Theory of p-adic Galois Representations

Transcription

1 Jean-Marc Fontaine and Yi Ouyang Theory of p-adic Galois Representations Springer

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3 Contents 0 Preliminary Inverse limits and Galois theory Inverse limits Galois theory Witt vectors and complete discrete valuation rings Nonarchimedean fields and local fields Witt vectors Structure of complete discrete valuation rings Cohen rings Galois groups of local fields Ramification groups of finite Galois extension Galois group of K s /K The functions Φ and Ψ Ramification groups of infinite Galois extension Different and discriminant Ramification in p-adic Lie extensions Sen s filtration Theorem Totally ramified Z p -extensions Continuous Cohomology Abelian cohomology Non-abelian cohomology l-adic representations of local fields: an overview l-adic Galois representations Linear representations of topological groups l-adic representations Representations arising from linear algebra Examples of l-adic Galois representations l-adic representations of finite fields l-adic Galois representations of finite fields l-adic geometric representations of finite fields

4 ii Contents 1.3 l-adic representations of local fields l-adic representations of local fields An alternative description The case of a finite residue field Geometric l-adic representations of G K p-adic Representations of fields of characteristic p B-representations and regular G-rings B-representations Regular (F, G)-rings Mod p Galois representations of fields of characteristic p > Étale ϕ-modules over E The functor M The inverse functor V p-adic Galois representations of fields of characteristic p > Étale ϕ-modules over E The field Ê ur OÊur and Z p representations p-adic representations Down to earth meaning of the equivalence of categories C-representations and Methods of Sen Krasner s Lemma and Ax-Sen s Lemma Krasner s Lemma Ax-Sen s Lemma Classification of C-representations Almost étale descent Decompletion Study of C-representations Sen s operator Θ Sen s operator Θ and the Lie algebra of ρ(g) Main Theorem Application of Sen s filtration Theorem Sen s method Tate-Sen s conditions (TS1), (TS2) and (TS3) Almost étale descent Decompletion Applications to p-adic representations C-admissible representations Notations for the rest of the book K-admissible p-adic representations C-admissible representations

5 Contents iii 4 The ring R and (ϕ, Γ )-module The ring R The ring R(A) Basic properties of the ring R The multiplicative group Fr R The action of Galois groups on R The action of Galois groups R(K cyc 0 /po K cyc), ε and π A fundamental theorem An overview of Galois extensions A summary of Galois extensions of K 0 and E The field B and its subrings (ϕ, Γ )-modules and p-adic Galois representations (ϕ, Γ )-modules The operator ψ de Rham representations Hodge-Tate representations de Rham representations The homomorphism θ The rings B + dr and B dr The element t Galois cohomology of B dr de Rham representations A digression Overconvergent rings and overconvergent representations The rings of Overconvergent elements Overconvergent representations Tate-Sen s method for Ã(0,r] The ring B ]0,r] Semi-stable p-adic representations The rings B cris and B st The ring B cris The Frobenius map ϕ on B cris The logarithm map The ring B st The operators ϕ and N on B st Some properties about B cris Some ideals of W (R) A description of A cris The filtration by I [r] Semi-stable p-adic Galois representations Filtered (ϕ, N)-modules Definitions

6 iv Contents t N (D) and t H (D) Admissible filtered (ϕ, N)-modules Statement of Theorem A and Theorem B de Rham implies potentially semi-stable Weakly admissible implies admissible Proof of Theorem A and Theorem B Admissible filtered (ϕ, N)-modules of dimension 1 and Hodge and Newton polygons The case when the filtration is trivial Tate s twist Admissible filtered (ϕ, N)-modules of dimension Admissible filtered (ϕ, N)-modules of dimension Proof of Proposition B Construction of the natural isomorphism Unramified representations Reduction to the algebraically closed residue field case Representations of dimension End of proof of Proposition B Q p r-representations and filtered (ϕ r, N)-modules Definitions Main properties The Q p r-representation V (r) Outline of the proof Reduction of Proposition B2 to Proposition B Outline of the Proof of Propositions A and B Proof of Proposition 2A and Proposition 2B Proof of Proposition 2A Fundamental complex of D Reduction to the case t H = The case for V The case for D Finish of proof The Fundamental Lemma of Banach-Colmez space Application of the Fundamental Lemma Recurrence of the Hodge polygon and end of proof References

7 0 Preliminary 0.1 Inverse limits and Galois theory Inverse limits. In this subsection, we always assume that A is a category with infinite products. In particular, one can let A be the category of sets, of (topological) groups, of (topological) rings, of left (topological) modules over a ring A. Recall that a partially ordered set I is called a directed set if for any i, j I, there exists k I such that i k and j k. Definition 0.1. Let I be a directed set. Let (A i ) i I be a family of objects in A. This family is called an inverse system(or a projective system) of A over the index set I if for every pair i j I, there exists a morphism ϕ ji : A j A i such that the following two conditions are satisfied: (1) ϕ ii = Id; (2) For every i j k, ϕ ki = ϕ ji ϕ kj. Definition 0.2. The inverse limit(or projective limit) of a given inverse system A = (A i ) i I is defined to be an object A in A { A = lim A i = (a i ) } A i : ϕ ji (a j ) = a i for every pair i j, i I i I such that the natural projection ϕ i : A A i, a = (a j ) j I a i is a morphism for each i I. Remark 0.3. One doesn t need the set I to be a directed set but only to be a partially ordered set to define an inverse system. For example, let I be a set with trivial ordering, i.e. i j if and only if i = j, then lim A i = A i. i I i I However, this condition is usually satisfied and often needed in application. By the inverse system condition, one can see immediately ϕ i = ϕ ji ϕ j for every pair i j. Actually, A is the solution of the universal problem:

