How To Understand The Theory Of Algebraic Geometry

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1 FORMAL AND RIGID GEOMETRY: AN INTUITIVE INTRODUCTION, AND SOME APPLICATIONS JOHANNES NICAISE Abstract. We give an informal introduction to formal and rigid geometry over complete discrete valuation rings, and we discuss some applications in algebraic and arithmetic geometry and singularity theory, with special emphasis on recent applications to the Milnor fibration and the motivic zeta function by J. Sebag and the author. 1. Introduction Let R be a complete discrete valuation ring, with quotient field K, and residue field k. We choose a uniformizing parameter π, i.e. (π) is the unique maximal ideal of R. Geometers may take R = C[[t]], the ring of formal power series over the complex numbers, with K = C((t)), k = C, π = t, while number theorists might prefer to think of R = Z p, the ring of p-adic integers, with K = Q p, k = F p, π = p. A formal scheme over R consists of an algebraic variety over k, together with algebraic information on an infinitesimal neighbourhood of this variety. If X is a variety over R, we can associate to X its formal completion X, a formal scheme over R, in a natural way. An important aspect of the formal scheme X is the following phenomenon. A closed point x on the scheme-theoretic generic fiber of X over K has coordinates in some finite extension of the field K, and there is no natural way to associate to the point x a point of the special fiber X 0 /k by reduction modulo π. However, by inverting π in the structure sheaf of X, we can associate a generic fiber Xη to the formal scheme X, which is a rigid variety over K. Rigid geometry provides a satisfactory theory of analytic geometry over non-archimedean fields. A point on X η has coordinates in the ring of integers of some finite extension K of K: if we denote by R the normalization of R in K, we can canonically identify X η (K ) with X(R ). By reduction modulo π, we obtain a canonical retraction of the generic fiber X η to the special fiber X 0. Roughly speaking, the formal scheme X has the advantage that its generic and its special fiber are tightly connected, and what glues them together are the R -sections on X, where R runs over the finite extensions of R. A main disadvantage of rigid geometry is the artificial nature of the topology on rigid varieties: it is not a classical topology, but a Grothendieck topology. In the nineties, Berkovich developed his spectral theory of non-archimedean spaces. His spaces carry a true topology, which allows to apply classical techniques from algebraic topology. In particular, the unit disc R becomes connected by arcs, while it is totally disconnected w.r.t. its π-adic topology. 1

2 2 JOHANNES NICAISE In Section 3, we give a brief survey of the basic theory of formal schemes, and Section 4 is a crash course on rigid geometry. Section 5 contains the basic defnitions of Berkovich approach to non-archimedean geometry. In the final Section 6, we briefly discuss some applications of the theory, with special emphasis on the relation with arc spaces of algebraic varieties, and the Milnor fibration. This intuitive introduction merely aims to provide some insight into the theory of formal schemes and rigid varieties. We do not provide proofs; instead, we chose to give a list of more thorough introductions to the different topics dealt with in this note. 2. Conventions and notation For any field F, we denote by F alg and F s its algebraic, resp. separable closure. If S is any scheme, a S-variety is a separated reduced scheme of finite type over S. Throughout this note, R denotes a complete discrete valuation ring, with residue field k, and quotient field K. We fix a uniformizing parameter π, i.e. a generator of the maximal ideal of R. For any integer n 0, we denote by R n the quotient R/(π n+1 ). The discrete valuation on K defines a non-archimedean absolute value. on K, which induces a topology on K, called the π-adic topology. The sets π n R, n 0, form a fundamental system of open neighbourhoods of the origin in K. The resulting topology is totally disconnected. The absolute value on K extends uniquely to an absolute value on K alg. For any integer m > 0, we endow (K alg ) m with the norm z := max i=1,...,m z i. We denote by R s the integral closure of R in K s, and by K s the completion of K s Likewise, R alg denotes the integral closure of R in K alg. 3. Formal geometry In this note, we will only consider formal schemes topologically of finite type over the complete discrete valuation ring R. This case is in many respects simpler than the general one, but it serves our purposes. For a more thorough introduction to the theory of formal schemes, we refer to [22, 10], [21, no. 182], [27], or [12]. Intuitively, a formal scheme X over R consists of its special fiber X 0, which is a scheme over k, endowed with a structure sheaf containing additional algebraic information on an infinitesimal neighbourhood of X Affine formal schemes. For any system of variables x = (x 1,..., x m ), we define a Noetherian R-algebra R{x} as the projective limit R{x} := lim R n [x] n The elements of R{x} are exactly the convergent power series over R, i.e. power series m c(x) = c i i=(i 1,...,i m) N m j=1 with coefficients c i in R, such that (c i ) converges to zero (w.r.t. the π-adic topology) as i tends to. This means that for each n N, there exists a value i 0 N x i j j

3 FORMAL AND RIGID GEOMETRY, AND SOME APPLICATIONS 3 such that c i is divisible by π n if i > i 0. Note that this is exactly the condition which guarantees that the images of c(x) in the truncations R n [[x]] are actually polynomials, i.e. belong to R n [x]. The algebra R{x} is the sub-algebra of R[[x]] consisting of those power series which converge on the closed unit disc R m = {z K m z 1}, since an infinite sum converges in a non-archimedean field iff its terms tend to zero. An algebra topologically of finite type over R, is an R-algebra A isomorphic to an algebra of the form R{x 1,..., x m }/I, for some integer m > 0 and some ideal I. For any integer n 0, we denote by A n the quotient A/(π n+1 ). To any such algebra A, we can associate the affine formal scheme Spf A. This object is a locally ringed space in R-algebras ( A, O A ), and it is defined as the direct limit Spf A := lim n Spec A n in the category of locally ringed spaces. Note that the transition morphisms Spec A m Spec A n, m n, are homeomorphisms on the underlying topological spaces. Hence, the underlying topological space Spf A is the set of open prime ideals J of A (i.e. prime ideals containing π), endowed with the Zariski topology, and it is homeomorphic to Spec A 0. The special fiber X 0 of the affine formal scheme X = Spec A is the k-scheme X 0 = Spec A 0. A morphism between affine formal schemes is by definition a local morphism of locally ringed spaces in R-algebras. The functor Spf induces an equivalence between the opposite category of R-algebras topologically of finite type, and the category of affine formal schemes 1, just like in the algebraic scheme case Formal schemes. A formal scheme X topologically of finite type (tft) over R is obtained by gluing a finite number of affine formal schemes. It is often convenient to describe X in terms of the direct system (X n := X R R n ), n 0, where the locally ringed space X n is a scheme of finite type over R n, for any n, and the transition map of R n -schemes u m,n : X m X n induces an isomorphism of R m -schemes X m = Xn Rn R m for any pair m n. Any such direct system (X n, u m,n ) determines a tft formal R-scheme X by putting X := lim n X n as a locally ringed space. A morphism f : X Y between tft formal R- schemes is nothing but a system (f n : X n Y n ), n 0, where f n is a morphism of R n -schemes, and all the squares X m f m X n f n Y m Y n commute. We call the space X 0 the special fiber of X. It is a k-scheme of finite type. The formal scheme X is called separated if the scheme X n is separated, for each n. In fact, this will be the case as soon as the special fiber X 0 is separated. We 1 A morphism of R-algebras topologically of finite type is automatically continuous, since it maps π to itself.

