Imperial College London Mathematics M4R - Master’s Research Project The Fontaine-Wintenberger Theorem Author Christophe Jefferies With thanks to my supervisor Professor Ambrus Pal June 9, 2020 CONTENTS Contents Introduction 2 1 A recap of the basics 3 2 Norm and trace maps 7 3 Local rings, valuations, and valuation rings 8 4 The fields in the isomorphism 11 5 Topological groups 14 6 Inductive and inverse limits 16 7 The Krull topology and the infinite Galois correspondence 18 8 Local fields 19 9 Separable closures and absolute Galois groups 21 10 Fractional ideals 22 11 Ramification 24 12 APF extensions and elementary extensions 26 13 The field of norms X K ( L ) 28 14 The map X K ( τ ) 34 15 The Fontaine-Wintenberger isomorphism 36 16 Conclusions and further discussion 39 17 Appendices 40 18 References 46 1 Introduction The aim of this paper is to present Jean-Pierre Wintenberger’s ‘field of norms’ construction, and the Fontaine-Wintenberger theorem, in a way that an undergraduate mathematician with basic knowledge of abstract algebra can understand. The theorem says that there is an isomorphism of absolute Galois groups Gal( K sep 0 /K 0 ) ∼ = Gal( K sep 1 /K 1 ) , where K 0 = Q p ( p ∞ √ p ) , K 1 = F p (( p ∞ √ t )) , and sep denotes separable closure. We will explain all of this notation in due course. This paper is useful for several reasons. First and foremost, it provides a direct route to understanding Wintenberger’s work, with all the prerequisites presented in a concise way. It also makes the subject matter accessible to an undergraduate-level reader, with helpful explanations around the more difficult parts. Furthermore, it serves as a translation from French in to English of Wintenberger’s original work, making the precise details of his constructions available to a wider audience. Wintenberger first presented his work in his thesis [1], under the supervision of Jean-Marc Fontaine. He later published the results in his paper Le corps des normes de certaines extensions infinies de corps locaux [2] ( The field of norms of certain infinite extensions of local fields ), and also stated the results without proof in two shorter releases, [3] and [4]. Wintenberger’s work is important for a number of reasons. It has motivated a lot of further research; for example, Peter Scholze’s theory of perfectoid spaces builds directly on Winten- berger’s constructions, and was a central reason for him winning a Fields medal in 2018. The isomorphism is also fundamental in constructing certain invariants within p -adic Hodge theory , a very active area of modern mathematics (see [5] for more context). We discuss the consequences of his work in more detail at the end. The earlier chapters of this paper are introductions to the topics needed to understand Win- tenberger’s work, illustrated with examples. Chapters 13 to 15 then provide an explanation of his constructions and proofs, with more explicit discussion than in the original papers, but omitting some of the longer calculations. This is my own work except where otherwise stated. 2 1 A RECAP OF THE BASICS 1 A recap of the basics This chapter is a reminder of fields, extensions, and the basics of Galois theory. The material is standard and can be found for example in [6]. We assume knowledge of basic ring theory. Fields Definition 1.1. A field is a commutative ring in which every nonzero element has a multiplicative inverse. Common fields include R , Q , and for each prime p , the field F p of integers mod p For convenience, we do not consider the zero ring to be a field. Definition 1.2. The characteristic of a field F is the smallest integer n ≥ 1 such that n · 1 = 0 , where n · 1 denotes 1 + . . . + 1 ( n times ) . If no such integer exists, then F has characteristic 0. R and Q have characteristic 0, whilst F p has characteristic p Here are two basic properties of fields. Proposition 1.3. Every field is an integral domain. Proof. Let a, b ∈ F with a 6 = 0 and ab = 0. Then b = a − 1 ab = a − 1 · 0 = 0, as required. Proposition 1.4. If a field has characteristic n 6 = 0 , then n is prime. Proof. n 6 = 1, otherwise 1 = 0 and F is the zero ring. So suppose n = lm for l, m > 1. Then 0 = n · 1 = lm · 1 = ( l · 1)( m · 1). Hence one of these brackets is zero, as F is an integral domain; but this would contradict the minimality of n . So n is prime. Note that if F has characteristic 0, then it contains a copy of Z , and hence of Q by existence of multiplicative inverses. If F has characteristic p , then it contains a copy of F p , namely the elements 0 , 1 , . . . , p − 1. Extensions and automorphisms Definition 1.5. A field extension is a pair of fields K ⊂ L which have the same operations when restricted to K For example, R is a field extension of Q with the usual operations. We often write a field extension K ⊂ L as L/K (this is just notation and not a quotient). 