Homotopy Theory for Rigid Analytic Varieties

Home , Berkovich space, Michel Raynaud, Nisnevich topology, Rigid analytic space

Helene Sigloch

Dissertation zur Erlangung des Doktorgrades der Fakultat¨ fur¨ Mathematik und Physik der Albert-Ludwigs-Universitat¨ Freiburg im Breisgau

Marz¨ 2016 Dekan der Fakultat¨ fur¨ Mathematik und Physik: Prof. Dr. Dietmar Kr¨oner

Erster Referent: Dr. Matthias Wendt

Zweiter Referent: Prof. Dr. Joseph Ayoub

Datum der Promotion: 25. Mai 2016 Contents

Introduction 3

1 Rigid Analytic Varieties 11 1.1 Deﬁnitions ...... 12 1.2 Formal models and reduction ...... 20 1.3 Flatness, smoothness and ´etaleness ...... 23

2 Vector Bundles over Rigid Analytic Varieties 27 2.1 A word on topologies ...... 28 2.2 Serre–Swan for rigid analytic quasi-Stein varieties ...... 30 2.3 Line bundles ...... 37 2.4 Divisors ...... 38

3 Homotopy Invariance of Vector Bundles 41 3.1 Serre’s problem and the Bass–Quillen conjecture ...... 41 3.2 Homotopy invariance ...... 42 3.3 Counterexamples ...... 44 3.4 Local Homotopy Invariance ...... 48 3.5 The case of line bundles ...... 56 3.5.1 B1-invariance of Pic ...... 57 1 3.5.2 Arig-invariance of Pic ...... 59

4 Homotopy Theory for Rigid Varieties 65 4.1 Model categories and homotopy theory ...... 65 4.2 Sites and completely decomposable structures ...... 70 4.3 Homotopy theories for rigid analytic varieties ...... 71 4.3.1 Relation to Ayoub’s theory ...... 75 4.3.2 A trip to the zoo ...... 76

5 Classification of Vector Bundles 79 5.1 Classification of vector bundles: The classical results ...... 79 5.2 H-principles and homotopy sheaves ...... 82 5.3 The h-principle in A1-homotopy theory ...... 86 5.4 Classifying spaces ...... 89 5.5 Homotopy invariance implies a classification of vector bundles . . . . 95 5.6 Discussion ...... 100

1 2 CONTENTS

Bibliography 103

Index 112 Introduction

The goal of this thesis is to develop tools to classify vector bundles over suitable rigid analytic varieties in a suitable homotopy theory. Our main result is: Theorem 1 (Theorem 5.41). Let X be a smooth rigid analytic variety over a complete, nonarchimedean, nontrivially valued ﬁeld k of characteristic zero. Let X be quasi-Stein. Then there is a natural bijection

∞ ∼ [X, ] 1 −→ {analytic k-line bundles over X}/ ∼= P Arig between motivic homotopy classes from X into the inﬁnite projective space P∞ over k and isomorphism classes of analytic k-line bundles over X. The assumption on the characteristic of k is superﬂuous if we use a theorem by Kerz–Saito–Tamme [KST16] which was attained after submission of this thesis.

Theorem 1 can be seen in the context of Morel’s A1-homotopy classification of algebraic vector bundles over a smooth affine variety and of Grauert’s homotopy classification of holomorphic vector bundles over a complex Stein space. The classical model for all of these theorems is Steenrod’s homotopy classification of continuous vector bundles over a paracompact Hausdorff space. Theorem 2 (F. Morel [Mor12], Asok–Hoyois–Wendt [AHW15a]). Let X be a smooth affine algebraic variety over a field K. There is a natural bijection ∼ ∼ [X, Grn/K ]A1 −→ {alg. K-vector bundles of rank n over X}/ =

1 between A -homotopy classes from X into the inﬁnite Grassmannian Grn over K and isomorphism classes of algebraic K-vector bundles of rank n over X. If n = 1, the theorem holds more generally for any smooth scheme X over a regular base [MV99, §4, Proposition 3.8].

Strikingly, if K = C, this classification still holds after analytification. The analytification of a smooth affine complex algebraic variety is a Stein manifold and the analytification of the complex affine line is C. Theorem 3 (Grauert [Gra57a, Gra57b]). Let X be a complex Stein space. There is a natural bijection ∼ ∼ [X, Grn/C] −→ {holomorphic vector bundles of rank n over X}/ = between homotopy classes of maps from X into the infinite Grassmannian Grn over C and isomorphism classes of holomorphic vector bundles of rank n over X.

3 4 CHAPTER 0. INTRODUCTION

Grauert’s theorem is stated in terms of classical (i. e., continuous) homotopy classes and with [0, 1] as an interval. Lárussonshowed that Grauert’s theorem can also be phrased in terms of motivic homotopy theory with C as an interval [Lár03, Lár04, Lár05]. Morel’s theorem and Grauert’s theorem seem to be linked via analytification. But as Morel’s theorem is valid without restrictions on the base field, it seems natural to ask if the same holds for other analytic fields, in particular for a complete nonarchimedean valued field k such as a field Qp of p-adic numbers or a field κ((T )) of formal Laurent series over another field κ.

The starting point for this project was Ayoub’s B1-homotopy theory for rigid analytic varieties. It is a version of Morel–Voevodsky’s homotopy theory of a site with interval where the “unit ball” B1 := Sp khT i ﬁgures as an interval. There are several approaches to nonarchimedean analytic geometry. Motivated by Ayoub’s work, we chose to work with rigid analytic varieties, too. His approach was well suited to deﬁne and work with motives of rigid analytic varieties, cf. [Ayo15, Vez14a, Vez14b, Vez15].

1 However, it turns out that for our purposes the analytification Arig of the affine line is a better choice of interval. It also fits better with Morel’s theorem.

An example

1 1 Theorem 1 becomes false if one just replaces the interval object Arig by B : Let k = Qp, p ≥ 3 and 2 X = Sp QphT1,T2i/(T1 − T2(T2 − p)(T2 − 2p)) . 1 Consider a line bundle L on X ×B and its restrictions Lr, Ls to X ×{r}, respectively 1 to X × {s} for some r, s ∈ B . Viewing Lr and Ls as line bundles over X, they are 1 ∞ in the same B -homotopy class in [X, P ]B1 . If we are to get a natural bijection ∞ ∼ [X, P ]B1 −→ {analytic k-line bundles over X}/ = , then for each line bundle L on X × B1 and each pair r, s ∈ B1 as above we would need Lr and Ls to be isomorphic. Consequently, we would need every line bundle L on X × B1 to be isomorphic to the pullback of a line bundle over X along 1 pr1 : X × B −→ X. It is a result by Gerritzen that this is not the case. Gerritzen shows that if k is discretely valued then the Picard group of line bundles on a distinguished smooth aﬃnoid variety X whose canonical model is regular and the Picard group of the canonical reduction X˜ are isomorphic [Ger77]. In our example, ˜ 2 3 X = Spec Fp[T1,T2]/(T1 − T2 ) . By a theorem by Bass and Murthy,

1 Pic(X˜) Pic(X˜ × A ) = Pic(X^× B1). Van der Put showed that the same problem occurs if k is algebraically closed instead of discretely valued [vdP82]. 5

The proof

It turns out that homotopy invariance with respect to the interval is really the key question for a homotopy classiﬁcation of vector bundles over (suitable) rigid analytic 1 1 varieties. For an interval object I ∈ {B , Arig} we say that “vector bundles of rank n are I-invariant” on a subcategory R of the category of rigid analytic varieties if for each object X ∈ R, the projection X × I → X induces a bijection on isomorphism classes of vector bundles of rank n. Let us give an outline of the proof of Theorem 1. Let k be a complete, nonarchimedean, nontrivially valued ﬁeld and X a smooth rigid analytic variety.

1 a) If char k = 0, then every line bundle over X × Arig has a local trivialisation 1 by subsets of the form {Ui × Arig}i∈I where {Ui}i∈J is an admissible covering of X. This is Corollary 3.35, a corollary of Theorem 3.32 by Tamme. After submission of this thesis, Tamme’s theorem was generalised to k of arbitrary characteristic by Kerz, Saito and Tamme [KST16].

1 b) Consider a line bundle on X × Arig and a local trivialisation of the form 1 1 1 {Ui × Arig}i∈I . Then every transition function from Ui × Arig to Uj × Arig on 1 Ui ∩ Uj × Arig is actually a transition function from Ui to Uj on Ui ∩ Uj, which 1 means that it is constant in the Arig-direction. This is Theorem 3.36. 1 c) Consequently, the projection X × Arig → X induces an isomorphism of groups ∼ 1 1 Pic(X) = Pic(X × Arig), i. e., line bundles are Arig-invariant on smooth rigid analytic varieties. This is Theorem 3.31. d) Every vector bundle over a quasi-Stein or a quasicompact rigid analytic variety admits a ﬁnite local trivialisation. These are Theorems 2.16 and 2.17. e) Let R be a subcategory of the category of smooth rigid analytic varieties. As- 1 1 sume that R has a strictly initial object ∅. Let I ∈ {B , Arig} be a representable interval object on R. Theorem 4 (Theorem 5.40). For some n ∈ N, assume that vector bundles of rank n are I-invariant on R. Assume that every vector bundle of rank n over an object X of R admits a ﬁnite local trivialisation. Then there is a natural bijection ∼ ∼ [X, Grn/k]I −→ {analytic k-vector bundles of rank n over X}/ = .

1 f) The assumptions of Theorem 4 are satisﬁed if n = 1, I = Arig, R is the subcategory of quasi-Stein spaces and char k = 0. Again, the assumption on the characteristic is only needed for Tamme’s theorem and can be removed using Kerz–Saito–Tamme [KST16]. Theorem 1 follows.

Outlook

We expect Theorem 1 to generalise to arbitrary characteristic and to vector bundles of higher rank: 6 CHAPTER 0. INTRODUCTION

Conjecture 1. Theorem 1 holds also if char k > 0. Conjecture 2. Let X be a smooth rigid analytic quasi-Stein variety over any complete, nonarchimedean, nontrivially valued ﬁeld k. Then there is a natural bijection

∼ [X, Gr ] 1 −→ {analytic k-vector bundles of rank n over X}/ ∼= n/k Arig between motivic homotopy classes from X into the infinite Grassmannian Grn/k and isomorphism classes of analytic k-vector bundles of rank n over X. Conjecture 1 is now known to be true, after the original submission of this thesis. It follows with Kerz–Saito–Tamme’s generalisation of Tamme’s Theorem [KST16]. Towards a generalisation of Theorem 1 to vector bundles of higher rank, it will probably be necessary to work with a Grothendieck topology on R which is different from the G-topology on rigid analytic varieties, for example with the Nisnevich topology. In fact, we prove more than we stated above as Theorem 4. The theorem we prove allows us to work with a Grothendieck topology different from the G- topology. For a proof of Conjecture 2, it remains to show that vector bundles over 1 smooth quasi-Stein varieties are Arig-invariant. This means, if X is smooth and 1 quasi-Stein, then every vector bundle over X × Arig is isomorphic to the pullback 1 along X × Arig → X of some vector bundle over X. We hope to do this in future work. Furthermore, homotopy invariance should be the only obstruction against generalising the theorem further to nonarchimedean Lie groups other than GLn. In the algebro- geometric setting, Asok–Hoyois–Wendt’s proof also works for principal bundles over an isotropic, reductive, algebraic group over an infinite field [AHW15b, Theorem 4.1.3]. In the complex setting, Grauert’s Oka principle even holds for every complex Lie group [Gra57b]. If Conjecture 2 holds, it should generalise to principal bundles under other suitable nonarchimedean Lie groups. One of our main interests now is to develop geometric applications of Theorem 1, or, more generally, of nonarchimedean motivic homotopy theory. Both Morel’s theorem and Grauert’s theorem set the stage for other theorems that are interesting on their own. Using Morel’s theorem and the theory of algebraic Euler classes also constructed by Morel [Mor12, §8], Asok and Fasel proved splitting theorems for algebraic vector bundles [AF14c, AF14a, AF14b, AF15]. Grauert’s theorem is strongly linked to Gromov’s h-principle, a powerful tool in differential topology. There should be geometric applications of Theorem 1 and Conjecture 2 along the same lines.

Structure of the thesis

In the first chapter we introduce rigid analytic varieties and prove two lemmas (1.50 and 1.51) for later use. The second chapter treats vector bundles over rigid analytic varieties. We prove Serre– Swan Theorems for rigid analytic varieties (Theorems 2.12 and 2.14). We conclude that over an affinoid variety, vector bundles with respect to the Zariski topology 7 and vector bundles with respect to the G-topology coincide up to isomorphism (Corollary 2.15). Furthermore, we show that every vector bundle over a quasicompact or quasi-Stein rigid analytic variety admits a finite local trivialisation (Theorems 2.16 and 2.17). The chapter finishes with two short sections on line bundles and divisors. The third chapter treats the question of homotopy invariance with respect to an interval object I, i. e. for which interval I and under which assumptions on X is it true that every vector bundle over X × I is isomorphic to the pullback of some vector bundle over X along X × I −→ X? The first question in this context is the case X = Sp k, the question whether all vector bundles over I are trivial. In the case of algebraic varieties and I = A1, the answer is given by the Quillen–Suslin Theorem. For schemes, the general case is the subject of the Bass–Quillen conjecture. We review the Quillen–Suslin Theorem, the Bass–Quillen Conjecture and possible versions in the rigid analytic setting in the first two sections of chapter three. In the third section we collect examples where I-invariance is violated. Section 4 contains local results on homotopy invariance. The fifth section first collects the known results on B1-invariance by Gerritzen, Bartenwerfer and van der Put. Then we state Tamme’s Theorem on pro-homotopy invariance of the Picard group of a smooth k-affinoid variety if char k = 0 and deduce Theorem 3.31: Line bundles over a smooth k-rigid 1 analytic variety are Arig-invariant if char k = 0. In the fourth chapter we define motivic homotopy theories for rigid analytic varieties in the spirit of Morel–Voevodsky’s A1-homotopy theory and Ayoub’s B1-homotopy 1 1 theory (Proposition 4.15). We show that Arig is contractible in the B -homotopy category (Lemma 4.18). In the fifth and last chapter we prove Theorem 1/Theorem 5.41. The chapter starts out by recalling the classical theorems by Steenrod, Grauert and Morel on homotopy classification of vector bundles. We introduce Gromov’s h-principle and explain how it can be seen as a homotopy sheaf property. This gives the link to excision, which is used both in Morel’s proof and in our proof and which can be seen as an algebraic variant of the h-principle. In the fourth section we give a short account of classifying 1 spaces and prove that the simplicial classifying space of GLn in the Arig-homotopy category is the infinite Grassmannian, a proof we learned from Arndt (Proposition 5.34). In the fifth section we prove Theorem 4/Theorem 5.40, following the proof of the corresponding theorem in A1-homotopy theory by Asok–Hoyois–Wendt. Now we can deduce Theorem 1/Theorem 5.41. The thesis finishes with a brief resuméand dicussion of open questions.

Acknowledgements

First of all, I want to thank Matthias Wendt for advise and patience, for giving me this intriguing question and making me learn a lot of very beautiful mathematics. Second, I thank Annette Huber-Klawitter for approving of this project and this way making the whole project possible in the ﬁrst place, for sharing her life experience and being a great boss. Third, I had the opportunity to discuss questions around this thesis with the following mathematicians: Peter Arndt, Aravind Asok, Joseph 8 CHAPTER 0. INTRODUCTION

Ayoub, Federico Bambozzi, Antoine Ducros, Carlo Gasbarri, Fritz Hörmann,Annette Huber-Klawitter, Marc Levine, Werner Lütkebohmert, Florent Martin, Vytautas Paˇsk¯unas,JérômePoineau, Dorin Popescu, Shuji Saito, Marco Schlichting, Georg Tamme, Konrad Völkel, Matthias Wendt. Thank you! Dear reader, if you do not occur in the list although we had an interesting dicussion, I apologise. I thank Maximilian Schmidtke and Eva Nolden for reading a draft of the first two chapters and pointing out typos and bad language. Once again, I thank Matthias Wendt for reading versions of this thesis carefully and for being severe with it. I thank Jens, Konrad, Matthias, Oliver and Shane for improving the Introduction. Big thanks to JérômePoineau for pointing out a very stupid mistake. I am very grateful to Joseph Ayoub for acting as referee for this thesis, for reading it thoroughly and for his detailed comments. Furthermore, I thank Florian Sigloch for fairness, love and caring. I thank Eva Nolden and Frieder Sigloch for helping out in cases of emergency. Eva Nolden took days off to look after our children, a precious gift to Florian and me. I also thank Elisabeth and Wolfgang Hochmuth for planning their holidays according to our needs. Thanks to Karin Wanzel and Paul Sigloch, to Eva, Lena, Frank, Pablo, Felix, Björn and Melli. Dear family and friends, I do not think we would have made it without you. Last, I thank the state Baden-Württemberg for a grant in the Brigitte Schlieben- Lange-Programm and the Deutsche Forschungsgemeinschaft (DFG) for funding me in the Graduiertenkolleg 1821. No research without money. Zu allerletzt danke ich Florian, Arthur, Jakoba und Birke dafür,dass wir eine großartige Familie sind. Conventions

We will use the following conventions: All rings are commutative with 1. All algebras are associative with 1. All valuations are nontrivial. All rigid analytic varieties are separated. k is a complete, nonarchimedean, nontrivially valued ﬁeld, unless stated otherwise. Residue ﬁelds are usually denoted by κ. Isomorphisms are denoted by ∼=. Weak equivalences are denoted by '. For other notation we refer to the index at the end of the thesis.

9 10 CHAPTER 0. INTRODUCTION Chapter 1

Rigid Analytic Varieties

The theory of rigid analytic varieties began with a seminar by John Tate at Harvard in 1961. Tate’s purpose was to understand degenerations of elliptic curves. His new approach soon led to a revival of nonarchimedean analysis. With the new theory at hand, complex analysts became interested in nonarchimedean analysis. Notes from Tate’s seminar circulated among them, but it took ten years before they finally were published [Tat71]. The area of nonarchimedean analysis was very active during the sixties and seventies. Several breakthroughs were achieved, for example Kiehl’s version of Cartan’s Theorems A and B for rigid analytic quasi-Stein spaces [Kie67], the Gerritzen–Grauert Theorem on locally closed immersions [GG69] or Raynaud’s theory which views a rigid analytic variety over a field as the generic fibre of a formal scheme over the valuation ring [Ray74].

Tate’s theory of rigid analytic varieties is stronlgy tied to algebraic and arithmetic geometry on the one hand via the analytification functor and GAGA theorems, on the other hand via the reduction functor on affinoid varieties. There are applications in singularity theory via the theory of Mumford curves and Schottky groups (cf. [FvdP04, 5.4]). Harbater proved Abhyankar’s conjecture in inverse Galois theory, using rigid analytic varieties. Stable reduction for curves over a complete discretely valued field can be proved using rigid analytic varieties [BL93a, vdP84].

Besides Tate’s theory, Berkovich’s theory of analytic spaces of seminorms [Ber90] is very fruitful. A third approach, Roland Huber’s theory of adic spaces [Hub93b, Hub93a], had a recent revival sparked by Scholze’s work [Sch12].

In this first chapter we review some aspects of rigid analytic geometry, mostly in order to fix terminology and notation. For a detailed exposition of affinoid algebras and the analytic aspects of the theory, we recommend the book by Bosch–Güntzer– Remmert [BGR84]. The other classic is Fresnel–van der Put [FvdP04]. It takes a more algebro-geometric point of view, is less detailed and covers more recent material than [BGR84]. In particular, it contains sections on formal models and reduction, on generalised points, on cohomology theories and on Abhyankar’s conjecture.

11 12 CHAPTER 1. RIGID ANALYTIC VARIETIES

1.1 Deﬁnitions

Let k be a complete nonarchimedean valued ﬁeld. Proposition/Deﬁnitions 1.1. a) A Banach k-algebra is a complete normed k-algebra. b) The Tate algebra     X i1 in khT1,...,Tni := aiT1 ··· Tn all ai ∈ k, |ai| −→ 0 i1,...,in n i=(i1,...,in)∈N  is a Banach algebra with respect to the Gauß norm

X i1 in aiT1 ··· Tn := sup |ai|. n i=(i1,...,in)∈N

All ideals of khT1,...,Tni are closed. A detailed exposition can be found in Bosch–G¨untzer–Remmert [BGR84, chapter 5]. c) An aﬃnoid k-algebra is a Banach k-algebra A such that there exists a continuous epimorphism

α: khT1,...,Tni → A from some Tate algebra onto A. As the map α is open, it induces an isomorphism of Banach k-algebras ∼ A = khT1,...,Tni/ ker α. Choosing different presentations α yields different norms. All of them are equivalent because they induce the same topology on A. Assume that A is reduced, i. e., has no nilpotent elements except zero. Then there is a distinguished norm kfk on A, the spectral norm, see Definition 1.3 below. d) A morphism of affinoid algebras is a continuous k-algebra homomorphism. Remark 1.2. The Tate algebras khT1,...,Tni and, more generally, affinoid algebras are noetherian [BGR84, 5.2.6 Theorem 1]. We can see an affinoid algebra A as an algebra of functions on its maximal spectrum Max A := {m ⊂ A | m maximal ideal} where f ∈ A maps a maximal ideal m to the residue class [f]m of f in A/m. The residue field A/m is a finite field extension of k. Thus, choosing an algebraic closure k¯ of k and embeddings of the residue fields A/m ,→ k,¯ the affinoid algebra A can be seen as an algebra of functions f : Max A → k¯

m 7→ [f]m. The absolute value | · | extends uniquely to the ﬁnite ﬁeld extension A/m. 1.1. DEFINITIONS 13

Definition 1.3 (spectral seminorm [FvdP04, Definition 3.3.1]). The spectral seminorm on an affinoid algebra A is

kfk := sup |f(x)|, f ∈ A. x∈Max A

Lemma 1.4 ([FvdP04, Corollary 3.4.4]). Let A be a reduced affinoid algebra. Then the spectral seminorm on A is a norm. The category of affinoid algebras has finite coproducts, given by the complete tensor product: Definition 1.5 (Complete tensor product [BGR84, 2.1.7]). Let R = (R, | |) be a normed ring and L, M normed R-modules. The function

| |: L ⊗R M −→ R+ deﬁnes a seminorm on L ⊗R M. The completion of L ⊗R M as a seminormed group with respect to this seminorm is denoted by L⊗ˆ RM and called the complete tensor product of L and M over R.

The complete tensor product L⊗ˆ RM of normed R-modules L, M is a normed R- module and a normed Rˆ-module [BGR84, p. 71]. Example 1.6. Let A be an aﬃnoid algebra. Then     X i1 in AhT1,...,Tni := aiT1 ··· Tn all ai ∈ A, kaikA −→ 0 i1,...,in n i=(i1,...,in)∈N  is aﬃnoid and ∼ AhT1,...,Tni = A⊗ˆ kkhT1,...,Tni [BGR84, p. 224].

For more details about the complete tensor product, see [BGR84, section 2.1.7 and p. 224]. Proposition 1.7 ([FvdP04, Lemma 3.7.1]). The category of affinoid k-algebras has finite coproducts, given by the complete tensor product over k. Definitions 1.8 (A◦, A◦◦). Let A be an affinoid algebra. a) The subring of power-bounded elements of A is ◦ r A = f ∈ A sup kf k < ∞ . r∈N

b) The set of topologically nilpotent elements of A is n o A◦◦ = f ∈ A f r −→ 0 . r→∞

The set A◦◦ is an ideal in A◦, cf. [BGR84, 1.2.4, 1.2.5]. 14 CHAPTER 1. RIGID ANALYTIC VARIETIES

Example 1.9. For the field Qp of p-adic numbers, the ring of power-bounded elements ◦ ◦◦ is the ring Qp = Zp of p-adic integers with maximal ideal Qp = pZp. Definitions 1.10 (affinoid subset, rational subset [FvdP04, 4.1.4, 4.1.1]). Let A be an affinoid algebra. A subset U ⊂ Max A is called affinoid if there is an affinoid algebra B and a morphism of affinoid algebras

ϕ: A → B such that the induced morphism

ϕ∗ : Max B → Max A maps ϕ∗(Max B) ⊂ U and is universal with respect to this property: For every aﬃnoid algebra C and morphism

ψ : A → C with ψ∗(Max C) ⊂ U there is a unique morphism of aﬃnoid algebras τ : B → C such that ψ = τ ◦ ϕ.

A subset U ⊂ Max A is called rational if there are elements f0, f1, . . . , fs ∈ A generating the unit ideal of A such that

U = {x ∈ Max A | |fi(x)| ≤ |f0(x)| for i = 1, . . . , s}.

Rational subsets are in particular aﬃnoid and if U ⊂ Max A is rational as above, we have ∼ U = Max AhT1,...,Tsi/(f1 − f0T1, . . . , fs − f0Ts) .