4 4 JOHANNES NICAISE will work in the category of separated formal schemes, topologically of finite type over R; we ll call these objects stf t formal R-schemes. An stft formal scheme X over R is flat if its structural sheaf has no π-torsion. A typical example of a non-flat stft formal R-scheme is one with an irreducible component concentrated in the special fiber. Any stf t formal R-scheme has a maximal flat closed formal subscheme, obtained by killing π-torsion Coherent modules. Let A be an algebra, topologically of finite type over R. An A-module N is coherent iff it is finitely generated. Any such module N defines a sheaf of modules on Spf A in the usual way. A coherent sheaf of modules N on a stft formal R-scheme X is obtained by gluing coherent modules on affine open formal subschemes. More conveniently, a coherent sheaf N on X is defined by a direct system (N n ), n 0, where N n is a coherent sheaf on X n, and the O Xn -linear transition map v m,n : N m N n induces an isomorphism of coherent O Xm -modules N m = u m,n N n for any pair m n The completion functor. Any separated R-scheme of finite type X defines an inductive system (X n = X R R n ), n 0, and this system defines, at its turn, an stft formal R-scheme X. This formal scheme is called the formal (π-adic) completion of the R-scheme X. Its special fiber X 0 coincides with the fiber of X over the closed point of Spec R. The formal scheme X is flat iff X is flat over R. Intuitively, X should be seen as the infinitesimal neighbourhood of X0 in X. Example 1. If X = Spec R[x 1,..., x n ]/(f 1,..., f l ), then its formal completion X is simply Spf R{x 1,..., x n }/(f 1,..., f l ). A morphism of separated R-schemes of finite type f : X Y induces a system of morphisms f n : X n Y n, and hence a morphism of formal R-schemes ˆf : X Ŷ between the formal π-adic completions of X and Y. We get a completion functor : (sft Sch/R) (stft F or/r) : X X where (sf t Sch/R) denotes the category of separated R-schemes of finite type, and (stf t F or/r) denotes the category of separated formal schemes, topologically of finite type over R. For a general pair of separated R-schemes of finite type X, Y, the completion map Hom (sft Sch/R) (X, Y ) Hom (stft F or/r) ( X, Ŷ ) : f f is not a bijection. It is a bijection, however, if X is proper over R: this is a corollary of Grothendieck s Existence Theorem; see [23, 5.4.1]. In particular, the completion map induces a bijection between R -sections of X, and R -sections of X, for any finite extension R of the complete discrete valuation ring R. If X is a separated R-scheme of finite type, and N is a coherent sheaf of modules on X, then N induces a direct system (N n ), n 0, where N n is a coherent sheaf of modules on X n. This system defines a coherent sheaf of modules N on X. If X is proper over R, it follows from Grothendieck s Existence Theorem that the functor N N is an equivalence between the category of coherent modules on X and the category of coherent modules on X; see [23, 5.1.6]. If a stft formal R-scheme Y is isomorphic to the π-adic completion Ŷ of a separated R-scheme Y of finite type, we call the formal scheme Y algebrizable,

5 FORMAL AND RIGID GEOMETRY, AND SOME APPLICATIONS 5 with algebraic model Y. The following theorem is the main criterion to recognize algebrizable formal schemes [23, 5.4.5]: if Y is proper over R, and L is an invertible O Y -bundle such that the pull-back L 0 of L to Y 0 is ample, then Y is algebrizable. Moreover, the algebraic model Y for Y is unique up to canonical isomorphism, there exists a unique line bundle M on Y with L = M, and M is ample Formal blow-ups. Let X be a stft formal R-scheme, and let I be an ideal sheaf on X (defined in the usual way), such that I contains a power of the uniformizing parameter π. We can define the formal blow-up of X at the center I. This formal blow-up is the limit of the system (π n : X n X n ), where π n is the blow-up of the scheme X n, with as center the pull-back of I to X n. It is a stft formal R-scheme, and if X is flat over R, then so is its blow-up at I. If X is algebrizable, with algebraic model X, and I is isomorphic to Ĵ, with J an ideal sheaf on X, then the formal blow-up of X at the center I is the formal completion of the blow-up of X at the center J. 4. Rigid geometry In this note, we ll only be able to cover the very basics of rigid geometry. We refer the reader to [33, 12, 7, 9, 19] for a more thorough introduction Analytic geometry over non-archimedean fields. Let us consider the complete discrete valuation field K (the reader may think of a field of Laurent series k((t)), with the t-adic valuation, or of the field of p-adic numbers Q p, with the p-adic valuation). The valuation induces a non-archimedean norm. on K, which extends uniquely to a norm on the algebraic closure K alg. The discrete valuation ring R (resp. the ring R alg ) consists of the elements z of K (resp. K alg ) with z 1 (the closed unit disc in K, resp. K alg ). Using this norm on K, we might try to develop a theory of analytic geometry over the base field K. Naïvely, we can define analytic functions as functions which are locally defined by a convergent power series. However, we are immediately confronted with some pathological phenomena. Consider, for instance, the p-adic unit disc The partition Z p = {x Q p x 1} {p Z p, 1 + p Z p,..., (p 1) + p Z p } is an open cover of Z p. Hence, the characteristic function of p Z p is analytic, according to our naïve definition. This contradicts some elementary properties that one expects an analytic function to have, such as uniqueness of analytic continuation. The cause of this and similar pathologies, is the fact that the unit disc Z p is totally disconnected with respect to the p-adic topology. Rigid geometry can be seen as a more refined approach to non-archimedean analytic geometry, turning the unit disc into a connected space. Rigid spaces are endowed with a certain Grothendieck topology, only allowing a special type of covers. We ll indicate two possible approaches to the theory of rigid varieties over K. The first one is due to Tate [35], the second one to Raynaud [33].