3 1 A RECAP OF THE BASICS Recall that an automorphism is an isomorphism from an object to itself. Definition 1.6. The automorphism group Aut ( L ) of a field L is the set of automorphisms of L under composition. Definition 1.7. The automorphism group Aut K ( L ) or Aut ( L/K ) of a field extension L/K is the subgroup of Aut ( L ) which fixes K ; that is, all the elements whose restrictions to K are the identity. As an example, one can show that Aut( C / R ) has two elements: the identity automorphism, and complex conjugation (see [7], page 138). So Aut( C / R ) ∼ = Z / 2 Z . Other extensions can have much more complicated automorphism groups. Properties of extensions We would like to state the finite Galois correspondence For this, we must define finite , algebraic , normal , and separable extensions. Given any extension L/K , we can consider L as an additive vector space over K Definition 1.8. If L is finite-dimensional as a vector space over K , then L/K is a finite extension (else infinite). Its degree [ L : K ] is this dimension. For example, C / R is a finite extension with degree [ C : R ] = 2, because C is a vector space over R with basis { 1 , i } . However R / Q is an infinite extension. Q ( √ 2 , √ 3 , √ 5 , √ 7 , . . . ) / Q is also an infinite extension. Here Q ( √ 2 , √ 3 , √ 5 , √ 7 , . . . ) is the smallest subfield of R containing Q and √ 2 , √ 3 , √ 5 , √ 7 , . . . Definition 1.9. Let α ∈ L α is algebraic over K if it is a root of a polynomial over K The extension L/K is algebraic if all elements of L are algebraic over K For example, Q ( √ 2 ) / Q is algebraic, because a general element a + b √ 2 (with a, b ∈ Q ) is a root of X 2 − 2 aX + ( a 2 − 2 b 2 ) ∈ Q [ X ]. However R / Q is certainly not algebraic. Note that every finite extension is automatically algebraic: if [ L : K ] = n ∈ N and α ∈ L , then the n + 1 ‘vectors’ 1 , α, . . . , α n cannot be linearly independent over K . So α is the root of some polynomial over K (of degree ≤ n ), in other words it is algebraic over K The following will be useful for defining normal and separable extensions. Definition 1.10. Let f ( X ) ∈ K [ X ] f splits (completely) over K if it is a product of linear polynomials over K For example, f ( X ) = X 2 − 2 ∈ Q [ X ] does not split over Q . However it splits completely over R , into ( X + √ 2)( X − √ 2). 4 1 A RECAP OF THE BASICS Recall that a polynomial is irreducible over K if it cannot be factored into two or more non-constant polynomials over K Definition 1.11. An extension L/K is normal if every irreducible polynomial over K with a root in L splits completely over L C / R is (trivially) a normal extension, since every polynomial over R splits completely over C , including the irreducible ones. Q ( 3 √ 2 ) / Q is not a normal extension. For example, X 3 − 2 ∈ Q [ X ] is irreducible, but Q ( 3 √ 2 ) only contains one of its roots (the real one). So the polynomial cannot be written as a product of linear factors over Q ( 3 √ 2 ) , and so does not split. We remind the reader of minimal polynomials before defining separable extensions. Definition 1.12. Let L/K be a field extension and α ∈ L . Suppose α is algebraic over K The minimal polynomial of α over K is the monic polynomial over K of least degree among those having α as a root. Proposition 1.13. The minimal polynomial of an algebraic element is unique. Proof. Suppose f 1 ( X ), f 2 ( X ) ∈ K [ X ] are distinct monic polynomials having α as a root, and are of least degree n amongst such polynomials. Then f 1 − f 2 is a nonzero polynomial of strictly smaller degree, also having α as a root, which contradicts the minimality of n Definition 1.14. Let L/K be algebraic. • An irreducible polynomial f ( X ) ∈ K [ X ] is separable if it has distinct roots in any extension of K where it splits. • α ∈ L is separable over K if its minimal polynomial