Example 1.11 (complete localisation). If A is an aﬃnoid k-algebra and f ∈ A with |f| ≤ 1, then

Ahf −1i := AhT i/(1 − fT )

−1 is called the complete localisation of A at f. The algebraic localisation Af =: A[f ] is dense in Ahf −1i [Bos14, §3.3]. Deﬁnition 1.12 (G-topology [FvdP04, Deﬁnition 2.4.1]). A G-topology on a set X consists of a family F of subsets of X (admissible subsets) and for each admissible subset U ∈ F a set Cov(U) of coverings by elements of F (admissible coverings) such that: a) ∅,X ∈ F . b) U, V ∈ F ⇒ U ∩ V ∈ F . c) U ∈ F ⇒ {U} ∈ Cov(U). d) U, V ∈ F,V ⊂ U, U ∈ Cov(U) ⇒ U ∩ V ∈ Cov(V ) where U ∩ V := {U 0 ∩ V | U 0 ∈ U}. 1.1. DEFINITIONS 15

e) Let U ∈ F, {Ui}i∈I ∈ Cov(U) and for each i ∈ I let Ui ∈ Cov(Ui). S S S 0 0 Then i∈I Ui ∈ Cov(U) where i∈I Ui := i∈I {U | ∃i ∈ I,U ∈ Ui}. In short: A G-topology on X is a Grothendieck pretopology T on a category C that has certain subsets of X as objects and inclusions as morphisms. Remark 1.13. Every topology induces a a G-topology: Every open set is admissible and every covering is admissible. A G-topology does not necessarily arise from a topology: First, an arbitrary union of admissible subsets does not need to be admissible. Second, a jointly surjective union of admissible subsets is not necessarily an admissible covering. Definition 1.14 ((very) weak G-topology [FvdP04, Definition 4.2.1], [BGR84, 9.1.4]). Let A be an affinoid algebra. The very weak G-topology on the set Max A of maximal ideals of A is defined as follows: • The admissible subsets are rational subsets of Max A. • Let U be an admissible subset. An admissible covering of U is a covering {Ui}i∈I by admissible subsets Ui for which there exists a finite subset J ⊂ I S such that U = i∈J Ui. The weak G-topology on Max A • has as admissible subsets finite unions of rational subsets of Max A. • Let U be an admissible subset. An admissible covering of U is a covering {Ui}i∈I by admissible subsets Ui for which there exists a finite subset J ⊂ I S such that U = i∈J Ui. Definition 1.15 (slightly finer ([FvdP04, p. 26], [BGR84, 9.1.2 Definition 1]). Let T and T 0 be G-topologies on a set X. Then T 0 is called slightly finer than T if a) every T -admissible set is T 0-admissible, b) every T -covering is a T 0-covering, c) every T 0-admissible set U has a T 0-covering by T -admissible sets, d) every T 0-covering of a T -admissible set can be refined by a T -covering. Definition 1.16 (strong G-topology [FvdP04, pp. 80f], [BGR84, 9.1.4]). Let A be an affinoid algebra. The strong G-topology TA on Max A is given by: The admissible subsets are the affinoid subsets of X. Let U be an admissible subset. An admissible covering of U is a covering {Ui}i∈I by admissible subsets Ui for which there exists a S finite subset J ⊂ I such that U = i∈J Ui. The strong G-topology is the finest G-topology which is slightly finer than the very weak G-topology [BGR84, 9.1.4]. For most purposes it is enough to keep in mind the weak G-topology. Besides those G-topologies, the set Max A carries two honest topologies: The topology Tk·k induced by the norm and the Zariski topology TZar. The Zariski topology is very coarse and ignores the analytic structure given by the norm. The topology induced by the norm is very fine: The topological space (Max A, Tk·k) is totally disconnected. The G-topology TA generated by rational subsets and finite coverings is coarser than Tk·k and finer than TZar. 16 CHAPTER 1. RIGID ANALYTIC VARIETIES

A G-topology (or more generally: a site) is enough to define the glueing condition for sheaves. Definition 1.17. Let (X, T ) be a set with a G-topology. Define X to be the category whose objects are the admissible subsets of X and whose morphisms are the inclusions. A presheaf of sets (or of rings, or of simplicial sets, or whatever) is a contravariant functor from X to the category of sets (or of rings, or of simplicial sets, or whatever). Let C be a category that has finite products. A presheaf F : X → C is a sheaf if sections over admissible coverings glue uniquely, i. e., for each admissible subset U ⊂ X and each admissible covering U of U, the diagram of restrictions Y Y F(U) → F(Ui) ⇒ F(Ui ∩ Uj) Ui∈U Ui,Uj ∈U is an equaliser diagram. Definition 1.18. A locally G-ringed space is a triple (X, TX , OX ) where X is a set, TX a G-topology on X and OX a presheaf of rings on X which is a sheaf with respect to TX . Definition 1.19 (affinoid variety). The affinoid variety associated to an affinoid algebra A is the locally G-ringed space

Sp A := (Max A, TA, OSp A) where the structure sheaf OSp A is the sheaf of Banach algebras on (Max A, TA) deﬁned by

OSp A(U) := B if U = Sp B admissible and

O(Sp B,→ Sp B0) = (res: B0 → B).

See [BGR84, 7.3.2] for more details. A morphism

Sp A → Sp B of affinoid varieties is the morphism of locally G-ringed spaces induced by a morphism B → A of affinoid algebras. For more details, see [BGR84, 7.1.4 and 7.2.2]. Remark 1.20. The category of affinoid varieties with the very weak G-topology and the category of affinoid varieties with the strong G-topology are equivalent [FvdP04, pp. 80f]. Definition 1.21. The affinoid unit ball B1 is defined as

1 B := Sp(khT i). Lemma 1.22 (Fibre product of affinoid varieties [BGR84, 7.1.4 Proposition 4]). The category of affinoid k-varieties has fibre products. Suppose we are given affinoid maps Sp A → Sp C and Sp B → Sp C. Then the fibre product of Sp A with Sp B over Sp C is given by

Sp A ×Sp C Sp B := Sp(A⊗ˆ C B). 1.1. DEFINITIONS 17

Deﬁnition 1.23 (rigid analytic variety [FvdP04, Deﬁnition 4.3.1]). A k-rigid analytic variety is a locally G-ringed space (X, TX , OX ) where:

a) The structure sheaf OX is a sheaf of k-algebras.

b) There exists an admissible covering {Xi}i∈I ∈ Cov(X) such that for each i ∈ I,

the space (Xi, TX |Xi , OX |Xi ) is an aﬃnoid variety.

c) Admissibility of subsets can be tested on aﬃnoids: U ∈ TX if and only if

U ∩ Xi ∈ TXi for all i.

The sheaf OX is called the structure sheaf of X. n Example 1.24 (The aﬃne n-space Arig [BGR84, 9.3.4 Example 1]). Let η ∈ k with |η| > 1 and deﬁne

−i −i Ai := khη T1, . . . , η Tni. n i This means that Sp(Ai) = B|η|i is the n-dimensional ball of polyradius |η| . The algebras Ai form a chain

khT1,...,Tni = A0 ) A1 ) A2 ) ··· ) k[T1,...,Tn] and the inclusions Ai ⊃ Ai+1 correspond to inclusions of rational subsets n n B|η|i ⊂ B|η|i+1 . n The aﬃne n-space Arig is deﬁned as the colimit

n [ n Arig := B|η|i i∈N of all those inclusions of rational subsets. It does not depend on the choice of the n element η. Affine space Arig is a rigid analytic variety which is not affinoid. It is an unbounded Stein space, see Definition 1.30 below. Definition 1.25 (Fibre product of rigid varieties [BGR84, 9.3.5 Theorem 2]). Let X → S and Y → S be morphisms of rigid analytic varieties. Then the fibre product X ×S Y of X with Y over S exists in the category of rigid analytic varieties. Furthermore, if X, Y and S are affinoid, then this fibre product coincides with the fibre product of X with Y over S in the category of affinoid varieties constructed in Lemma 1.22. Definition 1.26 (closed immersion [BGR84, 7.1.4 Definition 3 and 9.5.3, p. 388]). An affinoid map Sp A → Sp B is called a closed immersion if the corresponding map B → A on affinoid algebras is an epimorphism. A morphism ϕ: X → Y of rigid analytic varieties is called a closed immersion if there exists an admissible affinoid covering {Ui}i∈I of the target Y such that for all i ∈ I, the induced map −1 ϕ (Ui) → Ui is a closed immersion of affinoid varieties. Definitions 1.27. a) An affinoid algebra is reduced if it has no nonzero nilpotent elements. An affinoid variety Sp A is reduced if A is. b) A rigid analytic variety X is separated if the diagonal morphism

∆: X → X ×Sp k X is a closed immersion [BGR84, 9.6.1, Deﬁnition 1]. 18 CHAPTER 1. RIGID ANALYTIC VARIETIES

c) A rigid analytic variety is quasiseparated if the intersection of any two open affinoid domains is a finite union of open affinoid domains [FvdP04, Defini- tions 7.3.4]. Remark 1.28. Every affinoid variety is quasi-separated as the intersection of two affinoid subdomains of an affinoid variety is again an affinoid subdomain [BGR84, 7.2.2 Corollary 5]. There are three special classes of rigid analytic varieties that we are particularly interested in: Affinoid varieties, quasicompact varieties and quasi-Stein varieties. Definition 1.29 (quasicompact rigid analytic variety [FvdP04, Definitions 7.3.4]). A rigid analytic variety is called quasicompact if it has an admissible covering by finitely many affinoid subsets. Quasi-Stein and Stein varieties were defined by Kiehl in [Kie67] where he also proved the analogues of Cartan’s Theorem A and Theorem B for them. They are analogues of complex Stein spaces in the nonarchimedean world. Definition 1.30 ((quasi-)Stein [Kie67, Definition 2.3]). a) A rigid analytic space X is quasi-Stein if there is an admissible covering by open affinoid subspaces

U1 ⊂ U2 ⊂ U3 ⊂ · · ·

such that for all i, the image of OX (Ui+1) is dense in OX (Ui).

b) The space X is Stein if additionally, for each i there exists ai+1 ∈ k with (i+1) (i+1) 0 < |ai+1| < 1 and a system of topological generators f1 , . . . , fni+1 of O(Ui+1) such that

(i+1) Ui = {x ∈ Ui+1 | |fj (x)| ≤ |ai+1| for j = 1, . . . , ni+1}. Definition 1.31 (coherent sheaf [FvdP04, Definition 4.5.1]). Let X be a rigid analytic variety and F a sheaf of OX -modules. The sheaf F is called coherent if there exists an admissible covering by affinoids {Xi}i∈I ∈ Cov(X) and for every i a finitely generated OX (Xi)-module Mi such that the restriction of F to Xi is isomorphic to ˜ ˜ Mi as a sheaf of OXi -modules. Here the sheaf Mi is defined by

U 7→ Mi ⊗OX (Xi) OX (U) for U ⊂ Xi rational.

Theorem 1.32 (Kiehl [Kie67, Satz 2.4]). Let X be a quasi-Stein space and {Ui}i∈N an admissible covering as in Deﬁnition 1.30a). Let G be a coherent sheaf on X. Then the following hold:

a) The image of G(X) is dense in G(Ui) for all i. b) The cohomology groups Hi(X, G) vanish for i > 0 (Theorem B).

c) For each x ∈ X, the image of G(X) in the stalk Gx generates this stalk as an OX,x-module (Theorem A). i Theorem 1.33 (Lütkebohmert [Lüt73]). a) All fj can be chosen to be global i functions: fj ∈ OX (X) by [Lüt73,Korollar 4.2]. b) Stein varieties are separated [Lüt73,Satz 3.4]. 1.1. DEFINITIONS 19

Remark 1.34. a) Lütkebohmert proved moreover that a Stein variety X over an algebraically closed field k can be embedded in some kN if X is smooth or if the local embedding dimension of X is bounded [Lüt73, Theorem 4.21]. Here, an embedding is a finite, injective, proper analytic function onto a closed subvariety of kN such that its differentials generate the universal differential module of X. For precise definitions and statements we refer to Lütkebohmert’s article.

b) There is also a weaker notion of Steinness, introduced by Liu in [Liu88] and investigated further in [Liu89]. Liu calls a rigid analytic space X Stein if for every coherent sheaf F on X all higher cohomology groups Hi(X, F), i ≥ 1, vanish. Definition 1.35 (Stein algebra). Let X be a rigid analytic (quasi-)Stein variety. The k-algebra OX (X) of global functions on X is called a (quasi-)Stein algebra. (Quasi-)Stein algebras are Fréchet algebras, but in general not Banach. They are complete, but their topology does not necessarily arise from a norm. As the following example shows, (quasi-)Stein algebras are in general not noetherian. Example 1.36 (compare [GR04, Remark 3 on p. 179]). Let X be a (quasi-)Stein space that contains an infinite discrete subset D which does not have a limit point in X. Let A = OX (X) be the corresponding (quasi-)Stein algebra. The nonarchimedean analogue of Weierstrass’ product theorem holds: Given an infinite discrete subset D0 ⊂ X which does not have a limit point in X and an assignment

0 D −→ N d 7−→ m(d)

0 there exists a function f ∈ OX (X) such that for all d ∈ D , the function f has a zero in d of multiplicity m(d). The proof is the same as in the complex case, e. g. as in [Rud87]. We now construct a non-ﬁnitely generated ideal m ⊂ A. The elements of the ideal

I = {f ∈ A | f(x) = 0 for almost all x ∈ D} have no common zeroes. So a maximal ideal m containing I cannot be ﬁnitely generated.

In complex analysis, Stein algebras were introduced by Forster [For64, For67] and studied by several mathematicians since then. In nonarchimedean analysis they appear implicitly in Lütkebohmert’s 1973 article [Lüt73]. The more general Fréchet– Stein algebras and their coadmissible modules (corresponding to coherent sheaves on the Stein varieties) were introduced by Schneider and Teitelbaum [ST03] in the context of p-adic analytic groups and Langlands theory. There are approaches to unify the archimedean and the nonarchimedean theory. Poineau constructed Berkovich spaces over Z, capturing both the archimedean and the nonarchimedean theory, cf. [Poi13]. Poineau uses Liu’s definition of Stein spaces. Bambozzi–Ben Bassat–Kremnizer investigate the topology of Stein spaces over any valued base field [BBK15]. They use Kiehl’s definition of nonarchimedean Stein spaces. 20 CHAPTER 1. RIGID ANALYTIC VARIETIES

1.2 Formal models and reduction

Definition 1.37 (reduction of an affinoid algebra [BGR84, 6.3]). Let A be an affinoid k-algebra. We define its reduction by

A˜ := A◦/A◦◦ where A◦ ⊂ A is the subring of power-bounded elements and A◦◦ the set of topologically nilpotent elements of A. The set A◦◦ is an ideal in A◦. The reduction A˜ is an affine k˜-algebra, that is, a quotient of a polynomial ring over k˜ in finitely many variables. In fact, reduction defines a functor from the category of affinoid k-algebras to the category of affine k˜-algebras. Examples 1.38. a) The reduction of the p-adic numbers Qp is ∼ Qfp = Zp/pZp = Fp,

the ﬁeld with p elements. b) The power-bounded elements of the aﬃnoid algebra

A = khT1i

are the elements whose coeﬃcients are all power-bounded, i. e.,

◦ ◦ A = k hT1i.

Similarly, the topologically nilpotent elements are

◦◦ ◦◦ ◦ A = k · k hT1i.

Thus, the reduction of A is

˜ ˜ A = k[T1].

−1 c) Let A be aﬃnoid and f ∈ A with kfksup = 1. Then Ahf i ⊂ A is an aﬃnoid subset. Its reduction is

−1 ˜ ˜−1 ˜ A^hf i = A[f ] = Af˜

the ring-theoretic localisation of A˜ at f˜ [BGR84, 7.2.6 Proposition 3]. d) Assume char k˜ > 3. Let

2 A1 = khT1,T2i/(T1 − T2(T2 − 1)(T2 − 2)).

Its reduction is

˜ ˜ 2 A1 = k[T1,T2]/(T1 − T2(T2 − 1)(T2 − 2)). 1.2. FORMAL MODELS AND REDUCTION 21

e) Assume char k˜ > 3 and choose π ∈ k with 0 < |π| < 1. Let

2 A2 = khT1,T2i/(T1 − T2(T2 − π)(T2 − 2π)). Its reduction is

˜ ˜ 2 3 A2 = k[T1,T2]/(T1 − T2 ).

Although A2 is smooth, its reduction has a cuspidal singularity. f) Again, assume char k˜ > 3 and choose π ∈ k with 0 < |π| < 1. Let

2 A3 = khT1,T2i/(T1 − T2(T2 − 1)(T2 − 2π)). Its reduction is

˜ ˜ 2 2 A3 = k[T1,T2]/(T1 − T2 (T2 − 1)).

This time, A3 is smooth and its reduction has a nodal singularity. Remark 1.39. Examples e) and f) show that a smooth affinoid algebra can have a singular reduction. Assume that π ∈ k has a sixth root ζ in k. Then we can see the affinoid varieties corresponding to the algebras in examples d) and e) as pieces of different size cut out from the Stein variety

[ n 2 m m m X = Sp khT1,T2i/(η T1 − δ T2(δ T2 − 1)(δ T2 − 2)) n,m∈N for arbitrary η, δ ∈ k with |η|, |δ| > 1. Obviously, X does not depend on the choice 2 of η and δ as long as |η|, |δ| > 1. The Stein variety X embeds into Arig. Let

2 −n −m Bn,m = Sp khη T1, δ T2i

n m 2 −1 be the polydisc of polyradius (η , δ ) in Arig. Choosing η = δ = ζ gives

2 Sp A1 = X ∩ B0,0 2 Sp A2 = X ∩ B3,2.

Let A be an affinoid k-algebra and Ae its reduction. Then A◦ corresponds to a formal k◦-scheme X that has Sp A as its generic fibre and Spec Ae as its special fibre. Raynaud [Ray74] realised that in many cases one can work with formal schemes instead of rigid analytic varieties and hence apply tools from algebraic geometry. Definition 1.40. The formal scheme X is called the canonical model of X. The special fibre of X is an affine algebraic k˜-variety and denoted by X˜ c. It is called the canonical reduction of X. Remark 1.41. Obviously, X˜ c = Spm A˜. Here we denote by Spm R the algebraic variety defined by a ring R, that is, the maximal spectrum of R together with the Zariski topology and the structure sheaf. For nonaffinoid rigid analytic varieties, formal models can still exist, but usually there is no canonical choice any more. 22 CHAPTER 1. RIGID ANALYTIC VARIETIES

Deﬁnition 1.42 (formal model, analytic reduction). If the rigid analytic k-variety X is the generic ﬁbre of the formal k◦-scheme X , then X is called a formal model of X.

If a rigid analytic variety has a locally finite covering by affinoids, then it has a formal model. Raynaud’s approach was pursued further in the “Formal and rigid geometry” series [BL93a, BL93b, BLR95a, BLR95b] by Bosch, Lütkebohmert and Raynaud. Among other things, they develop the technique of admissible formal blowup in [BL93a]. Admissible formal blowup gives resolution of singularities in the case of equal characteristic zero, cf. Nicaise [Nic09, Proposition 2.43]. This follows from Temkin’s resolution of singularities for noetherian quasi-excellent schemes of characteristic zero [Tem08]. Compare also Hartl [Har03]. Resolving singularities in the special fibre, it leads to: Theorem 1.43 (Ayoub [Ayo15, Proposition 1.1.63]). Let k = κ((T )) for κ a field of char κ = 0. Every smooth k-affinoid variety has a semistable formal model (see Definition 1.46). Definition 1.44 (admissible formal scheme). a) A k◦-algebra A is called admissible if it has no k◦◦-torsion, i. e., if {a ∈ A | ∃c ∈ k◦◦ : ca = 0} = {0} [BL93a, p. 293].

b) An aﬃne formal k◦-scheme Spf A is admissible if A is admissible [BL93a, p. 296].

c) A formal k◦-scheme X is admissible if it has a covering by affine admissible formal schemes. Lemma 1.45 ([BL93a, Proposition 1.7]). Let A be a k◦-algebra that is complete and ◦◦ separated with respect to the k -adic topology and let {Spf Bi}i∈I be an affine open covering of Spf A. Then A is an admissible k◦-algebra if and only if for all i ∈ I, the ◦ k -algebra Bi is admissible. In this case the canonical maps A → Bi are flat.

Hence, Definition 1.44c) makes sense. Definition 1.46 (semistable model, [HL00, Definition 1.1]). An admissible formal scheme X is called strictly semistable if

a) its generic ﬁbre Xη is smooth over k,

b) its special ﬁbre Xσ is geometrically reduced, i. e., reduced and still so after base change to the separable closure k˜sep of k˜,

c) the special ﬁbre Xσ has (at most) normal crossings singularities.

∼ If X is a strictly semistable formal scheme with Xη = X, then X is called a strictly semistable formal model of X. Remark 1.47 ([Har03, Remark 1.1.1]). Such a formal scheme is regular. 1.3. FLATNESS, SMOOTHNESS AND ETALENESS´ 23

1.3 Flatness, smoothness and ´etaleness

Definition 1.48 (flat morphism). a) A morphism ϕ: A → B of affinoid algebras is flat if the functor

ModA −→ ModB

M 7−→ B ⊗A M

from A-modules to B-modules is exact. b) A morphism φ: Sp B → Sp A of affinoid varieties is flat if the corresponding morphism ϕ: A → B is flat. c) A morphism φ: X → Y of rigid analytic varieties is flat if for each x ∈ X there exists an affinoid subset U 3 x such that φ|U is flat. Lemma 1.49. Let A be an affinoid algebra. Then A → AhT i is flat. If |f| ≤ 1, then A → Ahf −1i is flat. Inclusion of an affinoid subdomain is flat.

Proof. The morphisms A → A[T ] and A → Af are flat. Completion is faithfully flat [Bou61, §3, no. 4 Théorème3 and no. 5 Proposition 9]. Inclusion of an affinoid subdomain is flat [Bos14, 4.1 Corollary 5].

We extend this to the coeﬃcient rings appearing in the deﬁnition of a quasi-Stein variety. Lemma 1.50. Let

X = colim(U1 ⊂ U2 ⊂ U3 ⊂ · · · ) be a quasi-Stein variety with Ui = Sp Ai. Then A := OX (X) → Ai is ﬂat for all i.

Proof. Using Kiehl’s Theorem B [Kie67, Satz 2.4], the proof works just as Gruson’s proof of [Gru68, V, Corollaire 1]: Let I ⊂ A be any ﬁnitely generated ideal. We want to show that the morphism

I ⊗A Ai −→ A ⊗A Ai (1.1) is injective. Then Ai is ﬂat over A by [sta, Tag 00H9, Lemma 10.38.5].

The ideal I gives rise to a coherent sheaf I on X. For each j ≥ i, Aj is ﬂat over Ai by Lemma 1.49. Therefore, for each j ≥ i, the morphism

I(Uj) ⊗Aj Ai −→ Aj ⊗Aj Ai

1 is injective. By Kiehl’s Theorem B [Kie67, Satz 2.4], the derived limit limj (I(Uj)) vanishes and hence the morphism 1.1 is injective, too.

We want to use Lemma 1.50 in combination with the following lemma. It is the Banach version of a well-known theorem in commutative algebra. 24 CHAPTER 1. RIGID ANALYTIC VARIETIES

φ Lemma 1.51. Let A −→ A0 be a flat morphism of commutative Banach algebras. Let M be a finitely presented complete normed A-module and N a finitely generated complete normed A-module. Then the natural map

0 0 0 σˆ : A ⊗ˆ A HomA(M,N) −→ HomA0 (A ⊗ˆ AM,A ⊗ˆ AN) is an isomorphism. To prove this, we want to use the fact that there is no essential difference between the category of finitely generated complete normed A-modules with continuous A-linear maps as morphisms and the category of finitely generated A-modules with A-linear maps as morphisms without demanding continuity [BGR84, §3.7.3]. During the proof, Hom-sets in the first named category are denoted by Homcont, morphisms being both linear and continuous. On the other hand, Hom-sets in the second named category consist of morphisms that only have to be linear and will be denoted by Homlin.

Proof. We only need to prove thatσ ˆ is bijective. There is a commutative diagram

0 ˆ cont σˆ cont 0 ˆ 0 ˆ A ⊗A HomA (M,N) / HomA0 (A ⊗AM,A ⊗AN) . O O o α o β

0 lin ∼ lin 0 0 A ⊗A HomA (M,N) σ / HomA0 (A ⊗A M,A ⊗A N)

The maps α and β are bijections by [BGR84, §3.7.3 Proposition 2, Proposition 6]. The map σ is an isomorphism by [Lam06, Proposition I.2.13] or [Eis95, Proposition 2.10]. Thus,σ ˆ is bijective, too. Definition 1.52 ([Ayo15]). a) An affinoid k-variety X = Sp A is called regular in x ∈ X if OX,x is regular. Furthermore, X is called smooth in x if for every 0 0 0 finite field extension k ⊂ k , the affinoid k -variety Sp(A⊗ˆ kk ) is regular in each point x0 lying above x. b) A rigid analytic variety X is called regular (smooth) if it is regular (smooth) in all x ∈ X. c) A morphism f : X → Y of rigid k-varieties is smooth if it is flat and for each y ∈ Y the morphism restricted to the fibre f −1(y) → κ(y) is smooth. d) The morphism f is étale if additionally all its non-empty fibres are of dimension zero. e) The morphism f is étalein x ∈ X if x has an affinoid neighbourhood U 3 x such that the restriction f|U is étale. Definition 1.53 (smRig). The category of smooth rigid analytic k-varieties with rigid analytic morphisms is denoted by smRig k. The subscript k will be suppressed in the notation except when there is a chance of confusion. As for schemes, there are several ways to define étaleness.Several characterisations of smooth and étalemaps between rigid analytic varieties can be found in the articles 1.3. FLATNESS, SMOOTHNESS AND ETALENESS´ 25 by de Jong–van der Put [dJvdP96, section 3] and Ayoub [Ayo15]. For example, if f : Y → X is a morphism, one can define the sheaf Ωf of relative differentials as in algebraic geometry. It is a coherent OY -module. Smoothness can as usual be characterised by looking at Ωf or the Jacobian matrix. Berkovich spaces and Huber’s adic spaces are very similar to rigid analytic varieties, especially if we only care for affinoid varieties. So we find further characterisations of étale,smooth, flat, unramified and finite maps in Berkovich’s paper [Ber93, section 3.3] and Roland Huber’s book [Hub96, section 1.7]. Theorem 1.54 (Berkovich [Ber93, Theorem 1.6.1]). The category of paracompact strictly k-analytic Berkovich spaces and the category of quasiseparated rigid analytic k-varieties that have an admissible covering of finite type are equivalent. An affinoid covering is said to be of finite type if each of its members intersects only finitely many of the other members. Remark 1.55. Berkovich’s definition of étaleness[Ber93, Definition 3.3.4] is equivalent to ours by [Ayo15, Theorème1.1.47]. Lemma 1.56 ([Ber93, Proposition 3.3.10]). Let ϕ: X → Y be a finite morphism of affinoid varieties. Then the set of points x ∈ X in which ϕ is ramified, respectively non-étale,is Zariski closed.

Proof. This is true for a quasiﬁnite morphism of Berkovich spaces by [Ber93, Propo- sition 3.3.10]. By Theorem 1.54 it also holds for aﬃnoid varieties. 26 CHAPTER 1. RIGID ANALYTIC VARIETIES Chapter 2

Vector Bundles over Rigid Analytic Varieties

A vector bundle over a space X can be thought of as a space E together with a projection to X such that, locally on X, the total space E looks like the product of X with a vector space. The definition depends on the choice of topology on X. In the first section we define different kinds of vector bundles.

In the second section we prove a Serre–Swan theorem for vector bundles over rigid analytic varieties. There are several situations in various fields of geometry where vector bundles are completely determined by their global sections. In those situations, the global sections of a vector bundle form a projective module over the ring of global sections of the structure sheaf of the underlying geometric object. The first instance of this principle was discovered by Serre [Ser55, §I.50, p. 242] and it was maybe the starting point to ask his famous question about projective modules. Serre showed that vector bundles of finite rank over an affine variety correspond to finitely generated projective modules over the coefficient ring. Swan [Swa62, Theorem 2] proved the same for topological vector bundles over compact Hausdorff spaces. Serre’s Theorem, Swan’s Theorem and other theorems of this type are usually referred to as Serre–Swan Theorems. A Serre–Swan Theorem for complex Stein spaces follows from Forster’s work [For64, For67]. Morye extracted the greatest common divisor of Serre–Swan Theorems, i. e., which features are responsible for a Serre–Swan Theorem to hold. We use her nice and very clear exposition [Mor09] to prove that a Serre–Swan Theorem also holds for vector bundles over rigid analytic quasi-Stein varieties. As a corollary we get that on an affinoid variety vector bundles with respect to the G-topology and vector bundles with respect to the Zariski topology coincide up to isomorphism.

The chapter concludes with a few remarks on line bundles and divisors.