6 6 JOHANNES NICAISE 4.2. Tate algebras. The basic objects in Tate s theory are the algebras T m := K{x 1,..., x m } = {f = m x i j j α i K, α i 0 as i } α i i N m j=1 The convergence condition means that, for any n N, there exists i 0 N such that for i > i 0, the coefficient c i belongs to R and is divisible by π n. The algebra T m is the algebra of power series over K which converge on the closed unit polydisc R m in K m (since an infinite sum converges in non-archimedean field iff its terms tend to zero). Note that T m = R{x1,..., x m } R K. Analogously, we can define an algebra of convergent power series B{x 1,..., x m } for any Banach algebra B. The algebra T m is a Banach algebra for the sup-norm f sup = max i α i. It is Noetherian, and any ideal I is closed, so that the quotient T m /I is again a Banach algebra. A Tate algebra, or affinoid algebra, is a K-algebra isomorphic to such a quotient T m /I. If we denote by R alg the normalization of R in an algebraic closure K alg of K, then the maximal ideals of T m correspond bijectively to G(K alg /K)-orbits of tuples (z 1,..., z m ), with z i R alg. This follows from [8, ], where it is shown that any morphism of K-algebras ϕ : T m K alg is contractive, in the sense that ϕ(a) a sup for any a in T m, so in particular, ϕ(x i ) belongs to R alg. Remark 1. Intuitively, the reason that we obtain tuples of elements in R alg, rather than K alg, is the following: if z is an element of K, then x z is invertible in K{x} iff z does not belong to R (since 1/(x z) converges on the closed unit disc R iff z / R). Its inverse is given by (1/z) i 0 (x/z)i ; note that this formal power series converges iff z > 1, i.e. z / R Affinoid spaces. The category of affinoid spaces over K is by definition the opposite category of the category of Tate algebras over K. For any affinoid space X, we ll denote the corresponding Tate algebra by A(X). Conversely, for any Tate algebra A, we denote the corresponding affinoid space by Spm A (some authors use the notation Sp instead). To any affinoid space X = Spm A over K, we can associate the set X of maximal ideals of the Tate algebra A = A(X). If we present A as a quotient T m /(f 1,..., f n ), then elements of Spm A corresponds bijectively to G(K alg /K)- orbits of tuples z = (z 1,..., z m ), with z i R alg, and f j (z) = 0 for each j. For any maximal ideal x of A, the quotient A/x is a finite extension of K, and carries a unique prolongation of the absolute value. on K. Hence, for any f A, we can speak of the value f(x) of f at x, and its absolute value f(x). The sup-norm (or rather, semi-norm) on A is defined by f sup := sup f(x) x X By the maximum principle [8, ], this supremum is, in fact, a maximum, i.e. there is a point x in X with f(x) = f sup. Moreover, for A = T m, this definition concides with the one in the previous section, by [8, 1.4.7]. The canonical topology on X is the inital topology w.r.t. the functions x f(x), where f varies in A. This topology is totally disconnected. If A = K{x}, we can view R as a subset of X, and the induced topology on R is precisely the π-adic topology.

7 FORMAL AND RIGID GEOMETRY, AND SOME APPLICATIONS 7 If h : X Y is a morphism of affinoid spaces, we ll denote by h the induced morphism on the level of the underlying sets. This notation should not be confused with the notation f(x) introduced above. A closed immersion h : X Y of affinoid spaces over K is a morphism of the form A(Y ) A(Y )/I = A(X), for some ideal I in A(Y ) Open covers. A morphism h : Y X of affinoid spaces over K is called an open immersion, if it satisfies the following two properties: h is an injection, and for any morphism g : Z X of affinoid spaces over K, such that g factors through h, there is a unique morphism g : Z Y such that g = h g. An open cover of X is a set of open immersions u i : U i X such that their images cover X as a set. A special kind of open covers is constructed as follows: take a set of elements f 1,..., f n in A(X), which generate the ideal A(X). Consider, for each i = 1,..., n, the affinoid space U i given by A(U i ) = A(X){T 1,..., T n }/(f j T j f i ) j=1,...,n An open subspace of this type is called a rational subspace. The obvious morphisms u i : U i X form an open cover. The image of u i is the set of points x of X, such that f i (x) is maximal among the f j (x), j = 1,..., n, since a morphism of K-algebras x : A(X) K alg factors through a morphism of K-algebras x : A(U i ) K alg iff T j (x) = f j (x)/f i (x) belongs to R alg, i.e. f j (x) f i (x). We call such an open cover a standard cover. By Tate s Acyclicity Theorem [8, 1.9], any finite open cover {u i : U i X} satisfies the gluing property (meaning that the algebras A(U i ) satisfy the sheaf properties). Moreover, any finite open cover can be refined by a standard cover. We can define a Grothendieck topology on an affinoid space X: the admissible open sets are rational subspaces, and the admissible covers are covers by rational subspaces which have a finite subcover 2. We can define a presheaf O X on X with respect to this Grothendieck topology, with O X (U i ) = A(U i ) for any rational subspace U i of X. By Tate s Acyclicity Theorem, O X is a sheaf. One can check that the affinoid space associated to T m is connected with respect to this Grothendieck topology, for any m Rigid varieties. Now, we can define a general rigid variety over K. It is a set X, endowed with a Grothendieck topology and a sheaf of K-algebras O X, such that X has an admissible cover {U i } i I, where each U i is isomorphic to an affinoid space. A morphism Y X of rigid varieties over K consists of a continuous morphism h : Y X, and a local homomorphism of sheaves of K-algebras h : O Y h O X. The category of affinoid spaces over K is a fully faithful subcategory of the category of rigid varieties over K. A rigid variety over K is called quasi-compact if it is a finite union of affinoid subspaces. It is called quasi-separated if the intersection of any pair of affinoid open subspaces is quasi-compact, and separated if the diagonal morphism is a closed immersion Rigid spaces and formal schemes. Finally, we come to a second approach to the theory of rigid spaces, due to Raynaud [33]. We ll follow the constructions given in [7]. We saw before that the underlying topological space of a stft formal R- scheme X coincides with the underlying space of its special fiber X 0. Nevertheless, 2 One can define several finer Grothendieck topologies on affinoid spaces, to obtain gluing properties for rigid varieties. We will not go into this matter here.