over K is separable. • L/K is separable if every element of L is separable over K We give some examples. X 3 − 2 has 3 distinct roots over C , as well as any extension of C , so it is separable over C But it is not separable as a polynomial over F 3 , because X 3 − 2 = ( X + 1) 3 ∈ F 3 [ X ] has a repeated root. Q ( √ 2 ) / Q is a separable extension: the minimal polynomial over Q of an element a + b √ 2 is X 2 − 2 aX + ( a 2 − 2 b 2 ) = ( X − a + b √ 2)( X − a − b √ 2), which has distinct roots. (If b = 0, then a has minimal polynomial X − a instead, also with no repeated roots.) F 2 ( √ t ) / F 2 ( t ) is not a separable extension. The minimal polynomial of √ t over F 2 ( t ) is X 2 − t , but this has a repeated factor X 2 − t = ( X − √ t ) 2 when it splits in F 2 ( √ t ) 5 1 A RECAP OF THE BASICS The Galois correspondence for finite extensions Definition 1.15. A field extension L/K is Galois if it is algebraic, separable, and normal. In this case we call its automorphism group the Galois group of the extension, and write Gal ( L/K ) instead of Aut ( L/K ) The following is an important correspondence relating the structure of a finite Galois exten- sion to its Galois group. Theorem 1.16 (Fundamental theorem of finite Galois theory) Let L/K be a finite Galois extension and G = Gal ( L/K ) . Then there is a one-to-one correspondence between subgroups H of G and intermediate extensions K ⊆ K ′ ⊆ L , given by: • H 7 −→ { x ∈ L : σ ( x ) = x ∀ σ ∈ H } , • K ′ 7 −→ { σ ∈ G : σ ∣ ∣ K ′ = id K ′ } So each subgroup of G corresponds to its ‘fixed field’ (the elements of L which it doesn’t affect), and conversely, each intermediate field extension corresponds to the subgroup of G which fixes it. The correspondence also holds only if L/K is Galois. A concise proof of both directions is given in [8]. We have only stated the correspondence for finite Galois extensions. However, Winten- berger’s work concerns itself largely with infinite extensions. If we apply the above mappings to an infinite extension, we can find that different subgroups have the same fixed field, and the correspondence fails. An example is given in appendix A. We will fix this later using the Krull topology on a Galois group (see chapter 7). 6 2 NORM AND TRACE MAPS 2 Norm and trace maps Norm maps are crucial for defining Wintenberger’s ‘field of norms’. Trace maps will also be useful in a few proofs. Let L/K be a finite extension, and [ L : K ] = n . Choose a basis b 1 , . . . , b n of L as a vector space over K . Then for any α ∈ L , multiplication by α is a linear transformation of L Definition 2.1. The norm N L/K ( α ) of α is the determinant of multiplication by α The trace T L/K ( α ) of α is the trace of multiplication by α More explicitly, for each basis vector b i , we can write α · b i = a i 1 · b 1 + . . . + a in · b n for some a ij ∈ K . Let A = ( a ij ) be the n × n matrix with these coefficients as entries. Then N L/K ( α ) is the determinant of A , and T L/K ( α ) is the trace of A Neither of these depend on the choice of basis. This follows from the determinant and trace of a matrix appearing (up to a sign change) as coefficients of the characteristic polynomial of A , which is an invariant under change of basis. Also note that both maps take values in K , because the entries of A are in K For example, let K = Q and L = Q ( √ 2 , √ 3 ) With some algebraic number theory, one can show that L is a 4-dimensional vector space over K , with basis { 1 , √ 2 , √ 3 , √ 6 } . (The vector space is additive, so even though √ 6 = √ 2 · √ 3, the element √ 6 is not a Q -linear combination of { 1 , √ 2 , √ 3 } , and we must include it in the basis.) Then what are N L/K ( √ 6 ) and T L/K ( √ 6 ) ? We look at the effect of √ 6 on our chosen basis: √ 6 · 1 = 0 +0 +0 +1 · √ 6 √ 6 · √ 2 = 0 +0 +2 · √ 3 +0 √ 6 · √ 3 = 0 +3 · √ 2 +0 +0 √ 6 · √ 6 = 6 · 1 +0 +0 +0 So A = 0 0 0 1 0 0 2 0 0 3 0 0 6 0 0 0 , with determinant N L/K ( √ 6) = 36 and trace T L/K ( √ 6) = 0. Note that these are both in K = Q as expected. 