8 2 0 Preliminary Proposition 0.4. Let (A i ) be an inverse system in A, A be its inverse limit and B be an object in A. If there exist morphisms f i : B A i for all i I such that for every pair i j, f i = ϕ ji f j, then there exists a unique morphism f : B A such that f j = ϕ j f. Proof. This is an easy exercise. By definition, if A is the category of topological spaces, i.e., if X i is a topological space for every i I and ϕ ij s are continuous maps, then X = lim X i is a topological space equipped with a natural topology, the weakest i I topology such that all the ϕ i s are continuous. Recall that the product topology of the topological space X i is the weakest topology such that the projection i I pr j : X i X j is continuous for every j I. Thus the natural topology i I of X is the topology induced as a closed subset of X i with the product topology. For example, if each X i is endowed with the discrete topology, then X is endowed with the topology of the inverse limit of discrete topological spaces. In particular, if each X i is a finite set endowed with discrete topology, then we will get a profinite set (inverse limit of finite sets). In this case, since lim X i X i is closed, and since X i, as the product space of compact i I i I spaces, is still compact, lim X i is compact too. In this case one can see that lim X i is also totally disconnected. If moreover, each X i is a (topological) group and if the ϕ ij s are (continuous) homomorphisms of groups, then lim X i is a group with ϕ i : lim X j X i j a (continuous) homomorphism of groups. If the X i s are finite groups endowed with discrete topology, the inverse limit in this case is a profinite group. Thus a profinite group is always compact and totally disconnected. As a consequence, all open subgroups of a profinite group are closed, and a closed subgroup is open if and only if it is of finite index. Example 0.5. (1) For the set of positive integers N, we define an ordering n m if n m. For the inverse system (Z/nZ) n N of finite rings where the transition map ϕ mn is the natural projection, the inverse limit is i I. Ẑ = lim n N Z/nZ (2) Let l be a prime number, for the sub-index set {l n : n N} of N, Z l = lim Z/l n Z n N

9 0.1 Inverse limits and Galois theory 3 is the ring of l-adic integers. The ring Z l is a complete discrete valuation ring with the maximal ideal generated by l, the residue field Z/lZ = F l, and the fraction field [ ] 1 Q l = Z l = l m Z l l m=0 being the field of l-adic numbers. If N 1, let N = l r1 1 lr2 2 lr h h be its primary factorization. Then the isomorphism h Z/NZ Z/l ri i Z i=1 induces an isomorphism of topological rings Ẑ l prime number Z l Galois theory. Let K be a field and L be a (finite or infinite) Galois extension of K. The Galois group Gal(L/K) is the group of the K-automorphisms of L, i.e., Gal(L/K) = {g : L L, g(γ) = γ for all γ K}. Denote by E the set of finite Galois extensions of K contained in L and order this set by inclusion, then for any pair E, F E, one has EF E and E, F EF, thus E is in fact a directed set and L = E. As a result, we can study the inverse limits of objects over this directed set. For the Galois groups, by definition, E E γ = (γ E ) lim Gal(E/K) if and only if (γ F ) E = γ E for E F E. E E Galois theory tells us that the following restriction map is an isomorphism Gal(L/K) lim Gal(E/K) E E g (g E ) : g E the restriction of g in E. From now on, we identify the two groups through the above isomorphism. Put the topology of the inverse limit with the discrete topology on each Gal(E/K), the group G = Gal(L/K) is then a profinite group, endowed with a compact and totally disconnected topology, which is called the Krull topology. We have Theorem 0.6 (Fundamental Theorem of Galois Theory). There is a one-one correspondence between intermediate field extensions K K L

10 4 0 Preliminary and closed subgroups H of Gal(L/K) given by K Gal(L/K ) and H L H where L H = {x L g(x) = x for all g H} is the invariant field of H. Moreover, the above correspondence gives one-one correspondences between finite extensions (resp. finite Galois extensions, Galois extensions) of K contained in L and open subgroups (resp. open normal subgroups, closed normal subgroups) of Gal(L/K). Remark 0.7. (1) Given an element g and a sequence (g n ) n N of Gal(L/K), the sequence (g n ) n N converges to g if and only if for all E E, there exists n E N such that if n n E, then g n E = g E. (2) The open normal subgroups of G are the groups Gal(L/E) for E E, and there is an exact sequence 1 Gal(L/E) Gal(L/K) Gal(E/K) 1. (3) A subgroup of G is open if and only if it contains an open normal subgroup. A subset X of G is an open set if and only if for all x X, there exists an open normal subgroup H x such that xh x X. (4) If H is a subgroup of Gal(L/K), then L H = L H with H being the closure of H in Gal(L/K). We first give an easy example: Example 0.8. Let K be a finite field with q elements, and let K be an algebraic closure of K with Galois group G = Gal(K/K). For each n N, n 1, there exists a unique extension K n of degree n of K contained in K s. The extension K n /K is cyclic with Galois group Gal(K n /K) Z/nZ = ϕ n where ϕ n = (x x q ) is the arithmetic Frobenius of Gal(K n /K). We have the following diagram G lim Ẑ Gal(K n /K) lim Z/nZ. Thus the Galois group G Ẑ is topologically generated by ϕ G: ϕ(x) = xq for x K s, i.e., with obvious conventions, any elements of G can be written uniquely as g = ϕ a with a Ẑ. The element ϕ is called the arithmetic Frobenius and its inverse ϕ 1 is called the geometric Frobenius of G. If K = F p, the arithmetic Frobenius (x x p ) is called the absolute Frobenius and denoted as σ. Moreover, for any field k of characteristic p, we call the endomorphism σ : x x p the absolute Frobenius of k. σ is an automorphism if and only if k is perfect. In the case K = Q, let Q be an algebraic closure of Q, and let G Q = Gal(Q/Q).