8 8 JOHANNES NICAISE the structure sheaf of X contains information on an infinitesimal neighbourhood of X 0, so one might hope to construct the generic fiber X η of X. As it turns out, this is indeed possible, but we have to leave the category of (formal) schemes: this generic fiber X η is a rigid variety over K The affine case. Let A be an algebra topologically of finite type over R, and consider the affine formal scheme X = Spf A. The tensor product A R K is a Tate algebra, and the generic fiber X η of X is simply the affinoid space associated to A R K. Let K be any finite extension of K, and denote by R the normalization of R in K. We will construct a bijection between the set of morphisms of formal R-schemes Spf R X, and the set of morphisms of rigid K-varieties Spm K X η. Consider a morphism of formal R-schemes Spf R X, or, equivalently, a morphism of R-algebras A R. Tensoring with K yields a morphism of K- algebras A R K R R K = K, and hence a K -point of X η. Conversely, for any morphism of K-algebras A R K K, the image of A will be contained in R, by [8, ]. To any R -section on X, we can associate a point of X, namely the image of the singleton Spf R. In this way, we obtain a specialization map sp : X η X = X The general case. Let X be any stft formal R-scheme. We define X η as the set of all integral closed formal subschemes of X, which are finite and flat over R. Any such formal subscheme is contained in an affine open formal subscheme of X, and by taking its unique closed point, we get a specialization map sp : X η X For any affine open formal subscheme U of X, there is a natural bijection between sp 1 ( U ) and U η. We can use this bijection to endow sp 1 ( U ) with the structure of an affinoid space. All these structures are compatible on X η, and define a rigid K-variety X η, the generic fiber of X. This structure makes the specialization map into a morphism of ringed sites. The generic fiber of a stft formal R-scheme is a separated, quasi-compact rigid K-variety. The formal scheme X is called a formal R-model for the rigid K-variety X η. Since the generic fiber is obtained by inverting π, it is clear that the generic fiber does not change if we replace X by its maximal flat closed formal subscheme (by killing π-torsion). Hence, we might as well restrict to flat stft formal R-schemes. The construction of the generic fiber is functorial: a morphism of stf t formal R-schemes h : Y X induces a morphism of rigid K-varieties h η : Y η X η, and the square commutes. We get a functor sp Y η Y h η Xη sp h X (. ) η : (F stft F or/r) (sqc Rig/K) : X X η

9 FORMAL AND RIGID GEOMETRY, AND SOME APPLICATIONS 9 from the category of flat stf t formal R-schemes, to the category of separated, quasi-compact rigid K-varieties. This functor is compatible with fibered products. For any locally closed subset Z of X 0, the inverse image sp 1 (Z) is called the tube of Z in X, and denoted by ]Z[. If Z is open in X 0, then ]Z[ is canonically isomorphic to the generic fiber of the open formal subscheme Z of X with underlying space Z ; this is a quasi-compact open subspace of X η, and ]Z[ X η is an open immersion. For any closed subscheme Z of X 0, the inverse image sp 1 (Z) is an open rigid subspace of X η in a natural way, not quasi-compact in general. Berthelot showed in [7, 0.2.6] how to construct the generic fiber of a broader class of formal R- schemes, not necessarily tft. In particular, using his construction, ]Z[ is canonically isomorphic to the generic fiber of the formal completion of X along Z (this formal completion is the locally topologically ringed space with underlying topological space Z and strucure sheaf lim O X /IZ n n where I Z is the defining ideal sheaf of Z in X ). In particular, if Z is a closed point x of X 0, then ]x[ is the generic fiber of the formal spectrum of the completed local ring ÔX,x Localization by formal blow-ups. The functor (. ) η is not an equivalence, since formal blow-ups are turned into isomorphisms; intuitively, it is clear that blowing-up at a center included in the special fiber, does not change the generic fiber of a formal scheme. In some sense, this is the only restriction. Denote by C the category of flat stft formal R-schemes, localized with respect to the formal blow-ups (this means that we formally add inverse morphisms for formal blow-ups, thus turning them into isomorphisms). The objects of C are simply the objects of (F stft F or/r), but a morphism in C from Y to X, is given by a formal blow-up π : Y Y, and a morphism of stft formal R-schemes Y X. The functor (. ) η factors though a functor C (sqc Rig/K). Raynaud showed that this is an equivalence of categories (a detailed proof is given in [12]). This means that the category of separated, quasi-compact rigid K-varieties, can be described entirely in terms of formal schemes. To give an idea of this dictionary between formal schemes and rigid varieties, we list some results. Let X η and Y η be separated, quasi-compact rigid varieties over K. [12, 4.4] For any finite cover U of X η by quasi-compact open subspaces, there exists a formal model X of X η, and a Zariski cover {U 1,..., U s } of X 0 such that U = { ]U 1 [,..., ]U s [ }. [12, 4.1(d)] Given an isomorphism ϕ : Y η X η, there exist a stft formal scheme Y and formal blow-ups f : Y Y and g : Y X, such that g η = ϕ f η. See [12, 13, 14, 15] for many others Analytification of a K-variety. For any separated K-scheme X of finite type, we can endow the set X o of closed points of X with the structure of a rigid K-variety. More precisely, by [7, 0.3.3] there exists a functor (. ) an : (sft Sch/K) (s Rig/K)

10 10 JOHANNES NICAISE from the category of separated K-schemes of finite type, to the category of separated rigid K-varieties, such that (1) For any separated K-scheme of finite type X, there exists a natural morphism of ringed sites i : X an X which induces a bijection between the set X an and the set X o of closed points of X. Moreover, if f : X X is a morphism of separated K- schemes of finite type, the square X i f X i (X ) an f an X an commutes. (2) If U is Zariski-open in X, then U an is a union of admissible opens in X an. (3) The functor (. ) an commutes with fibered products, and takes open (resp. closed) immersions of K-schemes to open (resp. closed) immersion of rigid K-varieties. We call X an the analytification of X. It is quasi-compact if X is proper over K, but not in general. The analytification functor has the classical GAGA properties: if X is proper over K, then analytification induces an equivalence between coherent sheaves on X and coherent sheaves on X an, and the cohomology groups agree; a closed rigid subvariety of X an is the analytification of an algebraic subvariety of X; and a map X an Y an is algebraic, for any proper K-variety Y. These results can be deduced from Grothendieck s Existence Theorem; see [28, 2.8] Proper R-varieties. Now let X be a separated scheme of finite type over R, and denote by X K its generic fiber. By [7, 0.3.5], there exists a canonical open immersion α : ( X) η (X K ) an. If X is proper over R, then α is an isomorphism. For a separated proper R-scheme of finite type R, we can describe the specialization map sp : (X K ) o = (X K ) an = X η X = X 0 as follows: let x be a closed point of X K, denote by K its residue field, and by R the normalization of R in K. The valuative criterion for properness states that the morphism Spec R Spec R lifts to a unique morphism h : Spec R X K with h(spec K ) = x. If we denote by 0 the closed point of Spec R, then sp(x) = h(0) X 0. In general, the open immersion α : Xη (X K ) an is strict. Consider, for instance, a proper R-variety X, and let X be the variety obtained by removing a point x from the special fiber X 0. Then X K = X K; however, by taking the formal completion X, we loose all the points in X η that map to x under sp, i.e. X η = X η \ ]x[. 5. Berkovich spaces We recall some definitions from Berkovich theory of analytic spaces over nonarchimedean fields. We refer to [2], or to [6] for a short introduction. A very nice survey of the theory, and some of its applications, is given in [18].

