ; cf. [BLR, Theorem 9.2.1] and [SGA 3, II, Exposé XVII, Théorème 7.2.1]. The torus part accounts for the multiplicative line bundles, i.e. H^{1}(X_{0}, ℤ) ≅ Hom(\overline{𝔾}_{m,R}, \overline{T}) .

The formal torus \overline{T}_{rig} = \overline{𝔾}_{m,K}^{r} is embedded into the affine Torus T_{K} := 𝔾_{m,K}^{r} . We define P̂_{K} to be the push-out of \overline{P}_{K} via this embedding

1 ⟶ \overline{T}_{rig} ⟶ \overline{P}_{K} ⟶ B_{rig} ⟶ 1
⊂⟶  ⊂⟶  =
1 ⟶ T_{K} ⟶ P̂_{K} ⟶ B_{rig} ⟶ 1 .

Due to its cubical structure the Poincaré bundle \overline{P}_{rig} on X_{K} ×_{K} \overline{P}_{K} extends to a line bundle P̂_{K} on X_{K} ×_{K} P̂_{K} ; cf. [Mo, Définition I.2.4.5] and [BL, Propositions 4.1 and 4.2]. We will now construct the Picard variety as a quotient of P̂_{K} by a suitable lattice. Let

Λ := { p ∈ P̂_{K} : ( id_{X_{K}} ×p)^{∗}P̂_{K} is trivial } .

Then Λ is a lattice in P̂_{K} and the quotient P_{K} := P̂_{K}/Λ is a smooth rigid analytic group variety. The lattice Λ also operates canonically on the Poincaré bundle P̂_{K} . We can divide out this action and obtain the Poincaré bundle P_{K} on X_{K} ×_{K} P_{K} . We claim

Theorem 3.1 The smooth connected rigid analytic group variety P_{K} represents the functor \underline{Pic}_{X_{K}/K}^{0} on the category ℭ_{K} .

Proof 1.) Let first be V a formal scheme as in Proposition 2.1 and L_{K} a line bundle on X_{K} ×_{K} V_{rig} trivialized along v ∈ V_{rig}(K) . Then L_{K} decomposes uniquely according to Proposition 2.1

L_{K} ≅ M_{K} ⊗ N_{rig}

into a multiplicative line bundle M_{K} and a formal line bundle N . The line bundle M_{K} gives rise to a uniquely determined morphism

V_{rig} ⟶ T_{K} ⟶ P̂_{K} ⟶ P_{K} .

The line bundle N gives rise to a uniquely determined morphism

V_{rig} ⟶ \overline{P}_{K} ⟶ P̂_{K} ⟶ P_{K} .

By adding these two morphisms in P_{K} we get a uniquely determined morphism f : V_{rig} ⟶ P_{K} with L_{K} ≅ ( id_{X_{K}} ×f)^{∗}P_{K} .

2.) Now let V_{K} be an arbitrary smooth rigid analytic space and L a line bundle on X_{K} ×_{K} V_{K} which is trivialized along some v ∈ V_{K}(K) . By the formal analog of de Jong’s alteration result there exists an étale covering Ṽ_{K} ⟶ V_{K} which admits a semi-stable formal model Ṽ . Further we can chose Ṽ to be of the type considered under 1.) and so we obtain a morphism f̃ : Ṽ_{K} ⟶ P_{K} . We can descend this morphism to the morphism f : V_{K} ⟶ P_{K} with L_{K} ≅ ( id_{X_{K}} ×f)^{∗}P_{K} we were looking for.

The rigid analytic group P_{K} will be the identity component of the representing space for the usual Picard functor \underline{Pic}_{X_{K}/K} .

As we have remarked in the beginning (Remark 1.6), P_{K} need not be proper. Nevertheless it is an extension of an abeloid variety by an affine torus. Namely using the lattice Λ we can split the torus T_{K} of P̂_{K} into a product of subtori

T_{K} = T′_{K} × T′′_{K}

with rk_{ℤ} Λ = dim T′′_{K} = r′′ . Due to the structure of P̂_{K} we obtain an extension via push-out with respect to the projection T_{K} ⟶ T′′_{K}

1 ⟶ T_{K} ⟶ P̂_{K} ⟶ B_{K} ⟶ 1
⟶  ⟶  =
1 ⟶ T′′_{K} ⟶ Q̂_{K} ⟶ B_{K} ⟶ 1 .