7 3 LOCAL RINGS, VALUATIONS, AND VALUATION RINGS 3 Local rings, valuations, and valuation rings Local rings Definition 3.1. A ring R is local if it has a unique maximal ideal m In this case its residue field is R/ m For example, every field F is local: any nonzero ideal of F contains a unit, and therefore contains 1, so it is the whole ring. So the only ideals of F are { 0 } and F , and { 0 } is the unique maximal ideal. Its residue field F/ { 0 } is isomorphic to F We can give more interesting examples after introducing valuations. As a warning, we will define local fields later on. This does not refer to a field which is local as a ring (that would be any field), but to a field with certain topological properties. Valuations Let K be a field. We would like to assign a number to each element of K in a useful way. Definition 3.2. An ordered abelian group is an abelian group ( G, +) equipped with a total order ≤ , such that x ≤ y = ⇒ x + z ≤ y + z for all x, y, z ∈ G The examples to have in mind are ( Z , +) and ( Q , +). Note that if K is a field, then K × = K \ { 0 } forms a group under multiplication. Definition 3.3. A valuation on a field K is a surjective homomorphism v : K × → G , where G is an ordered abelian group, such that v ( x + y ) ≥ min { v ( x ) , v ( y ) } for all x, y ∈ K × As usual, v being a homomorphism just means v ( x · y ) = v ( x ) + v ( y ) for all x, y ∈ K × We usually set v (0) = ∞ , extending G and its order to G ∪ {∞} in the obvious way. One example of a valuation is the trivial valuation v ( x ) = 0 x 6 = 0 ∞ x = 0 on any field. A more interesting valuation is the p -adic valuation v p on Q . Take G = Z . For each n ∈ Z , let n = p k r where p - r , and set v p ( n ) = k . Then extend this to Q by v p ( n m ) = v p ( n ) − v p ( m ) (i.e. extend v to a homomorphism). For example, v 2 (24) = 3, and v 3 ( 7 18 ) = − 2. We can also turn a real-valued valuation v in to an absolute value in the sense of Serre, [9], chapter 2. 8 3 LOCAL RINGS, VALUATIONS, AND VALUATION RINGS Definition 3.4. Fix a real number a ∈ (0 , 1) . An absolute value associated to a valuation v is a function || · || given by || x || = a v ( x ) ( x 6 = 0) 0 x = 0 This is also an example of a norm on the space; one can easily check that such an absolute value is multiplicative, and satisfies the triangle inequality. It therefore induces a metric on K , given by d ( x, y ) = || x − y || This in turn defines a topology on K , generated by the open balls B r ( x ) = { y : d ( x, y ) < r } ; one can check that this topology does not depend on the choice of a . We will make use of this topology in chapter 8. Definition 3.5. The p -adic norm | · | p is the absolute value arising from v p , with a = p − 1 More succinctly, | x | p = p − v p ( x ) So an element of Q (written in simplest form) has small p -adic norm when its numerator is very divisible by p Choosing a = p − 1 is simply a convention. As an example, the sequence a n = 10 n converges to 0 according to the 5-adic norm, because | a n − 0 | 5 = | 5 n · 2 n | 5 = | 5 n | 5 · | 2 n | 5 = | 5 n | 5 · 1 = (5 − 1 ) n −→ 0. p -adic norms will be useful when defining the p -adic numbers in chapter 4. We now to focus on valuation rings. Valuation rings Definition 3.6. Let v be a valuation on a field K . The valuation ring A K of K is the set { x ∈ K : v ( x ) ≥ 0 } , with ring operations inherited from K This is indeed a ring; from the definition of a valuation, one can easily check that it is closed under addition, multiplication and taking additive inverses. It also contains 1, as v (1) = 0 (because v is a homomorphism). Proposition 3.7. A valuation ring A K is local, with maximal ideal I = { x ∈ K : v ( x ) > 0 } Proof. We first check that I is an ideal. 0 ∈ I , so it is nonempty. Let x, y ∈ I Then v ( x + y ) ≥ min { v ( x ) , v ( y ) } > 0, so I is closed under addition. Now let x ∈ I, a ∈ A K . Then v ( a · x ) = v ( a ) + v ( x ) ≥ v ( x ) > 0, so a · x ∈ I as required. It is therefore enough to show that A × K = A K \ I ; then since any ideal containing a unit is the whole ring, and I contains all the non-units, I must be the unique maximal ideal. Suppose x ∈ A K is a unit, so xy = 1 for some y ∈ A K . Then 0 = v (1) = v ( x · y ) = v ( x )+ v ( y ), where v ( x ) , v ( y ) ≥ 0. So v ( x ) = 0, and x 6 ∈ I 9 3 LOCAL RINGS, VALUATIONS, AND VALUATION RINGS Conversely, suppose v ( x ) = 0. So v ( x − 1 ) + v ( x ) = v (1) = 0 (where x − 1 is the inverse in K ), so v ( x − 1 ) = 0, and x is a unit in A K as required. For example, the valuation ring of Q with the p -adic norm is { a b ∈ Q : p - b } . It is local with unique maximal ideal generated by p (all the elements where p | a ). Valuation rings