11 FORMAL AND RIGID GEOMETRY, AND SOME APPLICATIONS 11 For a commutative Banach ring with unity (A, ), the spectrum M (A ) is the set of all bounded multiplicative semi-norms : A R +. A point x of M (A ) gives rise to a bounded character χ x : A H (x), where H (x) is the completion of the fraction field of the quotient of A by the kernel of x. We denote the image of f A under χ x by f(x), and to stay close to the classical notation, we write f(x) for x(f) R +. We endow M (A ) with the initial topology with respect to all the morphisms x f(x), where f varies over A. We ll refer to this topology as the spectral topology on M (A ). It makes M (A ) into a non-empty compact Hausdorff topological space [2, 1.2.1]. Let L be a non-archimedean field. A strictly L-affinoid algebra is a quotient of the Banach algebra of restricted power series L{T 1,..., T n } (with the quotient norm, which depends on the chosen presentation), for some positive integer n. The spectrum M (A ) of a strictly L-affinoid algebra A is called a strictly L-affinoid space. General strictly L-analytic spaces are obtained by gluing strictly L-affinoid spaces. For a Hausdorff strictly L-analytic space X, the set of rigid points X rig := {x X [H (x) : L] < } can be endowed with the structure of a quasi-separated rigid variety over L. Moreover, the functor X X rig induces an equivalence between the category of paracompact strictly L-analytic spaces, and the category of quasi-separated rigid analytic spaces over L which have an admissible affinoid covering of finite type [3, 1.6.1]. The space X rig is quasi-compact if and only if X is compact. For any quasi-compact rigid variety X η over K, we denote by M (X η ) the corresponding K-analytic space. In particular, for any K-affinoid algebra A, M (Spm A) = M (A). The canonical (π-adic) topology on X η coincides with the induced topology on the set of rigid points M (X η ) rig = X η. The big advantage of Berkovich spaces is that they carry a true topology instead of a Grothendieck topology, with very nice features (Hausdorff, locally connected by arcs,... ). Berkovich obtains his spaces by adding points to the rigid points of X η (not unlike the generic points in algebraic geometry) which have an interpretation in terms of valuations. We refer to [2, 1.4.4] for a description of the points and the topology of the closed unit disc D := D(0, 1) = Spm K{x}. To give a taste of these Berkovich spaces, let us explain how two points of D rig can be joined by a path in D. A closed subdisc E rig = D(a, ρ) rig of D rig defines a multiplicative semi-norm. E on the Banach algebra K{x}, by mapping f = n=0 a n(t a) n to f E = max n a n ρ n, and hence a Berkovich-point of D. Now a path between two points a, b of D rig can be constructed as follows: γ : [0, 1] D : x { D(a, 2t), if 0 t 1/2, D(b, 2(1 t)), if 1/2 t 1. Let us mention that there are still alternative approaches to non-archimedean geometry, such as Schneider and Van der Put s prime filters, Fujiwara s Zariski- Riemann space [20], or Huber s adic spaces [26]. See [37] for a (partial) comparison. 6. Some applications 6.1. Relation to arc schemes and the Milnor fibration.

12 12 JOHANNES NICAISE Arc spaces. Let k be any field, and let X be a variety over k. Put R = k[[t]]. For each n 1, we define a functor F n : (k alg) (Sets) : A X(A k R n ) from the category of k-algebras to the category of sets. It is representable by a k-scheme L n (X) (this is nothing but the Weil restriction of X k R n to k). For any pair of integers m n 0, the truncation map R m R n induces by Yoneda s Lemma a morphism of k-schemes π m n : L m (X) L n (X) It is easily seen that these morphisms are affine, and hence, we can consider the projective limit L(X) := lim L n (X) n in the category of k-schemes. This scheme is called the arc scheme of X. It satisfies L(X)(k ) = X(k [[t]]) for any field k over k (these points are called arcs on X), and comes with natural projections π n : L(X) L n (X) In particular, we have a morphism π 0 : L(X) L 0 (X) = X. For any subscheme Z of X, we denote by L(X) Z the fiber of L(X) over Z. If X is smooth over k, the schemes L n (X) and L(X) are fairly well understood: if X has pure dimension d, then, for each pair of integers m n 0, πn m is a locally trivial fibration with fiber A d(m n) k (w.r.t. the Zariski topology). If x is a singular point of X, however, the scheme L(X) x is still quite mysterious. It contains a lot of information on the singular germ (X, x); interesting invariants can be extracted by the theory of motivic integration on L(X) (see [16, 17, 38]). The scheme L(X) x is not of finite type, in general, which complicates the study of its geometric poperties. Already the fact that it has only finitely many irreducible components is a non-trivial result (and only known if k has characteristic zero). We will show how rigid geometry allows to reduce question concerning the arc space to arithmetic problems on rigid varieties (which are finite type-objects). The relative case. Let k be any algebraically closed field of characteristic zero 3, and put R = k[[t]]. Let X be a smooth irreducible variety over k, endowed with a dominant morphism f : X A 1 k = Spec k[t]. We denote by X the formal completion of the R-scheme X R = X k[t] k[[t]]; we will also call this the t-adic completion of f. Its special fiber X 0 is simply the fiber of f over the origin. There exists a tight connection between the points on the generic fiber X η of X, and the arcs on X. Any finite extension of the quotient field K is of the form K(d) = k(( d t)), for some integer d > 0. We denote by R(d) the normalization k[[ d t]] of R in K(d). For any integer d > 0, we denote by X (d) the closed subscheme of L(X) defined by X (d) = {ψ L(X) f(ψ) = t d } We will construct a canonical bijection ϕ : X η (K(d)) X (d)(k) 3 This condition is only imposed to simplify the arguments.