27 28 CHAPTER 2. VECTOR BUNDLES OVER RIGID ANALYTIC VARIETIES

2.1 A word on topologies

Let us first discuss the concept of a vector bundle very generally and, for the beginning, informally. A vector bundle on a geometric object X is a projection from another geometric object p: E → X whose fibres have the structure of isomorphic vector spaces and such that there exists a local trivialisation. One often identifies the vector bundle p with its sheaf of local sections on X. Adopting this point of view, a vector bundle on a geometric object X is a locally free OX -module where OX is the sheaf of functions on X. We explain now what this means. Definition 2.1. a) Let C be a geometric category: A geometric object X ∈ Ob(C) is supposed to come with • a site X whose objects are morphisms (U → X) ∈ Mor(C) and whose morphisms are commutative triangles

U / X

of morphisms in C. By abuse of notation, we write U ∈ Ob(X) and (V → U) ∈ Mor(X).

• a sheaf OX of rings on the site X, the structure sheaf of X. b) A sheaf F on X is by definition a presheaf from X that satisfies the cocycle condition. Assuming that the category X has fibre products and that the presheaf has values in a category which admits products, the cocycle condition reads: For all U ∈ Ob(X) and all {Ui → U}i∈I ∈ Cov(U), the diagram of restrictions Y Y F(U) → F(Ui) ⇒ F(Ui ×X Uj) i∈I i,j∈I

is an equaliser diagram. c) By a sheaf on X we mean a sheaf on X.

d) An OX -module is a sheaf of modules such that for each U ∈ Ob(X), its value F(U) is an OX (U)-module.

e) It is locally free if there exists {Ui}i∈I ∈ Cov(X) such that the F(Ui) are free OX (Ui)-modules for all i.

f) A covering {Ui}i∈I such that all F(Ui) are free is called a local trivialisation of F. g) If K is a ﬁeld, a locally free sheaf F of K-algebras of rank n is called a K-vector bundle of rank n. h) Two vector bundles are called isomorphic if they are isomorphic as sheaves. 2.1. A WORD ON TOPOLOGIES 29

A local trivialisation {Ui}i∈I together with cocycle maps

i n ϕj ∈ Aut (OX (Ui ×X Uj)) = GLn(OX (Ui ×X Uj)), i, j ∈ I fulﬁlling the cocycle conditions

i ϕi = id i j i ϕjϕl = ϕl is enough to determine the isomorphism type of the vector bundle. To conclude, let us note: When we speak of a vector bundle on X, we make implicit choices: The choice of site and the choice of structure sheaf. We will be careful to always specify which (Grothendieck) topology we are referring to. On the contrary, there will be no chance of confusion about the structure sheaf.

The term “vector bundle” means “locally free OX -module” throughout this thesis. The two points of view, i. e. the projection map versus the sheaf of sections, reappear in section 5.2 in the context of fibre bundles and Gromov’s h-principle. We will see that if a partial differential relation on a manifold M satisfies Gromov’s h-principle, then this means that a presheaf of certain sections of the jet bundle of M satisfies the sheaf conditions up to homotopy. Such a presheaf is called a homotopy sheaf . In the world of rigid analytic varieties, there are in particular the following two types of vector bundles: Definitions 2.2 (analytic/Zariski vector bundle). a) Vector bundles with respect to the G-topology of a rigid analytic variety are called analytic vector bundles. b) Vector bundles with respect to the Zariski topology of an affinoid variety are called Zariski vector bundles. In fact the two notions are the same: We will see in Corollary 2.15 that the isomorphism classes of Zariski vector bundles and the isomorphism classes of analytic vector bundles coincide. The reason is that both Zariski and analytic vector bundles on affinoid varieties are determined by their global sections, and those coincide by definition. Remark 2.3. The definition of an analytic vector bundle coincides with the usual definition of a vector bundle on a rigid analytic variety [FvdP04, p. 87]. Furthermore, every quasiseparated rigid analytic variety can be seen as a Berkovich space by Theorem 1.54 by Berkovich. Berkovich spaces are topological spaces and thus come with a notion of continuous vector bundle. But often one wants to take the analytic structure into account. Definition 2.4 (Vector bundle on a Berkovich space). Let X be a Berkovich space.

a) Let OX be the structure sheaf of analytic functions on X. An analytic vector bundle on X is a locally free OX -module. b) Let C0(X) be the sheaf of continuous functions on X.A continuous vector bundle on X is a locally free C0(X)-module. 30 CHAPTER 2. VECTOR BUNDLES OVER RIGID ANALYTIC VARIETIES

Let X be a quasiseparated rigid analytic space and XB the corresponding Berkovich space. Do their categories of analytic vector bundles coincide? The answer depends on X. Proposition 2.5 (Berkovich). If every point x ∈ XB has an aﬃnoid neighbourhood, then the category of analytic vector bundles on X and the category of analytic vector bundles on XB are equivalent.

Proof. Berkovich calls an analytic space Y good if every point x ∈ XB has an affinoid neighbourhood [Ber93, Remark 1.2.16]. If Y is good, the category of analytic vector bundles with respect to the Berkovich topology on Y and the category of vector bundles with respect to the G-topology on Y are equivalent by [Ber93, Proposition 1.3.4]. The category of vector bundles with respect to the G-topology on Y and the category of analytic vector bundles on the associated rigid analytic space are equivalent by [Ber93, bottom of p. 37]. Berkovich’s G-topology on Y is defined on [Ber93, p. 25]. The G-topology on the corresponding rigid analytic space coincides with the one in Definition 1.16.

Example 2.6. If X is an affinoid space, the corresponding Berkovich space is called strictly affinoid. Strictly affinoid Berkovich spaces are obviously good. Each strictly affinoid Berkovich space is of the form XB for an affinoid rigid analytic space X by construction of the correspondence 1.54. Berkovich spaces corresponding to rigid analytic quasi-Stein spaces are good by construction. Remark 2.7. It is an open question under which conditions the category of analytic vector bundles and the category of continuous vector bundles on a Berkovich space are equivalent.

2.2 Serre–Swan for rigid analytic quasi-Stein varieties

In this section we prove the rigid analytic version of the Serre–Swan theorem: Isomorphism classes of vector bundles of rank n over a rigid analytic quasi-Stein variety X correspond to isomorphism classes of projective modules of rank n over the ring of global functions OX (X). There is a general principle for proving Serre–Swan theorems. This was analysed by A. Morye [Mor09] and we give a short account of her argument. Definition 2.8. Let (X, T , OX ) be a locally G-ringed space. A coherent sheaf of OX -modules is called finitely generated by global sections if there is a finite set of global sections whose images in the stalk at any point x ∈ X generate that stalk as an OX,x-module. The following notion will only be needed in this section: Definition 2.9 (admissible subcategory [Mor09, Definition 1.1]). Let (X, OX ) be a locally G-ringed space. A subcategory C of the category of OX -modules is called an admissible subcategory if 2.2. SERRE–SWAN FOR RIGID ANALYTIC QUASI-STEIN VARIETIES 31

C1 C is a full abelian subcategory of the category of OX -modules with internal

Hom, i. e., for F, G ∈ Ob(C) the sheaf HomOX (F, G) is also an object of C, C2 every sheaf in C is acyclic and generated by global sections, and

C3 C contains the category of locally free OX -modules. Theorem 2.10 (General Serre–Swan [Mor09, Theorem 2.1]). Let (X, OX ) be a locally G-ringed space such that a sequence of coherent sheaves

0 → E → F → G → 0 on X is exact if and only if for each x ∈ X the sequence of OX,x-modules

0 → Ex → Fx → Gx → 0 is exact (“coherent sheaves may be tested on stalks”).

Assume that the category of OX -modules contains an admissible subcategory C and that every locally free sheaf of bounded rank on X is ﬁnitely generated by global sections. Then the Serre–Swan theorem holds for (X, OX ), i. e., the category of locally free sheaves of bounded rank on X and the category of ﬁnitely generated projective OX (X)-modules are equivalent via the global sections functor. Morye’s Theorem is formulated for a locally ringed space, but she actually proves the slightly stronger statement given here. The site associated to a locally ringed space always has enough points, so all sheaves may be tested on stalks (see [MLM94, Chapter IX, §3, Proposition 3 on p. 480]).

Proof. Let (X, OX ) be a locally G-ringed space such that coherent sheaves may be tested on stalks. The proof proceeds in the following steps:

a) If D is a full subcategory of the category of OX -modules such that OX ∈ Ob(D) and such that every sheaf in D is generated by global sections, then the global sections functor is fully faithful on D [Mor09, Proposition 2.2].

b) Under the same assumptions, the composition of the “associated sheaf” functor

S : OX (X)-modules −→ OX -modules

M 7−→ (U 7→ M ⊗OX (X) OX (U)

with the global sections functor is essentially surjective: Every F ∈ Ob(D) is isomorphic to S(F(X)) [Mor09, Proposition 2.5].

c) Assume that the category of OX -modules contains an admissible subcategory. Let F be locally free sheaf of bounded rank which is ﬁnitely generated by global sections. Then F(X) is a ﬁnitely generated projective OX (X)-module [Mor09, Proposition 2.8]. 32 CHAPTER 2. VECTOR BUNDLES OVER RIGID ANALYTIC VARIETIES

Proof. By assumption there exist an n ∈ N and an epi- n morphism u: OX F. As HomOX (F, ker u) is acyclic, Ext1 (F, ker u) = 0 and hence the exact sequence OX

n u 0 → ker u → OX → F → 0

splits and the exact sequence induced on global sections

n u 0 → (ker u)(X) → (OX (X)) → F(X) → 0

splits, too.

Recall the deﬁnition of quasi-Stein spaces and Stein spaces from 1.30. S Lemma 2.11. Let X = i(Sp(Ai)) be a rigid analytic quasi-Stein variety of bounded dimension, i. e., supx∈X dim OX,x =: d < ∞. Let E be a vector bundle on X. Then E is ﬁnitely generated by global sections.

Proof. By Theorem A [Kie67, Satz 2.4.3], every coherent sheaf on a quasi-Stein space is generated by global sections. Let A := OX (X). We have to show that E(X) is finitely generated as an A-module. Gruson does this for Stein spaces which are unions of polydiscs [Gru68, V, Théorème1] and the same proof works for general quasi-Stein spaces. We give it now.

Let n the rank of the Ai-module E(Sp Ai). Both are independent of i. Let F := An(d+1). Of course there exist homomorphisms F → E(X). We show that the set E := {F E(X)} of epimorphisms from F onto E(X) is dense in HomA(F, E(X)). Then E is in particular nonempty and we know that E(X) is ﬁnitely generated.

The map Ai+1 → Ai is ﬂat by Lemma 1.49. For each i, set Fi := F ⊗ˆ AAi and ∼ ∼ Mi := E(Sp Ai). We have that Fi = Fi+1⊗ˆ AAi and Mi = Mi+1⊗ˆ AAi for all i. Moreover, Fi and Mi are ﬁnitely presented for all i. Using Lemma 1.51, we see that the projective system

· · · ← Ai ← Ai+1 ← · · · induces a projective system

· · · ←− HomAi (Fi, E(Sp Ai)) ←− HomAi+1 (Fi+1, E(Sp Ai+1)) ←− · · ·

The limit is HomA(F, E(X)) by Kiehl’s Theorem B [Kie67, Satz 2.4.2]. As ﬂat base change maps epimorphisms to epimorphisms, we can look at the projective system

· · · ←− Ei ←− Ei+1 ←− · · · where Ei := {Fi E(Sp Ai)}. We claim that its limit is E. It is clear that E ⊂ lim(Ei)i∈N. We have to show that the limit of a system (φi)i ∈ (Ei)i of epimorphisms is an epimorphism: lim(φi)i ∈ E. As the category of coherent sheaves 2.2. SERRE–SWAN FOR RIGID ANALYTIC QUASI-STEIN VARIETIES 33 on a rigid analytic variety is abelian [FvdP04, p. 87], the sheaf cokernel D of F → E is coherent. By Kiehl’s Theorem B [Kie67, Satz 2.4.2], it is enough to test surjectivity of a map of coherent sheaves on an aﬃnoid covering. But we have D(Sp Ai) = 0 for all i, because there the morphisms F(Sp Ai) E(Sp Ai) are epi. Thus, the cokernel sheaf D is zero and we have lim(φi)i ∈ E. A lemma by Gruson [Gru68, V, Lemme 2] states that if B is any Banach algebra of topologically ﬁnite type of Krull dimension d and if P is a projective B-module of rank p and L is the free B-module B(d+1)p, then the set of epimorphisms L → P is open and dense in the topological B-module Hom(L, P ).

This applies in particular to the affinoid algebras Ai. We get that Ei is open and dense in HomAi (Fi, E(Sp Ai)) for each i. By the general Mittag-Leffler Theorem [Bou66, II.3.5, Théorème1], E is dense in HomA(F, E(X)) with respect to the limit topologies. Hence, E is nonempty and E(X) is finitely generated by global sections. S Theorem 2.12 (rigid analytic Serre–Swan). Let X = i(Sp(Ai)) be a quasi-Stein variety of bounded dimension, i. e., supx∈X dim OX,x < ∞. Set A := OX (X). Then the global sections functor

ﬁn,proj Γ: VBX −→ A-Mod from vector bundles over X to ﬁnitely generated projective A-modules is an equivalence of categories.

Proof. Let us verify the assumptions of Theorem 2.10 (general Serre–Swan). • A sequence of coherent sheaves

0 → E → F → G → 0

on a rigid analytic variety X is exact if and only if for each x ∈ X the sequence of OX,x-modules

0 → Ex → Fx → Gx → 0 is exact [FvdP04, p. 192].

• The category of coherent OX -modules on X is abelian [FvdP04, p. 87] and an

admissible subcategory of ModOX [Kie67, Satz 2.4]. • Vector bundles are ﬁnitely generated by global sections. This is true for aﬃnoid varieties by Kiehl [Kie67, Satz 2.2] and for general quasi-Stein varieties of bounded dimension by Lemma 2.11. Consequently the Serre–Swan Theorem holds for rigid analytic quasi-Stein varieties.

Remark 2.13. If X = Sp A is affinoid itself, this follows directly from Kiehl [Kie67], as pointed out by Gruson [Gru68] and Lütkebohmert [Lüt77].If

[ n [ −1 −1 X = Bs = khη T1, . . . , η Tni s

alg ﬁn,proj Γ: VBX −→ A-Mod from Zariski vector bundles over X to ﬁnitely generated projective A-modules is an equivalence of categories.

Proof. Again we need to verify the assumptions of Theorem 2.10 (general Serre– Swan). • A sequence of coherent sheaves

0 → E → F → G → 0

on an aﬃnoid variey X is exact if and only if for each x ∈ X the sequence of the Zariski stalks

Zar Zar Zar 0 → Ex → Fx → Gx → 0 is exact [sta, Tag 00EN, Lemma 10.23.1(3)].

Zar • As (X, Zar, OX ) is a locally ringed space (not only locally G-ringed), the category of coherent Zariski OX -modules on X is abelian by the classical arguments [sta, Tag 01BU, Lemma 17.12.4]. It is an admissible subcategory of

ModOX : C1 Internal Hom between coherent sheaves is coherent [sta, Tag 01CM, Lemma 17.20.5]. C2 Every coherent sheaf is acyclic by a theorem by Grothendieck (see [Har77, Theorem 2.7]) and generated by global sections by noetherianity (cf. the argument in the next point below). C3 is clear. • Vector bundles are ﬁnitely generated by global sections. Let E be a vector bundle of rank n on X and dim X = d. As A is noetherian, E has a ﬁnite

local trivialisation of the form {Sp(Afi )}i=1,...,N with fi ∈ A for all i. That is,

{Sp(Afi )}i=1,...,N is a ﬁnite covering of X by aﬃnoid domains Xi := Sp(Afi ) such that for all i the restriction E(Xi) is trivial. In other words,

∼ n E(Xi) = (Afi ) .

Let fi1 , . . . , fin ∈ Afi be generators of E(Xi) as an Afi -module. Now if some

fij is not in the image of the localisation A ⊂ Afi , this means that fij has s1 sn a ﬁnite power sj of fi in the denominator. The elements fi1 fi , . . . , fin fi 2.2. SERRE–SWAN FOR RIGID ANALYTIC QUASI-STEIN VARIETIES 35

are now in the image of the localisation A ⊂ Afi . As fi is a unit in Afi , the s1 sn fi1 fi , . . . , fin fi still generate E(Xi) as an Afi -module. They lift to global ˜ ˜ sections fi1 ,..., fin ∈ A and hence E is generated by the global sections ˜ ˜ ˜ ˜ f11 ,..., f1n ,..., fN1 ,..., fNn . By induction over the dimension we can even show that N can be chosen less than or equal to d + 1. Let E be a Zariski vector bundle on X. We need to show that E is ﬁnitely generated by global sections. As A is noetherian, there is a local trivialisation of E by aﬃnoid Zariski-open subspaces

Xi = Sp(Afi ), i = 1, . . . , r with fi ∈ O(X) for i ∈ {1, . . . , r}. By Kiehl’s Theorem A, that is, Theorem 1.32c) in this thesis, every stalk Ex is generated by global sections as an OX,x-module. Set n := dim X. For each x ∈ X, choose global sections fx1, . . . , fxn whose images generate Ex as an OX,x-module. For each i ∈ {1, . . . , r}, the images of the global sections from

Si := {fxj ∈ G(X) | x ∈ Xi, j = 1, . . . , n} in G(Xi) generate G(Xi) as an OX (Xi)-module. As G(Xi) is a free OX (Xi)-module of dimension n, already n of those sections are suﬃcient. Choose fi1, . . . , fin ∈ Si such that their images in G(Xi) generate it as an OX (Xi)-module. In particular, for any x ∈ Xi, the images of the fij in Gx generate it as an OX,x-module. The set

S := {fij ∈ G(X) | i = 1, . . . , r; j = 1, . . . , n} of global sections of G(X) is finite. The images of its elements generate each stalk of G. Corollary 2.15. The categories of analytic vector bundles and of Zariski vector bundles on an affinoid variety are equivalent. Recall from Definition 1.29 that a rigid analytic variety is quasicompact if it can be covered by finitely many affinoids. Proposition 2.16. Every vector bundle on a quasicompact rigid analytic variety has a finite local trivialisation.

Proof. Let X be a quasicompact variety and let E be a vector bundle on X. Assume that X is affinoid. Then E is isomorphic to a Zariski vector bundle E 0 on X. It has a finite local trivialisation because X is noetherian. If X is not affinoid, it has a finite admissible covering {U1,...,Ur} by affinoid subsets. The restrictions of E to those affinoid subsets each have a finite local trivialisation. Putting them together yields a finite local trivialisation of E.

The same holds for quasi-Stein varieties: Theorem 2.17 (Ben’s Theorem). Every vector bundle on a rigid analytic quasi-Stein variety has a ﬁnite local trivialisation. 36 CHAPTER 2. VECTOR BUNDLES OVER RIGID ANALYTIC VARIETIES

The author asked for a proof or counterexample on mathoverflow.net (question 219140). The following proof was given by the user Ben. He proved that, over a locally ringed space, any vector bundle which is ﬁnitely generated by global sections admits a ﬁnite local trivialisation. This easily carries over to a rigid analytic variety with the G-topology.

Proof. Let X be a quasi-Stein variety and E be a vector bundle of rank r on X. By Lemma 2.11, it is generated by ﬁnitely many global sections s1, . . . , sd. Let XB be the Berkovich space associated to X. As X is quasi-Stein, every x ∈ XB has an aﬃnoid neighbourhood (Example 2.6). By Proposition 2.5, the category of vector bundles with respect to the Berkovich topology on XB and the category of analytic vector bundles on X are equivalent. The proof is done by explicit construction. For 1 ≤ j ≤ r, let e be the section of Or that is given by the constant function j XB (0,..., 0, 1, 0,..., 0) where the 1 is located in the j-th entry. Clearly, e1, . . . , er are generators of Or . For each XB

I = {i1, . . . , ir} ⊂ {1, . . . , d} deﬁne the morphism ϕ : Or −→ E I XB

ej 7−→ sij . Then the subsets

UI = {x ∈ XB | (ϕI )x → Ex is an iso } form a ﬁnite trivialisation of E. By deﬁnition of UI , the map ϕ | : Or (U ) → E(U ) I UI XB I I induces isomorphisms on stalks. As E is coherent, it has to be an isomorphism.

We still need to check that {UI | I = {i1, . . . , ir} ⊂ {1, . . . , d}} is a covering of XB. Let x ∈ XB. The stalk Ex is generated by the residue classes of s1, . . . , sd. As it is of dimension r, already r of them suffice. Thus, there is an I such that x ∈ UI . Hence, all the UI together cover XB. Each UI is open because the set of x ∈ XB for which the residue classes of si1 , . . . , sir in Ex satisfy a nontrivial relation is closed. We have shown that every vector bundle over the Berkovich space associated to a quasi-Stein rigid analytic variety has a finite local trivialisation with respect to the Berkovich topology. We need a finite local trivialisation with respect to the G-topology.

Let X1 ⊂ X2 ⊂ · · · be a strictly aﬃnoid covering of XB.

Let V ∈ Cov(XB) be an admissible G-covering which is a local trivialisation of E such that for any V ⊂ V there exists an I with V ⊂ UI . Let W ⊂ V be a subcovering whose restriction to each Xi is ﬁnite. This exists because the Xi are aﬃnoid and hence compact. Set ˜ [ UI := V. V ∈W, V ⊂UI 2.3. LINE BUNDLES 37

˜ Now {UI | I = (i1, . . . , ir) ⊂ {1, . . . , d}} is a ﬁnite admissible G-covering of X which trivialises E. It gives rise to a ﬁnite local trivialisation of the original vector bundle over the rigid analytic space X.

2.3 Line bundles

Definition 2.18 (Line bundle). Let X be a rigid analytic variety. A locally free OX -module of rank one is called a line bundle. As usual, line bundles form a group, the Picard group. Proposition/Definition 2.19 (Picard group, cf. [Ger77, 1.1, 1.2]). Let X be a rigid analytic variety. The set of isomorphism classes of line bundles on X forms a group under ⊗. It is called the Picard group and denoted by Pic(X). For an affinoid algebra A we write Pic(A) := Pic(Sp(A)) for the group of isomorphism classes of invertible A-modules of rank one.

Proof. Compare [Ger70]. Let L, L0 be line bundles on X. Then their tensor product 0 0 L ⊗ L is again a line bundle on X. Let {Ui}i∈I and {Uj}j∈J be local trivialisations 0 0 of L and L , respectively. Then a common reﬁnement {Ui ∩ Uj | i ∈ I, j ∈ J} yields a local trivialisation of L ⊗ L0. Hence, L ⊗ L0 is a line bundle on X. Let L be a line −1 bundle on X. Then the sheaf L := Hom(L, OX ) is a line bundle on X and ∼ ∼ Hom(L, OX ) ⊗ L = Hom(L, L) = OX . ∼ Obviously, L ⊗ OX = L.

The Picard group of an affinoid variety is strongly linked to the variety’s canonical formal model and canonical reduction: Theorem 2.20 (Gerritzen [Ger77, §4]). Let k be a complete discretely, nontrivially valued field. Let A be a reduced k-affinoid algebra whose ring of integers A◦ is regular. Assume kAk = |k|. Then there is an isomorphism

Pic(A) ∼= Pic(A˜).

More generally, Urs Hartl and Werner Lütkebohmert proved the following. Theorem 2.21 (Hartl–Lütkebohmert [HL00]). Let k be a complete discretely, nontrivially valued field. Let X be a smooth, proper, connected rigid analytic k-variety that admits a strict semistable formal model over k◦. Assume furthermore that there exists a k-rational point x: Sp k → X. Denote by smRig the category of smooth, quasicompact, separated rigid analytic k-varieties. Then the Picard functor

smRig → Sets, n o ∼ ∗ ∼ V 7→ (L, λ) L line bundle over X × V, λ: OV → (x, id) L iso / = is representable. 38 CHAPTER 2. VECTOR BUNDLES OVER RIGID ANALYTIC VARIETIES

2.4 Divisors

Deﬁnition 2.22 (Divisor in a ring [Eis95, p. 262]). Let R be a ring. A divisor in R is an element of the free abelian group Div(R) generated by those prime ideals of R that are of height one. Example 2.23. If R is of dimension one, then

Div(R) = Max(R)Z. In other words, a divisor on a curve X is a formal sum of closed points of X. Deﬁnition 2.24 (Divisor class group of a ring). The divisor of a principal ideal is called a principal divisor, cf. [Bou06, p. 196]. The set of principal divisors of a ring R is denoted by P (R). The divisor class group of a ring R is the quotient

Cl(R) := Div(R)/P (R).

Definition 2.25 (Divisor class group of an affinoid variety). For an affinoid variety X = Sp(A) we denote

Div(X) := Div(A) and Cl(X) := Cl(A). S Proposition 2.26 ([Gru68, p. 87, Remarque 2]). If X = s

The corresponding statement for closed discs follows from Bourbaki [Bou06], cf. [Gru68, Remarque 2 on p. 87]. We name a few results on the class group. Definition 2.27 (spherically complete, see [Laz62, Définition 5.2] or [BGR84, 2.4.4 Definition 1]). A complete valued field k is called spherically complete (also: maximally complete) if it satisfies the embedded discs principle: For r ∈ R and x ∈ k denote the disc of radius r with centre x by Br(x) ⊂ k. Now look at sequences of discs Br (xn) such that for all n ∈ we have Br (xn+1) ⊂ Br (xn). The n n∈N N n+1 n embedded discs principle is satisfied if each such sequence of discs has a common point: T B (x ) 6= ∅. n∈N rn n Examples 2.28. a) Complete discretely valued fields are spherically complete [Sch02, Lemma 1.6].

b) The field Cp, defined as the completion of the algebraic closure of Qp, is not spherically complete [Sch02, p. 4f]. S Theorem 2.29 (Lazard [Laz62, Théorème2 in §7]). Let X = s

Proposition 2.30 (Denneberg [Den69, Korollar 1]). Let X = Sp A be an affinoid curve over a complete, algebraically closed, nonarchimedean field k, i. e., a k-affinoid variety of dimension one. Let A be reduced. If the reduction X˜ is nonsingular and (f) ∈ Div(X˜) is a principal divisor, then every preimage of (f) under the reduction is a principal divisor in Div(X). Corollary 2.31 (Denneberg [Den69, Korollar 2]). Let X be an affinoid curve over a complete, algebraically closed, nonarchimedean field k. If its reduction X˜ is irreducible, ˜ then the natural epimorphism Div(X) Div(X) induces an epimorphism of the class groups ˜ ϕ: Cl(X) Cl(X). If additionally X˜ is nonsingular, then ϕ is an isomorphism. In the context of the class group, van der Put proved among other things: Theorem 2.32 (Van der Put [vdP80, Theorem 2.1]). Let k be algebraically closed and X = Sp A a connected affinoid curve. Then A is a unique factorisation domain 1 if and only if X is an affinoid subspace of Pk. Furthermore he asserts that for a normal, connected, affinoid curve X over a complete, algebraically closed, nonarchimedean field k,

1 ∗ ∼ H (X, OX ) = Div(X)/ P(X).