have very nice properties when the v is integer-valued. Definition 3.8. A discrete valuation is a valuation with G = Z If v is a discrete valuation on K , then A K is a discrete valuation ring. There are many equivalent characterisations of discrete valuation rings (DVRs), as seen for example in [10]. Here we just state some properties that will be useful. Every DVR is a principal ideal domain. It also has a unique irreducible element π up to multiplication by units, which generates its maximal ideal; we call π a uniformizer This implies that all non-zero proper ideals of a DVR have the form ( π n ) with n > 0. So every nonzero x ∈ R can be written uniquely in the form uπ n , where u is a unit. We will now describe the fields which appear in the Fontaine-Wintenberger isomorphism, as stated in the introduction. 10 4 THE FIELDS IN THE ISOMORPHISM 4 The fields in the isomorphism We first describe the fields Q p and K (( X )), and define algebraic closures. Then we construct the fields K 0 and K 1 appearing in the Fontaine-Wintenberger isomorphism. Completions and Q p Recall that a metric space M is complete if every Cauchy sequence in M converges in M R and C are examples of complete spaces, whereas Q (with the usual Euclidean norm) is not; for example, there are sequences of rational numbers converging to π 6 ∈ Q In fact Q is not complete with respect to any p -adic norm either. An argument is given in appendix B. We can form the completion of a field K with respect to a norm | · | as follows; see [10], chapter 10 for the formal details. Definition 4.1. Let C be the ring of Cauchy sequences in K (according to | · | ), with the ring operations + and · defined component-wise. Let I be the ideal of sequences converging to 0. Then the completion of K with respect to | · | is ˆ K = C / I As proved in [10], I is a maximal ideal of C , so C / I is a field. Intuitively, the completion process ‘adds in’ limit points for all the Cauchy sequences in K We consider ˆ K to contain K , by identifying α ∈ K with the equivalence class of the constant sequence ( α, α, . . . ). If we complete Q with respect to the Euclidean norm, we end up with R . But we can also use the p -adic norm. Definition 4.2. The field of p -adic numbers Q p is the completion of Q with respect to | · | p The p -adic valuation v p and norm | · | p extend naturally to Q p (see [11], section 4). Note that Q p contains a copy of Q , and so has characteristic 0. Formal Laurent series For any field K , the polynomial ring K [ X ] is not a field, since X has no multiplicative inverse; nor is the ring of formal power series K [[ X ]] = { ∞ ∑ i =0 a i X i : a i ∈ K } for the same reason. We would still like to put a field structure on polynomial-like objects, which motivates the next definition. 11 4 THE FIELDS IN THE ISOMORPHISM Definition 4.3. The field of formal Laurent series over a field K is K (( X )) = { ∞ ∑ i = − n a i X i : n ∈ Z , a i ∈ K, a − n 6 = 0 } , with the ring operations defined as follows; let A ( X ) = ∞ ∑ i = − n a i X i and B ( X ) = ∞ ∑ i = − m b i X i be elements of K (( X )) . Then • the coefficient of X i in A ( X ) + B ( X ) is a i + b i , • the coefficient of X i in A ( X ) · B ( X ) is ∑ n ∈ Z a n b i − n In other words, we treat the formal sums like polynomials. Note that we only allow finitely many negative exponents. We also ignore any idea of convergence - these sums are just formal symbols. K (( X )) is indeed a field: the inverse of a given non-zero element can be worked out induc- tively, by comparing coefficients. We leave the reader to check the details. As a word of warning, K (( X )) is not to be confused with classical Laurent series, which can have infinitely many negative exponents, and do not in general form a field. Algebraic closures and the fields in the isomorphism Definition 4.4. A field K is algebraically closed if every non-constant polynomial in K [ X ] has a root in K For example, C is algebraically closed (this is the fundamental theorem of algebra), but R is not, as X 2 + 1 ∈ R [ X ] has no roots in R Definition 4.5. An algebraic closure K of a field K is an algebraic extension of K which is algebraically closed. A standard result (see [12]) is that any field K has an algebraic closure K , and that it is unique up to isomorphism. So we usually refer to ‘the’ algebraic closure of a field. Therefore, a