The canonically induced map P̂_{K} ⟶ Q̂_{K} gives rise to an isomorphism of lattices

P̂_{K} ⟶ Q̂_{K}
⟵⊃  ⟵⊃
Λ ⟶^{∼} Λ′′

where Λ′′ is now a lattice of Q̂_{K} of full rank r′′ . In particular, the quotient

Q_{K} = Q̂_{K} / Λ′′

is an abeloid variety, i.e. a smooth proper connected rigid analytic group variety. Thus we obtain the following commutative diagram

Λ = Λ′′
⊂⟶  ⊂⟶
1 ⟶ T′_{K} ⟶ P̂_{K} ⟶ Q̂_{K} ⟶ 1
=  ⟶  ⟶
1 ⟶ T′_{K} ⟶ P_{K} ⟶ Q_{K} ⟶ 1

which makes P_{K} an extension of an abeloid variety Q_{K} by an affine torus T′_{K} . This proves Theorem 1.5 as far as the identity component is concerned.

4 The Néron-Severi Group

We next study the Néron-Severi group of X_{K} which is by definition the quotient

NS_{X_{K}/K} = \underline{Pic}_{X_{K}/K}(𝕂) / Pic_{X_{K}/K}^{0}(𝕂)

where \underline{Pic}_{X_{K}/K}(𝕂) denotes the group of isomorphism classes of line bundles on X_{K} ˆ⊗_{K}𝕂 and where Pic_{X_{K}/K}^{0}(𝕂) is the set of 𝕂 -rational points of P_{K} . We want to relate this group to the Néron-Severi group of the special fiber X_{0} ; i.e., to the group

NS_{X_{0}/k} = Pic_{X_{0}/k}(\overline{k}) / Pic_{X_{0}/k}^{0}(\overline{k})

where \overline{k} is an algebraic closure of the residue field k of R . Therefore we note that every element of NS_{X_{K}/K} can in fact be represented by a line bundle L_{K} on X_{K} ⊗_{K} K̃ for a finite field extension K̃ of K . We try to extend L_{K} to a formal line bundle L on X̃ := X ⊗_{R} R̃ where R ⊆ R̃ is the associated extension of discrete valuation rings. If π̃ is the uniformizer of R̃ and e ∈ ℕ the ramification index then

π = ũ · π^{e}

for a unit ũ of R̃ . In general the formal line bundle L does not exist, since X̃ is not regular. But we can proceed similarly to the case of X ×_{R} V we discussed in Proposition 2.1:

(a) There is a desingularization procedure to obtain a regular formal model X̃ ⟵ Ỹ of X̃_{rig} = Ỹ_{rig} . The line bundle L_{K} extends to a formal line bundle L̃ on Ỹ .

(b) With respect to a suitable covering {U^{i}} of X the line bundle L̃ descends to line bundles L^{i} on Ũ^{i} := U^{i} ⊗_{R} R̃ .

(c) In order to glue these local extensions of L_{K} we study the invertible functions on the intersections W̃ := Ũ ∩ Ũ′ .

This time OX̃_{rig}(W̃_{rig})^{×} contains the following subgroup

OX̃_{rig}(W̃_{rig})^{×} ⊇ OX̃(W̃)^{×} ⊕ ξ Γ(W̃_{0}, ℤ)_{1} ⊕ … ⊕ ξ Γ(W̃_{0}, ℤ)_{s} .

The quotient by this subgroup is Γ(W̃_{0}, ℤ/eℤ) , coming from the fact that π̃ is an invertible function on W̃_{rig} . This means we can associate to L_{K} an element

cyc(L_{K}) ∈ H^{1}(X_{0}, ℤ/eℤ) .

We consider the exact sequence

0 ⟶ ℤ ⟶ ℤ ⟶ ℤ/eℤ ⟶ 0

of constant sheaves on X_{0} . The image of cyc(L_{K}) under the connecting homomorphism δ : H^{1}(X_{0}, ℤ/eℤ) ⟶ H^{2}(X_{0}, ℤ) is zero if and only if cyc(L_{K}) is the image of some n ∈ H^{1}(X_{0}, ℤ) . This is the case if and only if there exists a multiplicative line bundle M_{K} := (π^{n}) on X̃_{rig} such that L_{K} ⊗ MK^{∨} extends to a formal line bundle L on X̃ . Since M_{K} corresponds to a K̃ -rational point of P̂_{K} it plays no role in the investigation of the Néron-Severi group. So we see that the obstruction for extending L_{K} globally to a line bundle L on X̃ is given by an element of

im(δ : H^{1}(X_{0}, ℤ/eℤ) ⟶ H^{2}(X_{0}, ℤ)) ,

i.e. by a torsion element of H^{2}(X_{0}, ℤ) . Since the later cohomology group is finitely generated, it contains only finitely many torsion elements.