This assertion is used in the proof of Theorem 2.33 (Van der Put). Let X be a normal, connected, affinoid curve over a complete, algebraically closed, nonarchimedean field k. a) There exists a complete, nonsingular algebraic curve C over k such that X is an affinoid subspace of the analytification of C. b) Such an embedding into an algebraic curve C fixed, there is an isomorphism

1 ∗ ∼ H (X, OX ) = J(C)/H

where J(C) denotes the Jacobian of C and H is the subgroup consisting of the images of those divisors of degree zero on C whose support lies in C \ X.

Proof. a) This is [vdP80, Theorem 1.1]. b) This is [vdP80, Proposition 3.1].

Van der Put used this theorem to compute the Picard groups of some aﬃnoid curves [vdP80, pp. 163f]. 40 CHAPTER 2. VECTOR BUNDLES OVER RIGID ANALYTIC VARIETIES Chapter 3

Homotopy Invariance of Vector Bundles

3.1 Serre’s problem and the Bass–Quillen conjecture

In [Ser55], Serre asked whether all finitely generated projective modules over a polynomial ring over a field are free. The question was answered affirmatively and independently in 1976 by Quillen [Qui76] and by Suslin [Sus76]. The theorem is now known as the Quillen–Suslin Theorem. We warmly recommend Lam’s wonderful book [Lam06] about Serre’s problem. The analogue of the Quillen–Suslin Theorem in rigid analytic geometry is also true: Theorem 3.1 (Lütkebohmert, Kedlaya). Every finitely generated projective khT1,...,Tni-module is free.

Proof. Lütkebohmert [Lüt77, Theorem 1] uses Weierstraß preparation and GAGA to reduce the proof to the original Quillen–Suslin Theorem. Kedlaya [Ked04, Proposition 6.6] proves a more general version of the theorem. He uses Weierstraß preparation to reduce the proof to classical statements about unimodular vectors and finite free resolution of modules. Definition 3.2 (extended module). Let R be a ring. An R[T ]-module M is called extended from R if there exists an R-module N such that ∼ M = N ⊗R R[T ].

When Serre’s problem was still open, Bass generalised the question to the following: Question 3.3 (Bass [Bas73, Problem IX on page 21]). Let R be a commutative regular ring. Is every ﬁnitely generated projective R[T ]-module extended from R? For his proof of Serre’s problem [Qui76], Quillen developed a method to patch ﬁnitely presented modules. With Quillen patching at hand, Quillen reformulated

41 42 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

Bass’ question: For an affirmative answer to question 3.3, it is enough to prove the statement for a commutative regular local ring R. The problem became known as the Bass–Quillen Conjecture. Hartmut Lindel showed in [Lin82] that the Bass–Quillen Conjecture is true if R is a regular algebra over a field K such that R is essentially of finite type over K. Here, essentially of finite type means that R is a localisation of a K-algebra R0 which is of finite type over K. Lindel’s theorem was generalised by Dorin Popescu [Pop89, Corollary 4.4] and others, e. g. Sankar Prasad Dutta [Dut95, Theorem 3.4]. For the most general form of the theorem, see [AHW15a, Theorem 5.2.1]. Theorem 3.4 (Lindel–Popescu, [Lin82, Pop89]). The Bass–Quillen Conjecture is true if R is a regular local ring containing a field. We are concerned with the corresponding question in the rigid analytic world: Question 3.5. Let A be a smooth affinoid k-algebra and M a finitely generated projective AhT i-module. Is M extended from A? That is, does there exist a finitely generated projective A-module N such that ∼ M = N⊗ˆ AAhT i?

One motivation to be interested in the Bass–Quillen conjecture is its geometric meaning. By the Serre–Swan theorem, the Bass–Quillen conjecture is a conjecture about vector bundles.

3.2 Homotopy invariance

By the Serre–Swan Theorem 2.12, Question 3.5 becomes: Question 3.6. Let X = Sp A be a smooth aﬃnoid k-variety, B1 = Sp khT i the unit ball and

1 pr1 : X × B −→ X the projection onto the ﬁrst factor. Is every vector bundle E over X × B1 the pullback along pr1 of some vector bundle F over X? This immediately becomes a subquestion of the following question: Question 3.7. Let X be a smooth rigid analytic k-variety and

1 pr1 : X × B −→ X the projection onto the first factor. Is every vector bundle E over X × B1 the pullback along pr1 of some vector bundle F over X? In the sequel we will refer to this as B1-invariance. We will see that the answer is no in general, for example if X = P1. So we will have to refine the question. The corresponding questions in algebraic geometry and in complex analysis are well understood. In algebraic geometry, affineness is a necessary condition for homotopy 3.2. HOMOTOPY INVARIANCE 43 invariance of vector bundles of rank ≥ 2. In complex analysis, the space X has to be Stein. In both cases, the projective line P1 provides a counterexample for vector bundles of rank at least two. Examples using Grothendieck’s classification of line bundles on P1 and Serre’s GAGA are well known. We will see one of those examples in 3.10, adapted to the rigid analytic situation. A complex Stein space is a complex space for which Cartan’s Theorem B holds, see also Definition 5.4. For a detailed exposition on complex Stein spaces we recommend Grauert–Remmert [GR04]. For now, let us just say that a complex Stein space has many global functions. As a consequence every holomorphic vector bundle over a complex Stein space is entirely determined, up to isomorphism, by its global sections [For64, For67]. In algebraic geometry, the varieties that have many global functions are exactly the affine varieties. By Serre’s Serre–Swan Theorem [Ser58] every vector bundle on an affine variety is entirely determined, up to isomorphism, by its global sections. In our case, rigid analytic quasi-Stein varieties have many global functions. By Theorem 2.12, the isomorphism type of a vector bundle over a quasi-Stein variety is completely determined by its global sections. Thus rigid analytic quasi-Stein varieties should be the right class of spaces to look at. Another obvious generalisation of question 3.6 is 1 Question 3.8. Let X be a smooth rigid analytic quasi-Stein variety, Arig the rigid analytic affine line (see Example 1.24) and

1 pr1 : X × Arig −→ X

1 the projection onto the ﬁrst factor. Is every vector bundle E over X ×Arig the pullback along pr1 of some vector bundle F over X? 1 In the sequel we will refer to this as Arig-invariance. The ﬁrst thing to check is whether the Quillen–Suslin Theorem still holds. That is, 1 are all vector bundles over Arig trivial? Proposition 3.9 ([Gru68, V, Proposition 2]). Let Xr be an open polydisc of polyradius r ∈ (|k| ∪ {∞})n, that is

[ −1 −1 Xr = Sp(khη1 T1, . . . , ηn Tni).

|ηi|=si

The Picard group Pic(Xr) of line bundles over Xr is trivial if and only if one of the following holds: a) the ﬁeld k is spherically complete (see Deﬁnition 2.27) or

n b) the polyradius is r = (∞,..., ∞), that is, Xr = Arig is the analytic aﬃne space. The “only if” direction was proven by Lazard [Laz62, Proposition 6]: Let k be spherically incomplete and r ∈ |k|. Lazard constructs a divisor on Xr which is not a principal divisor. This implies that open discs Xr which are bounded in at least one direction have nontrivial line bundles. The reason is that the divisor class group and the Picard group coincide on open polydiscs D by Gruson’s Proposition 2.26. 44 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

3.3 Counterexamples

Homotopy invariance cannot hold in full generality. In order to stake the limits we assemble examples where homotopy invariance is violated in this section. The examples are:

1 1 • Neither B -invariance nor Arig-invariance hold for vector bundles of rank at 1 least two over Prig (Example 3.10). 1 1 • Neither B -invariance nor Arig-invariance hold for vector bundles over an aﬃnoid variety with a cusp singularity (Example 3.12 by Gerritzen).

• B1-invariance can fail for a smooth aﬃnoid variety whose reduction has a cusp singularity (Example 3.14 by Gerritzen or Bosch). 1 1 Example 3.10. We show that neither B - nor Arig-invariance hold for vector bundles 1 1 of rank at least two over Prig. Every counterexample to A -invariance of algebraic vector bundles over algebraic k-varieties works. The example we present here is the analytiﬁed version of Asok–Morel [AM11, Example 3.2.9].

1 1 Introducing coordinates (t, x) on (Prig \{0, ∞}) × Arig, let E be the vector bundle on 1 1 1 1 1 1 Prig × Arig which is free on (Prig \{0}) × Arig and free on (Prig \ {∞}) × Arig and has the transition matrix

t2 xt 0 1

1 1 1 on (Prig \{0, ∞}) × Arig. We claim that it is not the pullback of any bundle on Prig. 1 1 Restricting E to x ∈ A yields a bundle Ex on Prig. For x = 0 we get E0, described by

t2 0 . 0 1

t2 t For x = 1 we get E1, described by 0 1 . Another representative of this cocycle in −1 GL2(kht, t i) −1 GL2(khti) \ /GL2(kht i) is

t−1 0 . 0 t

The bundles E0 and E1 are the analytifications of bundles Ee0, respectively Ee1 on the algebraic variety P1 that are described by the same transition matrices. By GAGA [FvdP04, Proposition 9.3.1], the analytic bundles E0 and E1 are isomorphic if and only if the algebraic bundles Ee0 and Ee1 are isomorphic. We show that they are not isomorphic. Grothendieck classified holomorphic (and, by complex GAGA, algebraic) vector bundles over 1 [Gro57]. Hazewinkel and Martin gave an elementary proof for PC algebraic vector bundles over any field K: 3.3. COUNTEREXAMPLES 45

Theorem 3.11 ([HM82, Theorem 4.1]). Let K be any ﬁeld. Let F be a vector bundle 1 1 of rank m over PK . Denote by O(s) the line bundle on P given by the transition matrix (ts) on A1 \{0} = Spec K[t, t−1]. Then F is isomorphic (over K) to a direct sum of line bundles ∼ F = O(s1) ⊕ · · · ⊕ O(sm), s1 ≥ · · · ≥ sm, all si ∈ Z, and the tuple (s1, . . . , sm) is uniquely determined by the isomorphism class of F.

In particular, we can apply this to our example and see that the bundles E0 and E1 are not isomorphic. Denote the projection by

1 1 1 pr1 : Prig × Arig → Prig and the zero section, respectively, the one section by

1 1 1 1 ι0 : Prig → Prig × {0} ⊂ Prig × Arig and 1 1 1 1 ι1 : Prig → Prig × {1} ⊂ Prig × Arig

1 For every vector bundle F on Prig we get ∗ ∗ ∼ ∼ ∗ ∗ ι0 pr1 F = F = ι1 pr1 F. As

∗ ∗ ι0E = E0 E1 = ι1E, ∼ ∗ 1 our bundle E is not of the form E = pr1 F for any bundle F on Prig. 1 1 Thus neither B -invariance nor Arig-invariance hold for bundles of rank at least two 1 on Prig. Example 3.12 (Gerritzen [Ger77, §3.3]). There exists an aﬃnoid variety with a cusp 1 1 singularity such that neither B -invariance nor Arig-invariance hold for vector bundles over this aﬃnoid variety. First consider B1-invariance. The space

2 3 X = Sp khT1,T2i/(T1 − T2 ) is an aﬃnoid neighbourhood of the cusp of the analytiﬁcation of the algebraic 2 3 k-variety Spec k[T1,T2]/(T1 − T2 ). Choose c ∈ k with 0 < |c| < 1 and cover X by

2 Uc = {(x1, x2) ∈ X | kx1k ≤ |c| } and 2 Vc = {(x1, x2) ∈ X | kx1k ≥ |c| }.

Then Uc = {Uc,Vc} is an admissible covering. The sheaf OX (< 1) on X is deﬁned by

OX (< 1)(U) := {f ∈ OX (U) | kfk < 1}.

The ﬁrst Cechˇ cohomology group of X with respect to this covering with coeﬃcients in OX (< 1) is:

ˇ 1 ∼ ◦ ◦ H ({Uc,Vc}, OX (< 1)) = k /ck . 46 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

We have injections

1 ˇ 1 ∼ 1 H (Uc, OX (< 1)) ,→ H (X, OX (< 1)) = H (X, OX (< 1)).

1 In particular, H (X, OX (< 1)) is non-trivial.

Deﬁne a sheaf G0 on X by ∞ 1 X i G0(U) := f ∈ OX× 1 (U × B ) = OX (U)hT i f = 1 + fiT B i=1 with |fi(u)| < 1 ∀i ≥ 1 ∀u ∈ U for U ⊂ X admissible.

By [Ger77, Satz 2] there is a decomposition

1 ∼ 1 Pic(X × B ) = Pic(X) ⊕ H (X, G0). If char k = 0, then

1 1 H (X, OX (< 1)) = 0 ⇔ H (X, G0) = 0 by [Ger77, Satz 4] and hence if char k = 0, then

2 3 1 2 3 Pic Sp khT1,T2i/(T1 − T2 ) × B Pic Sp khT1,T2i/(T1 − T2 ) . Remark 3.13 (Concerning the notation). Gerritzen’s notation differs from ours. Our ∗ sheaf G0 is his sheaf G1 whereas he denotes the sheaf OX by G0. The sheaf that we ˇ call OX (< 1) is called HX in Gerritzen’s article. We chose a notation similar to that of van der Put [vdP82]. The following example occured first in Gerritzen [Ger77, Beispiel 1, pp. 36f] or Bartenwerfer [Bar78, p. 1]. Bartenwerfer attributes it to Bosch. Example 3.14 (Bad reduction). B1-invariance can fail for a smooth affinoid variety whose reduction has a cusp singularity. Assume that k is discretely valued and choose π ∈ k with 0 < |π| < 1. Let

2 A = khT1,T2i/(T1 − T2(T2 − π)(T2 − 2π)) be the aﬃnoid algebra that we already examined in Example 1.38e). Its reduction A˜ is an algebraic k˜-variety with a cusp singularity:

˜ ˜ 2 3 A = k[T1,T2]/(T1 − T2 ). Now the ring of integers

◦ ◦ 2 A = k hT1,T2i/(T1 − T2(T2 − π)(T2 − 2π)) is regular. By Gerritzen’s Theorem 2.20,

Pic(A) ∼= Pic(A˜) and Pic(A]hT i) ∼= Pic(AhT i). 3.3. COUNTEREXAMPLES 47

A theorem by Bass and Murthy [BM67, Theorem 8.1] entails that √ Pic(A˜) ∼= Pic(A˜[T ]) ⇔ b = b where b denotes the conductor of the integral closure of A in the quotient ﬁeld Quot A. In our case the conductor is

T 2 b = 1 , T2 clearly not a radical. Hence,

Pic(A˜) Pic(A˜[T ]) and, as A]hT i ∼= A˜[T ], this implies

Pic(A) Pic(AhT i). Remark 3.15. a) The same counterexample works if k is algebraically closed instead of discretely valued [vdP82, Remark 3.24]. b) We see that the choice of analytic reduction matters. If kAk = |k|, then the canonical reduction knows all the vector bundles of a smooth aﬃnoid variety. Let k = C((T )) with |T | = p−1 and p ≥ 5 prime. Take X as in Example 3.14. Its canonical formal model

2 X = C[[T ]]hξ1, ξ2i/ ξ1 − ξ2(ξ2 − T )(ξ2 − 2T ) is aﬃne and contains all information about the vector bundles of X. Similarly, the canonical formal model for Y := X × B1 is 2 Y = C[[T ]]hξ1, ξ2, ξi/ ξ1 − ξ2(ξ2 − T )(ξ2 − 2T ) .

Blowing up three times in (T, ξ1, ξ2) yields a resolution of singularities, giving rise to regular formal schemes X˜, Y˜ which are models for X, respectively ˜ for Y , but no longer affine. Look at the special fibres: Xσ is the preimage of 2 3 C[ξ1, ξ2]/(ξ1 −ξ2 ) under the triple blowup of C[ξ1, ξ2] in the point (ξ1 = 0, ξ2 = 0) ˜ 2 3 and Yσ is the preimage of C[ξ1, ξ2, ξ]/(ξ1 − ξ2 ) under the triple blowup of ˜ C[ξ1, ξ2, ξ] in the divisor (ξ1 = 0, ξ2 = 0). We are in the situation where Xσ is ˜ ˜ 1 regular and Yσ = Xσ × A . They have the same isomorphism classes of line bundles by Theorem 3.25 (Bass–Murthy). Hence, some formal models contain the information about their generic fibres’ vector bundles while other formal models do not. c) For the divisor class group the situation is different. The divisor class group of the reduction of an affinoid variety does not necessarily contain the entire information about the affinoid variety’s divisor class group. Recall from section 2.4: Denneberg showed in [Den69, Korollar 2] that if the reduction X˜ of an affinoid curve X is irreducible and nonsingular, then

Cl(X) ∼= Cl(X˜). 48 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

On the other hand he gave an example of a nonsingular k-aﬃnoid curve X whose reduction has a cusp singularity where the natural map ˜ Cl(X) Cl(X) is an epimorphism, but no isomorphism [Den69, Beispiel 1].

3.4 Local Homotopy Invariance

There are several ways to attack the question of B1-invariance locally. We present two of them in this section. Unfortunately, these local approaches are of no help with proving actual B1-invariance. The first local result is a theorem by Werner Lütkebohmert, stating the following: Let X be a rigid analytic variety and E a vector bundle over X × B1. Then a local trivialisation of E can be chosen to come from X, i. e., it is of product form 1 {Ui × B }i∈I with {Ui}i∈I ∈ Cov(X). 1 We would have liked to prove the corresponding statement for Arig-invariance: Every 1 vector bundle over X × Arig has a local trivialisation coming from X. At least for 1 line bundles and if char k = 0, this is true. It follows from Arig-invariance for line bundles over a smooth affinoid variety, Corollary 3.35 at the end of this chapter. The second local result stems from an attempt to prove Quillen Patching in the affinoid setting. All we could prove, however, was Patching in a tubular neighbourhood of an affinoid variety Sp A embedded as the zero section in Sp(AhT i):

1 1 Sp(A) × {0} −→ Sp(A) × Bε(0) −→ Sp(A) × B . The proof of Patching in a tubular neighbourhood uses some lemmas which are assembled at the end of the section. Theorem 3.16 (L¨utkebohmert). Let X be a rigid analytic variety and E a vector n bundle over X × B . Then there is an admissible covering {Ui}i∈I of X such that n {Ui × B }i∈I provides a local trivialisation of E. This means that for all i, the n restriction E(Ui × B ) is a free OX (Ui)hT1,...,Tni-module.

Proof. This is [Lüt77, §1 Satz 1] or, for rank one, [Ger77, §2 Satz 1]. Theorem 3.17 (Patching in a tubular neighbourhood). Let A be an affinoid algebra and M a finitely presented AhT i-module. Assume that there exist f1, . . . , fs ∈ A with |fi| ≥ 1 for i = 1, . . . , s that generate the unit ideal of A such that for each i the restriction ˆ −1 M⊗AhT iAhfi ihT i −1 is extended from Ahfi i. Then there exists an ∈ k with 0 < || ≤ 1 such that the restriction

−1 M⊗ˆ khT ikh T i is extended from A. 3.4. LOCAL HOMOTOPY INVARIANCE 49

Remark 3.18. This is a version of Quillen Patching [Qui76, Theorem 1]. We are interested in the case where M is the module of global sections of a vector bundle E over Sp A × B1. Theorem 3.17 says that if, locally on A, E comes from a bundle 1 over Sp A, then there exists a tubular neighbourhood Sp A × B|| of the “zero section” 1 1 Sp A × {0} in Sp A × B such that the whole restriction E(Sp A × B||) of E to the tubular neighbourhood comes from a bundle on Sp A. Before coming to the proof we need some lemmas. Lemma 3.19 ([FvdP04, Lemma 4.5.3, p. 88]). Let X = Sp A be an affinoid space. Let {X0,X1} be a covering of X by subsets X0 = {x ∈ X | |f(x)| ≤ 1} and X1 = {x ∈ X | |f(x)| ≥ 1}, for some f ∈ A. There is a constant c > 0 such that every matrix M ∈ GLn(O(X0 ∩ X1)) with kM − 1k < c can be written as a product M0 · M1 with Mi, i = 0, 1 lying in the image of GLn(O(Xi)) −→ GLn(O(X0 ∩ X1)). Here, the matrix norm is defined as k(aij)i,jk := max|aij|. i,j The following lemma is a slight generalisation of a Patching Theorem by Harbater: Lemma 3.20. Let A be a k-affinoid algebra. Let g0, g1 ∈ A such that A = Ag0 +Ag1. Then the following hold: a) The sequence

0 → A[[T ]] −→∆ A [[T ]]× A [[T ]] −→− A [[T ]] → 0 g0 A[[T ]] g1 g0g1 (3.1) f 7−→ (f, f), (f, g) 7−→ f − g

is exact. b) For every ﬁnitely presented A-module N, consider the commutative diagram

Aut(Ng0 ) / Aut(Ng0 ) × Aut(Ng1 ) / Aut(Ng0g1 )

α ˆ ˆ ˆ β ˆ Aut(Ng0 ⊗A[[T ]]) / Aut(Ng0 ⊗A[[T ]]) × Aut(Ng1 ⊗A[[T ]]) / Aut(Ng0g1 ⊗A[[T ]]).

ˆ Every element of Aut(Ng0g1 ⊗A[[T ]]) is the product of an element in the image of α with an element in the image of β.

In particular, let θ ∈ Aut(Mg0g1 ) be an automorphism that reduces modulo T to the ˆ ˆ identity. Then there are φ ∈ Aut(Ng0 ⊗A[[T ]]) and ψ ∈ Aut(Ng1 ⊗A[[T ]]) such that θ = φ−1ψ.

Proof. a) If we remove the T ’s in (3.1), the sequence is exact by Tate’s Acyclicity Theorem. As A[[T ]] is ﬂat over A, we are done.

b) We show that Harbater’s proof for the GLn case works in general. It is done ˆ by explicit construction: Given an element θ ∈ Aut(Ng0g1 ⊗A[[T ]]), we construct −1 ˆ −1 −1 ˆ preimages θ0 ∈ α (Aut(Ng0g1 ⊗A[[T ]])) and (φ , ψ) ∈ β (Aut(Ng0g1 ⊗A[[T ]])) −1 such that θ = α(θ0)β(φ , ψ).

Let θ0 be the reduction of θ modulo T . As θ is invertible, its reduction θ0 has

to be invertible, too. This means that θ0 lies in the image of Aut(Ng0g1 ) in 50 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

−1 Aut(Mg0g1 ). We may as well replace θ by θ0 θ and thus assume that θ0 = 1. × We now have to show that every automorphism θ ∈ (End(Ng0g1 )[[T ]]) that reduces modulo T to the identity can be factorised as θ = φ−1ψ

× × with φ ∈ (End(Ng0 )[[T ]]) and ψ ∈ (End(Ng1 )[[T ]]) .

In the GLn case, Harbater proves [Har93, Lemma 1] that there is a formal −1 factorisation θ = φ ψ with φ ∈ GLn(Ag0 [[T ]]) and ψ ∈ GLn(Ag1 [[T ]]). Like in Harbater’s proof, we construct sequences ˆ (φm)m∈N ⊂ End(Ng0 ⊗ A[T ]) ∩ Aut(Ng0 ⊗A[[T ]]), ˆ (ψm)m∈N ⊂ End(Ng0 ⊗ A[T ]) ∩ Aut(Ng0 ⊗A[[T ]]) such that for all m

m+1 φm+1 ≡ φm, ψm+1 ≡ ψm, φm+1θ ≡ ψm mod T . As N is ﬁnitely presented,

l n A → A N,

we can as well do the construction upstairs on GLn(A), but we need to check independence of choices. The identities will be preserved under the morphism n n A N. So assume for now that N = A and for • ∈ {g0, g1, g0g1, 1} identify End(N•) with

Mn×n(A•) := {n × n matrices with entries in A•}.

Set φ0 = ψ0 = 1. For all m ∈ N, we inductively construct φm and ψm such m+1 that that ψm ≡ 1 mod T and the matrix ηm = φmθ − ψm is divisible by T . Let m ∈ N. By the inductive hypothesis, we already have φm−1 and ψm−1 such that that ψm−1 ≡ 1 mod T and the matrix ηm−1 = φm−1θ − ψm−1 is m m divisible by T . Let wm,i,j be the coeﬃcient of T in the (i, j)-th entry of

ηm−1. Apply Tate’s Acyclicity Theorem to the entries of wm ∈ Mn×n(A)g0g1

and write wm = am + bm with am ∈ Mn×n(A)g1 and bm ∈ Mn×n(A)g0 . Deﬁne m µ ∈ End(Ng0 ⊗ A[T ]) as µ = 1 − bmT and φm := µφm−1 ∈ End(Ng0 ⊗ A[T ]). ˆ Then φm ≡ 1 mod T and so φm ∈ Aut(Ng0 ⊗A[[T ]]). We have

m φmθ = µφm−1θ ≡ φm−1θ ≡ ψm−1 mod T .

m The coeﬃcient of T in the (i, j)-th entry of φmθ−ψm−1 is wm −bm = am ∈ Ag1 . m ˆ We set ψm = ψm−1 + amT ∈ End(Ng1 ⊗A[T ]) and get ψm ≡ 1 mod T , ˆ m+1 ψm ∈ End(Ng1 ⊗A[[T ]]) and ψm ≡ φmθ mod T . This gives us ∞ X i ψ = 1 + aiT i=1 ∞ X i φ = 1 + −biT . i=1 3.4. LOCAL HOMOTOPY INVARIANCE 51

Definition 3.21 (Artin approximation property). Let R be a ring and m ⊂ R a left ideal. The m-adic completion Rˆ of R, Rˆ := lim mn \ R , n∈N is said to have the Artin approximation property with respect to (R, m) if every system of polynomial equations with coefficients in R that has a solution in Rˆ also has a solution in R and if moreover, given any c ∈ N, the solution over R can be chosen to coincide with the solution over Rˆ modulo mc. Lemma 3.22 (Artin approximation for endomorphism rings). Let R be a commutative noetherian ring and m ⊂ R a principal ideal such that the m-adic completion Rˆ of R has the Artin approximation property with respect to (R, m). Let M be a finitely presented R-module. Then m · End(M) is an ideal in End(M) and End(M ⊗ Rˆ) has the Artin approximation property with respect to (End(M), m · End(M)). The proof was suggested by Dorin Popescu.