field is algebraically closed if and only if it equals its own algebraic closure. Now, Q p is not algebraically closed; an argument is outlined in appendix C. As an example, x p n − p ∈ Q p [ X ] has no roots in Q p for any n ≥ 1. For each n ≥ 1, we can adjoin such a root pn √ p to Q p . If we adjoin all of these at once, then 12 4 THE FIELDS IN THE ISOMORPHISM the result is the first field in the isomorphism: K 0 = Q p ( p ∞ √ p ) := ⋃ n ∈ N Q p ( pn √ p ) This is by definition the smallest subfield of Q p which contains Q p and pn √ p for every n . So it contains not only these new elements, but also their inverses, and all combinations of finite sums and products of these and other elements of Q p ( K 0 is still not the full algebraic closure of Q p , which is harder to describe explicitly.) Similarly, the field of formal Laurent series F p (( t )) over a finite field F p is not algebraically closed, as X p n − t has no roots over F p (( t )). We can repeatedly adjoin p n -th roots of t to form the second field: K 1 = F p (( t )) ( p ∞ √ t ) := ⋃ n ∈ N F p (( t ))( pn √ t ) The two constructions at first seem quite similar. However, the resulting fields still differ in some important ways: the first is built on a p -adic field, and has characteristic 0, whereas the second is built on a function field, over a finite field of characteristic p . They are certainly not isomorphic, as they have different characteristics. The Fontaine-Wintenberger isomorphism says that these two fields nonetheless have the same Galois theory - specifically, isomorphic absolute Galois groups (see chapter 9). In fact, K 1 is the ‘field of norms’ of K 0 , and this isomorphism is a special case of a much more general theorem (15.3), which we prove towards the end of the paper. 13 5 TOPOLOGICAL GROUPS 5 Topological groups Our next goal is to define local fields , as well as the Krull topology on a Galois group. Both of these make use of topological groups and inductive and inverse limits , which we cover in these next two chapters. Let G be a group, with group operation ∗ : G × G → G . Let i : G → G be the inversion function i ( g ) = g − 1 Suppose we also have a topological structure on the set G . That is, a set I of subsets of G which: • contains ∅ and G , • is closed under finite intersection, • is closed under any union. Recall that a function between topological spaces is continuous if the pre-image of any open set is open. Definition 5.1. ( G, I ) is a topological group if ∗ and i are continuous as functions between topological spaces. In the above definition, we implicitly endow G × G with the product topology : Definition 5.2. A basis for a topological space ( X, I ) is a subset B ⊆ I such that any open set in X is a union of elements of B The product topology on a Cartesian product ∏ i ∈ I X i of topological spaces ( X i , I i ) is the topology with basis B = {∏ i ∈ I U i : U i ∈ I i } So the open subsets of G × G are all unions of sets of the form U × V , with U, V ∈ I . A product of topological groups is clearly also a topological group, taking the group operations component-wise. Some authors (e.g. [13]) also require topological groups to be Hausdorff as topological spaces: Definition 5.3. A topological space ( X, I ) is Hausdorff if for any distinct points x, y ∈ X , there are disjoint open sets U, V with x ∈ U , y ∈ V We will also assume that topological groups are Hausdorff from now on. 14 5 TOPOLOGICAL GROUPS Here are some examples of (Hausdorff) topological groups: • ( R , +) with the Euclidean topology is a topological group, because the addition and negation of a (fixed) number are continuous functions. The same applies to R n • The unit circle S 1 as a multiplicative group, with the topology inherited from C , is a topological group. • The discrete topology (where all sets are open) trivially makes any group in to a topological group. We often consider two topological spaces ‘the same’ if there is a homeomorphism between them: Definition 5.4. A map f : X → Y between topological spaces is a homeomorphism if it is a continuous bijection with continuous inverse. If such an f exists, then X and Y are homeomorphic. For example, the open interval ( a, b ) ⊂ R (where a < b ) is homoeomorphic to R , via the homeomorphism f : ( a, b ) → R mapping x 7 −→ 1 a − x + 1 b − x The following is an important property carried