11 0.2 Witt vectors and complete discrete valuation rings 5 The structure of G Q is far from being completely understood. An open question is: Let J be a finite groups. Is it true that there exists a finite Galois extension of Q whose Galois group is isomorphic to J? There are cases where the answer is known(eg. J is abelian, J = S n, J = A n, etc). For each place p of Q (i.e., a prime number or ), let Q p be a chosen algebraic closure of the p-adic completion Q p of Q (for p =, we let Q p = R and Q p = C). Choose for each p an embedding σ p : Q Q p. From the diagram Q Q p Q Q p one can identify G p = Gal(Q p /Q p ) to a closed subgroup of G Q, called the decomposition subgroup of G at p. To study G Q, it is necessary and important to know properties about each G p. This phenomenon is not unique. There is a generalization of the above to number fields, i.e., a finite extension of Q, whose completions are finite extensions of Q p. There is also an analogue for global function fields, i.e., finite extensions of k(x) with k a finite field, whose completions are of the type k ((y)), where k is a finite extension of k. As a consequence, we are led to study the properties of local fields. 0.2 Witt vectors and complete discrete valuation rings Nonarchimedean fields and local fields. First let us recall the definition of valuation. Definition 0.9. Let A be a ring. If v : A R {+ } is a function such that (1) v(a) = + if and only if a = 0, (2) v(ab) = v(a) + v(b), (3) v(a + b) min{v(a), v(b)}, and if there exists a A such that v(a) / {0, + }, then v is called a (nontrivial) valuation on A. If v(a) is a discrete subset of R, v is called a discrete valuation. The above definition of valuation is usually called a valuation of height 1. For a ring A with a valuation v, one can always define a topology to A with a neighborhood basis of 0 given by {x : v(x) > n}, then A becomes a topological ring. The valuation v on A defines an absolute value: a = e v(a). For any a A, then a is small a is small v(a) is big.

12 6 0 Preliminary If v 1 and v 2 are valuations on A, then v 1 and v 2 are equivalent if there exists r R, r > 0, such that v 2 (a) = rv 1 (a) for any a A. Thus v 1 and v 2 are equivalent if and only if the induced topologies in A are equivalent. If A is a ring with a valuation v, then A is always a domain: if ab = 0 but b 0, then v(b) < + and v(a) = v(ab) v(b) = +, hence a = 0. Let K be the fraction field of A, we may extend the valuation to K by v(a/b) = v(a) v(b). Then the ring of valuations (often called the ring of integers) O K = {a K v(a) 0} (0.1) is a local ring, with the maximal ideal m K given by and k K = O K /m K being the residue field. m K = {a K v(a) > 0}, (0.2) Definition A field K with a valuation v is called a valuation field. A valuation field is nonarchimedean: the absolute value defines a metric on K, which is ultrametric, since a + b max( a, b ). Let K denote the completion of K of the valuation v. Choose π O K, π 0, and v(π) > 0, let O K = lim O K /(π m ). Then O K is again a domain and K = O K[1/π]. Remark The ring O K does not depend on the choice of π. Indeed, if v(π) = r > 0, v(π ) = s > 0, for any n N, there exists m n N, such that π mn π n O K, so lim O K/(π m ) lim O K /(π n ). Definition A field complete with respect to a valuation v is called a complete nonarchimedean field. We quote the following well-known result of valuation theory: Proposition If F is a complete nonarchimedean field with a valuation v, and F is any algebraic extension of F, then there is a unique valuation v on F such that v (x) = v(x), for any x F. Moreover, F is complete if and only if F /F is finite. If α, α F are conjugate, then v(α) = v(α ). Remark By abuse of notations, we will set the extended valuation v = v. Let F be a complete field with respect to a discrete valuation, let F be any algebraic extension of F. We denote by v F the unique valuation of F extending the given valuation of F such that v F (F ) = Z. v F is called the normalized valuation of F. If F is a field with a valuation, for any a m F, a 0, let v a denote the unique valuation of F equivalent to the given valuation such that v a (a) = 1.

13 0.2 Witt vectors and complete discrete valuation rings 7 Definition A local field is a complete discrete valuation field whose residue field is perfect of characteristic p > 0. Thus a local field is always a complete nonarchimedean field. A p-adic field is a local field of characteristic 0. Example A finite extension of Q p is a p-adic field. In fact, it is the only p-adic field whose residue field is finite. Let K be a local field with the normalized valuation and perfect residue field k, char k = p > 0. Let π be a uniformizing parameter of K. Then v K (π) = 1 and m K = (π). One has an isomorphism O K lim n O K /m n K = lim n O K /(π n ), the topology defined by the valuation for O K is the same as the topology of the inverse limit with the discrete topology in each O K /m n K. Thus we have the following propositions: Proposition The local field K is locally compact (equivalently, O K is compact) if and only if the residue field k is finite. Proposition Let S be a set of representatives of k in O K. Then every element x O K can be uniquely written as x = i 0 s i S and x K can be uniquely written as x = i n s i S s i π i (0.3) s i π i. (0.4) As p m K, by the binomial theorem, for a, b O K, we have the following fact: a b mod m K = a pn b pn mod m n+1 K for n 0. (0.5) Proposition For the natural map O K k, there is a natural section r : k O K which is unique and multiplicative. Proof. Let a k. For any n N, there exists a unique a n k such that a pn n = a, a p n+1 = a n. Let â n be a lifting of a n in O K. By (0.5), â p n+1 â n mod m K implies that â pn+1 n+1 âpn n mod m n+1 K. There- exists. By (0.5) again, r(a) is found to be independent fore r(a) := lim n âpn n of the choice of the liftings of the â n s. It is easy to check that r is a section of ρ and is multiplicative. Moreover, if t is another section, we can always choose â n = t(a n ), then hence the uniqueness follows. r(a) = lim n âpn n = lim t(a n) pn = t(a), n