Proof. The set m End(M) := {aϕ | a ∈ m, ϕ ∈ End(M)} is a left ideal in End(M): Let m = (f), let g, h ∈ m and A, B ∈ End(M). We need to show that gA + hB ∈ m End(M). There exist g0, h0 ∈ R such that g = fg0 and h = fh0. Now gA + hB = f(g0A + h0B) ∈ m End(M). We want to show that End(M ⊗ Rˆ) has the Artin approximation property with respect to (End(M), m End(M)). First case: The module M ∼= Rn is free. Identify n End(R ) with Mn×n(R). Every system Σ of polynomial equations with coefficients in M can be written as a larger system Θ of polynomial equations with coefficients in R by writing down one system of equations for each position (i, j), i, j ∈ {1, . . . , n}. ˆ ˆ If Σ has a solution in Mn×n(R), then Θ has a solution in R. Artin approximation yields a solution to Θ in R and thus a solution to Σ in Mn×n(R). Moreover, for any given c, the solution to Θ in R can be chosen to coincide with the solution in Rˆ c modulo m by Artin approximation for (R, m). Thus, the solution to Σ in Mn×n(R) ˆ c also coincides with the solution in Mn×n(R) modulo m . Second case: The module M is finitely presented, but not necessarily free. Let

ϕ Rn −→ Rm −→ M → 0 ˆ be a presentation of M and deﬁne Mc := M ⊗R R. Every endomorphism f ∈ End(M) can be lifted to the presentation:

ϕ Rn / Rm / M / 0

h g f ϕ Rn / Rm / M / 0

n m Identify End(R ) with Mn×n(R) and End(R ) with Mm×m(R). Denote the matrix associated to h by H ∈ Mn×n(R), the matrix associated to g by G ∈ Mm×m(R) and the matrix associated to ϕ by Φ ∈ Mn×m(R). Then f is encoded in the equation ΦH = GΦ. (3.2) 52 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

Let Σ be a system of polynomial equations in End(M). Choose a lift of Σ to End(Rm). Add to it one equation of the type (3.2) for each summand of every equation in Σ. The result is a system Θ of polynomial equations in End(Rm) which encodes Σ. Proceed as in the ﬁrst case.

Now we can prove Patching in a tubular neighbourhood.

Proof of Theorem 3.17. Put N = M/T M. We have to show that there exists an ◦ ∼ −1 ∈ k \{0} such that M = N⊗ˆ AAh T i. Let us give an outline. The proof is very much like Quillen’s proof of the Patching Theorem [Qui76, Theorem 1], very clearly presented in Lam [Lam06, V, §1]. We show that the set S of functions g with |g| ≥ 1 and

−1 ∼ Mhg i = Nhgi⊗ˆ AhgiAhgihT i contains 1.

The main step is a variant of Cartan’s Lemma: For g0, g1 ∈ S with 1 ∈ Ag0 +Ag1 and −1 ˆ ∼ −1 for both i ∈ {0, 1} let ui : Ahgi ihT i⊗AN = Mhgi i be an isomorphism reducing −1 ˆ −1 −1 modulo T to the canonical isomorphism Ahgi i⊗AN = Mhgi i/T Mhgi i. Then there exists an ∈ k× with || ≤ 1 such that the automorphism

−1 −1 u¯0 u¯1 of Ahg0, g1ih T i⊗ˆ AN

−1 is the identity. Here, u¯i denotes the restriction of ui to Ahg0, g1ih T i⊗ˆ AN. The proof of this main step uses Artin Approximation. Now we start with the proof. Let S be the set of g in A with |g| ≥ 1 such that ∼ −1 Mg = Ng⊗ˆ Ahg−1iAhg ihT i. We try to show that 1 ∈ S. First we show that 1 is in the ideal generated by S in A and then we show that S is something like an ideal. If S generates a proper ideal, it has to be contained in some maximal ideal m. By assumption, there is an f ∈ A \ m with |f| ≥ 1 such that there is an isomorphism

−1 ∼ −1 −1 N⊗ˆ AAhf ihT i = M⊗ˆ AhT iAhf ihT i =: Mhf i which reduces modulo T to the canonical isomorphism

−1 ∼ −1 −1 N⊗ˆ AAhf i = Mhf i/T Mhf i. Hence f ∈ S, contradicting the assumption that f ∈ A \ m ⊂ A \ S. Thus the ideal generated by S in A has to be all of A.

It remains to prove that if g0, g1 ∈ S, then every v ∈ Ag0 + Ag1 with |v| ≥ 1 is also in S. That means, S is almost an ideal, except that it is only closed under multiplication by elements of norm at least 1. Replacing A, M by Av,Mv, we can assume that 1 ∈ Ag0 + Ag1. We have to show that 1 ∈ S, i. e., that M is extended from A. −1 ˆ ∼ −1 For i ∈ {0, 1}, let ui : Ahgi ihT i⊗AN = Mhgi i be an isomorphism reducing −1 ˆ −1 −1 modulo T to the canonical isomorphism Ahgi i⊗AN = Mhgi i/T Mhgi i. We want 3.4. LOCAL HOMOTOPY INVARIANCE 53

∼ to patch them to get a global isomorphism u: AhT i⊗ˆ AN = M reducing modulo T to N = M/T M. Since M is finitely presented, N is finitely presented, too. The map A → Ahg−1ihT i is flat by the following argument: The map A → Ahg−1i is flat by [BGR84, 7.3.2 Corollary 6] and the map Ahg−1i → Ahg−1ihT i is flat because k → khT i is flat. Furthermore, both A and Ahg−1ihT i are commutative. By Lemma 1.51, the map

−1 −1 −1 Ahg ihT i⊗ˆ A HomA(N,L) −→ HomAhg−1ihT i(Ahg ihT i⊗ˆ AN,Ahg ihT i⊗ˆ AL) is an isomorphism for any ﬁnitely generated A-Banach module L. In particular,

−1 ∼ −1 ∼ −1 EndAhg−1ihT i(Ahg ihT i⊗ˆ AN) = Ahg ihT i⊗ˆ A EndA(N) = EndA(N) hg ihT i.

The automorphism

−1 −1 −1 ˆ (u0)g1 (u1)g0 of Ahg0 , g1 ihT i⊗AN (3.3)

−1 −1 × can thus be identiﬁed with an element θ in 1 + T EndA(N) hg0 , g1 ihT i .

We want θ to be 1. That would mean (u0)g1 = (u1)g0 and would allow us to patch u0 and u1 to get a global isomorphism u: AhT i⊗ˆ AN → M. We may replace ui by −1 ˆ −1 another isomorphism of Ahgi ihT i⊗AN → Mhgi i that reduces modulo T to the −1 ˆ −1 −1 canonical isomorphism Ahgi i⊗AN → Mhgi i/T Mhgi i. 1 For all (x0, x1) ∈ X := Sp A × B = Sp AhT i, deﬁne −1 f(x) := g0(x0) · min |g0(x0)| . x0:g1(x0)=0

Then X0 := {x ∈ X | |f(x)| ≥ 1} and X1 := {x ∈ X | |f(x)| ≤ 1} cover X and for −1 i = 0, 1, Xi ⊂ Sp Ahgi i is an aﬃnoid domain. We ﬁx c to be the constant for the covering X0 ∪ X1 = X from Lemma 3.19. −1 −1 n Free case: Assume Nhgi i = Ahgi i for i = 0, 1. Then we also have

−1 × −1 End −1 (Nhgi i)hT i = GLn(Ahg ihT i). Ahgi i i If we had kθ − 1k < c, we could use Lemma 3.19 to get a decomposition. We know that θ(0) = 1 and thus |θ(0) − 1| = 0. Furthermore, we know that kθ − 1k < ∞. As a power series, θ is of the form

∞ X i θ(T ) = 1 + θiT i=1 and for all a ∈ B1(0) we have

∞ X i i i |θ(a) − 1| = θia ≤ max(|θia |) ≤ max(|θi||a| ) ≤ max(|θi||a|) = |a| · kθ − 1k. i≥1 i≥1 i≥1 i=1 54 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

As the valuation on k is non-trivial, there is an ε > 0 in the value group of k such that |θ(a) − 1| < c for all a ∈ Bε(0) ⊂ Ag0g1 . Here Bε(0) means the “closed” ball around 0 of radius ε. For example if (kθ − 1k)−1 is in the value group of k, we could take ε = (kθ − 1k)−1. Choose some ∈ k with || = ε. We deﬁne Y := Sp Ah−1T i, Y0 = X0 ∩ Y and Y1 = X1 ∩ Y . By Lemma 3.19, there are ψi ∈ GLn(O(Yi)) for i = 0, 1 such that

−1 θ = (ψ0)g1 (ψ1 )g0 .

Again we can consider ψi as a power series.We may assume that its zeroeth coeﬃcient −1 pi is 1. Otherwise, pi had to be a unit and we would substitute ψi by pi ψi. We may now replace ui|Yi by ui|Yi ψi to get

u0|Y0∩Y1 = u1|Y0∩Y1 .

These may be patched to give an isomorphism

−1 −1 u: Ah T i⊗ˆ AN → Ah T i⊗ˆ AhT iM.

General case: By Lemma 3.20, our endomorphism × θ ∈ 1 + T EndA(N) hT i g0g1 decomposes into two formal power series

0 0 −1 θ = (ψ0)g1 (ψ1 )g0 .

0 × with ψ ∈ 1 + T EndA(N) [[T ]] for i = 0, 1. i gi We apply the Artin Approximation Lemma 3.22 with

R = Ag0 ⊕ Ag1

−1 (where we identify each Ahgi i with its image in Ag0g1 ) and the system of equations

xy = θ x · (1, 0) = 0 y · (0, 1) = 0 xu = 1 yv = 1.

The system can be solved over Rˆ with

0 x = ψ0 0 −1 y = (ψ1) 0 −1 u = (ψ0) 0 v = ψ1. 3.4. LOCAL HOMOTOPY INVARIANCE 55

By the Artin Approximation Lemma 3.22 there exists a solution over R, i. e., a decomposition

−1 θ = (ψ0)g1 (ψ1 )g0 . (3.4) × where the ψi ∈ 1 + T EndA(N) {T } are convergent power series. For i = 0, 1 gi ◦ let εi be the radius of convergence of ψi. Choose i ∈ k \{0} such that

−1 × ψi ∈ 1 + T EndA(N) h T i , i = 0, 1 gi and set ( if | | ≤ | | and := 0 0 1 1 otherwise.

As in the ﬁrst case, we may replace ui|Yi by ui|Yi ψi to get

u0|Y0∩Y1 = u1|Y0∩Y1 which may be patched to give an isomorphism

−1 −1 u: Ah T i⊗ˆ AN → Ah T i⊗ˆ AhT iM. Remark 3.23. To get a Patching theorem like [Qui76, Theorem 1] we would need a factorisation as in (3.4) where the ψi are strictly convergent power series, i. e.,

−1 × ψi ∈ 1 + T EndA(N) h T i . (3.5) gi

We know that the zeroes, poles and essential singularities of ψ0 and ψ1 have to −1 cancel out on the unit ball because θ = (ψ0)g1 (ψ1 )g0 is strictly convergent. Thus −1 the zeroes and singularities have to be deﬁned both over EndA(N) hg0 i and over −1 EndA(N) hg1 i. Imagine there was a function

−1 × ζ ∈ 1 + T EndA(N) { T } gathering all the zeroes, poles and essential singularities, which means that

−1 −1 × ψiζ ∈ 1 + T EndA(N) h T i

−1 for i = 0, 1. Then we could replace each ψi by ψiζ and get (3.5). Assuming the ψi to be algebraic power series (which is possible by Artin approximation 3.22), we tried to exclude essential singularities and show that we are only left with a finite number of zeroes and poles which are already defined over EndA(N). We failed because we couldn’t get the zeroes and poles under control. In nonarchimedean geometry a convergent power series always diverges on all points of the boundary of its domain of convergence. One example: Let π ∈ k◦◦ and f = T − π ∈ k{T }×. The inverse f −1 diverges on all u ∈ k with |u| ≥ |π|, but the product of f −1 with the polynomial f is convergent. Is it true in general that it is enough to remove a finite number of zeroes and poles in order to remove all of them? 56 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

If M is locally free over AhT i there is a much shorter proof of Theorem 3.17 as Joseph Ayoub pointed out in his referee report for this thesis. It goes as follows.

Ayoub’s proof of Theorem 3.17. Assume that M is locally free over AhT i. Let

M0 = M ⊗AhT i A be the restriction to the zero section Sp(A) ,→ Sp(AhT i). The tensor product is taken via AhT i → A, T 7→ 0 which is a ﬂat morphism of rings. Set

0 M := AhT i ⊗A M0.

Then M and M 0 coincide over Sp(A) ⊂ Sp(AhT i). By ﬂatness of AhT i → A we get

∼ 0 HomA(M0,M0) = HomAhT i(M,M ) ⊗AhT i A.

0 In particular, the identity idM0 ∈ HomA(M0,M0) lifts to morphisms f : M → M and g : M 0 → M. Both of them restrict to the identity over Sp(A) ⊂ Sp(AhT i), Hence, the supports of coker(f) and coker(g), respectively, are Zariski closed subsets of Sp(AhT i) that do not meet the zero section. Thus there exists an F ∈ AhT i which vanishes on the supports of coker(f) and coker(g), respectively, and such that F (0) ∈ A is invertible. We may even assume that F (0) = 1. That is, F is of the form

X i F (T ) = 1 + aiT . i≥1

Evaluating it, F (x) is invertible for all

1 x ∈ A := Sp(AhT i) with |x| < max |ai| =: m. B i

1 In particular F is invertible on the tubular neighbourhood B 1 of Sp(A) in Sp(AhT i). A, 2m 1 Therefore f and g are isomorphisms over B 1 . A, 2m

3.5 The case of line bundles

Recall from Proposition/Deﬁnition 2.19 that the isomorphism classes of line bundles over a rigid analytic variety X form a group Pic(X), the Picard group. We saw in Examples 3.12 and 3.14 that the Picard group of a curve with a cuspidal singularity and the Picard group of a curve of bad reduction fail to be B1-invariant. The ﬁrst part of this section contains theorems about when the Picard group of a space is B1-invariant. This part about B1-invariance contains no new material. In the second part of this section, we prove that the Picard group of a smooth quasicompact or 1 quasi-Stein variety is Arig-invariant. 3.5. THE CASE OF LINE BUNDLES 57

3.5.1 B1-invariance of Pic

Theorem 3.24 (Gerritzen). Let k be a complete discretely valued ﬁeld. Let A be a reduced k-aﬃnoid algebra with regular reduction A˜. Assume kAk = |k|. Then the Picard group is B1-invariant on Sp(A):

? ? Pic(A) ∼= Pic(A˜) ∼= Pic(A]hT i) ∼= Pic(AhT i).

Proof. By Gerritzen’s Theorem 3.24, for each aﬃnoid algebra B with kBk = |k| and whose ring of integers B◦ is regular, there is an isomorphism

Pic(B) ∼= Pic(B˜).

This applies in particular when B˜ is regular and proves the isomorphisms ? (keeping in mind that kAk = kAhT ik). The isomorphism follows from the following theorem by Bass and Murthy. Theorem 3.25 (Bass–Murthy). Let X be a separated reduced scheme over k and assume that it has (at most) normal crossings singularities. Then

∼ 1 Pic(X) = Pic(X × A ).

The original reference is Bass–Murthy [BM67]. There is a very clearly presented proof in Hartl–L¨utkebohmert [HL00, Proposition 2.2 (1)].

If X is aﬃne and onedimensional, there is an explicit decomposition of Pic(X × A1): Theorem 3.26 (Endo [End63, Theorem 4.3], Bass–Murthy [BM67, Proposition 7.12]). Let R be a reduced ring of dimension ≤ 1 and assume that its integral closure R¯ is ﬁnitely generated as an R-module. Let b be the conductor of R in R¯. Then ∼ Pic(R[T ]) = Pic(R) ⊕ G0 where √ G0 = 0 ⇔ b = b, that is, if the conductor is a radical ideal.

B1-invariance or noninvariance is measured by the following decomposition: Theorem 3.27 (Gerritzen [Ger77, §3, Satz 2]). Let X = Sp A be an aﬃnoid variety. Then

1 ∼ 1 Pic(X × B ) = Pic(X) ⊕ H (X, G0) where the sheaf G0 on X is deﬁned as follows. For U ⊂ X admissible, 1 G0(U) := f ∈ OX× 1 (U × B ) = OX (U)hT i B ∞ X i f = 1 + fiT with |fi(u)| < 1 ∀i ≥ 1 ∀u ∈ U . i=1 58 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

Let

1 ι1 : X → X × B x 7→ (x, 1)

∗ be the 1-section. The sheaf G0 on X is the subsheaf of ι1OX×B1 consisting of the normalised invertible elements. Remark 3.28. As before, our notation differs from Gerritzen’s. The sheaf G0 is called G1 in Gerritzen’s article. For k algebraically closed, there are results by van der Put: Theorem 3.29 (van der Put). Let k be a nonarchimedean valued, complete and algebraically closed field. Let A be an affinoid k-algebra.

a) There exists a sheaf G0 on Sp(A) such that

i ∗ ∼ i ∗ i H Sp(AhT i), OSp(AhT i) = H Sp(A), OSp(A) ⊕ H Sp(A), G0 for all i.

In particular,

∼ i Pic(AhT i) = Pic(A) ⇔ H (Sp(A), G0) = 0.

b) If char k˜ = 0, then

i i H (Sp(A), G0) = 0 ⇔ H Sp(A), OSp(A)(< 1) = 0

where

OSp(A)(< 1)(U) := {f ∈ OSp A(U) | |f(x)| < 1 ∀x ∈ U}.

c) If Sp(A) is either a generalised polydisc (a polydisc with ﬁnitely many holes) or monomially convex (for a deﬁnition, see [vdP82, p. 186]), then

i H Sp(A), OSp(A)(< 1) = 0 for i 6= 0.

d) If Sp(A) is a reduced aﬃnoid k-variety of good reduction and k is algebraically closed, then

1 H Sp(A), OSp(A)(1) = 0.

1 e) There exists an aﬃnoid k-algebra A such that H (Sp(A), G0) 6= 0. Concerning the terms in c) that we did not deﬁne, let us just remark that if Sp(A) is a generalised polydisc or monomially convex, it is in particular of good reduction.

Proof. a) This is [vdP82, Proposition 3.32 (3)]. The number s in van der Put’s theorem is the number of holes in the disc D. In our case, s = 0. b) This is [vdP82, Proposition 3.32 (4)]. 3.5. THE CASE OF LINE BUNDLES 59

c) This is [vdP82, Theorem 3.15]. d) This is Bartenwerfer [Bar78, Theorem 2’]. The proof is at the very end of the article. It uses Grauert–Remmert’s ﬁniteness theorem [GR66, §3]. e) Take a variety of bad reduction like in Example 3.14. Van der Put proves that 1 H (Sp(A), G0) 6= 0 [vdP82, Remark 3.24].

A special case is Proposition 3.30 ([vdP82, Corollary 3.29]). Let Sp A be a k-aﬃnoid space such that for all r ∈ R and all constant sheaves G we have i H Sp(A), OSp(A)(< r) = 0 for i 6= 0 and Hi Sp(A),G = 0 for i 6= 0.

Then

Hi Sp(AhT i), O∗ ∼= Hi Sp(A), O∗ for i 6= 0. In particular,

Pic(AhT i) ∼= Pic(A).

There is a generalisation by Hartl and L¨utkebohmert [HL00, Proposition 2.6]. As a special case they show that if X is a smooth rigid analytic variety X that has a strict semistable formal model X , then

Pic0(X) ∼= Pic0(X ). Here k is assumed to be discretely valued and the residue field k˜ is assumed to be separably closed. The variety X does not need to be affinoid. However, their theorem does not immediately imply any statement about B1-invariance. If we had (khT i)× = k×, the question on B1-invariance of the Picard group would be local on X and we could deduce B1-homotopy invariance from this theorem. But as (khT i)× ) k×, we cannot. To summarise, the Picard group of a smooth affinoid variety of semistable reduction is B1-invariant under additional assumptions on the base field and the value group. We have seen examples of a singular variety, a proper variety and a variety of bad reduction whose respective Picard groups are not B1-invariant. We do not know whether the assumptions on the base field and on the value group are necessary or at least can be weakened. Neither do we know anything about B1-invariance of line bundles over a smooth (quasi-)Stein space yet.

1 3.5.2 Arig-invariance of Pic

1 In this section we prove that the Picard group is Arig-invariant on all smooth k-rigid analytic varieties if char k = 0. We state the proposition ﬁrst and prove it at the end of the section after collecting the ingredients that we need in the proof. 60 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

Proposition 3.31. Let k be a complete nonarchimedean, nontrivially valued ﬁeld of characteristic zero and X be a smooth k-rigid analytic variety. Then

∼ 1 Pic(X) = Pic(X × Arig).

The next theorem is the main ingredient in the proof of Proposition 3.31. Let us fix notation. Let X = Sp A be a smooth affinoid k-variety. Let π ∈ k× with |π| < 1. Define r := |π|−1 and

1 j Brj = Sp(khπ T i).

The inductive system

1 1 ··· ,→ X × j ,→ X × j+1 ,→ · · · Br Br j∈N induces a projective system of groups

1 1 · · · → Pic(X × j ) → Pic(X × j ) → · · · , Br +1 Br j∈N

1 called a pro-group and denoted by “lim” Pic(X × Brj ). j∈N Theorem 3.32 (Tamme [Tam15]). Let X = Sp A be a smooth aﬃnoid k-variety × −1 1 and assume char k = 0. Let π ∈ k with |π| < 1. Deﬁne r := |π| and Brj as above. 1 Then the projection X × Arig → X induces an isomorphism of pro-groups

∼ 1 Pic(X) = “lim” Pic(X × Brj ). j∈N

Idea of proof. Van der Put proved that Pic(X × B1) decomposes into Pic(X) and a sheaf G0 (cf. Theorem 3.29). Using the logarithm, Tamme relates these sheaves to Bartenwerfer’s “metric sheaves” OX (s) of functions bounded by s. Bartenwerfer shows [Bar79, Folgerung 3] that there exists a global constant c ∈ k× such that

1 c · H (X, OX (s)) = 0.

The assumption that char k = 0 is used here. Choosing l big enough, Tamme deduces that

1 1 H (X, Gj+l) → H (X, Gj) is the zero map for every j. Remark 3.33. After submission of this thesis Kerz, Saito and Tamme proved this theorem for fields of arbitrary characteristic [KST16]. Therefore our results automatically generalise to arbitrary characteristic, too. Corollary 3.34. Let X = Sp A be a smooth affinoid variety over a field of charac- 1 teristic zero. Then the projection X × Arig → X induces an isomorphism

∼ 1 Pic(X) = Pic(X × Arig). 3.5. THE CASE OF LINE BUNDLES 61

Proof. The isomorphism of pro-systems in Theorem 3.32 induces an isomorphism of their limits. Applying the Serre–Swan Theorem 2.12, we may replace line bundles 1 1 over X, respectively, X × Brj , respectively, X × Arig by invertible modules over A, j 1 respectively over Ahπ T i, respectively over O(X × Arig). We need to show that

j j limj Pic(Ahπ T i) = Pic(limj Ahπ T i).

This means, we have to show that

(1) j lim Pic(Ahπ T i)j = 0.

This holds by Kiehl’s Theorem B 1.32. Corollary 3.35. Let X = Sp A be a smooth k-rigid analytic variety and assume 1 that k is of characteristic zero. Then every line bundle over X × Arig has a local 1 trivialisation of the form {Ui × Arig}i∈I such that {Ui}i∈I is an admissible covering of X.

Proof. Let {Vj}j∈J be an admissible covering of X by aﬃnoid subsets. Let L be a 1 line bundle over X × Arig. As usual, consider the projection

1 pr1 : X × Arig −→ X.

By Corollary 3.34, there exist line bundles Lj over X such that for each j ∈ J, we have

1 ∼ ∗ L(Vj × Arig) = pr1 Lj.

1 For each j, let {Uij }ij ∈Ij be a local trivialisation of Lj. Then {Uij × Arig}ij ∈Ij is 1 S 1 a local trivialisation of L(Vj × Arig). Setting I := j∈J Ij, the set {Ui × Arig}i∈I provides a local trivialisation of L of the desired form. Lemma 3.36. Let A be a reduced aﬃnoid algebra. Then

1 × ∼ × O(Sp(A) × Arig) = A . In particular,

1 × × O(Arig) = k . (3.6)

Proof. In the proof, we consider an invertible formal power series f ∈ A[[T ]]. We compute the condition on the coeﬃcients of the power series f and its formal inverse f −1 for f and f −1 to converge on a tubular neighbourhood around the zero section

1 Sp A × {0} ⊂ Sp(A) × Arig. The condition is that the coefficients’ norms become small as the radius of the tubular neighbourhood becomes big. We conclude that if both f and f −1 are to converge on 1 −1 the whole space Sp(A) × Arig, then all coefficients of f and f except the constant coefficient have to vanish. 62 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES

Let

∞ X i 1 f = fiT ∈ O(Sp(A) × Arig) ⊂ A[[T ]], all fi ∈ A. i=0

As a formal power series, f is invertible if and only if its constant coeﬃcient f0 is P∞ i invertible in A. Assume without loss of generality that f = 1 + i=1 fiT . The formal power series inverse of f is

∞ −1 X ˜ i f = 1 + fiT i=1 where

i ˜ X ˜ fi = − fifi−j. j=1

We want

−1 1 f, f ∈ O(Sp(A) × Arig) which means

f, f −1 ∈ Ahη−lT i (3.7) for some η ∈ k with |η| > 1 and for all l ∈ N. Fix l ∈ N and set r := |η|l. For this l, the relation (3.7) holds if and only if the coeﬃcients satisfy

i kfikr < 1 ∀i ≥ 1. (3.8)

Let us check this claim. As A is reduced, its norm k · k is power-multiplicative. i Assume that there exists an i with kfikr ≥ 1. Choose i minimal with respect to this property. Then

i−1 ˜ i X ˜ i i kfikr = fi − fjfi−j r = kfikr ≥ 1

j=1 because for 1 ≤ j ≤ i − 1 we have

˜ i ˜ i i ˜ i i kfjfi−jkr ≤ kfjkkfi−jkr ≤ kfjkr · kfi−jkr < 1 ≤ kfikr . r≥1 ∗

The inequality marked with an asterisk ∗ holds because i was chosen such that j kfjkr < 1 for all j < i. Similarly, we get

2i ˜ 2i X ˜ 2i ˜ 2i i 2 kf2ikr = fjf2i−j r = kfifikr = (kfikr ) ≥ 1

j=1 3.5. THE CASE OF LINE BUNDLES 63 and for all n ∈ N ˜ ni kfnikr ≥ 1.

Thus, f −1 ∈/ Ahη−lT i, which contradicts the assumptions. We have proven that (3.7) and (3.8) are equivalent for a ﬁxed l. To get back to our problem, we have

−l × f ∈ Ahη T i ∀l ∈ N which is equivalent to 1 1 kf k < = ∀i ≥ 1 ∀l ∈ . i ri |η|il N

As |η| > 1, this means that

( × 1 × f0 ∈ A and f ∈ O Sp(A) × Arig ⇐⇒ fi = 0 for i ≥ 1.