13 FORMAL AND RIGID GEOMETRY, AND SOME APPLICATIONS 13 such that the square X η (K(d)) X (d)(k) sp π 0 X 0 (k) ϕ = X 0 (k) commutes. The specialization morphism of ringed sites sp : X η X induces a bijection X η (K(d)) X(R(d)), and the morphism sp maps a point of X η (K(d)) to the reduction modulo d t of the corresponding point of X(R(d)). By Grothendieck s Existence Theorem, the completion functor induces a bijection (X R )(R(d)) X(R(d)). Finally, a reparametrization t d t yields a bijection (X R )(R(d)) X (d)(k). In other words, if we take an arc ψ : Spec R X with f(ψ) = t d, then the generic point of the completion ψ is a K(d)-point on X η, and this correspondence defines a bijection between X (d)(k) and X η (K(d)). Moreover, the image of ψ under the projection π 0 : L(X) X, is nothing but the image of the corresponding element of X η (K(d)) under the specialization map sp : X η X = X 0. The Galois group G(K d /K) = µ d (k) acts on X η (K(d)), and its action on the level of arcs is easy to describe: if ψ is an arc Spec R X with f(ψ) = t d, and ξ is an element of µ d (k), then ξ.ψ(t) = ψ(ξ.t). The absolute case. This case is easily reduced to the previous one. Let X be any k-variety, and consider its base change X R = X k R. We denote by X the formal completion of X R. There exists a canonical bijection between the sets L(X)(k) and X R (R). Hence, k-rational arcs on X correspond to K-points on the generic fiber X η of X, by a canonical bijection and the square ϕ : L(X)(k) X η (K) L(X)(k) π 0 ϕ X η (K) sp X(k) = X(k) commutes. So the rigid counterpart of the space L(X) Z := L(X) X Z of arcs with origin in some closed subscheme Z of X, is the tube ]Z[ of Z in X η (or rather, its set of K-rational points). Of course, the scheme structure L(X) is very different from the analytic structure on X η. Nevertheless, the structure on X η seems to be much richer than the one on L(X), and one might hope that some essential properties of the infinite typeobject L(X) are captured by the finite type-object X η. Moreover, there exists a satisfactory theory of étale cohomology for rigid K-varieties (in fact, there exist several such theories, see for instance [3]), making it possible to apply cohomological techniques to the study of the arc space.

14 14 JOHANNES NICAISE The analytic Milnor fiber. Let g : C m C be an analytic map, and denote by Y 0 the variety defined by g = 0. Let x be a complex point of Y 0. Consider a disc D := B(0, η) of radius η around the origin in C, and a disc B := B(x, ε) in C m. We denote by D the punctured disc obtained by removing the origin. We put ]x[ := B g 1 (D ) Then, for 0 < η ε 1, the induced map g : ]x[ D is a locally trivial fibration, called the Milnor fibration of g at x (and it does not depend on η and ε). Its fiber at a point t is denoted by F x (t), and it is called the (topological) Milnor fiber of g at x (w.r.t. t). To remove the dependency on the base point, one constructs the canonical Milnor fiber F x by considering the fiber product F x :=]x[ D D where D is the universal covering space of D. Since this covering space is contractible, F x is homotopically equivalent to F x (t). The fundamental group π 1 (D ) = Z acts on the singular cohomology spaces of F x ; the action of the canonical generator of π 1 (D ) is called the monodromy transformation of g at x. The Milnor fibration g contains a lot of information on the topology of Y 0 near x; for instance, g is trivial iff x is a smooth point of Y 0. We return to the algebraic setting: let k be an algebrically closed field of characteristic zero, suppose that X is a smooth irreducible variety over k, and let f : X A 1 k = Spec k[t] be a dominant morphism. As before, we denote by X the formal t-adic completion of f, with generic fiber X η. For any closed point x on X 0, we put F x :=]x[, and we call this rigid K-variety the analytic Milnor fiber of f at x. This object was introduced and studied in [30, 32]. We consider it as a bridge between the topological Milnor fibration and arc spaces; a tight connection between these data is predicted by the motivic monodromy conjecture. See [29] for more on this point of view. The topological intuition behind the construction is the following: the formal neighbourhood Spf R of the origin in A 1 k = Spec k[t], corresponds to a sufficiently small disc around the origin in C. Its inverse image under f is realized as the t-adic completion of the morphism f; the formal scheme X should be seen as a tubular neighborhood of the special fiber X 0 defined by f on X. The inverse image of the punctured disc becomes the complement of X 0 in X ; this complement makes sense in the category of rigid spaces, and we obtain the generic fiber X η of X. The specialization map sp can be seen as a canonical retraction of X η on X 0, such that F x corresponds to the topological space ]x[ considered above. Note that this is not really the Milnor fiber yet: we had to base-change to a universal cover of D, which corresponds to considering F x K Ks instead of F x. The monodromy action is translated into the Galois action of G(K s /K) = Ẑ(1)(k) on F x K Ks. As we ve seen above, for any integer d > 0, the points in F x (K(d)) correspond canonically to the arcs ψ : Spec k[[t]] X satisfying f(ψ) = t d and π 0 (ψ) = x. Moreover, by Berkovich comparison result in [5, 3.5] (see also Section 6.3), there are canonical isomorphisms H í et(f x K Ks, Q l ) = R i ψ η (Q l ) x