14 8 0 Preliminary Remark This element r(a) is usually called the Teichmüller representative of a, often denoted as [a]. If char(k) = p, then r(a + b) = r(a) + r(b) since (â n + b n ) pn = â pn pn n + b n. Thus r : k O K is a homomorphism of rings. We can use it to identify k with a subfield of O K. Then Theorem If O K is a complete discrete valuation ring, k is its residue field and K is its quotient field. Let π be a uniformizing parameter of O K. Suppose that O K (or K) and k have the same characteristic, then O K = k[[π]], K = k((π)). Proof. We only need to show the case that char(k) = 0. In this case, the composite homomorphism Z O K k is injective and the homomorphism Z O K extends to Q, hence O K contains a field Q. By Zorn s lemma, there exists a maximal subfield of O K. We denote it by S. Let S be its image in k. We have an isomorphism S S. It suffices to show that S = k. First we show k is algebraic over S. If not, there exists a O K whose image ā k is transcendental over S. The subring S[a] maps to S[ā], hence is isomorphic to S[X], and S[a] m K = 0. Therefore O K contains the field S(a) of rational functions of a, contradiction to the maximality of S. Now for any α k, let f(x) be the minimal polynomial of S(α) over S. Since char(k) = 0, f is separable and α is a simple root of f. Let f S[X] be a lifting of f. By Hensel s Lemma, there exists x O K, f(x) = 0 and x = α. One can lift S[α] to S[x] by sending α to S. By the maximality of S, x S. and thus k = S. If K is a p-adic field, char(k) = 0, then r(a + b) r(a) + r(b) in general. Witt vectors are useful to describe this situation Witt vectors. Let p be a prime number, A be a commutative ring. Let X i, Y i (i N) be indeterminates and let A[X, Y ] = A[X 0, X 1,, X n, ; Y 0, Y 1,, Y n, ]. Lemma For all Φ Z[X, Y ], there exists a unique sequence {Φ n } n N in Z[X, Y ] such that Φ(X pn 0 + p X pn p n X n, Y pn 0 + Y pn p n Y n ) = (Φ 0 (X, Y )) pn + p (Φ 1 (X, Y )) pn p n Φ n (X, Y ). (0.6) Moreover, Φ n Z[X 0, X 1,, X n ; Y 0, Y 1,, Y n ].

15 0.2 Witt vectors and complete discrete valuation rings 9 Proof. First we work in Z[ 1 p ][X, Y ]. Set Φ 0(X, Y ) = Φ(X 0, Y 0 ) and define Φ n inductively by ( ) Φ n (X, Y ) = 1 ( n n ) n 1 p n Φ p i X pn i i, p i Y pn i i p i Φ i (X, Y ) pn i. i=0 i=0 Clearly Φ n exists, is unique in Z[ 1 p ][X, Y ], and is in Z[ 1 p ][X 0,, X n ; Y 0,, Y n ]. We only need to prove that Φ n has coefficients in Z. This is done by induction on n. For n = 0, Φ 0 certainly has coefficients in Z. Assuming Φ i has coefficients in Z for i n, to show that Φ n+1 has coefficients in Z, we need to prove that Φ(X pn p n X n ; Y pn p n Y n ) Φ 0 (X, Y ) pn + pφ 1 (X, Y ) pn p n 1 Φ n 1 (X, Y ) p mod p n. One can verify that LHS Φ(X pn p n 1 X p n 1 ; Y pn p n 1 Y p n 1 ) mod pn Φ 0 (X p, Y p ) pn 1 + pφ 1 (X p, Y p ) pn p n 1 Φ n 1 (X p, Y p ) mod p n. By induction, Φ i (X, Y ) Z[X, Y ], hence Φ i (X p, Y p ) (Φ i (X, Y )) p mod p, and p i Φ i (X p, Y p ) pn 1 i p i Φ i (X, Y ) pn i mod p n. i=0 Putting all these congruences together, we get the lemma. Remark The polynomials W n = n i=0 p i X pn i i (n N) are called the Witt polynomials for the sequence (X 0,, X n, ). One can easily see that X n Z[p 1 ][W 0,, W n ] for each n. For n 1, let W n (A) = A n as a set. Applying the above lemma, if Φ = X+ Y, we set S i Z[X 0, X 1,, X i ; Y 0, Y 1,, Y i ] to be the corresponding Φ i ; if Φ = XY, we set P i Z[X 0, X 1,, X i ; Y 0, Y 1,, Y i ] to be the corresponding Φ i. For two elements a = (a 0, a 1,, a n 1 ), b = (b 0, b 1,, b n 1 ) W n (A), put a + b = (s 0, s 1,, s n 1 ), a b = (p 0, p 1,, p n 1 ), where s i = S i (a 0, a 1,, a i ; b 0, b 1,, b i ), p i = P i (a 0, a 1,, a i ; b 0, b 1,, b i ). Remark It is clear that S 0 = X 0 + Y 0, P 0 = X 0 Y 0. (0.7)

16 10 0 Preliminary From (X 0 + Y 0 ) p + p S 1 = X p 0 + p X 1 + Y p 0 + p Y 1, we get p 1 S 1 = X 1 + Y 1 i=1 Also from (X p 0 + p X 1) (Y p 0 + p Y 1) = X p 0 Y p 0 + p P 1, we get 1 p ( ) p X0 i Y p i 0. (0.8) i P 1 = X 1 Y p 0 + Xp 0 Y 1 + p X 1 Y 1. (0.9) But for general n, it is too complicated to write down S n and P n explicitly. Consider the map ρ W n (A) A n (a 0, a 1,, a n 1 ) (w 0, w 1,, w n 1 ) where w i = W i (a) = a pi 0 + p api p i a i. Then w i (a + b) = w i (a) + w i (b) and w i (ab) = w i (a) w i (b). We notice the following facts: (1) If p is invertible in A, ρ is bijective and therefore W n (A) is a ring isomorphic to A n. (2) If A has no p-torsion, by the injection A A[ 1 p ], then W n(a) W n (A[ 1 p ]). Thus W n(a) is a subring with the identity 1 = (1, 0, 0, ), as a, b W n (A) implies that a b W n (A), when applying Lemma 0.22 to Φ = X Y. (3) In general, any commutative ring can be written as A = R/I with R having no p-torsion. Then W n (R) is a ring, and W n (I) = {(a 0, a 1,, a n ) a i I} is an ideal of W n (R). Then W n (R/I) is the quotient of W n (R) by W n (I), again a ring itself. For the sequence of rings W n (A), consider the maps W n+1 (A) W n (A) (a 0, a 1,, a n ) (a 0, a 1,, a n 1 ). This is a surjective homomorphism of rings for each n. Define W (A) = lim n N W n (A). Put the topology of the inverse limit with the discrete topology on each W n (A), then W (A) can be viewed as a topological ring. An element in W (A) is written as (a 0, a 1,, a i, ).