The line bundles for which this obstruction vanishes form a subgroup NS_{X_{K}/K}^{1} of NS_{X_{K}/K} . By reducing mod π we get a map

NS_{X_{K}/K}^{1} ⟶ NS_{X_{0}/k} / M

where M is the image of the set of formal line bundles on X̃ which induce multiplicative line bundles on the generic fiber. Elements in the kernel of this map can be represented by formal line bundles L on X̃ such that the reduction L_{0} := L ⊗R̃ k̃ belongs to Pic_{X_{0}/k}^{0}(k̃) . There k̃ is the residue field of R̃ . Due to our construction, the class of L is therefore an R̃ -rational point of \overline{P′} . Thus we obtain an exact sequence

\overline{P′}_{rig}(𝕂) / \overline{P}_{K}(𝕂) ⟶ NS_{X_{K}/K}^{1} ⟶ NS_{X_{0}/k} / M

Since \overline{P}_{K} is a subgroup of finite index in \overline{P′}_{rig} we conclude that NS_{X_{K}/K}^{1} is an extension of a subquotient of the Néron-Severi group of the special fiber by a finite group. NS_{X_{0}/k} is finitely generated; cf. [SGA 6, Exposé XIII, Thm. 5.1]. Thus the Néron-Severi group NS_{X_{K}/K} of X_{K} is finitely generated.

The rigid analytic group variety

Pic_{X_{K}/K} := ∐_{[L_{K}] ∈ NS_{X_{K} /K}} L_{K} ⊗ Pic_{X_{K}/K}^{0}

represents the Picard functor \underline{Pic}_{X_{K}/K} on the category of smooth rigid analytic spaces. This finishes the proof of Theorem 1.5.

References

[Ar 1] Artin, M.: Algebraization of formal moduli I, Global Analysis, Papers in Honor of K.

Kodaira, Princeton University Press 1969, 21 - 71.

[Ar 2] Artin, M.: The Implicit Function Theorem in Algebraic Geometry, Arithmetical Algebraic Geometry, Proceedings Paper of the Conference on Arithmetic Algebraic Geometry at Purdue University in 1963, Harper and Row, New York 1965, 13 - 34.

[BGR] Bosch, S., Güntzer, U., Remmert, R.: Nonarchimedian Analysis, Grundl. 261, Springer-Verlag, Berlin-Heidelberg 1984.

[BL] Bosch, S., Lütkebohmert, W.: Degenerating Abelian Varieties, Topology Vol. 30, No. 4 (1991), 653 - 698.

[BLR] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models, Ergeb. 3. Folge, Band 21, Springer-Verlag, Berlin-Heidelberg 1990.

[BS] Bosch, S., Schlöter, K.: Néron models in the setting of formal and rigid geometry, Mathematische Annalen 301 (1995), 339 - 362.

[dJ] de Jong, A.J.: Smoothness, Semi-stability and Alterations, Publications Mathématiques IHES 83 (1996), 51 - 93.

[FGA] Grothendieck, A.: Fondements de la Géométrie Algébrique, Séminaire Bourbaki 1957 - 62, Secrétariat Mathématiques, Paris 1962.

[FRG] Bosch, S., Lütkebohmert, W., Raynaud, M.: Formal and Rigid Geometry I. Rigid Spaces, Mathematische Annalen 295 (1993), 291 - 317. II. Flattening Techniques, Mathematische Annalen 296 (1993), 403 - 429. III. The Relative Maximum Principle, Mathematische Annalen 302 (1995), 1 - 29. IV. The Reduced Fiber Theorem, Inventiones Mathematicae 119 (1995), 361 - 398.

[HL] Hartl, U., Lütkebohmert, W.: On rigid-analytic Picard varieties, to appear in Journal für die reine und angewandte Mathematik, 43 p.

[L1] Lütkebohmert, W.: Formal-algebraic and rigid-analytic geometry, Mathematische Annalen 286 (1990), 341 - 371.

[L2] Lütkebohmert, W.: The structure of proper rigid groups, Journal für die reine und angewandte Mathematik 468 (1995), 167 - 219.

[Mo] Moret-Bailly, L.: Pinceaux de Variétés Abéliennes, Astérisque 129 (1985).

[Mu] Mustafin, G.A.: p -adic Hopf varieties, Functional Analysis and its Applications 11 (1977), 234 - 235.

[Ra] Raynaud, M.: Géométrie analytique rigide d’après Tate, Kiehl, …. Table ronde d’analyse non archimedienne, Mémoires de la Société Mathématique de France 39/40 (1974), 319 - 327.

[SGA 3] Demazure, M., Grothendieck, A., et al.: Séminaire de Géométrie Algébrique 3: Schémas

en Groupes I, II, III, LNM 151, 152, 153, Springer-Verlag, Berlin-Heidelberg 1970.

[SGA 6] Grothendieck, A., et al.: Séminaire de Géométrie Algébrique 6: Théorie des Intersections

et Théorème de Riemann-Roch, LNM 225, Springer-Verlag, Berlin-Heidelberg 1971.

[Ta] Tate, J.: Rigid analytic spaces, Inventiones Mathematicae 12 (1971), 257 - 289.

[TE] Kempf, G.; Knudsen, F.; Mumford, D.; Saint-Donat, B.: Toroidal embeddings I, LNM 339, Berlin-Heidelberg-New York: Springer 1973