Proof of Proposition 3.31. Let X be a smooth k-rigid analytic variety. Let L be a 1 line bundle over X × Arig. By Corollary 3.35, the bundle L has a local trivialisation 1 of the form {Ui × Arig}i∈I such that {Ui}i∈I is an admissible covering of X. For 1 i, j ∈ I and Ui ∩ Uj 6= ∅, the transition function on (Ui ∩ Uj) × Arig is given by an 1 element f ∈ O 1 ((Ui ∩ Uj) × ). By Lemma 3.36, this function is in fact an X×Arig Arig element of OX (Ui ∩ Uj).

1 We assert that in characteristic zero, Question 3.8 about Arig-invariance is answered for line bundles by Proposition 3.31 and Example 3.12 (an aﬃnoid curve with a cusp). 64 CHAPTER 3. HOMOTOPY INVARIANCE OF VECTOR BUNDLES Chapter 4

Homotopy Theory for Rigid Varieties

Our goal is a classification of vector bundles over rigid analytic varieties in terms of abstract homotopy theory. In the current chapter we first recall simplicial model categories and Bousfield localisation. Then we define the categories that we want to use and relate them to one another.

4.1 Model categories and homotopy theory

Abstract homotopy theory was born with Quillen’s book “Homotopical Algebra” [Qui67] in 1967. Quillen deﬁnes the notion of a model category, formalising the features from classical homotopy theory that are actually suﬃcient to get many of the classical theorems. His way to abstractify homotopy theory in order to make its tools accessible to other areas of mathematics has proved very fruitful since then. Simplicial model categories are a particularly powerful variant. One example is Morel– Voevodsky’s A1-homotopy theory of schemes. It is the prototype for our homotopy theories of rigid analytic varieties. Nowadays there are also other approaches to abstract homotopy theory, for example Lurie’s ∞-categories.

We give a short introduction to model categories and simplicial model categories after recommending our favourite books. A very accessible introduction is Dwyer–Spalinski [DS95]. Hirschhorn’s book [Hir03] is very comprehensive and clear and contains all the proofs. For simplicial model categories, we recommend “Simplicial Homotopy Theory” by Goerss and Jardine [GJ09]. Definition 4.1 (model category [Hir03, 7.1, 13.11]). A model category is a category C together with three distinguished classes of morphisms, called cofibrations, fibrations and weak equivalences, such that the following axioms hold:

M1: (Limits and colimits) The category C contains all small limits and all small colimits.

65 66 CHAPTER 4. HOMOTOPY THEORY FOR RIGID VARIETIES

M2: (Two out of three) Let f, g ∈ Mor(C) be composable morphisms:

f g A −→ B −→ C. If two of the morphisms f, g, gf are weak equivalences, then so is the third. M3: (Retracts) Let f, g ∈ Mor(C) such that f is a retract of g [Hir03, Definition 7.1.1]. If g is a cofibration, fibration or weak equivalence, then so is f. M4: (Lifting) Let i: A → B be a cofibration and p: X → Y be a fibration such that we get a commutative diagram:

A / X

i p B / Y. If i or p is also a weak equivalence, then there exists a lift

A / X > i p B / Y.

M5: (Factorisation) Every morphism f ∈ Mor(C) can be factorised functorially in two ways: a) f = qi where i is a cofibration and q both a fibration and a weak equivalence, and b) f = pj where j is both a cofibration and a weak equivalence and p is a fibration. A morphism which is both a cofibration and a weak equivalence is called a trivial cofibration (also: acyclic cofibration). A morphism which is both a fibration and a weak equivalence is called a trivial fibration (also: acylic fibration). A model category C is proper if every pushout of a weak equivalence along a cofibration is a weak equivalence and every pullback of a weak equivalence along a fibration is a weak equivalence. Remark 4.2. Note that Quillen’s original definition of a closed model category is a bit weaker than this. Goerss and Jardine [GJ09] work with Quillen’s definition. By axiom M1, a model category is in particular closed under pullbacks, pushouts, inductive limits and projective limits and contains an initial object and a terminal object. Definition 4.3 ((co)fibrant). We denote the initial object of a model category by ∅ and the terminal object by ∗. An object X ∈ Ob(C) is cofibrant if the morphism ∅ −→ X is a cofibration. It is called fibrant if the morphism X −→ ∗ is a fibration. 4.1. MODEL CATEGORIES AND HOMOTOPY THEORY 67

Remark 4.4 ((co)fibrant replacement). By the factorisation axiom M5, there is a cofibrant replacement functor replacing each object X by a weakly equivalent cofibrant object X˜. It is given by factorising the map

∅ −→ X into a coﬁbration followed by a trivial ﬁbration

∅ −→cof X˜ we−→ ﬁb X.

Similarly, there is a fibrant replacement functor replacing each object by a weakly equivalent fibrant object. Remark 4.5 (LLP, RLP). The model structure is already determined by the class of weak equivalences together with either the class of cofibrations or the class of fibrations. Assume we are given the weak equivalences and the cofibrations. Then the fibrations are exactly those maps p that have the right lifting property (RLP) with respect to all trivial cofibrations, i. e., for every solid arrow commutative diagram

A / X > we cof p B / Y the dotted arrow exists. The trivial fibrations are exactly those maps that have the RLP with respect to all cofibrations. Similarly, the cofibrations are exactly those maps i that have the left lifting property (LLP) with respect to all trivial fibrations and the trivial cofibrations are exactly the maps that have the LLP with respect to all fibrations. One direction is provided by the lifting axiom M4. The other direction uses factorisation and retracts. See Hirschhorn [Hir03, Proposition 7.2.3] for a proof. Definition 4.6 (homotopy category [GJ09, p. 81]). Let C be a model category. Its homotopy category is the category theoretic localisation of C at the class W of weak equivalences, i. e., a category Ho C together with a functor

γ : C −→ Ho C such that a) if f is a weak equivalence, then γ(f) is an isomorphism, and b) each functor ϕ: C −→ D to another category D that maps all weak equivalences to isomorphisms factors uniquely via γ. We will suppress γ in the notation. The homotopy category of a model category exists by [GJ09, II, Theorem 1.11]. It is comparatively easy to put a model structure on the category of simplicial sets S which is rather explicit and suitable for computations [GJ09, chapter I]. The idea of a simplicial model category is to exploit this feature as much as possible. 68 CHAPTER 4. HOMOTOPY THEORY FOR RIGID VARIETIES

Deﬁnition 4.7 (simplicial model category [GJ09, II.2, II.3]). A simplicial model category is a model category C with a mapping space functor

Map : Cop × C −→ S fulﬁlling

a) Map(X,Y )0 = Hom(X,Y ) for X,Y ∈ Ob(C), b) for each X ∈ Ob(C), the functor Map(X, ·): C → S has a left adjoint, c) for each Y ∈ Ob(C), the functor Map(·,Y ): Cop → S has a left adjoint, d) axiom SM7: For each coﬁbration j : A → B and each ﬁbration q : X → Y , the induced morphism of simplicial sets

∗ (j ,q∗) Map(B,X) −→ Map(A, X) ×Map(A,Y ) Map(B,Y ) (4.1)

is a fibration in S which is trivial if j or q is trivial. More about the mapping space functor and about axiom SM7 can be found in Goerss–Jardine [GJ09, II.2, II.3]. Definition 4.8 (homotopy cartesian square, homotopy pullback, homotopy fibre, [GJ09, II.9]). Let C be a proper simplicial model category. a) A commutative square

X / Y

f W / Z in C is homotopy cartesian if for any factorisation f = pi into a weak equivalence i and a ﬁbration p, the map i∗ deﬁned by

X / Y

i∗ $ i ˜ ˜ W ×Z Y / Y f

p z W / Z is a weak equivalence. We call X “the” homotopy pullback of the diagram

f W −→ Z ←− Y,

although it is only unique up to weak equivalence. b) The homotopy ﬁbre of a morphism f : Y → Z is the homotopy pullback of the diagram

f ∗ −→ Z ←− Y. 4.1. MODEL CATEGORIES AND HOMOTOPY THEORY 69

Definition 4.9 (left Bousfield localisation, [Hir03, 3.1, 3.3]). Let C be a model category and S ⊂ Mor(C) a class of morphisms. a) An S-local object X ∈ Ob(C) is a fibrant object such that for every morphism

f : A → B ∈ S,

the induced map of mapping spaces

Map(B,X) −→ Map(A, X)

is a weak equivalence of simplicial sets. b) An S-local equivalence is a morphism

g : A → B ∈ Mor(C)

such that for each S-local object X the induced map of mapping spaces

Map(B,X) −→ Map(A, X)

is a weak equivalence of simplicial sets. c) Assume that there exists a model structure on the underlying category of C where i) the weak equivalences are the S-local weak equivalences, ii) the cofibrations are the cofibrations from C and iii) the fibrations are defined via the RLP. Then the resulting model category is called the left Bousfield localisation of C with respect to S. The Bousfield localisation of a model category does not necessarily exist. If it exists, Bousfield localisation of a simplicial model category yields again a simplicial model category [Hir03, 4.1.1]. A lot of research has been going into the existence of localisation in various situations. We can only mention some of it. First of all, there is the foundational work by Hirschhorn [Hir03], investigating both left and right Bousfield localisation. Hirschhorn proved that left Bousfield localisation of a left proper cellular model category at an arbitrary set of morphisms exists. Dugger [Dug01] has a very elegant and general approach to localisation, a generators-and-relations construction via a so-called universal homotopy category. Much more specifically, Morel and Voevodsky [MV99] show the existence of the homotopy category of a site with interval. They show that left Bousfield localisation of a simplicial model category of sheaves on a site at the morphism mapping the interval to the final element exists. Several mathematicians obtained very general results about the localisation of model structures on topoi and presheaf categories. Let us list a few of them. Goerss and Jardine [GJ98] show that Jardine’s model 70 CHAPTER 4. HOMOTOPY THEORY FOR RIGID VARIETIES category of simplicial presheaves on a small Grothendieck site can be localised at any cofibration of simplicial presheaves. Blander [Bla01] shows that the local projective model structure on an essentially small site is proper and cellular and hence its left Bousfield localisation at any set of morphisms exists. His results were generalised further by Toënand Vezzosi [TV05]. Cisinski [Cis02] shows that any model structure on a topos whose cofibrations are the monomorphisms may be localised at any set of morphisms.

4.2 Sites and completely decomposable structures

We give a short account on Grothendieck topologies and completely decomposable structures as they will be important in the sequel. In particular, we introduce several Grothendieck topologies on rigid analytic varieties. Deﬁnition 4.10 (site). A site (C, T ) is a category C equipped with a Grothendieck topology T . For the deﬁnition of Grothendieck topology, we refer to Mac Lane– Moerdijk [MLM94, chapter 3]. Examples 4.11. a) Let X be a rigid analytic variety with G-topology T . The category X has as objects the admissible open subsets of X and inclusions as morphisms. With admissible coverings as covering sieves, it becomes a site, also denoted by X.

b) Let X = Sp A be an aﬃnoid variety. The site XZar has as objects Zariski open subsets of X, as morphisms inclusions and as covering sieves the coverings by Zariski open subsets.

c) Let X be a separated rigid analytic variety. The ´etalesite Xet consists of the following. • The objects are ´etalemorphisms Y → X from separated rigid analytic varieties Y to X. • The morphisms are commutative triangles

Y / X >

• The covering sieves are families of ´etalemorphisms whose images form admissible coverings.

d) The “big sites” are deﬁned analogously: The big G site smRig G is deﬁned as follows. The underlying category is the category smRig of smooth rigid analytic varieties. Covering sieves are given by families of open immersions whose images form admissible coverings.

e) The big ´etalesite smRig et is deﬁned by the following data. The underlying category is the category smRig of smooth rigid analytic varieties. Covering 4.3. HOMOTOPY THEORIES FOR RIGID ANALYTIC VARIETIES 71

sieves are given by families of étalemorphisms whose images form admissible coverings. Some Grothendieck topologies are particularly easy to describe. They are called completely decomposable topologies. Definition 4.12 (cd structure, [Voe10, Definition 2.1]). Let C be a category with an initial object ∅.A cd structure on C is a collection P of commutative squares

B / Y (4.2)

p A e / X

such that if Q ∈ P and Q0 is isomorphic to Q, then Q0 ∈ P . The Grothendieck topology tP generated by the cd structure P is the coarsest topology on C such that • the empty sieve covers the initial object ∅ and • for every Q ∈ P as in (4.2), the sieve generated by the morphisms p and e is a covering sieve. A Grothendieck topology which is generated by a cd structure is called completely decomposable. Examples 4.13. Consider the same examples as above.

b) The Grothendieck topology of the Zariski site XZar of an aﬃnoid variety X is generated by squares

U ∩ V / V

∩ U ⊂ / X

where U → X and V → X are the inclusions of Zariski open subsets of X that jointly cover X. a), c) The Grothendieck topologies of X and Xet are usually not completely decomposable. The reason is that there may be inﬁnite covering sieves S such that no ﬁnite subcollection S0 ⊂ S covers X. The Grothendieck topology of X is completely decomposable if and only if X is quasicompact.

4.3 Homotopy theories for rigid analytic varieties

The first to do motivic homotopy theory for rigid analytic varieties was Joseph Ayoub. Ayoub followed the most natural approach for such a homotopy theory [Ayo15]: His homotopy category of rigid analytic varieties is Morel–Voevodsky’s homotopy category of a site with interval [MV99] where the site is the category of rigid analytic varieties equipped with the Grothendieck topology coming from the G-topology (Example 4.11d) and with the unit ball B1 as an interval object. Ayoub used stable versions of 72 CHAPTER 4. HOMOTOPY THEORY FOR RIGID VARIETIES this theory to construct motives for rigid analytic varieties [Ayo15]. Further results in this line were obtained by Alberto Vezzani [Vez14a, Vez14b, Vez15]. We will use a more flexible definition allowing three parameters: • A subcategory R of the category of rigid analytic varieties, • a Grothendieck topology T on R, usually the one coming from the G-topology or the Nisnevich topology, and

1 1 • an interval object I, usually I = B or I = Arig. For now, we ﬁx none of these parameters. Deﬁnition 4.14 (representable interval object, [AHW15a, 4.1.1]). A (representable) interval object on a small category C is a quadruple (I, m, ι0, ι1) consisting of a representable presheaf of (simplicial) sets I on C, a morphism (“multiplication”) m: I × I → I and two morphisms (“endpoints”) ι0, ι1 : ∗ → I such that the following hold:

a) For every X ∈ Ob(C), the presheaf HomC( · ,X) × I is representable. b) Let p: I → ∗. Then the following morphisms I → I coincide:

m(ι0 × id) = m(id ×ι0) = ι0p and

m(ι1 × id) = m(id ×ι1) = id .

` ` c) The morphism ι0 ι1 : ∗ ∗ → I is a monomorphism. We will sometimes drop the word “representable” and we will suppress the morphisms m, ι0, ι1 in the notation. Proposition 4.15. Let C be a small category equivalent to smRig. Let R ⊂ C be a subcategory. Let sPSh(R) a small category equivalent to the category of simplicial presheaves on R. Let I be a representable interval object and T a Grothendieck topology on R. Then there are simplicial model structures on sPSh(R) as follows. a) The injective model structure M(R) where • the cofibrations are the pointwise monomorphisms, • the weak equivalences are sectionwise weak equivalences, i. e., two simplicial presheaves F, G ∈ sPSh(R) are weakly equivalent if and only if F(X) 'G(X) in S for all X ∈ Ob(R), and • the fibrations are defined via the right lifting property.

b) The model structure MI,T (R) obtained by left Bousﬁeld localisation of M(R) at the set {I × F → F | F ∈ sPSh(R)} ∪ Cov(T ) where ∗ denotes the terminal object and where Cov(T ) is the set of all covering sieves of T . 4.3. HOMOTOPY THEORIES FOR RIGID ANALYTIC VARIETIES 73

Proof. The existence of these model categories was shown by Cisinski [Cis02].

Deﬁnition 4.16 (HI,T (R)). The homotopy category of MI,T (R) is denoted by HI,T (R). If the Grothendieck topology T is generated by a cd structure τ (see Deﬁnition 4.12), we also denote the resulting categories by MI,τ (R), respectively by HI,τ (R).

We view the category R ⊂ smRig as a subcategory of MI,T (R) via the Yoneda embedding. Deﬁnition 4.17. Let R be a small category that has an initial object ∅ as in Proposition 4.15. Let I be a representable interval object, τ a cd structure on sPSh(R), let MI,τ (R) be the model category constructed in Proposition 4.15 and HI,τ (R) the resulting homotopy category. Let E, F ∈ Ob(sPSh(R)). The set of homotopy classes from E to F is

[E, F] := Map (E, F) := Hom (E,RF)/homotopy I,τ HI,τ (R) MI,τ (R) where R is a fibrant replacement functor. In the model category MI,τ (R), all objects are cofibrant, so we do not need to replace E cofibrantly. Lemma 4.18. As in Proposition 4.15, let R be a small category equivalent to a 1 1 subcategory of smRig such that B and Arig ∈ sPSh(R) are representable interval objects. Let the Grothendieck topology T be equal to or finer than the Grothendieck 1 topology coming from the G-topology. Then the rigid analytic affine line Arig is contractible in MB1,T (R).

Proof. We may assume without loss of generality that T = G. If T 6= G, then

MB1,T (R) is obtained by further localisation of MB1,G(R). Localisation preserves 1 weak equivalences, thus if Arig is contractible in MB1,G(R), it is still so in MB1,T (R). Let c ∈ k with |c| > 1 and set

1 −1 −2 Arig = colim Sp(khξi) ,→ Sp(khc ξi) ,→ Sp(khc ξi) ,→ ... as in Example 1.24. The colimit of simplicial sheaves coincides with the simplicial 1 sheaf represented by the rigid analytic variety Arig. 1 We need to show that for each rigid analytic variety X, the map X × Arig → X is a weak equivalence in MB1 (R), i. e. that the projection 1 X × Arig → X is a B1-local equivalence in M(R). That is, if Y is any B1-local object, the induced map of mapping spaces

1 Map(X,Y ) → Map(X × Arig,Y ) is a weak equivalence of simplicial sets.

Let Y be a B1-local object. The monomorphisms Sp(khc−iξi) ,→ Sp(khc−(i+1)ξi) are coﬁbrations in M(R). As Y is ﬁbrant in M(R), the maps

Map(X × Sp khc−(i+1)ξi,Y ) → Map(X × Sp khc−iξi,Y ) 74 CHAPTER 4. HOMOTOPY THEORY FOR RIGID VARIETIES are ﬁbrations of simplicial sets by axiom SM7 [GJ09, Chapter II, Proposition 3.2]. The isomorphisms

khc−iξi → khξi ξ 7→ ciξ a 7→ a for a ∈ k give rise to isomorphisms

Sp khξi → Sp khc−iξi which in particular are weak equivalences. As the morphism

Sp khξi → ∗ is a weak equivalence in MB1 (R) by deﬁnition, the two-out-of-three axiom gives that Sp khc−iξi → ∗ is a weak equivalence in MB1 (R). Thus we get weak equivalences of simplicial sets −i fi : Map(X,Y ) → Map(X × Sp khc ξi,Y ) for all i. Similarly, the inclusions

khc−(i+1)ξi → khc−iξi ξ 7→ ξ a 7→ a for a ∈ k give rise to coﬁbrations

Sp(khc−iξi) → Sp(khc−(i+1)ξi) in MB1 (R) and the maps −(j+1) −j jn : Map(X × Sp khc ξi,Y ) → Map(X × Sp khc ξi,Y ) are ﬁbrations in S. We put this together to get a sequence of maps

Map(X,Y )

f2 f0 f t 1 ... r / Map(X × Sp khc−1ξi,Y ) / Map(X × Sp khξi,Y ). j0

−i The spaces X × Sp khc ξi are cofibrant and Y is fibrant in MB1 (R), hence all appearing mapping spaces are fibrant in S by [GJ09, Chapter II, Proposition 3.2]. All horizontal maps jn are fibrations and all vertical maps fn for n ≥ 0 are weak equivalences by Axiom SM7 [GJ09, Proposition 3.2]. By [Hir03, Proposition 15.10.12(2)], the limit map

1 lim fn : Map(X,Y ) → Map(X × Arig,Y ) 4.3. HOMOTOPY THEORIES FOR RIGID ANALYTIC VARIETIES 75 is a weak equivalence of simplicial sets. Thus, the map

1 X × Arig → X 1 1 is a B -local equivalence and X and X ×Arig are isomorphic in the homotopy category HB1 (R).

Corollary 4.19. For each R as above, the model category MB1 (R) can be obtained 1 by left Bousﬁeld localisation of M 1 (R) at the set { × F → F | F ∈ sPSh(R)}. Arig B

We say that the homotopy theory defined by the model structure MB1 (R) is finer than the homotopy theory defined by the model structure M 1 (R). Arig

4.3.1 Relation to Ayoub’s theory

To be able to compare these categories to Ayoub’s B1-categories, let us first review Ayoub’s construction from [Ayo15, 1.3]. Ayoub starts with the category of presheaves from smRig into a so-called coefficient category M. A coefficient category is a stable model category with nice properties, cf. [Ayo07, Définition4.4.23] or [Ayo15, Définition1.2.31]. Examples include the category of complexes of abelian groups or categories of simplicial spectra, endowed with the stable projective model strucure [Ayo07, Exemple 4.4.24]. Denote the category of presheaves from smRig into the coefficient category M by PSh(smRig, M). Ayoub endows it with the projective model structure where a morphism of presheaves F → G is • a weak equivalence if and only if for each X ∈ Ob(smRig), it induces a weak equivalence F(X) 'G(X) in M, • a fibration if and only if for each X ∈ Ob(smRig), it induces a fibration F(X) 'G(X) in M and • a cofibration if and only if it satisfies the LLP with respect to all trivial fibrations. Now Ayoub localises this model structure with respect to the Nisnevich topology and the unit ball B1. The homotopy category of the resulting stable model category is called the unstable B1-homotopy category [Ayo15, 1.3.1]. There are several differences to our constructions: We start with a subcategory R of smRig and presheaves from R to simplicial sets. The category of simplicial sets is not a coefficient category in Ayoub’s sense. We first endow the resulting category sPSh(R) with the injective model structure and then localise with respect to an interval object I and a Grothendieck topology T . • Let us stabilise the simplicial model category which we get with R = smRig, I = B1 and T = Nis in the simplicial direction. • Let us choose the category of S1-spectra as a coefficient category in Ayoub’s construction. 76 CHAPTER 4. HOMOTOPY THEORY FOR RIGID VARIETIES

This gives two diﬀerent model structures on PSh(smRig, M) which have the same homotopy category.

Ayoub also has a stable B1-homotopy category [Ayo15, 1.3.3]: As before, he starts with PSh(smRig, M), additionally requiring M to be symmetric monoidal and unital. Endowing PSh(smRig, M) with the projective model structure as above yields a stable model category which is again symmetric monoidal and unital. Localising it with respect to the Nisnevich topology and the unit ball B1 and stabilising with 1 1 respect to Prig yields the stable B -homotopy category. It is still further away from our model categories.

4.3.2 A trip to the zoo

Let us take a trip to the zoo of homotopy theories for rigid analytic varieties. Let us observe the relations between them. The following subcategories of smRig are of interest for us: The category Afnd of smooth aﬃnoids, its subcategory stabAfnd of smooth aﬃnoids of semistable reduction, the category qStein of smooth quasi-Stein varieties and the whole category smRig. These categories are related as follows.

stabAfnd ⊂ Afnd ⊂ qStein ⊂ smRig.

The occurring Grothendieck topologies are • the Grothendieck topology induced by the G-topology, denoted by G, • the completely decomposable Grothendieck topology where only finite coverings with respect to the G-topology are allowed, denoted by fG, and • the Nisnevich topology, denoted by Nis. The Grothendieck topology induced by the G-topology is finer than its completely decomposable variant. They coincide on quasicompact rigid analytic varieties. The Nisnevich topology is finer than the completely decomposable Grothendieck topology where only finite coverings with respect to the G-topology are allowed. For R ∈ {smRig, qStein} we have the following relations between the homotopy categories:

H 1 (R) o H 1 (R) Arig,G loc Arig,fG

loc H 1 (R) o H 1 (R). B ,G loc B ,fG

Arrows labelled “loc” denote localisation of the model structure, i. e., adding isomorphisms in the homotopy category. 4.3. HOMOTOPY THEORIES FOR RIGID ANALYTIC VARIETIES 77

Let R ∈ {Afnd, stabAfnd}. Denote Ayoub’s unstable B1-homotopy category obtained 1 with the category of S -spectra as coeﬃcient category by HB1,S1 and stabilisation in the simplicial direction by an arrow labelled “stab”. Then we have the relations

HB1,G(R) = HB1,fG(R)

loc HB1,Nis(R)

stab HB1,S1 (R).

The homotopy categories H 1 (qStein) and H 1 (stabAfnd) will turn up in the Arig,fG B ,fG next chapter. 78 CHAPTER 4. HOMOTOPY THEORY FOR RIGID VARIETIES Chapter 5

Classiﬁcation of Vector Bundles

This chapter contains the main theorem of this thesis: Isomorphism classes of line bundles over a smooth and either quasicompact or (quasi-)Stein rigid analytic variety 1 over a field of characteristic zero are represented in the Arig-homotopy category. At the beginning of the chapter we review the known theorems that served as models for our theorem and its proof: Steenrod’s homotopy classification of continuous vector bundles over paracompact Hausdorff spaces, Grauert’s homotopy classification of holomorphic vector bundles over complex Stein spaces and Fabien Morel’s A1- homotopy classification of algebraic vector bundles over smooth affine varieties. Grauert’s proof builds on his generalisation of the Oka principle, an instance of what was later called an h-principle by Gromov. As a variant of an h-principle also occurs in Morel’s proof of the algebraic theorem, we give a short account on h-principles. A section on classifying spaces clarifies our use of the term. Then we finally come to our main theorem and its proof. The geometric part of the proof was done in chapter 3. The homotopy theoretic part is done here. We conclude the chapter by a discussion, collecting what we know now and which questions remain open.