17 0.2 Witt vectors and complete discrete valuation rings 11 Definition The ring W n (A) is called the ring of Witt vectors of length n of A, an element of it is called a Witt vector of length n. The ring W (A) is called the ring of Witt vectors of A (of infinite length), an element of it is called a Witt vector. By construction, W (A) as a set is isomorphic to A N. For two Witt vectors a = (a 0, a 1,, a n, ), b = (b 0, b 1,, b n, ) W (A), the addition and multiplication laws are given by The map a + b = (s 0, s 1,, s n, ), a b = (p 0, p 1,, p n, ). ρ : W (A) A N, (a 0, a 1,, a n, ) (w 0, w 1,, w n, ) is a homomorphism of commutative rings and ρ is an isomorphism if p is invertible in A. Example One has W (F p ) = Z p. W n and W are actually functorial: let h : A B be a ring homomorphism, then we get the ring homomorphisms W n (h) : W n (A) W n (B) (a 0, a 1,, a n 1 ) (h(a 0 ), h(a 1 ),, h(a n 1 )) for n 1 and similarly the homomorphism W (h) : W (A) W (A). Remark In fact, W n is represented by an affine group scheme over Z: W n = Spec(B), where B = Z[X 0, X 1,, X n 1 ]. with the comultiplication given by m : B B Z B Z[X 0, X 1,, X n 1 ; Y 0, Y 1,, Y n 1 ] X i X i 1, Y i 1 X i, m X i = S i (X 0, X 1,, X i ; Y 0, Y 1,, Y i ). Remark If A is killed by p, then So ρ is given by W n (A) wi A (a 0, a 1,, a n 1 ) a pi 0. W n (A) ρ A n (a 0, a 1,, a n 1 ) (a 0, a p 0,, apn 1 0 ). In this case ρ certainly is not an isomorphism. Similarly ρ : W (A) A N is not an isomorphism either.

18 12 0 Preliminary Maps related to the ring of Witt vectors. Let A be a commutative ring. We can define the following maps ν, r and ϕ related to W (A). (1) The shift map ν. We define ν : W (A) W (A), (a 0,, a n, ) (0, a 0,, a n, ), which is called the shift map. It is additive: it suffices to verify this fact when p is invertible in A, and in that case the homomorphism ρ : W (A) A N transforms ν into the map which sends (w 0, w 1, ) to (0, pw 0, ). By passage to the quotient, one deduces from ν an additive map of W n (A) into W n+1 (A). There are exact sequences 0 W k (A) νr W k+r (A) W r (A) 0. (0.10) (2) The Teichmüller map r. We define a map r : A W (A), x [x] = (x, 0,, 0, ). When p is invertible in A, ρ transforms r into the mapping that sends x to (x, x p,, x pn, ). One deduces by the same reasoning as in (1) the following formulas: r(xy) = r(x)r(y), x, y A (0.11) (a 0, a 1, ) = ν n (r(a n )), a i A (0.12) n=0 r(x) (a 0, ) = (xa 0, x p a 1,, x pn a n, ), x, a i A. (0.13) (3) The Frobenius map ϕ. Suppose k is a ring of characteristic p. The homomorphism k k, x x p induces a ring homomorphism: ϕ : W (k) W (k), (a 0, a 1, ) (a p 0, ap 1, ), which is called the Frobenius map. If moreover, k is a perfect field, the Frobenius on W (k) is often denoted as σ.

19 0.2 Witt vectors and complete discrete valuation rings Structure of complete discrete valuation rings with unequal characteristic. As an application of Witt vectors, we discuss the structure of complete discrete valuation rings in the unequal characteristic case. The exposition in this subsection follows entirely that in Serre [Ser80], Chap. II, 5. Definition We say that a ring A of characteristic p is perfect if the endomorphism x x p of A is an automorphism, i.e., every element of x A has a unique p-th root, denoted x p 1. When A is a field, this is the usual definition of a perfect field. Definition If A is a ring which is Hausdorff and complete for a decreasing filtration of ideals a 1 a 2 such that a m a n a m+n, and if the ring A/a 1 is perfect of characteristic p, then A is called a p-ring. If furthermore the filtration is the p-adic filtration {p n A} n N, with the residue ring k = A/pA perfect, and if p is not a zero-divisor in A, then A is called a strict p-ring. Proposition Let A be a p-ring, then: (1) There exists one and only one system of representatives f : k A which commutes with p-th powers: f(λ p ) = f(λ) p. (2) In order that a A belong to S = f(k), it is necessary and sufficient that a be a p n -th power for all n 0. (3) This system of representatives is multiplicative, i.e., one has f(λµ) = f(λ)f(µ) for all λ, µ k. (4) If A has characteristic p, this system of representatives is additive, i.e., f(λ + µ) = f(λ) + f(µ). Proof. The proof is very similar to the proof of Proposition We leave it as an exercise. Proposition 0.31 implies that when A is a p-ring, it always has the system of multiplicative representatives f : A/a 1 A, and for every sequence α 0,, α n,, of elements of A/a 1, the series f(α i )p i (0.14) i=0 converges to an element a A. If furthermore A is a strict p-ring, every element a A can be uniquely expressed in the form of a series of type (0.14). Let β i = α pi i, then a = f(β p i i )p i. We call {β i } the coordinates of i=0 a. Example Let X α be a family of indeterminates, and let S be the ring of p -polynomials in the X α with integer coefficients, i.e., S = Z[X p n n 0 α ]