15 FORMAL AND RIGID GEOMETRY, AND SOME APPLICATIONS 15 such that the Galois action of G(K s /K) on the left hand side corresponds to the monodromy action of G(K s /K) on the right. Here Hét is étale l-adic cohomology, and Rψ η denotes the l-adic nearby cycle functor associated to f. In particular, if k = C, this implies that Het í (F x K Ks, Q l ) is canonically isomorphic to the singular cohomology of the canonical Milnor fiber F x of f at x, and that the action of the canonical topological generator of G(K s /K) = Ẑ(1)(C) corresponds to the monodromy trasformation, by Deligne s classical comparison theorem for étale and analytic nearby cycles [1]. Applications to motivic zeta functions. The relationship between arc schemes and rigid varieties can be used in the study of motivic zeta functions and the monodromy conjecture, as is explained in [29, 32, 31] Deformation theory and lifting problems. Suppose that R has mixed characteristic, and let X 0 be scheme of finite type over the residue field k. Illusie sketches in [27, 5.1] the following problem: is there a flat scheme X of finite type over R such that X 0 = X R k? Grothendieck suggested the following approach: first, we try to construct an inductive system X n of flat R n -schemes of finite type such that X n = Xm Rm R n for m n 0. He shows how the obstructions to lifting X n to X n+1 live in a certain cohomology group on X 0, and that, when these obstructions vanish, the isomorphism classes of possible X n+1 correspond to elements in another approriate cohomology group on X 0. Once we found such an inductive system, its direct limit is a flat formal R-scheme X, topologically of finite type. Next, we need to know if this formal scheme is algebrizable, i.e. if there exists an R-scheme X whose completion is X. This scheme X would be a solution to our lifting problem. A useful criterion to prove the existence of X is the one quoted in Section 3.4: if X 0 is proper and carries an ample line bundle that lifts to a line bundle on X, then X is algebrizable. Moreover, the algebraic model X is uniqe op to isomorphism by Grothendieck s existence theorem in Section 3.4. For more concrete applications of this approach, we refer to Section 5 of [27] Nearby cycles for formal schemes. Berkovich used his étale cohomology theory for non-archimedean spaces, developed in [3], to construct nearby and vanishing cycles functors for formal schemes [4, 5]. His formalism applies, in particular, to stft formal R-schemes X, and to formal completions of such formal schemes along closed subschemes of the special fiber X 0. Let us denote by Rψ t η the functor of tame nearby cycles, both in the algebraic and in the formal setting. Suppose that k is separably closed, for simplicity. Let X be a variety over R, and denote by X its formal completion, with generic fiber X η. Let Y be a closed subscheme of X 0, and let F be an étale constructible sheaf of abelian groups on X R K, with torsion orders prime to the characteristic exponent of k. Then Berkovich associates to F in a canonical way an étale constructible sheaf F on X η, and an étale constructible sheaf F/Y on the tube ]Y [. His comparison theorem [5, 3.1] states that there are canonical quasi-isomorphisms Rψ t η(f) = Rψ t η( F) and Rψ t η(f) Y = Rψ t η ( F/Y ) Moreover, by [5, 3.5] there is a canonical quasi-isomorphism RΓ(Y, Rψ t η(f) Y ) = RΓ(]Y [ K Kt, F/Y )

16 16 JOHANNES NICAISE where K t denotes the completion of the tame closure of K. In particular, if x is a closed point of X 0, then, R i ψ t η(q l ) x is canonically isomorphic to the i-th l-adic cohomology space of the tube ]x[ K Kt. This proves a conjecture of Deligne s, stating that Rψη(F) t Y only depends on the formal completion of X along Y. In particular, the stalk of Rψη(F) t at a closed point x of X 0 only depends on the completed local ring ÔX,x Semi-stable reduction for curves. Bosch and Lütkebohmert show in [11, 10] how rigid geometry can be used to construct stable models for smooth projective curves over K, and uniformizations for Abelian varieties. Let us briefly sketch their approach to semi-stable reduction of curves (which does not use resolution of singularities). If A is a Tate algebra, then we define A o = {f A f sup 1} A oo = {f A f sup < 1} Note that A o is a subring of A, and that A oo is an ideal in A o. The quotient à := A o /A oo is an algebra of finite type over k, and X := Spec à is called the canonical reduction of the affinoid space X := Spm A. There is a reduction map X X mapping points of X to closed points of X. The inverse image of a closed point x of X is called the formal fiber of X at x; it is an open rigid subspace of X. Let C be a projective smooth curve over K, and consider its analytification C an. The idea is to construct a cover U of C an by affinoid open subspaces, whose canonical reductions are semi-stable. Then these pieces can be used to construct an (algebraic) semi-stable model for C over R. The advantage of passing to the rigid world is that the Grothendieck topology on C an is much finer then the Zariski topology on C, allowing finer patching techniques. To construct the cover U, it is proven that smooth and ordinary double points x on X can be recognized by looking at their formal fiber in X. For instance, x is smooth iff its formal fiber is isomorphic to an open disc of radius 1. An alternative (but similar) proof based on rigid geometry is given in [19, 5.6] Constructing étale covers, and Abhyankar s Conjecture. Formal and rigid patching techniques can also be used in the construction of Galois covers; see [25] for an introduction to this subject. This approach generalizes the classical Riemann Existence Theorem for complex curves to a broader class of base fields. Riemann s Existence Theorem states that, for any smooth connected complex curve X, there is an equivalence between the category of finite étale covers of X, the category of finite analytic covering maps of X, and the category of finite topological covering spaces of X. So the problem of constructing an étale cover is reduced to the problem of constructing a topological covering space, where we can proceed locally w.r.t. the complex topology and glue the resulting local covers. In particular, it can be shown in this way that any finite group is the Galois group of a finite Galois extension of C(x), by studying the ramified Galois covers of the affine line. The strategy in rigid geometry is quite similar: given a smooth curve X over our complete discretely valued field K, we consider its analytification X an. We construct an étale cover Y of X an by constructing covers locally, and gluing them to a rigid variety. Then we use a GAGA-theorem to show that Y is algebraic, i.e. Y = Y an for some curve Y over K; Y is an étale cover of X.

