Introduction to Rigid Analytic Geometry Marc Masdeu Abstract These are notes from a 90-minutes talk given in the study group at Warwick in November 2016, on an introduction to rigid analytic geometry. The main references used where [Sch98], [BCD+08, Chapter 1] and [Bos14]. 1 Motivation When studying algebraic varieties over a p-adic field (say over Cp), we would like to use analytic techniques. This is very fruitful when the base field is C, which allows many results to be transferred back and forth (using Serre's GAGA) between the algebraic and analytic worlds. The main problem is that the (non-archimedean) topology on X = Cp is totally discon- nected. This makes the analysis difficult (for example, for any open U ⊆ Cp, the space of locally constant functions on U is of infinite dimension over Cp!). We will restrict the opens that we consider and, more importantly, we will also restrict types of \open coverings" that we allow. Let us recall how schemes are constructed: 1. For each ring R, we consider X = Spec R = fp prime ideals of Rg, first as a set. We topologize X = Spec R by decreeing that a basis of closed sets is given by V (I) = fp j I ⊆ pg ∼= Spec(R=I). There is a convenient basis for this topology. Namely, for each f 2 R, we consider V (f) = Spec(R=(f)) ⊂ X and Xf = X n V (f) = fp j f 62 pg. (the sets Xf are called \principal opens"). 2. Note that X n V (I) = [f2I Xf . 3. One remarks that R can be thought of as the ring of functions on X = Spec(R): given g 2 R, there is a map [ X ! k(p); p 7! image of g in k(p): p2X For example, if k(x) = Rx=mx = C for all x 2 X, then this would be a function to C. In a similar fashion, for each f 2 R we obtain Xf = Spec(Rf ), and so Rf can be thought of as the ring of functions of a principal open. 1 4. Our goal is to define, for each open set U ⊂ X, a ring OX (U) of \functions on U". So far, we have defined OX (X) = R and OX (Xf ) = Rf (localizations). However, although Xfg = Xf \ Xg and so the set of principal opens is closed under finite intersection, it is not closed under unions, so we need to extend the definition of OX a bit more. This is quite easy to do, provided that we have what is called the sheaf axiom: given a basic open Xf and a covering by it by basic opens Xf = [i2I Xfi , then the following diagram (of sets, or of abelian groups) must be exact: Y Y OX (Xf ) ! OX (Xfi ) ! OX (Xfi \ Xfj ): i2I i;j One checks that this is in fact true. The crucial fact that one needs to use is that Xf = Spec(Rf ) is quasi-compact. This allows one to extract a finite subset J ⊂ I and work with only those opens, to glue the sections. 5. Finally, one defines a scheme as a topological space X and a sheaf of rings OX on X S such that X = i2I Ui with (Ui; OX jUi ) = Spec(Ai). We will try to mimic this construction in the non-archimedean world (so the rings to be considered will be rings of convergent power series, instead of the usual polynomial rings used in schemes). 2 Affinoid algebras Fix a non-archimedean field k with valuation ring R and residue field k~ = R=m. The Tate algebra over k in n variables is {︂ }︂ X J X Tn = Tn(k) = khX1;:::;Xni = aJ X j jaJ j −! 0 ; kJk = ji: kJk!1 The Gauss norm on Tn is: X J k aJ X k = max jaJ j: J It enjoys the following basic properties: ∙k f + gk ≤ max(kfk; kgk). ∙k cfk = jcjkfk. ∙k fgk = kfkkgk. 2 ∙ Tn is complete for the induced metric. ∙k fk = supx2B(k) jf(x)j = maxx2B(k) jf(x)j. (Here B(k) = fx 2 k j jxij ≤ 1g). The first properties make (Tn; k · k) into a k-Banach algebra. Note that the function f(x) = x − xp has the property that kfk = 1, but sup jf(x)j = 1=p. So for the maximimum x2Zp principle to hold it is important to allow evaluation at elements in the algebraic closure. The Tate algebra Tn is a \free object" in the sense that fcontinuous k-algebra maps Tn ! kg $ Bn(k) The next step is to prove a form of Weierstrass Preparation in this setting, from which one can