20 14 0 Preliminary If one provides S with the p-adic filtration {p n S} n 0 and completes it, one obtains a strict p-ring that will be denoted Ŝ = Z[X α p ]. The residue ring Ŝ/pŜ = F p[xα p ] is perfect of characteristic p. Since X α admits p n -th roots for all n, we identify X α in Ŝ with its residue ring. Suppose X 0,, X n, and Y 0,, Y n, are indeterminates in the ring, Y p ]. Consider the two elements Z[X p i i x = X i p i, y = Y i p i. i=0 If is one of the operations +,,, then x y is also an element in the ring and can be written uniquely of the form x y = i=0 i=0 f(q i )p i, with Q i F p [X p i, Y p i ]. As Q i are p -polynomials with coefficients in the prime field F p, one can evaluate it in a perfect ring k of characteristic p. More precisely, Proposition If A is a p-ring with residue ring k and f : k A is the system of multiplicative representatives of A. Suppose {α i } and {β i } are two sequences of elements in k. Then f(α i )p i f(β i )p i = f(γ i )p i i=0 i=0 with γ i = Q i (α 0, α 1, ; β 0, β 1, ). Proof. One sees immediately that there is a homomorphism θ : Z[X p i i=0, Y p i ] A which sends X i to f(α i ) and Y i to f(β i ). This homomorphism extends by continuity to Z[X p i, Y p i ] A, which sends x = X i p i to α = f(α i )p i and y = Y i p i to β = f(β i )p i. Again θ induces, on the residue rings, a homomorphism θ : F p [X p i, Y p i ] k which sends X i to α i and Y i to β i. Since θ commutes with the multiplicative representatives, one thus has f(αi )p i f(β i )p i =θ(x) θ(y) = θ(x y) = θ(f(q i ))p i = f( θ(q i ))p i, this completes the proof of the proposition, as θ(q i ) is nothing but γ i.

21 0.2 Witt vectors and complete discrete valuation rings 15 Definition Let A be a complete discrete valuation ring, with residue field k. Suppose A has characteristic 0 and k has characteristic p > 0. The integer e = v(p) is called the absolute ramification index of A. A is called absolutely unramified if e = 1, i.e., if p is a local uniformizer of A. Remark If A is a strict p-ring, and its residue ring A/pA is a field, then A is a complete discrete valuation ring, absolutely unramified. Proposition Suppose A and A are two p-rings with residue rings k and k, suppose A is also strict. For every homomorphism h : k k, there exists exactly one homomorphism g : A A such that the diagram g A A k h k is commutative. As a consequence, two strict p-rings with the same residue ring are canonically isomorphic. Proof. For a = f A (α i )p i A, if g is defined, then i=0 g(a) = g(f A (α i )) p i = f A (h(α i )) p i, i=0 hence the uniqueness. But by Proposition 0.33, g defined by the above way is indeed a homomorphism. Theorem For every perfect ring k of characteristic p, there exists a unique strict p-ring H with residue ring k. In fact H = W (k). Proof. The uniqueness follows from Proposition For the existence, if k = F p [Xα p ], then H = Ŝ satisfies the condition. In general, as every perfect ring is a quotient of a ring of the type F p [Xα p ], we just need to show if h : k k is a surjective homomorphism and if there exists a strict p-ring H k with residue ring k, then there exists a strict p-ring H k with residue ring k. Indeed, for a, b H k, we say a b if the images of their coordinates by h are equal. This is an equivalence relation, and if a b, a b, then a a b b by Proposition Let H k be the quotient of H k modulo this equivalence relation. It is routine to check H k is a strict p-ring with residue ring k. Now for the second part, let H be the strict p-ring with residue ring k, and let f : k H be the multiplicative system of representatives of H. Define θ : W (k) H, i=0 (a 0,, a n, ) i=0 f(a p i i )p i.

22 16 0 Preliminary It is a bijection. When H = Ŝ, a = (X 0, ), b = (Y 0, ), we have n i=0 f(x p i i )p i + n i=0 f(y p i i )p i = W n (X p n) + W n (Y p n ) = W n (S 0 (X p n, Y p n ), ), Since n f(s i (a, b) p i )p i = W n (f(s i (a, b) p n )). i=0 S i (X p n, Y p n ) f(s i (X p n, Y p n )) = f(s i (a, b) p n ) mod p, we get θ(a)+θ(b) θ(a+b) mod p n+1, for any n 0. Therefore, θ(a)+θ(b) = θ(a + b). Similarly, θ(a)θ(b) = θ(ab). It follows that the formulas are valid without any restriction on H, a and b. So θ is an isomorphism. By the above theorem and Proposition 0.36, we immediately have: Corollary For k, k perfect rings of characteristic p, Hom(k, k ) = Hom(W (k), W (k )). Corollary If k is a field, perfect or not, then νϕ = p = ϕν. Proof. It suffices to check this when k is perfect; in that case, applying the isomorphism θ above, one finds: θ(ϕνa) = which gives the identity. i=0 f(a p i i )p i+1 = pθ(a) = θ(pa), Now we can state the main theorems of the unequal characteristic case. Theorem (1) For every perfect field k of characteristic p, W (k) is the unique complete discrete valuation ring of characteristic 0 (up to unique isomorphism) which is absolutely unramified and has k as its residue field. (2) Let A be a complete discrete valuation ring of characteristic 0 with a perfect residue field k of characteristic p > 0. Let e be its absolute ramification index. Then there exists a unique homomorphism of ψ : W (k) A which makes the diagram W (k) A k commutative, moreover ψ is injective, and A is a free W (k)-module of rank equal to e.

23 0.2 Witt vectors and complete discrete valuation rings 17 Proof. (1) is a special case of Theorem For (2), the existence and uniqueness of ψ follow from Proposition 0.36, since A is a p-ring. As A is of characteristic 0, ψ is injective. If π is a uniformizer of A, then every a A can be uniquely written as a = f(α i )π i for α i k. Replaced π e by p (unit), then a is uniquely written as a = i=0 e 1 f(α ij ) π j p i, α ij k. i=0 j=0 Thus {1, π,, π e 1 } is a basis of A as a W (k)-module. Remark From now on, we denote the Teichmüller representative r(a) of a k by [a], then by the proof of Theorem 0.37, the homomorphism ψ : W (k) A in the above theorem is given by ψ((a 0, a 1, )) = n=0 p n [a p n n ]. For the case A = W (k), for a k, the Teichmüller representative r(a) is the same as the element r(a) = (a, 0, ), we have Cohen rings. (a 0, a 1, ) = n=0 p n [a p n n ]. (0.15) We have seen that if k is a perfect field, then the ring of Witt vectors W (k) is the unique complete discrete valuation ring which is absolutely unramified and with residue field k. However, if k is not perfect, the situation is more complicated. We first quote two theorems without proof from commutative algebra (cf. Matsumura [Mat86], 29, pp ): Theorem 0.42 (Theorem 29.1, [Mat86]). Let (A, πa, k) be a discrete valuation ring and K an extension of k; then there exists a discrete valuation ring (B, πb, K) containing A. Theorem 0.43 (Theorem 29.2, [Mat86]). Let (A, m A, k A ) be a complete local ring, and (R, m R, k R ) be an absolutely unramified discrete valuation ring of characteristic 0 (i.e., m R = pr). Then for every homomorphism h : k R k A, there exists a local homomorphism g : R A which induces h on the ground field. Remark The above theorem is a generalization of Proposition However, in this case there are possibly many g inducing h. For example, let k = F p (x) and A = Z p (x), then the homomorphism x x + α in A for any α pz p induces the identity map in k.