by all topological groups. Definition 5.5. A topological space X is homogeneous if for any x, y ∈ G , there is a homeomorphism G → G sending x to y For any x, y in a topological group G , one can easily check that the map g 7 −→ yx − 1 g is a homeomorphism sending x to y . So every topological group is homogeneous. This means that all elements of a topological group G are ‘the same’ as far as its topological structure is concerned. As a result, to specify the topological structure on a topological group, it is enough to specify only the open sets U containing the identity 1. Then the open sets containing an element g are all the sets gU = { gu : u ∈ U } Better still, we can just specify a basis of open sets around 1, from which we can easily figure out all other open sets. We will do this soon when defining the Krull topology 15 6 INDUCTIVE AND INVERSE LIMITS 6 Inductive and inverse limits Inductive and inverse limits are two ways of building a new object from a family of related objects. We follow [14] and [15] as appropriate. Inductive limits Let I be a partially-ordered set, and ( A i ) i ∈ I a family of objects (all from a chosen category , e.g. groups, rings, topological spaces, ...). Suppose we also have morphisms φ ij : A i → A j whenever i ≤ j , such that: • φ ii = id i for all i , • φ jk · φ ij = φ ik whenever i ≤ j ≤ k We define an equivalence relation ∼ on the disjoint union ⊔ i ∈ I A i . For x i ∈ A i , x j ∈ A j , x i ∼ x j ⇐⇒ φ ik ( x i ) = φ jk ( x j ) for some k ∈ I such that i ≤ k, j ≤ k. In other words, two elements are equivalent whenever their images are eventually equal as you move ‘up’ the family of objects. Definition 6.1. The pair ( A i ) , ( φ ij ) is an inductive system, and its inductive limit is lim − → A i := (⊔ i ∈ I A i ) / ∼ So two distinct elements of the inductive limit correspond to two elements of ∏ i ∈ I A i which are distinct ‘all the way up’ the family of objects. The inductive limit is still an object from the chosen category (see [15] for details). We will actually only need inductive limits once later on. However, we will need to make heavy use of inverse limits , which we focus on now. Inverse limits Let I be a partially-ordered set, and ( A i ) i ∈ I a family of objects from a chosen category. This time suppose we have morphisms φ ij : A i → A j whenever i ≥ j , such that: • φ ii = id i for all i , • φ jk · φ ij = φ ik whenever i ≥ j ≥ k Note that the morphisms map ‘down’ the family according to I 16 6 INDUCTIVE AND INVERSE LIMITS Definition 6.2. The pair ( A i ) , ( φ ij ) is an inverse system, and its inductive limit is lim ← − A i := { ( a i ) i ∈ I : φ ij ( a i ) = a j whenever i ≥ j } In other words, lim ← − A i is all the elements of ∏ i ∈ I A i which respect the maps φ ij , in that mapping an entry ‘down’ always matches the other entries. We give an example shortly. We will be especially interested in the case where the objects A i are topological groups. Theorem 6.3. If ( G i ) , ( φ ij ) is an inverse system of topological groups, then lim ← − G i is a closed subgroup of ∏ i ∈ I G i Proof. We show that its complement is open. Let ( g i ) ∈ ( ∏ i ∈ I G i ) \ ( lim ← − G i ) . Therefore φ ij ( g i ) 6 = g ( j ) for some i ≥ j . We build an open neighbourhood of ( g i ) disjoint from lim ← − G i Let U and V be disjoint open neighbourhoods in G j of g j and φ ij ( g i ) respectively (this is possible as topological groups are assumed Hausdorff). Let V ′ be an open neighbourhood in G i of g i such that φ ij ( V ′ ) ⊆ V . Then the set W := ∏ r ∈ I U r , where U r = U r = j, V ′ r = i, G i otherwise , is an open neighbourhood of ( g i ). It is also disjoint from ( lim ← − G i ) : any element ( w i ) ∈ W has φ ij ( w i ) ∈ V and w j ∈ U , which are disjoint, so ( w i ) cannot satisfy the definition of an element of the inverse limit. So W is the neighbourhood required. As a useful example, we can characterise the valuation ring Z p of Q p as an inverse limit. (In fact this is sometimes given as the definition, for instance in [16], page 5.) Take I = N as our partially-ordered set and let A n = Z /p n Z . Let the maps φ ij be the natural remainder maps. Then Z p = lim ← − A n So choosing an element of Z p is equivalent to choosing a sequence of integers ( a n ) n ∈ N such that a i ≡ a j (mod p j ) whenever i ≥ j . It can therefore be useful to consider an element of Z p as a power series a 0 + a 1 p + a 2 p 2 + . . . with ‘variable’ p . Such a series always converges according to the p -adic norm. With a little more work, one can show that every element of Q p can be uniquely written in the form ∑ n ≥− k a n p n , where k ∈ Z , a k 6 = 0, and a i ∈ { 0 , . . . , p − 1 } for all i . So there is a clear analogy with F p (( t )), though their underlying structures are still different. 