Lectures on Analytic Geometry Peter Scholze (all results joint with Dustin Clausen) Contents Analytic Geometry 5 Preface 5 1. Lecture I: Introduction 6 2. Lecture II: Solid modules 11 3. Lecture III: Condensed R-vector spaces 16 4. Lecture IV: M-complete condensed R-vector spaces 20 Appendix to Lecture IV: Quasiseparated condensed sets 26 + 5. Lecture V: Entropy and a real BdR 28 6. Lecture VI: Statement of main result 33 Appendix to Lecture VI: Recollections on analytic rings 39 7. Lecture VII: Z((T ))>r is a principal ideal domain 42 8. Lecture VIII: Reduction to \Banach spaces" 47 Appendix to Lecture VIII: Completions of normed abelian groups 54 Appendix to Lecture VIII: Derived inverse limits 56 9. Lecture IX: End of proof 57 Appendix to Lecture IX: Some normed homological algebra 65 10. Lecture X: Some computations with liquid modules 69 11. Lecture XI: Towards localization 73 12. Lecture XII: Localizations 79 Appendix to Lecture XII: Topological invariance of analytic ring structures 86 Appendix to Lecture XII: Frobenius 89 Appendix to Lecture XII: Normalizations of analytic animated rings 93 13. Lecture XIII: Analytic spaces 95 14. Lecture XIV: Varia 103 Bibliography 109 3 Analytic Geometry Preface These are lectures notes for a course on analytic geometry taught in the winter term 2019/20 at the University of Bonn. The material presented is part of joint work with Dustin Clausen. The goal of this course is to use the formalism of analytic rings as defined in the course on condensed mathematics to define a category of analytic spaces that contains (for example) adic spaces and complex-analytic spaces, and to adapt the basics of algebraic geometry to this context; in particular, the theory of quasicoherent sheaves. October 2019, Peter Scholze 5 6 ANALYTIC GEOMETRY 1. Lecture I: Introduction Mumford writes in Curves and their Jacobians: \[Algebraic geometry] seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate." For some reason, this secret plot has so far stopped short of taking over analysis. The goal of this course is to launch a new attack, turning functional analysis into a branch of commutative algebra, and various types of analytic geometry (like manifolds) into algebraic geometry. Whether this will make these subjects equally esoteric will be left to the reader's judgement. What do we mean by analytic geometry? We mean a version of algebraic geometry that (1) instead of merely allowing polynomial rings as its basic building blocks, allows rings of convergent power series as basic building blocks; (2) instead of being able to define open subsets only by the nonvanishing of functions, one can define open subsets by asking that a function is small, say less than 1; (3) strictly generalizes algebraic geometry in the sense that the category of schemes, the theory of quasicoherent sheaves over them, etc., all embed fully faithfully into the corresponding analytic category. The author always had the impression that the highly categorical techniques of algebraic geom- etry could not possibly be applied in analytic situations; and certainly not over the real numbers.1 The goal of this course is to correct this impression. How can one build algebraic geometry? One perspective is that one starts with the abelian category Ab of abelian groups, with its symmetric monoidal tensor product. Then one can consider rings R in this category (which are just usual rings), and over any ring R one can consider modules M in that category (which are just usual R-modules). Now to any R, one associates the space Spec R, essentially by declaring that (basic) open subsets of Spec R correspond to localizations R[f −1] of R. One can then glue these Spec R's together along open subsets to form schemes, and accordingly the category of R-modules glues to form the category of quasicoherent sheaves. Now how to build analytic geometry? We are basically stuck with the first step: We want an abelian category of some kind of topological abelian groups (together with a symmetric monoidal tensor product that behaves reasonably). Unfortunately, topological abelian groups do not form an abelian category: A map of topological abelian groups that is an isomorphism of underlying abelian groups but merely changes the topology, say (R; discrete topology) ! (R; natural topology); has trivial kernel and cokernel, but is not an isomorphism. This problem was solved in the course on condensed mathematics last semester, by replacing the category of topological spaces with the much more algebraic category of condensed sets, and accordingly topological abelian groups with abelian group objects in condensed sets, i.e. condensed abelian groups. Let us recall the definition. 