24 18 0 Preliminary Applying A = Z p to Theorem 0.42, then if K is a given field of characteristic p, there exists an absolutely unramified discrete valuation ring R of characteristic 0 with residue field K. By Theorem 0.43, this ring R is unique up to isomorphism. Definition Let k be a field of characteristic p > 0, the Cohen ring C(k) is the unique (up to isomorphism) absolutely unramified discrete valuation ring of characteristic 0 with residue field k. We now give an explicit construction of C(k). Recall that a p-basis of a field k is a set B of elements of k, such that (1) [k p (b 1,, b r ) : k p ] = p r for any r distinct elements b 1,, b r B; (2) k = k p (B). If k is perfect, only the empty set is a p-basis of k; if k is imperfect, there always exists nonempty sets satisfying condition (1), then any maximal such set (which must exist, by Zorn s Lemmma) must also satisfy (2) and hence is a p-basis. Let B be a fixed p-basis of k, then k = k pn (B) for every n > 0, and B p n = {b p n b B} is a p-basis of k p n. Let I n = B {0,, pn 1}, then { T n = b α = } b α b, α = (α b ) b B I n b B generates k as a k pn -vector space, and in general T pm n is a basis of k pm over k pn+m. Set C n+1 (k) = the subring of W n+1 (k) generated by W n+1 (k pn ) and [b] for b B. For x k, we define the Teichmüller representative [x] = (x, 0,, 0) W n+1 (k). We also define the shift map V on W n+1 (k) by V ((x 0,, x n )) = (0, x 0,, x n 1 ). Then every element x W n+1 (k) can be written as x = (x 0,, x n ) = [x 0 ] + V ([x 1 ]) + + V n ([x n ]). We also has [y]v r (x) = V r ([y pr ]x). Then C n+1 (k) is nothing but the additive subgroup of W n+1 (k) generated by {V r ([(b α ) pr x]) b α T n r, x k pn, r = 0,, n}. By Corollary 0.39, one sees that V r (ϕ r ([x])) = p r [x] mod V r+1. Let U r be ideals of C n+1 (k) defined by U r = C n+1 (k) V r (W n+1 (k)).

25 0.2 Witt vectors and complete discrete valuation rings 19 Then U r is the additive subgroup generated by {V m ([(b α ) pm x]) b α T n m, x k pn, m r}. Then we have C n+1 (k)/u 1 k and the multiplication p r : C n+1 (k)/u 1 U r /U r+1 induces an isomorphism for all r n. Thus U n is generated by p n and by decreasing induction, one has U r = p r C n+1 (k). Moreover, for any x C n+1 (k) U 1, let y be a preimage of x 1 C n+1 (k)/u 1, then xy = 1 z with z U 1 and xy(1 + z + + z n ) = 1, thus x is invertible. Hence we proved Proposition The ring C n+1 (k) is a local ring whose maximal ideal is generated by p, whose residue field is isomorphic to k. For every r n, the multiplication by p r induces an isomorphism of C n+1 (k)/pc n+1 (k) with p r C n+1 (k)/p r+1 C n+1 (k), and p n+1 C n+1 (k) = 0. Lemma The canonical projection pr : W n+1 (k) W n (k) induces a surjection π : C n+1 (k) C n (k). Proof. By definition, the image of C n+1 (k) by pr is the subring of W n (k) generated by W n (k pn ) and [b] for b B, but C n (k) is the subring generated by W n (k pn 1 ) and [b] for b B, thus the map π is well defined. For n 1, the filtration W n (k) V (W n (k)) V n 1 (W n (k)) V n (W n (k)) = 0 induces the filtration of C n (k) pc n (k) p n 1 C n (k) p n C n (k) = 0. To show π is surjective, it suffices to show that the associate graded map is surjective. But for r < n, we have the following commutative diagram p r C n+1 (k)/p r+1 C n+1 (k) j V r W n+1 (k)/v r+1 W n+1 (k) k gr π p r C n (k)/p r+1 C n (k) j gr pr=id V r W n (k)/v r+1 W n (k) k Since the inclusion j(resp. j ) identifies p r C n+1 (k)/p r+1 C n+1 (k) (resp. p r C n (k)/p r+1 C n (k)) to k pr, thus gr π is surjective for r < n. For r = n, p n C n (k) = 0. Then gr π is surjective at every grade and hence π is surjective. By Proposition 0.46, we thus have Theorem The ring lim n C n (k) is the Cohen ring C n (k) of k. Remark (1) By construction, C(k) is identified as a subring of W (k); moreover, for k 0 = k pn the largest perfect subfield of k, C(k) contains n N W (k 0 ). (2) As C(k) contains the multiplicative representatives [b] for b B, it contains all elements [B α ] and [B α ] for n N and α I n.