5.1 Classiﬁcation of vector bundles: The classical results

The existence of a classifying space for vector bundles is a fundamental result in the studies of topological vector bundles and K-theory: Theorem 5.1 (Steenrod). Let X be a paracompact Hausdorff space. There is a one-to-one correspondence 1:1 ∼ [X, Grn,R] ←→ {real vector bundles of rank n over X}/ = between homotopy classes of maps from X into the infinite Grassmannian and isomorphism classes of real vector bundles over X. Definition 5.2 (infinite Grassmannian). Let K be a field and for n, m ∈ N, n ≤ m, m Grn,m,K := {n-dimensional subvectorspaces of K }

79 80 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES the ﬁnite Grassmannians. For m < m0, the inclusion

Km −→∼ Km ⊕ 0 ⊕ · · · ⊕ 0 ⊂ Km0 induces maps

Grn,m,K → Grn,m0,K . As a set, the inﬁnite Grassmannian is the colimit over all those maps: [ Grn,K := Grn,m,K . m≥n

Remark 5.3 (Versions of the Grassmannian). a) Topological: If K is R or C, then the topology given by the archimedean norm induces a topology on each of the Grn,m and the ﬁnal topology on Grn.

b) Algebraic: As an algebraic variety, the finite Grassmannian Grn,m,K,alg is defined as a quotient of GLm,K by the parabolic subgroup that stabilises the flag 0 ⊂ Kn ⊂ Km. As such, it is smooth and projective [Hum75, 1.8]. The 0 maps Grn,m,alg → Grn,m0,alg for m < m are also algebraic. The colimit Grn,alg is then called an ind-variety. If K is algebraically closed, the underlying set of points of Grn,m,K,alg coincides with Grn,m,K as defined in 5.2.

c) Differentiable: By the same argument, Grn,m,R and Grn,m,C admit a structure of differentiable manifold [BtD95, 4.12]. The maps Grn,m,K → Grn,m0,K are differentiable and this way, the Grn,K become ind-manifolds for K = R, C.

d) Complex analytic: Similarly, Grn,m,C admits a structure of complex manifold, cf. [BtD95, 4.12]. The maps Grn,m,C → Grn,m0,C are analytic and this way, Grn,C becomes a complex ind-manifold.

e) Rigid analytic: If K = k is nonarchimedean, the same principle applies: Grn,m,k admits a structure of rigid analytic variety. It is the analytification of the algebraic structure. The resulting ind-rigid analytic variety is denoted by Grn,rig, by Grn,k,rig or simply by Grn. Again, if k is algebraically closed, the underlying set of points of Grn,m,k,rig coincides with Grn,m,k as defined in 5.2. Theorem 5.1 is proven as follows. There is a universal vector bundle E over the Grassmannian such that every vector bundle of rank n over a topological space X is the pullback of the universal bundle along some map X → Grn,R. The fabulous feature is that if X is paracompact and Hausdorff, then any two homotopic maps f ∗ ∗ X ⇒ Grn give rise to isomorphic bundles f E, g E. g Similar results hold for holomorphic vector bundles over complex Stein spaces and for algebraic vector bundles over smooth affine varieties. Definition 5.4 (Complex Stein space [Ste51], [For11, Definition 2.2.1]). A complex space X of dimension n is Stein (also: holomorphically complete) if a) it is holomorphically separable: For all x 6= y ∈ X there exists a function f ∈ O(X) with f(x) 6= f(y). 5.1. THE CLASSICAL RESULTS 81

b) For any x ∈ X there exist functions f1, . . . , fn ∈ O(X) whose diﬀerentials d fj at x are complex-linearly independent. c) X is holomorphically convex, i.e. for each compact set K ⊂ X, its O(X)-hull

{p ∈ X | |f(p)| ≤ max |f(x) ∀f ∈ O(X)} x∈K

is convex. This class of complex spaces was introduced by Karl Stein in 1951. In short, a complex Stein space is a complex space that has many global functions. Complex Stein spaces satisfy Cartan’s theorems A and B. They are the model for Kiehl’s deﬁnition of nonarchimedean Stein spaces. We introduced nonarchimedean Stein spaces in Deﬁnition 1.30. For further reading about complex Stein spaces, we refer to Grauert–Remmert [GR04] and Forstneriˇc[For11]. Theorem 5.5 (Grauert [Gra57b]). Let X be a complex Stein space. Then there is a bijection

∼ ∼ [X, Grn,C] −→ {holomorphic complex vector bundles of rank n over X}/ =

between homotopy classes of holomorphic maps X → Grn,C and isomorphism classes of holomorphic vector bundles on X. The main ingredient in the proof is Grauert’s Oka principle. Theorem 5.6 (Grauert’s Oka principle, [For11, Theorem 5.3.1]). Let X be a complex Stein space. a) Every continuous complex vector bundle over X is continuously isomorphic to a holomorphic vector bundle. b) Two holomorphic vector bundles over X that are continuously isomorphic are also holomorphically isomorphic. Grauert and Remmert describe the philosophy behind the Oka principle as follows: “On a reduced Stein space X, problems which can be cohomologically formulated have only topological obstructions” [GR04, p. 145]. The corresponding result for algebraic varieties is the following theorem by Morel. Here, an algebraic vector bundle of rank n over an algebraic variety X is defined as a locally free OX -module of rank n. Theorem 5.7 (Morel). Let X be a smooth affine variety over a field k. Then there is a natural bijection

∼ ∼ [X, Grn,k]A1 −→ {algebraic k-vector bundles of rank n over X}/ =

1 between A -homotopy classes from X to the infinite Grassmannian Grn (as an ind- variety) and isomorphism classes of algebraic vector bundles on X. Remark 5.8. Fabien Morel proved the theorem in the case that the field k is infinite and perfect and n 6= 2 in [Mor12]. The theorem was proven in general only recently by Aravind Asok, Marc Hoyois and Matthias Wendt [AHW15a]. 82 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

Homotopy invariance is given by Lindel’s solution to the geometric case of the Bass– Quillen conjecture (Theorem 3.4). The other important ingredient in the proof is an algebraic variant of Gromov’s h-principle, called the Brown–Gersten property by Morel–Voevodsky and excision by Asok–Hoyois–Wendt. We will come back to it in section 5.3. The Oka principle is an h-principle, too. As h-principles play an important role in both the holomorphic and the algebraic classiﬁcation of vector bundles, we elaborate a bit on h-principles in the following section.

5.2 H-principles and homotopy sheaves

When looking for solutions to a partial differential equation (I), one approach is to first look for formal solutions: All partial derivatives occuring in the differential equation are replaced by new variables. This yields a new equation (II). A solution to equation (II) is called a formal solution to equation (I). Clearly, the existence of a formal solution is necessary for the existence of a genuine solution. One says that a PDE (or, more generally, a partial differential relation) satisfies an h-principle if every formal solution can be deformed into a genuine solution. A similar principle applies in other settings: For a problem in complex analysis, the existence of a continuous solution is necessary for the existence of a holomorphic solution. Because of this, h-principles have been playing an important role in differential geometry and complex analysis since many decades. The first h-principle occured in Oka’s solution of the second Cousin problem in 1939. Oka showed that the second Cousin problem is solvable holomorphically if and only if it is solvable continuously. Other famous h-principles turned up in the work by Nash, Grauert, Smale, Hirsch and, in great generality, in Gromov’s work. Gromov also coined the term h-principle [Gro86]. We recommend the book [EM02] by Eliashberg and Mishachev as a starting point. Forstneriˇc’sbook [For11] is a very comprehensive account on the Oka principle. Our short exposition of the h-principle was also inspired by Weiß [Wei11]. Definition 5.9 (Jet bundle, holonomic sections, [EM02, 1.2—1.5]). a) Let f (s) be the tuple of all partial derivatives Dαf with |α| = s, ordered (n+r−1)! lexicographically. Let dn,r = (n−1)!r! denote the number of all partial derivatives α n D of order r of a function R → R. Set Nn,r = 1 + dn,1 + ··· + dn,r. With this notation, the r-jet of a smooth function f : Rn → Rq is the function

r n qNn,r Jf : R −→ R x 7−→ (f(x), f 0(x), . . . , f (r)(x)).

b) Smooth functions Rn → Rq correspond bijectively to smooth sections of the trivial ﬁbre bundle Rn × Rq → Rn. If f˜: Rn → Rn × Rq is a smooth section corresponding to a function f : Rn → Rq, then

r r Jf˜ := Jf 5.2. H-PRINCIPLES AND HOMOTOPY SHEAVES 83

is also called the r-jet of the smooth section f˜. One often does not distinguish between the function and the section and denotes both by f.

c) The space of r-jets of sections Rn → Rn × Rq is the space

r n q n qN J (R , R ) := R × R n,r

of all “a priori possible jets” of maps Rn → Rq. d) Let M be an n-dimensional manifold and p: X → M a smooth fibre bundle with fibre F of dimension q. Let y be a point of M and y ∈ U ⊂ M a small open neighbourhood of it. Then two local sections f, g : U → X are called r r r-tangential at y if f(y) = g(y) and Jϕ∗f (ϕ(y)) = Jϕ∗g(ϕ(y)) for some local trivialisation ϕ: V → Rn × Rq of some neighbourhood X ⊃ V 3 f(y) = g(y). Note that we may replace U and V by smaller neighbourhoods without changing the definition and that, by the chain rule, the definition is independent of the choice of trivialisation ϕ. e) Let f : M → X be a section of p. Then the r-tangency class of f in y ∈ M is r called the r-jet of f at y and denoted by Jf (y). f) The set of al r-jets of sections M → X is denoted by X(r). If p: M × F → M is a trivial fibre bundle, then we also write X(r) = J r(M,F ) as in c). Choosing (sufficiently small) charts of M, this endows X(r) with a smooth structure and thus we get a smooth fibre bundle

pr : X(r) → M,

the r-jet bundle of the ﬁbre bundle p. g) Let f : M → X be a section. The section

r (r) Jf : M −→ X r y 7−→ Jf (y)

of the r-jet bundle is called the r-jet of f. h) The space of sections M → X(r) of the r-jet bundle is denoted by Sec X(r). Similarly, the space of sections of X → M is denoted by Sec X. The space of holonomic sections of the r-jet bundle is deﬁned as

(r) (r) r Hol X = {s ∈ Sec X | ∃f ∈ Sec X, s = Jf }.

Definition 5.10 (Partial differential relation, [EM02, 5.1], [Gro86]). a) Let p: X → M be a smooth fibre bundle. A partial differential relation (PDR) of order r on sections M → X is a subset D of the jet space X(r). b) As before, we denote the space of sections of the PDR D by

Sec D := {s: pr(D) → D}. 84 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

c) Similarly, we denote the space of holonomic sections by

∗ r r Hol D := {s ∈ Sec D | ∃f ∈ Sec p p D ⊂ X, s = Jf }.

n n Definition 5.11 (Man ). For n ∈ N, the big site Man of n-manifolds consists of • objects smooth n-dimensional manifolds without boundary, • morphisms smooth maps and • coverings jointly surjective families of embeddings. The category Mann is enriched over topological spaces: For each pair of objects M,N ∈ Ob(Mann) the sets of morphisms Hom(M,N) becomes a topological space with the compact-open topology and composition of morphisms is continuous with respect to these topologies. Applying simplicial resolution (see next definition), it is also enriched over S. Definition 5.12 (compare [Hir03, Definition 1.1.6]). There is a functor Sing[0,1], called simplicial resolution, from the category of topological spaces and continuous maps to the category of simplicial sets. Denoting the topological realisation of the standard n-simplex by |∆[n]|, simplicial resolution is defined by

Sing[0,1] : Top −→ S X 7−→ Hom(|∆[n]|,X) n∈N and the corresponding diagrams on morphisms. Applying Sing[0,1] to each Hom-space, equipped with the compact-open topology, enriches Mann over simplicial sets. We denote Hom(M,N), equipped with the simplicial structure, by Map(M,N).

r n Look at a PDR D ⊂ J0 (R ,N) that is invariant under diﬀeomorphism germs in the origin. We see it as a “universal” partial diﬀerential relation because it applies to all smooth manifolds M ∈ Ob(Mann): For each x ∈ M, let

n ϕx : R → M, 0 7→ x be a coordinate chart around x. As the PDR D is invariant under diffeomorphism germs in the origin, the choice of coordinate chart does not matter and D defines a r ∞ PDR DM ⊂ J (M,N) on C (M,N). Definition 5.13 (Parametric h-principle [EM02, 6.2A]). A PDR D ⊂ X(r) is said to satisfy the parametric h-principle if for all k ≥ 0

πk(Sec D, Hol D) = 0.

Setting k = 0, this implies that every formal solution of D is homotopic in Sec D to an actual solution and that if two actual solutions of D are homotopic in Sec D, then they are even homotopic in Hol D. The following formulation of Gromov’s theorem is taken from Eliashberg–Mishachev’s book [EM02, Theorem 7.2.3]. 5.2. H-PRINCIPLES AND HOMOTOPY SHEAVES 85

Theorem 5.14 (Gromov [Gro69]). Let M be an open manifold and X → M a fibre bundle. Then any open Diff M-invariant PDR D ⊂ X(r) satisfies the parametric h-principle. Gromov’s h-principle specialises to several important classical theorems in various fields and proves many more. We pick a few examples. Examples 5.15. a) The h-principle of Smale–Hirsch: Let M,N be smooth mani- 1 folds. By definition, a C -map f : M → N is an immersion if rank Df = dim M everywhere on M. This condition gives rise to an open PDR on X(1) where we set p: X := M × N → M. Smale proved that immersions Sn → Rq with q ≥ n + 1 satisfy the h-principle [Sma58, Sma59]. Hirsch proved that immersions M → N satisfy the h-principle if either dim N > dim M [Hir59] or dim N = dim M and M is open [Hir61]. The h-principle of Smale–Hirsch is now a special case of Gromov’s h-principle [Gro86, 1.1.3]. By the same argument, submersions and, more generally, k-mersions are examples of Gromov’s 0 h-principle [Gro86, 1.3.1, Theorems (A) and (A1)]. b) Thom’s transversality theorem [Tho56]: Let X → M be a smooth fibre bundle. Let S ⊂ X(r) be a C∞-submanifold. The space of C∞-sections f : M → X r (r) such that Jf : M → X is transversal to S is open and dense in the space of all C∞-sections [Gro86, 1.3.2 Theorem (D)]. c) Grauert’s Oka principle (Theorem 5.5) is implied by Gromov’s h-principle: Let X,M be manifolds with a complex analytic structure. Let X → M be a complex analytic fibre bundle, for example a principal bundle under a complex Lie group. The Cauchy–Riemann equations give rise to an open PDR of order 1 on X(r). By Gromov’s h-principle, every continuous section M → X is homotopic to a holomorphic section [Gro86, 1.1.2]. d) Applications in symplectic geometry: For instance isotropic immersions, immersions transversal to a contact structure, isocontact immersions or isosymplectic immersions satisfy the h-principle. This is proven in Gromov’s book [Gro86, 3.4], see also [EM02, Part 3]. Definition 5.16 (homotopy sheaf [Wei11, Definition 1.1]). A continuous presheaf F of simplicial sets on Mann is a homotopy sheaf if it satisfies the sheaf conditions up to homotopy: a) F(∅) is contractible in S, S n b) if M = i Ui∈I in Man where all Ui are open in M, the diagram Y Y F(M) → F(Ui) ⇒ F(Ui ∩ Uj) i∈I i,j∈I is a homotopy equaliser in S, i. e., F(M) is the homotopy limit of the diagram Q Q i F(Ui) ⇒ i,j F(Ui ∩ Uj), n c) if Mj ∈ Ob(Man ) for all j in some index set J, then ! a Y F Mj ' F(Mj) j∈J j∈J 86 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

in S. Now let p: X → M be a fibre bundle with fibre F and let dim M = n, dim F = q as before. Let D ⊂ X(r) be a PDR. For any open submanifold U ⊂ M we define

U˜ := p−1(U) ∩ D.

We deﬁne presheaves on M, given on an open submanifold U ⊂ M by

˜ (r) FormD : U 7−→ Sec(U ) ˜ (r) SolD : U 7−→ Hol(U ).

The presheaf FormD of formal solutions to D is a homotopy sheaf if D is open. By the parametric h-principle, FormD(U) is weakly equivalent to SolD(U) for any open submanifold U ⊂ M. Hence, SolD is a homotopy sheaf, too. As holonomic sections ˜ (r) ˜ of U correspond to sections of p|U˜ , also the presheaf U 7→ SecD(U) is a homotopy sheaf. The point of view that the homotopy sheaf property is a good way to look at h-principles was adopted from lecture notes by Michael Weiss [Wei11, Theorem 1.5].

5.3 The h-principle in A1-homotopy theory

In A1-homotopy theory, the question of understanding the geometric content of the mapping space Mapτ,A1 (X,Y ) between two smooth varieties X and Y is very similar to the classical h-principle: Does every class in π0(Mapτ,A1 (X,Y )) have an algebraic representative? As before a simplicial presheaf is said to satisfy the h-principle if it is a homotopy sheaf with respect to a given cd structure τ (usually the Zariski or the Nisnevich topology). This property runs under many names: Brown–Gersten property with respect to τ [MV99, Mor12], τ-flasque [Voe10], Mayer–Vietoris property with respect to τ [CHSW08] or τ-excision [AHW15a]. We stick to the term excision, as suggested by Asok–Hoyois–Wendt. Asok–Hoyois–Wendt [AHW15a] show that only homotopy descent with respect to the Nisnevich topology and homotopy invariance are needed in order for Nisnevich excision to be satisfied (following a trick by Schlichting [Sch15]). Furthermore, they show that only excision and homotopy invariance are needed to get a classifying space for vector bundles. This way they obtain an elegant proof of Morel’s theorem 5.7 which makes the technical assumptions on the base field and the rank of the vector bundles superfluous. The base, which had to be an infinite perfect field in Morel’s proof, may now be any field or any ring which is ind-smooth over a Dedekind ring with perfect residue fields [AHW15a, Theorem 5.2.3]. Their approach even generalises to the case of principal bundles under other algebraic groups [AHW15b]. The homotopy theoretic part of the proof by Asok–Hoyois–Wendt carries over to the rigid analytic situation. The interesting question becomes then: For which subcategory R of the category of rigid varieties, for which cd structure τ and which 5.3. THE H-PRINCIPLE IN A1-HOMOTOPY THEORY 87 interval object I can we show homotopy invariance and excision? Once homotopy invariance and excision are ensured, one immediately gets a classifying space in the appropriate homotopy theory HI,τ (R). It then remains to determine how this homotopy theory relates to the corresponding homotopy theory HI,τ (smRig) on the full category of (smooth) rigid analytic varieties. Before coming to the theorems we need more definitions. Definition 5.17 (homotopy descent, [AHW15a, Definition 3.1.1]). Let (C, T ) be a small site. A simplicial presheaf F on C satisfies homotopy descent with respect to T if for every X ∈ Ob(C) and every covering sieve U ∈ Cov(X) the map

F(X) −→ holimU∈U F(U) induced by the restrictions F(X) → F(U) is a weak equivalence. Remarks 5.18. a) Homotopy sheaves satisfy homotopy descent by definition. b) Cortiñas–Haesemeyer–Schlichting–Weibel [CHSW08] show that for a complete, bounded, regular cd structure, homotopy descent is equivalent to quasifibrancy. They call a simplicial presheaf F quasifibrant if the local injective fibrant replacement H(·, F) gives rise to a weak equivalence on all spaces of sections: F(U) ' H(U, F) for all objects U. This is useful when one wants to compute homotopy classes. Usually, either cofibrant replacement or fibrant replacement is difficult to compute. A simplicial presheaf which satisfies homotopy descent is almost as good as fibrant because it still gives rise to the right computations. If the Grothendieck topology arises from a cd structure, one can consider excision instead: Definition 5.19 (excision, [AHW15a, Definition 3.2.1]). Let C be a small category with an initial object ∅ and τ a cd structure (see Definition 4.12) on C. A simplicial presheaf F on C satisfies τ-excision if a) F(∅) ' ∗ in S and b) for each distinguished square

B / Y

A / X in τ, the square

F(X) / F(A)

F(Y ) / F(B)

is a homotopy pullback square in S. Excision is easier to check than homotopy descent and if the cd structure is nice enough, the two notions are equivalent by Theorem 5.21 below. In nice cases excision is a weak version of ﬁbrancy as we explained in Remark 5.18b). 88 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

Deﬁnition 5.20 (strictly initial object [Bor94, Deﬁnition 3.4.6]). Let C be a category with an initial object ∅. The object ∅ is called strictly initial if every morphism X → ∅ is an isomorphism. Theorem 5.21 (Voevodsky, Asok–Hoyois–Wendt [AHW15a, Theorem 3.2.5]). Let C be a small category with a strictly initial object and τ a cd structure on C. Assume that a) every square

B / Y (5.1)

p A e / X in τ is cartesian, b) pullbacks of squares in τ exist and belong to τ, c) for every square (5.1) in τ the morphism e is a monomorphism, d) for every square (5.1) in τ, also the square

B / Y

∆ ∆ B ×A B / Y ×X Y is in τ. Then F ∈ sPSh(C) satisfies excision with respect to τ if and only if it satisfies homotopy descent with respect to the Grothendieck topology defined by τ. Definition 5.22 (simplicial resolution, [MV99, p. 88]). Let C be a small category and I a representable interval object (Definition 4.14) on C. Denote by ∆ the simplex category [GJ09, I.1, p. 3]. a) The cosimplicial presheaf I• is defined on objects [n] ∈ Ob(∆) by I• : ∆ −→ PSh(C), [n] 7−→ In.

n Let f ∈ Hom∆([n], [m]). Denote by prk : I → I the projection to the k-th coordinate. Then I•(f) is deﬁned by  pr if 1 ≤ N := min{l ∈ {0, . . . , n} | f(l) ≥ k} ≤ n  N •  n ι0 prk ◦I (f) = I → ∗ → I if {l ∈ {0, . . . , n} | f(l) ≥ k} = ∅  ι  In → ∗ →1 I if min{l ∈ {0, . . . , n} | f(l) ≥ k} = 0.

b) The simplicial resolution SingI F of a simplicial presheaf F is the simplicial presheaf deﬁned by I n Sing F n = F(X × I ) n. That is, it is the diagonal of the bisimplicial presheaf F( · × I•). 5.4. CLASSIFYING SPACES 89

Lemma 5.23 (Properties of SingI , [AHW15a, p. 15]). a) The construction of the simplicial resolution SingI gives rise to a natural transformation id → SingI on simplicial presheaves. Thus, for every F ∈ sPSh(C) there is a map

F −→ SingI F.

b) The simplicial resolution of a simplical presheaf is always I-invariant. c) For any I-invariant presheaf G, the map F → SingI F induces a weak equivalence

Map(SingI F, G) ' Map(F, G). (5.2)

d) Let τ be a cd structure. If F satisﬁes τ-excision and if π0F is I-invariant, then SingI F also satisﬁes excision.

Proof. b) Morel–Voevodsky prove it for simplicial sheaves on a site with interval under the general assumption that the site has enough points [MV99, §2 Corollary 3.5]. Neither the sheaf property nor the assumption that the site has enough points go into that particular proof, so we refrain from repeating it here. c) This is proven by Morel–Voevodsky, again without using the sheaf property or the assumption that the site has enough points [MV99, §2 Corollary 3.8]. d) This is [AHW15a, Theorem 4.2.3].

We saw that Gromov’s h-principle can be expressed in terms of homotopy sheaves. In analogy to this, we think of excision as an A1-homotopy theoretic version of the h-principle. This intuition also works in the complex analytic case: Lárussonshows that satisfying the Oka principle means being fibrant in a suitable model category [Lár03, Lár04, Lár05]. He calls manifolds (or, more generally, complex spaces) that satisfy an Oka principle Oka manifolds (Oka spaces). He constructs a model category of simplicial presheaves containing the category of complex manifolds, such that Stein manifolds are cofibrant and Oka manifolds are fibrant. It is an interesting question to find other Oka manifolds (or Oka spaces). In other words: One is looking for representable simplicial presheaves that satisfy the Oka principle. Or: Which holomorphic partial differential relations have a moduli space? The branch of complex analysis investigating this question is called Oka theory. For Oka theory, we again recommend Forstneriˇc’sbook [For11].

5.4 Classifying spaces

In this section we deﬁne the classifying space of a topological category, the classifying space of a topological group and the classifying space of a simplicial sheaf of groups. 90 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

1 We show that the inﬁnite Grassmannian Grn is Arig-homotopy equivalent to the simplicial classifying space of GLn. This will be needed for our main theorem, Theorem 5.40 in section 5.5. First recall the nerve construction: Deﬁnitions 5.24 ([Seg68, §2]). a) For n ∈ N, denote by n the category with objects 0, . . . , n and for each pair i, j ∈ Ob(n) with i ≤ j a morphism i → j. b) The nerve of a category C is the simplicial set given by the set of covariant functors

(NC)n = Fun(n, C)

in each simplicial degree n ∈ N.

In other words, (NC)0 = Ob C and for each morphism X → Y we get a 1-simplex between the corresponding vertices. For each commutative triangle of morphisms we get a 2-simplex and higher simplices if the corresponding bigger diagrams commute. Deﬁnition 5.25 (Classifying space of a topological category [Seg68, §2]). Let C be a topological category. Its classifying space is deﬁned as the topological realisation of its nerve:

BC := |NC|.

Deﬁnition 5.26 (Classifying space of a topological group, cf. [May99, chapter 23, §8]). Let G be a topological group and EG a contractible topological space with a continuous, free right G-action. Then EG is called a universal space for G and the quotient BG := EG/G is called a classifying space for G. The principal G-bundle EG → BG is called a universal G-bundle. Remark 5.27 ([May99, chapter 23, §8]). The classifying space of a topological group is not unique, but its homotopy type is. Example 5.28 (cf. [May99, Chapter 16, §5 and chapter 23, §8] or [Seg68, §3]). Let G be a topological group. Deﬁne simplicial topological spaces by

n+1 EnG = G n BnG = G where n ∈ N denotes the simplicial degree. Letting G act on EnG from the right by

(g0, . . . , gn).g := (g0, . . . , gn−1, gng) makes E∗G a simplicial G-space and BnG = EnG/G for all n. Deﬁne

EG := |E∗G|

BG := |B∗G| as the topological realisations. Now EG is contractible and inherits a free right G-action from E∗G. Furthermore, BG = EG/G. These are explicit constructions of a universal space EG and a classifying space BG as deﬁned in 5.26. 5.4. CLASSIFYING SPACES 91

This point of view carries over easily to sheaves of simplicial groups: Deﬁnition 5.29 (Classifying space of a simplicial sheaf of groups [MV99, §4.1]). Let G be a sheaf of simplicial groups (or, more generally: monoids) on a site C. Hence, to each X ∈ Ob(C) and each n ∈ N, the sheaf G assigns a group (monoid) GX,n. Viewing this group (monoid) as a category, it has a nerve N(GX,n) . Now the classifying space BG of G is the diagonal simplicial sheaf of the bisimplicial sheaf X 7−→ n 7→ N(GX,n) .

This means, for X ∈ Ob(C),

n BG(X) = (GX,n) n∈N

0 where (GX,0) := ∗. Remark 5.30 (Universal torsor, cf. [MV99, §4, Example 1.11], [Mor06, p. 12]). For a simplicial sheaf of groups G, deﬁne a simplicial sheaf of groups EG as the diagonal of the bisimplicial sheaf X 7−→ (n, m) 7→ E(GX,n)m .

The sheaf G is now a subsheaf of groups of EG. For X ∈ Ob(C), n ∈ N we again have an action of (GX,n)n on E(GX,n)n given by

(g0, . . . , gn).g := (g0, g1, . . . , gn−1, gng).

