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Homotopy Theory for Rigid Analytic Varieties

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Homotopy Theory for Rigid Analytic Varieties Helene Sigloch Homotopy Theory for Rigid Analytic Varieties 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 1 ∼ [X; ] 1 −! fanalytic k-line bundles over Xg= ∼= P Arig between motivic homotopy classes from X into the infinite projective space P1 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 −! falg. K-vector bundles of rank n over Xg= = 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, x4, Proposition 3.8]. Strikingly, if K = C, this classification still holds after analytification. The analyti- fication 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] −! fholomorphic vector bundles of rank n over Xg= = 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. L´arussonshowed that Grauert's theorem can also be phrased in terms of motivic homotopy theory with C as an interval [L´ar03, L´ar04, L´ar05]. 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 ×frg, respectively 1 to X × fsg for some r; s 2 B . Viewing Lr and Ls as line bundles over X, they are 1 1 in the same B -homotopy class in [X; P ]B1 . If we are to get a natural bijection 1 ∼ [X; P ]B1 −! fanalytic k-line bundles over Xg= = ; then for each line bundle L on X × B1 and each pair r; s 2 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 2 fB ; Arigg 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 2 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 fUi × Ariggi2I where fUigi2J 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 fUi × Ariggi2I . 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 2 fB ; Arigg be a representable interval object on R. Theorem 4 (Theorem 5.40). For some n 2 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 −! fanalytic k-vector bundles of rank n over Xg= = : 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.
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