1In the nonarchimedean case, the theory of adic spaces goes a long way towards fulfilling these goals, but it has its shortcomings: Notably, it lacks a theory of quasicoherent sheaves, and in the nonnoetherian case the general formalism does not work well (e.g. the structure presheaf fails to be sheaf). Moreover, the language of adic spaces cannot easily be modified to cover complex manifolds. This is possible by using Berkovich spaces, but again there is no theory of quasicoherent sheaves etc. 1. LECTURE I: INTRODUCTION 7 ∼ Definition 1.1. Consider the pro-´etalesite ∗pro´et of the point ∗, i.e. the category ProFin = Pro(Fin) of profinite sets S with covers given by finite families of jointly surjective maps. A condensed set is a sheaf on ∗pro´et; similarly, a condensed abelian group, ring, etc. is a sheaf of 2 abelian groups, rings, etc. on ∗pro´et. As for sheaves on any site, it is then true that a condensed abelian group, ring, etc. is the same as an abelian group object/ring object/etc. in the category of condensed sets: for example, a condensed abelian group is a condensed set M together with a map M ×M ! M (the addition map) of condensed sets making certain diagrams commute that codify commutativity and associativity, and admits an inverse map. More concretely, a condensed set is a functor X : ProFinop ! Sets such that (1) one has X(;) = ∗, and for all profinite sets S1, S2, the map X(S1 t S2) ! X(S1) × X(S2) is bijective; (2) for any surjection S0 ! S of profinite sets, the map 0 ∗ ∗ 0 0 X(S) ! fx 2 X(S ) j p1(x) = p2(x) 2 X(S ×S S )g 0 0 0 is bijective, where p1; p2 : S ×S S ! S are the two projections. How to think about a condensed set? The value X(∗) should be thought of as the underlying set, and intuitively X(S) is the space of continuous maps from S into X. For example, if T is a topological space, one can define a condensed X = T via T (S) = Cont(S; T ); the set of continuous maps from S into T . One can verify that this defines a condensed set.3 Part (1) is clear, and for part (2) the key point is that any surjective map of profinite sets is a quotient map. For example, in analysis a central notion is that of a sequence x0; x1;::: converging to x1. This is codified by maps from the profinite set S = f0; 1;:::; 1g (the one-point compactification N [ f1g of N) into X. Allowing more general profinite sets makes it possible to capture more subtle convergence behaviour. For example, if T is a compact Hausdorff space and you have any sequence of points x0; x1;::: 2 T , then it is not necessarily the case that it converges in T ; but one can always find convergent subsequences. More precisely, for each ultrafilter U on N, one can take the limit along the ultrafilter. In fact, this gives a map βN ! T from the space βN of ultrafilters on N. The space βN is a profinite set, known as the Stone-Cechˇ compactification of N; it is the initial compact Hausdorff space to which N maps (by an argument along these lines). 2As discussed last semester, this definition has minor set-theoretic issues. We explained then how to resolve them; we will mostly ignore the issues in these lectures. 3There are some set-theoretic subtleties with this assertion if T does not satisfy the separation axiom T 1, i.e. its points are not closed; we will only apply this functor under this assumption. 8 ANALYTIC GEOMETRY We have the following result about the relation between topological spaces and condensed sets, proved last semester except for the last assertion in part (4), which can be found for example in [BS15, Lemma 4.3.7].4 Proposition 1.2. Consider the functor T 7! T from topological spaces to condensed sets. (1) The functor has a left adjoint X 7! X(∗)top sending any condensed set X to its underlying set X(∗) equipped with the quotient topology arising from the map G S ! X(∗): S;a2X(S) (2) Restricted to compactly generated (e.g., first-countable, e.g., metrizable) topological spaces, the functor is fully faithful. (3) The functor induces an equivalence between the category of compact Hausdorff spaces, and quasicompact quasiseparated condensed sets. (4) The functor induces a fully faithful functor from the category of compactly generated weak Hausdorff spaces (the standard \convenient category of topological spaces" in algebraic topology), to quasiseparated condensed sets. The category of quasiseparated condensed sets is equivalent to the category of ind-compact Hausdorff spaces \ lim "T where all transition −!i i maps Ti ! Tj are closed immersions.