show: Proposition 1. ∙ Tn is a Noetherian, regular and UFD. ∙ For every maximal m, the localization (Tn)m has dimension n and its residue field Tn=m is a finite extension of k. ∙ Tn is Jacobson: every prime ideal is the intersection of maximal ideals. ∙ Every ideal in Tn is closed with respect to the Gauss norm. As a consequence, we get kfk = sup jf(x)j; x2Max(Tn) which allows us to avoid speaking of k, and also shows that kfk is an intrinsic norm, inde- pendent of the choice of \coordinates" x1; : : : ; xn. Affinoid algebras An affinoid algebra isa k-algebra A isomorphic to Tn=I for some ideal I ⊂ Tn. Write M(A) = Max(A). Choosing one such isomorphism we get an embedding M(A) ,! M(Tn), with image fx 2 M(Tn) j f(x) = 0 for all f 2 Ig: Note that things start getting a bit more complicated, though: firstly, M(Tn) (and also M(A)) have a lot more points than those coming from Rn. Remark. Note that Ahxi=(x−a) is not the same as A in general! In fact M(Ahai) is identified with the subset of M(A) consisting of elements x such that ja(x)j ≤ 1. 3 However, some sanity is preserved. Every k-affinoid algebra A is noetherian and Jacobson of finite Krull dimension, and A=m = k(m) is a finite extension of k (for each m 2 M(A)). Also, a function a 2 A is nilpotent if and only if a(x) = 0 for all x 2 M(A) (by Jacobson property), and a 2 A× if and only if a(x) 6= 0 for all x 2 M(A). So we can think of elements of A as functions on M(A) (valued in fields that vary at each points). Affinoid algebras admit a k-Banach algebra structure (using the residue norm) that is ∼ defined using a presentation A = Tn=I. In fact, this definition depends on the presentation, but the topology induced from it does not. So concepts like boundedness still make sense. There is a maximum modulus principle, which can be interpreted as saying that M(A) \wants to be compact": for all f 2 A, we have kfksup := sup jf(x)j = max jf(x)j < 1: x2M(A) x2M(A) If A is reduced, the above gives a k-Banach structure on A which is intrinsic, and recovers the Gauss norm when A = Tn. The only catch is that it is not multiplicative, in this case (only power-multiplicative). Lemma 2. If a 2 A, then a is power-bounded if and only if ja(x)j ≤ 1 for all x 2 M(A). The canonical topology We want to put a Hausdorff topology on M(A). This can be done as follows: ∙ Let A(k) denote the set of k-algebra maps A ! k with image contained in a finite extension of k. The Galois group Gk = Gal(k=k) acts on it by composition. ∙ For each x 2 M(A), choosing an embedding i: k(x) ,! k gives an element in A(k). Changing the embedding gives an element which is in the same Gk-orbit. Hence this gives a well-defined map M(A) ! A(k)=Gk, which turns out to be a bijection. All this is also functorial in A. ∙ Put on A(k) a topology by taking as basis of open sets the sets of the form fx 2 A(k) j jfi(x)j ≥ 휖i; jgj(x)j ≤ 휂j; 8i; jg: ∙ Give M(A) the quotient topology. One checks that the resulting topology is Hausdorff, totally disconnected, and functorial in A. This is called the canonical topology. 4 Laurent, Weierstrass and rational domains If X = M(A) is an affinoid domain (that is, A is an affinoid algebra), then a Laurent domain is a set of the form −1 Xhf; g i = fx 2 X j jfi(x)j ≤ 1; jgj(x)j ≥ 1g: This in fact is M(Ahf; g−1i), where AhX;Y i Ahf; g−1i = : (X1 − f1; : : : ; g1Y1 − 1;:::) These have a universal property, in the same way that the Tate algebras Tn have as well: for any map of k-Banach algebras 휑: A ! B, the diagram A / Ahf; g−1i 휑 $ B can be filled in in at most one way, and such an arrow exists if and onlyif 휑(fi) are power- × −1 bounded for all i, and 휑(gj) 2 B and 휑(gj) are power-bounded for all j. The set X = M(A) is called a Weierstrass domain in the special case that there are no gj's. All of these play the role of \opens" in our theory, as we will see. Next, let f1; : : : ; fn and g be elements of A with no common zero.