17 FORMAL AND RIGID GEOMETRY, AND SOME APPLICATIONS 17 We list some results that were obtained by means of these techniques, and references to their proofs. (Harbater) For any finite group G, there exists a ramified Galois cover f : X P 1 K with Galois group G. Moreover, we can find such a cover such that X is absolutely irreducible, smooth, and projective, and there exists a point x in X(K) at which f is unramified. An accessible proof by Q. Liu is given in [36, 3]. (Abhyankar s Conjecture for the projective line) Let k be an algebraically closed field of characteristic p > 0. A finite group G is the Galois group of a covering of P 1 k, only ramified over, iff G is generated by its elements having order p n with n 1. This conjecture was proven by Raynaud in [34]. This article also contains an introduction to rigid geometry and étale covers. (Abhyankar s Conjecture) Let k be an algebraically closed field of characteristic p > 0. Let X be a smooth connected projective curve of genus g, let ξ 0,..., ξ r (r 0) be closed points on X, and let Γ g,r be the topological fundamental group of the complex Riemann surface of genus g minus r + 1 points. Put U = X \ {ξ 0,..., ξ r }. A finite group G is the Galois group of an unramified Galois cover of X, iff every prime-to-p quotient of G is a quotient of Γ g,r. This conjecture was proven by Harbater in [24] (he even proved that we can find a G-cover which is tamely ramified away from ξ 0 ). The group Γ g,r can be described explicitly: it is generated by elements a 1,..., a g, b 1,..., b g, c 0,..., c r, subject to the relation g i=1 [a i, b i ] r j=0 c j = 1, where [a i, b i ] denotes the commutator of a i and b i. References [1] Groupes de monodromie en géométrie algébrique. II. Springer-Verlag, Berlin, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 7 II), Dirigé par P. Deligne et N. Katz, Lecture Notes in Mathematics, Vol [2] V. G. Berkovich. Spectral theory and analytic geometry over non-archimedean fields, volume 33 of Mathematical Surveys and Monographs. AMS, [3] V. G. Berkovich. Étale cohomology for non-archimedean analytic spaces. Publ. Math., Inst. Hautes Étud. Sci., 78:5 171, [4] V. G. Berkovich. Vanishing cycles for formal schemes. Invent. Math., 115(3): , [5] V. G. Berkovich. Vanishing cycles for formal schemes II. Invent. Math., 125(2): , [6] V. G. Berkovich. p-adic analytic spaces. Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. II, pages , [7] P. Berthelot. Cohomologie rigide et cohomologie rigide à supports propres. Prepublication, Inst. Math. de Rennes, [8] S. Bosch. Lectures on formal and rigid geometry. preprint, [9] S. Bosch, U. Güntzer, and R. Remmert. Non-Archimedean analysis. A systematic approach to rigid analytic geometry., volume 261 of Grundlehren der Mathematischen Wissenschaften. Springer Verlag, [10] S. Bosch and W. Lütkebohmert. Stable reduction and uniformization of abelian varieties. II. Invent. Math., 78: , [11] S. Bosch and W. Lütkebohmert. Stable reduction and uniformization of abelian varieties. I. Math. Ann., 270: , [12] S. Bosch and W. Lütkebohmert. Formal and rigid geometry. I. Rigid spaces. Math. Ann., 295(2): , 1993.

18 18 JOHANNES NICAISE [13] S. Bosch and W. Lütkebohmert. Formal and rigid geometry. II. Flattening techniques. Math. Ann., 296(3): , [14] S. Bosch, W. Lütkebohmert, and M. Raynaud. Formal and rigid geometry. III: The relative maximum principle. Math. Ann., 302(1):1 29, [15] S. Bosch, W. Lütkebohmert, and M. Raynaud. Formal and rigid geometry. IV: The reduced fibre theorem. Invent. Math., 119(2): , [16] J. Denef and F. Loeser. Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math., 135: , 1999, arxiv:math.ag/ [17] J. Denef and F. Loeser. Geometry on arc spaces of algebraic varieties. Progr. Math., 201: , 2001, arxiv:math.ag/ [18] A. Ducros. Espaces analytiques p-adiques au sens de Berkovich. Exposé 958 du séminaire Bourbaki (mars 2006). [19] J. Fresnel and M. van der Put. Rigid analytic geometry and its applications., volume 218 of Progress in Mathematics. Boston, MA: Birkhäuser, [20] K. Fujiwara. Theory of tubular neighborhood in étale topology. Duke Math. J., 80(1):15 57, [21] A. Grothendieck. Fondements de la géométrie algébrique. Extraits du Séminaire Bourbaki Paris: Secrétariat mathématique, [22] A. Grothendieck and J. Dieudonné. Eléments de Géométrie Algébrique, i. Publ. Math., Inst. Hautes Étud. Sci., 4:5 228, [23] A. Grothendieck and J. Dieudonné. Eléments de Géométrie Algébrique, iii. Publ. Math., Inst. Hautes Étud. Sci., 11:5 167, [24] D. Harbater. Abhyankar s conjecture on Galois groups over curves. Invent. Math., 117(1):1 25, [25] D. Harbater. Patching and Galois theory, in Schneps, L. (ed.),galois groups and fundamental groups, volume 41 of Math. Sci. Res. Inst. Publ., pages Cambridge: Cambridge University Press, [26] R. Huber. A generalization of formal schemes and rigid analytic varieties. Math. Z., 217(4): , [27] L. Illusie. Grothendieck s Existence Theorem in formal geometry. In B.Fantechi, L. Goettsche, L. Illusie, S. Kleiman, N. Nitsure, and A. Vistoli, editors, Fundamental Algebraic Geometry, Grothendieck s FGA explained, Mathematical Surveys and Monographs. AMS, [28] W. Lütkebohmert. Formal-algebraic and rigid-analytic geometry. Math. Ann., 286(1-3): , [29] J. Nicaise and J. Sebag. Rigid geometry and the monodromy conjecture. to appear in the Conference Proceedings of Singularités, CIRM, Luminy, 24 January-25 February [30] J. Nicaise and J. Sebag. Invariant de Serre et fibre de Milnor analytique. C.R.Ac.Sci., 341(1):21 24, [31] J. Nicaise and J. Sebag. Galois action on motivic Serre invariants, and motivic zeta functions. preprint, [32] J. Nicaise and J. Sebag. Motivic Serre invariants, ramification, and the analytic Milnor fiber. to appear in Invent. Math., [33] M. Raynaud. Géométrie analytique rigide d après Tate, Kiehl,.... Mémoires de la S.M.F., 39-40: , [34] M. Raynaud. Revêtements de la droite affine en caractéristique p > 0 et conjecture d Abhyankar. Invent. Math., 116(1-3): , [35] J. Tate. Rigid analytic geometry. Invent. Math., 12: , [36] M. van der Put. Valuation theory in rigid geometry and curves over valuation rings. In F.V. Kuhlmann, editor, Valuation theory and its applications. Volume I. Proceedings of the international conference and workshop, University of Saskatchewan, Saskatoon, Canada, July 28-August 11, 1999, volume 32 of Fields Inst. Commun., pages Providence, RI: American Mathematical Society (AMS), [37] M. van der Put and P. Schneider. Points and topologies in rigid geometry. Math. Ann., 302(1):81 103, [38] W. Veys. Arc spaces, motivic integration and stringy invariants. In Proceedings of the conference Singularity Theory and its applications, Sapporo (Japan), september 2003, Advanced Studies in Pure Mathematics, to appear.

19 FORMAL AND RIGID GEOMETRY, AND SOME APPLICATIONS 19 Université Lille 1, Laboratoire Painlevé, CNRS - UMR 8524, Cité Scientifique, Villeneuve d Ascq Cédex, France address: johannes.nicaise@math.univ-lille1.fr