This deﬁnes an isomorphism from the quotient of sheaves EG/G to BG. The morphism

EG −→ BG (5.3) is called the universal G-bundle or universal G-torsor. The simplicial classifying space BG in fact classiﬁes principal G-bundles: Deﬁnitions 5.31 ([MV99, pp. 127f]). a) Let C be a site and G a sheaf of simplicial groups on C. A right action of G on a simplicial sheaf F is called (categorically) free if the morphism

F × G −→ F × F (g, x) 7−→ (x.g, x)

is a monomorphism. b) If G acts on F from the right, then the quotient F/G is deﬁned as the coequaliser of

pr1 F × G ⇒ F. action

c) A principal G-bundle or G-torsor over F is a morphism p: E → F together with a free right action of G on E over F, such that E/G → F is an isomorphism. 92 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

Lemma 5.32 ([MV99, §4, Lemma 1.12]). Let C be a site and G a sheaf of simplicial groups on C. Let p: E → F be a principal G-bundle. Then there are morphisms • ω : F 0 → F such that F 0(U) → F(U) is both a ﬁbration and a weak equivalence in S for all U ∈ Ob C and • ϕ: F 0 → BG such that the pullback of the universal G-torsor along ϕ and the pullback of p along ω coincide:

EG o ϕ∗EG = ω∗E / E

0 BGFo ϕ ω / F

This means that in a model structure on sSh(C) where • morphisms F 0 → F such that F 0(U) → F(U) is a weak equivalence for all U ∈ Ob C are themselves weak equivalences and • morphisms F 0 → F such that F 0(U) → F(U) is a fibration for all U ∈ Ob C are themselves fibrations, the following hold: a) The classifying space BG is in fact the homotopy quotient ∗/hG. By this, we mean that the total space EG of the universal G-bundle is contractible and EG → BG is a fibration. b) The classifying space BG really classifies principal G-bundles in sSh(C).

We want to identify the inﬁnite Grassmannian as the classifying space of GLn. To be precise, we want to prove the rigid analytic version of the following A1-homotopy theoretic result. Proposition 5.33 ([MV99, p. 138, Proposition 3.7]). In the A1-homotopy category, ∼ we have B GLn = Grn. A proof was written down by Peter Arndt. It generalises the proof for n = 1 by Naumann–Spitzweck–Østvær [NSØ09, Corollary 2.5] and works also in the setting of rigid analytic varieties. One writes Grn as a quotient of the Stiefel variety by a free GLn-action and shows that the Stiefel variety is contractible. This approach is originally due to Morel, cf. [Mor06, Example 2.1.8, Lemma 4.2.5]. ∼ 1 Proposition 5.34 (Arndt). We have B GLn = Grn in the Arig-homotopy category. ∼ 1 Consequently, B GLn = Grn in the B -homotopy category. As Arndt’s proof is unpublished, we include it here.

Proof. Let R be a subcategory of the category of smooth rigid analytic varieties containing the aﬃnoids of good reduction. We write Grn as a quotient of the Stiefel space Frn by a free GLn-action and show that Frn is contractible in MB1 (R) and M 1 (R). Arig 5.4. CLASSIFYING SPACES 93

By [Lam06, Proposition I.2.13], we know that for each smooth k-algebra A we have ∼ GLn(A) = GLn(k) ⊗k A, hence the sheaf GLn is representable by GLn(k). m Let Grn,m be a ﬁnite Grassmannian. We deﬁne the space Frn,m of n-frames in Arig next, building it up from pieces MJ . We think of the space of n-frames as the space of n × m-matrices of rank n. During this proof we denote l O(Arig) =: khhT1,...,Tlii.

For each subset J = {j1, . . . , jn} ⊂ {1, . . . , m} of order n we define a rigid analytic space MJ as the analytification an MJ = Spec k[T11,...,T1m,T21,...,T2m,...,Tnm,T ]/((det((Tij)J )T ) − 1) where (Tij)J is defined as the n × n minor of (Tij) i=1,...,n given by J, i. e., j=1,...,m

(Tij)J := (Tij) i=1,...,n . j=j1,...,jn

The spaces MJ , the index set J running through all subsets of {1, . . . , m} with n elements, are supposed to cover the space of n-frames that we want to define. We think of MJ as the space of n × m-matrices such that the n × n-minors specified by J are of rank n. For a rigorous construction, we need to define the “intersections” of

MJ1 ,...,MJr . For r = 2, we define 0 0 an MJ,J0 = Spec k[T11,...,Tnm,T,T ]/ (det((Tij)J )T ) − 1, (det((Tij)J )T ) − 1 which we think of as the n × m-matrices such that both the n × n-minors specified by 0 J and the n × n-minors specified by J are of rank n. Higher “intersections” MJ1,...,Jr are defined analogously. We have open immersions MJ,J0 → MJ , given on the level of rings by

Tij 7→ Tij T 7→ T, and MJ,J0 → MJ0 , given on the level of rings by

Tij 7→ Tij T 7→ T 0 and analogously MJ,J0,J00 → MJ,J0 , MJ,J0,J00 → MJ,J00 , MJ,J0,J00 → MJ0,J00 and so on. Altogether, they ﬁt into a diagram D:

/ MJ1J2J2 / MJ1J2 / MJ1 < E 9 C ;

% # MJ1J2J4 / MJ1J3 MJ2 9 ;

# MJ2J3 / MJ3 : ...... 94 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

Now we deﬁne Frn,m := colim D as the “union” of all the spaces MJ .

The group GLn operates on Frn,m as follows. The action of GLn on the space of nm n × m-matrices by multiplication from the left deﬁnes an action of GLn on Arig . It leaves the ideals ((det((Tij)J )T ) − 1) invariant, hence it restricts to each of the MJ .

As the morphisms MJ1,...,Jr → MJ1,...,Jr−1 are GLn-equivariant, we get an action of GLn on Frn,m. This action is free because of the “full rank” condition.

There are GLn-equivariant embeddings Frn,m → Frn,m+n given by

nm nm nn ιm : Arig 7−→ Arig × Arig x 7−→ (x, 0).

As ιm “does not change the rank of a matrix”, which means it leaves the ideals ((det((Tij)J )T ) − 1) untouched, it restricts to Frn,m → Frn,m+n. Equivariance under the action by GLn is immediate. Fix m mod n and deﬁne

Frn := colim(Frn,m → Frn,m+n → Frn,m+2n → · · · ).

The inﬁnite space of n-frames Frn depends on the choice of m mod n, but this does not bother us. As the embeddings Frn,m → Frn,m+n are GLn-equivariant, the colimit Frn inherits the free GLn-action.

The only statement that is left to show is that Frn is contractible. We show that 1 each embedding ιm : Frn,m → Frn,m+n is Arig-homotopic to a constant map. Then, as the colimit over a chain of constant maps is contractible, we are done.

1 We can write down an Arig-homotopy from ιm to a constant map:

1 H : Arig × Frn,m −→ Frn,m+n

If k is algebraically closed, H is given on matrices by

1 0 . H :(t, (xij)ij) 7−→ t · (xij)ij (1 − t) · .. . 0 1

In general, it is given on the MJ (and similarly on the MJ,J0 and so forth) by

0 khhT11,...,Tn(m+n),T ii/(detJ ) −→ khhT11,...,Tnm,T ii 0 Tij 7−→ T Tij if j = 1, . . . , m,

Tim+j 7−→ 0 if i 6= j, 0 Tim+i 7−→ (1 − T )Tim+i.

Here, detJ := ((Tij)J )T ) − 1. We need to check that H really ends up in Frn,m+n. The reason is that, locally, one of t and 1 − t is always a unit, hence one of t · (xij)ij 1 0 1 0 . . and (1 − t) · .. is of full rank, hence t · (xij)ij (1 − t) · .. is of full 0 1 0 1 rank. 5.5. HOMOTOPY INVARIANCE IMPLIES A CLASSIFICATION OF VECTOR BUNDLES95

1 The map H deﬁnes an Arig-homotopy between idFrn and Frn → ∗. Consequently the Stiefel variety Frn is contractible in M 1 (smRig) and hence Arig

Frn / GLn ' B GLn in M 1 (smRig). It is clear that Frn / GLn = Grn, hence Arig

Grn ' B GLn in M 1 (smRig). Therefore the same holds for the sheaves represented by Grn, Arig respectively by B GLn and still so after restriction to R:

Grn(X) ' B GLn(X) for all X ∈ R. Hence

Grn ' B GLn in M 1 (R). As weak equivalences in M 1 (R) remain weak equivalences after Arig Arig further localisation of the model structure, we also get Grn ' B GLn in MB1 (R).

Remark 5.35. Please note that the “spaces” Frn,m, Frn, Grn,m and Grn are in fact presheaves that may not be representable on R. The spaces Frn,m and Grn,m are at least smooth rigid analytic varieties, the ﬁnite Grassmannians Grn,m are even quasicompact. The inﬁnite spaces Frn and Grn are only ind-rigid analytic varieties. They are thus “spaces” in some sense, albeit not necessarily objects of R.

5.5 Homotopy invariance implies a classiﬁcation of vector bundles

Deﬁnition 5.36. The functor Vectn assigns to a rigid analytic variety X the set of isomorphism classes of vector bundles on X, with the trivial vector bundle of rank n as a base point. To a morphism f : X → Y of rigid analytic varieties it assigns the pullback map f ∗ : Vectn(Y ) → Vectn(X). Without taking isomorphism classes it is not a functor any more: For each rigid analytic variety X let

Φn : X 7−→ {E vector bundle of rank n over X}.

Let E ∈ Φn(X) be a vector bundle and

f g X −→ Y −→ Z be morphisms of rigid analytic varieties. The vector bundles f ∗g∗E and (gf)∗E are isomorphic but in general not equal. As we will need a functor of vector bundles, we n replace the pseudofunctor Φn by Grayson’s functor V of big vector bundles. 96 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

Deﬁnition 5.37 (big vector bundle, big principal bundle, cf. [FS02, C.4], [Gra95]). Let R be a subcategory of a small category equivalent to smRig. In short, a big vector bundle on a rigid analytic variety X ∈ Ob(R) is a family of vector bundles together with isomorphism data. The construction is as follows. The overcategory R/X has as objects morphisms (f : Y → X) ∈ Mor(R). The morphisms of R/X are commutative triangles. Now a big vector bundle over X is a family of vector bundles

{EY }(f : Y →X)∈Ob(R/X) together with compatible isomorphisms ∗ {ϕ: g EZ → EY | (g : Y → Z) ∈ Mor(R/X)}. The assignment Vn : X 7−→ {big vector bundles of rank n over X} defines a functor from R to groupoids. Defining big principal bundles analogously, also n P : X 7−→ {big GLn-principal bundles over X} defines a functor from R to groupoids. Remark 5.38. It is quite obvious that for each X ∈ Ob(R) the cateory of big vector bundles over X and the category of vector bundles over X are equivalent. The following proposition is [AHW15a, Theorem 5.1.3] transferred into our setting. Proposition 5.39. Let R be a subcategory of the category of rigid analytic varieties. Assume that R is small and has a strictly initial object ∅. Let I be a representable interval object on R and τ a cd structure on R. Let F ∈ sPSh(R) be a simplicial presheaf satisfying τ-excision and assume that the associated presheaf of pointed sets π0F is I-invariant on R. Let Rτ be a fibrant replacement functor in Mτ (R). Then I Rτ Sing F is fibrant in MI,τ (R) and for all U ∈ Ob(R) the canonical map

π0F(U) −→ [U, F]I,τ is a bijection of pointed sets.

Proof. The proof is the same as in [AHW15a, Theorem 5.1.3], only easier: Since I I I Sing F is I-invariant by Lemma 5.23b) and since Rτ Sing F(U) = Sing F(U), also I Rτ Sing F is I-invariant. Being fibrant in Mτ (R) and I-invariant, it is I-local in Mτ (R) by Definition 4.9, hence fibrant in MI,τ (R). Therefore, for each U ∈ Ob(R) we get a bijection of pointed sets ? I ∼ I π0 Sing F(U) = π0Rτ Sing F(U). As, by assumption, ∼ I π0F(U) = π0 Sing F(U), we get the following identifications: ? ∼ I ∼ I ∼ I π0F(U) = π0 Sing F(U) = π0Rτ Sing F(U) = π0 Hom(U, Rτ Sing F)

= π0MapI,τ (U, F) = [U, F]I,τ . 5.5. HOINVARIANCE IMPLIES CLASSIFICATION 97

The next theorem is the rigid analytic version of [AHW15a, Theorem 5.2.3]. It basi- cally states that if homotopy invariance holds, then there is a homotopy classification of vector bundles. Theorem 5.40. Let R be a subcategory of the category of rigid analytic varieties. 1 1 Assume that R is small and has a strictly initial object ∅. Let I ∈ {B , Arig} be a n representable interval object on R. For some n ∈ N, assume that Vect satisfies I-invariance on R. Let τfG be the cd structure which consists of finite admissible coverings with respect to the G-topology. Assume that

• τ = τfG or • τ is a cd structure satisfying the assumptions of Theorem 5.21, τ is finer than n τfG and Vect satisfies excision with respect to τ. Then the infinite Grassmannian classifies Vectn in the (I, τ)-homotopy category of R: For each X ∈ Ob(R) there is a natural bijection n ∼ Vect (X) = [X, Grn]I,τ .

Proof. Let Vn be the functor “big vector bundles” as defined in 5.37. By assumption, n n Vect satisfies excision with respect to τ. If τ = τfG, then Vect satisfies descent, hence also excision, with respect to τ by Bosch–Görtz[BG98, Theorem 3.1]. We claim that the assignment U 7→ BVn(U) satisfies τ-excision. Let

U ×X V / U

f V g / X be a τ-distinguished triangle in R. Apply Quillen’s Theorem B in the version [BK12, Theorem B1 (3.5)] to the zigzag of groupoids

∗ ∗ n f n g n V (U) −→ V (U ×X V ) ←− V (V ).

Barwick–Kan’s condition B1 is trivially fulﬁlled for groupoids. Hence [BK12, Theorem B1 (3.5)] gives that

f ∗Vn(U) ↓ g∗Vn(V ) / Vn(U)

n n V (V ) / V (U ×X V ) is homotopy cartesian (which means, homotopy cartesian after taking nerves). Here, f ∗Vn(U) ↓ g∗Vn(V ) is the comma category whose objects are isomorphisms ∗ n ∼ ∗ n f V (U) −→ g V (V ) and whose morphisms are commutative diagrams. If τ = τfG, ∗ U then f and g are the inclusions of U, respectively of V , into X and f = resUV and ∗ V g = resUV are the restrictions of vector bundles on U, respectively V , to vector bundles on U ×X V = U ∩ V . In this case there are isomorphisms ? U n V n ∼ n n ∼ n resUV V (U) ↓ resUV V (V ) = V (U) ×Vn(U∩V ) V (V ) = V (X) 98 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

by deﬁnition. If τ =6 τfG then ? remains an isomorphism and, by excision, is at least a weak equivalence (after taking nerves). Putting everything together, the diagram

BVn(X) / BVn(U)

n n BV (V ) / BV (U ×X V ) is homotopy cartesian, proving the claim.

n n By assumption, Vect is I-invariant on R, hence π0BV is I-invariant on R. Applying Proposition 5.39 to the simplicial presheaf BVn gives

n n Vect (X) = [X,BV ]I,τ for all X ∈ Ob(R).

n Let now P be the functor “big GLn-principal bundles” deﬁned in 5.37. By Lemma 5.32, the simplicial classifying space B GLn really classiﬁes GLn-principal bundles on R. Hence, for all X ∈ R, we have

n B GLn(X) ' BP . Furthermore, Vn(X) and Pn(X) are equivalent as topological groupoids for each X ∈ R. In particular, we get that BPn(X) ' BVn(X).

As vector bundles are τfG-locally trivial by assumption and thus τ-locally trivial, this gives a τ-local weak equivalence

n B GLn ' BV .

By Proposition 5.34, B GLn ' Grn in MI,τ (R). Hence, we get for all X ∈ Ob(R) that n ∼ Vect (X) = [X, Grn]I,τ .

We saw in Corollary 4.19 that the homotopy theory defined by MB1,T (R) is finer than the homotopy theory defined by M 1 (R). Thus, for a simplicial presheaf Arig,T 1 F it is easier to satisfy Arig-invariance on a class of rigid analytic varieties than to satisfy B1-invariance. This fits with the observation we made in section 3.5: For the 1 1 the Picard group, satisfying Arig-invariance is easier than satisfying B -invariance. 1 With Arig as an interval, we get the following classification theorem: Theorem 5.41. Let k be of characteristic zero and let R be the category of smooth k-rigid analytic quasi-Stein varieties. Let τfG be the cd structure generated by squares

U ∩ V / V

∩ U ⊂ / X 5.5. HOINVARIANCE IMPLIES CLASSIFICATION 99 where U, V, X ∈ Ob(R) and {U → X,V → X} is an admissible covering of X. Then 1 the assumptions of Theorem 5.40 are satisﬁed for n = 1 and I = Arig. Consequently there is a natural bijection

∞ Pic(X) ∼= [X, ] 1 . Prig Arig,τfG

1 Proof. The object Arig is a representable interval object on the category of smooth rigid analytic quasi-Stein varieties. Vector bundles satisfy descent with respect to the G-topology by Bosch–Görtz[BG98, Theorem 3.1], hence they also satisfy homotopy descent with respect to the G-topology. Every vector bundle on a quasi-Stein variety has a finite local trivialisation by Ben’s Theorem 2.17. Homotopy invariance holds by Proposition 3.31. Hence, by Theorem 5.40, ∼ Pic(X) = [X, Gr1]I,τ . ∞ As Gr1 = Prig, we are finished.

On the other hand, choosing B1 as an interval object, homotopy invariance only holds over smooth affinoid varieties of sufficiently good reduction. The resulting data do not satisfy the assumptions of Theorem 5.40, as the following examples show. Examples 5.42. a) Let n = 1,I = B1 and k algebraically closed of residue characteristic zero. Let R be the category of smooth affinoid varieties of semistable canonical reduction. The interval object B1 is representable: If X ∈ Ob(R) is a smooth affinoid variety of semistable canonical reduction, then X × B1 also has these properties. Let τ = Zar be the cd structure generated by squares

U ∩ V / V

∩ U ⊂ / X where X ∈ Ob and the maps U → X and V → X are inclusions of Zariski open subsets. We saw in Corollary 2.15 that on an affinoid variety, every vector bundle with respect to the G-topology is isomorphic to a vector bundle with respect to the Zariski topology. As every Zariski vector bundle is in particular a G-vector bundle, the notions of Zariski vector bundle and G-vector bundle coincide on affinoid varieties. By Theorem 3.29, Pic is B1-invariant on smooth affinoid varieties of semistable canonical reduction. But unfortunately it does not seem likely that every vector bundle over a smooth affinoid variety of good reduction admits a finite trivialisation by affinoids that are also of good reduction. (We certainly cannot achieve this by refining coverings: Antoine Ducros pointed out that if U is an affinoid of bad reduction then every finite admissible affinoid covering of U will contain an affinoid which is again of bad reduction because the Shilov boundary of the Berkovich space associated to U also needs to be covered.) Let now X be a smooth affinoid variety of good reduction and E a line bundle over X which does not have a finite trivialisation by smooth affinoids of good reduction. Denote by X the site of admissible subsets of X and admissible coverings. The restriction of Pic to X ∩ R does not allow to trivialise the line bundle E. 100 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES

b) Analogously, let n = 1,I = B1, τ = Zar and k discretely valued. Let R be the category of smooth affinoid varieties of the form Sp A with kAk = |k| and with semistable canonical reduction. The interval object B1 is representable in R: As above, if Sp(A) ∈ Ob(R) is smooth, affinoid and of semistable canonical reduction, then Sp(A) × B1 = Sp(AhT i) shares these properties. As kAhT ik = kAk = |k|, the interval object B1 is representable. The Picard group is B1-invariant on smooth affinoid varieties of semistable canonical reduction by Theorem 3.24. But again a vector bundle over a smooth affinoid variety of good reduction might have no finite trivialisation by sets that are also of good reduction. Remark 5.43 (Relation to Ayoub’s theory). We already discussed the differences between the model categories defined in 4.15 and Ayoub’s B1-categories in detail in section 4.3.1. Recall that the main difference between the categories appearing in the examples and Ayoub’s unstable B1-category is that we do not localise with respect to the Nisnevich topology. Stabilising the category HB1,Nis(smRig) in the simplicial direction yields a different model structure on Ayoub’s category of presheaves giving rise to the same homotopy theory.

5.6 Discussion

By Theorem 5.40 a homotopy classiﬁcation of vector bundles follows directly from homotopy invariance, under two conditions: Firstly, the Grothendieck topology that we work with needs to be completely decomposable. Secondly, if homotopy invariance holds for rigid analytic varieties with some property (A), then we need that every vector bundle over an object with property (A) has a local trivialisation by subsets that also have the property (A). In other words, the conditions under which homotopy invariance holds should be local on the base space.

Therefore there are four possible ways to extend our results: One way is to try to get rid of the ﬁniteness assumptions that arise from the condition that the topology needs to be completely decomposable. This is a question in simplicial homotopy theory. One would need a diﬀerent form of Quillen’s Theorem B and one would have to redo the parts of Voevodsky’s and Asok–Hoyois–Wendt’s work where they only work with completely decomposable topologies.

1 Secondly, Arig-homotopy representability of line bundles over quasicompact rigid analytic varieties, or at least over projective rigid analytic varieties, should also be 1 true. However, as the interval Arig is not quasicompact, one has to be careful to work in the right category. The naive approach of taking the category of rigid analytic l varieties of the form X × Arig with X quasicompact does not work as Ayoub pointed out. The problem is the following: Let qcA be the full subcategory of smRig whose l objects are products of smooth quasicompact rigid analytic varieties with Arig for all l l l ∈ N. Let X × Arig ∈ qcA. Only allowing coverings of the form {Xi × Arig}i∈I where {Xi}i∈I is an admissible covering of X does not work because these are not stable 5.6. DISCUSSION 101 under pullback. For example, let X = B1. Pulling back along the isomorphism

1 1 1 1 B × Arig −→ B × Arig (x, y) 7−→ (x, xy) is not possible. Therefore the same construction as for quasi-Stein varieties does not give us a well-defined homotopy category in this case. Nevertheless the classification should still be true at least for projective varieties. The author plans to investigate 1 Arig-homotopy representability of line bundles over quasicompact rigid analytic varieties in future work. A main open question is when homotopy invariance is satisfied. This question is of entirely geometric nature. We examined some instances where homotopy invariance holds and some instances where homotopy invariance does not hold in chapter 3, but 1 much remains unknown. While Arig-invariance of line bundles is well understood now, the picture for B1-invariance is very patchy. We do not know under which conditions the Picard group of a rigid analytic Stein variety is B1-invariant. It is unknown whether the Picard group of a smooth affinoid variety of good reduction over a general complete, nonarchimedean, nontrivially valued field is B1-invariant (see section 3.5). For vector bundles of higher rank we have a few counterexamples to homotopy invariance, but no positive results yet. We hope that Tamme’s Theorem 3.32 (and its generalisation to positive characteristic by Kerz–Saito–Tamme) can be generalised to vector bundles of higher rank. The fourth question is about the case where homotopy invariance only holds under necessary conditions that are not local on the base. This is the case for the interval object B1: Homotopy invariance of the Picard group of an affinoid variety depends on the canonical reduction of the affinoid variety. We do not have an idea right now how to approach this question.

1 Putting everything together, it seems that the questions about an Arig-classification of vector bundles and about a B1-classification of vector bundles are in fact quite different and have to be answered with different methods. The method we chose 1 1 seems to be good for the interval object Arig but less so for the interval object B . 102 CHAPTER 5. CLASSIFICATION OF VECTOR BUNDLES Bibliography

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A◦, 13 admissible formal scheme, 22 A◦◦, 13 admissible subset, 14 J r(M,F ), 79 affinoid algebra, 12 r Jf , 78 reduced, 17 S-local equivalence, 65 reduction of, 19 S-local object, 65 affinoid algebras X(r), 79 morphism of, 12 ∗, 62 flat, 22 ∅, 62 affinoid subset, 14 κ, 9 affinoid varieties 1 Arig-invariance, 41, 57 morphism of, 16 n Arig, 17 affinoid variety, 16 B1, 16 canonical model of, 21 B1-invariance, 40, 54, 55 canonical reduction of, 21 local, 46 Artin approximation property, 49 1-invariance, 41 Br Banach algebra, 12 H (R), 69 I,T Berkovich space M(R), 68 good, 30 M (R), 68 I,T big vector bundle, 92 Mann, 80 Bousfield localisation, 65 Map(X,Y ), 64 Brown–Gersten property, 78 Grn, 75 Div(R), 35 category Grn,rig, 76 topological, 86 (r) Hol X , 79 cd structure, 67 Hol D, 79 classifying space Max A, 12 of a topological category, 86 Pic, 34 of a topological group, 86 Sec X, 79 simplicial, 87 Sec X(r), 79 closed immersion, 17 Sec D, 79 cofibrant object, 62 I Sing F, 84 cofibrant replacement, 63 [0,1] Sing , 80 cofibration, 61 Spm R, 21 trivial, 62 smRig, 24 complete localisation, 14 ⊗ˆ , 13 complete tensor product, 13 r-tangential, 79 completely decomposable topology, 67 admissible covering, 14 descent

112 INDEX 113

homotopy, 83 mapping space functor, 64 divisor maximally complete, 35 in a ring, 35 model category, 61 in an affinoid variety, 35 proper, 62 divisor class group simplicial, 64 of a ring, 35 module of an affinoid variety, 35 extended, 39 morphism of affinoid algebras, 12 excision, 83 extended module, 39 nerve, 86

fibrant object, 62 Oka principle, 77 fibrant replacement, 63 partial differential relation, 79 fibration, 61 PDR, 79 trivial, 62 Picard group, 34 flat morphism, 22 pro-group, 57 formal model, 21 semistable, 22 quasi-Stein, 18 formal solution of a PDR, 78 quasi-Stein algebra, 19

G-topology, 14 rational subset, 14 strong, 15 rigid analytic variety, 16 very weak, 15 formal model of, 21 weak, 15 semistable, 22 Gauß norm, 12 quasi-Stein, 18 Grassmannian, 75, 76, 88 quasicompact, 18 finite, 76 quasiseparated, 18 infinite, 76 separated, 17 group action Stein, 18 (categorically) free, 87 RLP, 63 h-principle, 78 section holonomic section, 79 holonomic, 79 homotopy cartesian diagram, 64 semistable, 22 homotopy category, 63 simplicial model category, 64 homotopy descent, 83 simplicial resolution, 80, 84 homotopy fibre, 64 slightly finer, 15 homotopy pullback, 64 spectral norm, 13 homotopy sheaf, 29, 81 spherically complete, 35 Stein algebra, 19 interval object, 68 Stein variety, 18 Stiefel space, 88 jet subset of a function, 78 admissible, 14 of a section, 78 affinoid, 14 rational, 14 line bundle, 34 LLP, 63 Tate algebra, 12 114 INDEX universal G-bundle, 87 universal torsor, 87 vector bundle, 28 weak equivalence, 61