17 7 THE KRULL TOPOLOGY AND THE INFINITE GALOIS CORRESPONDENCE 7 The Krull topology and the infinite Galois correspondence Let L/K be a (possibly infinite) Galois extension. We would like to put a topology on G = Gal( L/K ) in a useful way. We follow [15]. Let F = { K i : K ⊆ K i ⊆ L, and K i /K is finite and Galois } . By basic Galois theory, each extension L/K i is still Galois. Let σ i = Gal( L/K i ). Definition 7.1. The Krull topology on G = Gal ( L/K ) is the one with B = { σ i : K i ∈ F } as a basis of open sets around 1. So a basis of the open sets around any element ρ ∈ G is the set of cosets { ρσ i : K i ∈ F } (Take note that σ i is a subgroup of G , whilst ρ is an element of G .) Hence the open sets in G are ∅ , and any union of cosets ⋃ i ρ i σ i where ρ i ∈ G, σ i ∈ B If L/K is finite, then the Krull topology coincides with the discrete topology: any singleton set is a coset of the trivial subgroup { e } , and so is open. Therefore any set is open, being a union of singletons. Another important property is that every open subgroup H of G is also closed: the comple- ment G \ H of an open subgroup is the union all other cosets of H , as cosets are disjoint and cover G . Each coset is open, so G \ H is open, and H is closed. We can now state the fundamental theorem of infinite Galois theory. Let L/K , G , and F be as above, and let C be the set of σ i which are closed in G Theorem 7.2 (Fundamental theorem of infinite Galois theory) There is a bijection between F and C , given by: • K i 7 −→ σ i , • σ i 7 −→ { x ∈ L : ρ i ( x ) = x ∀ ρ ∈ σ i } Again, each subgroup of G maps to its fixed field, and each intermediate finite extension maps to the elements that fix it (but since each L/K i is Galois, this is exactly σ i ). So the correspondence is the same as in the finite case, except we only consider closed subgroups of G , and we only consider finite intermediate Galois extensions. From now on, every Galois group is implicitly equipped with the Krull topology. 18 8 LOCAL FIELDS 8 Local fields A local field is a type of topological field . Let F be a field with operations + , ∗ : F × F → F , and let i : F \ { 0 } → F \ { 0 } be the inversion function i ( x ) = x − 1 . Suppose F also has a topological structure I Definition 8.1. F is a topological field if + , ∗ , and i are continuous as functions between topological spaces. Again, we implicitly endow F × F with the product topology. We also implicitly endow F \ { 0 } with the subspace topology : Definition 8.2. Let ( X, I ) be a topological space and Y a subset of X . The subspace topology on Y is { Y ∩ U : U ∈ I } So the open sets in F \ { 0 } are just the open sets in F with 0 removed. After a reminder of compactness and local compactness , we can define a local field Definition 8.3. A topological space X is compact if every open cover of X has a finite sub-cover. A subset Y ⊆ X is compact if it is compact as a subspace with the subspace topology. A topological space X is locally compact if every x ∈ X has a compact neighbourhood. Definition 8.4. A local field is a topological field which is a locally compact topological space. Again, this is not to be confused with the concept of a local ring The above definition of a local field is concise, but not always useful. We now give a different characterisation in the case where we are given a discrete valuation, following Serre’s Local fields [9]. We can then more easily give some examples. Let K be a field with a discrete valuation v and valuation ring A K . As seen in chapter 3, v gives rise to an absolute value || · || , which induces a metric on K , and hence a topology on K Let π be a uniformizing element of A K . As usual, A K has unique maximal ideal πA generated by π Proposition 8.5. K is locally compact as a topological space if and only if it is complete and its residue field A/πA is finite. We refer the reader to [9] (page 27) for a proof of this equivalence. 19