26 20 0 Preliminary 0.3 Galois groups of local fields In this section, we let K be a local field with the residue field k = k K perfect of characteristic p and the normalized valuation v K. Let O K be the ring of integers of K, whose maximal ideal is m K. Let U K = OK = O K m K be the group of units and UK i = 1 + mi K for i 1. Replacing K by L, a finite separable extension of K, we get corresponding notations k L, v L, O L, m L, U L and UL i. Recall the following notations: e L/K N : the ramification index defined by v K (L ) = 1 e L/K Z; e L/K : the prime-to-p part of e L/K; p r L/K : the p-part of el/k ; f L/K : the index of residue field extension [k L : k]. From previous section, if char(k) = p > 0, then K = k((π)) for π a uniformizing parameter of m K ; if char(k) = 0, let K 0 = Frac W (k) = W (k)[1/p], then [K : K 0 ] = e K = v K (p), and K/K 0 is totally ramified Ramification groups of finite Galois extension. Let L/K be a Galois extension with Galois group G = Gal(L/K). Then G acts on the ring O L. We fix an element x of O L which generates O L as an O K -algebra. Lemma Let s G, and let i be an integer 1. Then the following three conditions are equivalent: (1) s operates trivially on the quotient ring O L /m i+1 L. (2) v L (s(a) a) i + 1 for all a O L. (3) v L (s(x) x) i + 1. Proof. This is a trivial exercise. Proposition For each integer i 1, let G i be the set of s G satisfying conditions (1), (2), (3) of Lemma Then the G i s form a decreasing sequence of normal subgroups of G. Moreover, G 1 = G, G 0 is the inertia subgroup of G and G i = {1} for i sufficiently large. Proof. The sequence is clearly a decreasing sequence of subgroups of G. We want to show that G i is normal for all i. For every s G and every t G i, since G i acts trivially on the quotient ring O L /m i+1 L, we have sts 1 (x) x mod m i+1 L, namely, sts 1 G i. Thus, G i is a normal subgroup for all i. The remaining part follows just by definition. Definition The group G i is called the i-th ramification group of G (or of L/K). We denote the inertia subgroup G 0 by I(L/K) and its invariant field by L 0 = (L/K) ur ; we denote by G 1 = P (L/K) and call it the wild inertia subgroup of G, and denote its invariant field by L 1 = (L/K) tame.

27 0.3 Galois groups of local fields 21 Remark The ramification groups define a filtration of G. The quotient G/G 0 is isomorphic to the Galois group Gal(k L /k) of the residue extension. The field L 0 is the maximal unramified subextension inside L. In Proposition 0.57, we shall see that L 1 is the maximal tamely ramified subextension inside L. Remark Let H be a subgroup of G and K = L H. If x O L is a generator of the O K -algebra O L, then it is also a generator of the O K -algebra O L. Then H i = G i H. In particular, the higher ramification groups of G are equal to those of G 0, therefore the study of higher ramification groups can always be reduced to the totally ramified case. We shall describe the quotient G i /G i+1 in the following. Let π be a uniformizer of L. Proposition Let i be a non-negative integer. In order that an element s of the inertia group G 0 belongs to G i, it is necessary and sufficient that s(π)/π = 1 mod m i L. Proof. Replacing G by G 0 reduces us to the case of a totally ramified extension. In this case π is a generator of O L as an O K -algebra. Since the formula v L (s(π) π) = 1 + v L (s(π)/π 1), we have s(π)/π 1 mod m i L s G i. We recall the following result: Proposition (1) UL 0/U L 1 = k L ; (2) For i 1, the group UL i i+1 /UL is canonically isomorphic to the group m i L /mi+1 L, which is itself isomorphic (non-canonically)to the additive group of the residue field k L. Back to the ramification groups, then the equivalence in Proposition 0.55 can be translated to s G i s(π)/π U i L. We have a more precise description of G i /G i+1 following Proposition 0.56: Proposition The map which to s G i, assigns s(π)/π, induces by passage to the quotient an isomorphism θ i of the quotient group G i /G i+1 onto a subgroup of the group UL i i+1 /UL. This isomorphism is independent of the choice of the uniformizer π. (1) The group G 0 /G 1 is cyclic, and is mapped isomorphically by θ 0 onto a subgroup of µ(k L ), the group of roots of unity contained in k L. Its order is prime to p, the characteristic of the residue field k L. (2) If the characteristic of k L is p 0, the quotients G i /G i+1, i 1, are abelian groups, and are direct products of cyclic groups of order p. The group G 1 is a p-group, the inertia group G 0 has the following property: it is the semi-direct product of a cyclic group of order prime to p with a normal subgroup whose order is a power of p.

28 22 0 Preliminary Remark The group G 0 is solvable. If k is a finite field, then G is also solvable. In fact, we can describe the cyclic group G 0 /G 1 = I(L/K)/P (L/K) more explicitly. Let N = e L/K = [L 1 : L 0 ]. The image of θ 0 in kl is a cyclic group of order N prime to p, thus k L = k L0 contains a primitive N th -root of 1 and Im θ 0 = µ N (k L ) = {ε k L ε N = 1} is of order N. By Hensel s lemma, L 0 contains a primitive N-th root of unity. By Kummer theory, there exists a uniformizing parameter π of L 0 such that L 1 = L 0 (α) with α a root of X N π. The homomorphism θ 0 is the canonical isomorphism Gal(L 1 /L 0 ) µ N (k L ) g ε if g α = [ε] α, where [ε] is the Teichmüller representative of ε. By the short exact sequence 1 Gal(L 1 /L 0 ) Gal(L 1 /K) Gal(k L /k) 1, Gal(L 1 /K) acts on Gal(L 1 /L 0 ) by conjugation. Because the group Gal(L 1 /L 0 ) is abelian, this action factors through an action of Gal(k L /k). The isomorphism Gal(L 1 /L 0 ) µ N (k L ) then induces an action of Gal(k L /k) over µ N (k L ), which is the natural action of Gal(k L /k) Galois group of K s /K. Let K s be a separable closure of K and G K = Gal(K s /K). Let L be the set of finite Galois extensions L of K contained in K s, then Let K ur = K s = L, G K = lim Gal(L/K). L L L L L L L/K unramified L, K tame = L L L/K tamely ramified Then K ur and K tame are the maximal unramified and tamely ramified extensions of K contained in K s respectively. The valuation of K extends uniquely to K s, but the valuation on K s is no more discrete, actually v K ((K s ) ) = Q, and K s is no more complete for the valuation. The field k = O K ur/m K ur is an algebraic closure of k. We use the notations L.