Last revised 12:50 p.m. April 12, 2018

Witt vectors

Bill Casselman University of British Columbia [email protected]

A is one that possesses a filtration whose associated graded module is a direct sum of copies of Ga. In characteristic 0 all commutative ones are isomorphic to products of Ga, but in characteristic p> 0 there are others. Most important are the groups associated to Witt vectors. Contents 1. Introduction 2. Unramified p•adic extensions 3. The Witt 4. The ring of power series 5. The Artin•Hasse exponential 6. References

1. Introduction

Suppose p to be prime. The additive group of Z/p2 has the same number of elements as the vector space of dimension two over the field Fp = Z/p, but of course is not isomorphic to it as a group. The following curious question however arises: can we introduce Fp•coordinates on the first group so that the group 2 structure becomes algebraic? Or, equivalently, can we find polynomials defining on Fp the structure of an 2 algebraic group whose set of rational points over Fp is isomorphic to Z/p ? In fact, we shall require and obtain something a bit stronger, as I shall explain soon. We can begin in an elementary fashion. Every element x of Z/p2 can be written uniquely in the form x = a + bp where 0 ≤ a,b < p; thus one natural coordinate system is to associate to x the vector (a,b).I define this coordinate map a little more precisely, Let π be the canonical projection from Z/p2 to Z/p, and let θ be the splitting of π which takes an element a of Z/p to the image in Z/p2 of the unique integer m in [0,p) 2 with π(m)= a. For any x in Z/p its first coordinate is x0 = π(x). The difference x − θ(x0) is a multiple of p, and its second coordinate is the image in Z/p of the quotient: x − θ(x ) x = π 0 . 1 p 2 Suppose x and y to be elements of Z/p with coordinates (x0, x1) and (y0,y1). What are the coordinates of the sum x + y? Its first coordinate is z0 = π(x + y)= x0 + y0, and according to the prescription above the second is x + y − θ(x + y ) z = π 0 0 1 p x + y − θ(x ) − θ(y ) θ(x )+ θ(y ) − θ(x + y ) = π 0 0 + π 0 0 0 0 p p x − θ(x )+ y − θ(y ) θ(x )+ θ(y ) − θ(x + y ) = π 0 0 + π 0 0 0 0 p p θ(x )+ θ(y ) − θ(x + y ) = x + y + π 0 0 0 0 . 1 1 p At first sight the question of algebraicity therefore amounts to the question: does there exist a polynomial f(x0,y0) with coefficients in Fp such that θ(x )+ θ(y ) − θ(x + y ) f(x ,y )= π 0 0 0 0 ? 0 0 p Witt vectors 2

It is eventually more illuminating to consider a more general question: suppose q to be a power of p, o a discrete valuation ring with maximal ideal p = (p) such that o/p is Fq. Let θ be any splitting of the canonical 2 2 projection π, and we ask the same question about the group o/p of q elements, except that now f(x0,y0) is allowed to have coefficients in Fq. q n Since x = x in Fq, every polynomial over Fq determines the same function on Fq as one which has only monomials of degree < q in any one variable. I call this a reduced polynomial. We then have this result, n which provides an answer to the previous question: Any Fq•valued function on Fq may be expressed by a unique reduced polynomial over Fq in n variables. Proving this remark is straightforward. It suffices to follow the method of Lagrange interpolation, and construct for each α in Fq the reduced indicator polynomial Pα(x) in one variable x, satisfying the conditions

0 (β = α) P (β)= . α 1 (β = α) This is because we can take a product of such one•variable polynomials to get indicator polynomials in several variables. The formula of Lagrange is valid in any field:

β=α(x − β) Pα(x)= β=α(α − β) The situation is not yet very satisfactory, since the polynomial we get depends strongly on the splitting θ we have chosen.

In fact, there is in every case a canonical splitting: For any element x0 of o/p there exists a unique element x 2 2 of o/p with the two properties (a) π(x)= x0; (b) x is a p•th power in o/p . 1/p This element is called the Teichmuller¨ representative τ(x0) of x0. Tofindit: let x0 be given, and let a = x0 . Choose α in o/p2 with π(α) = a, and let x = αp. It turns out that x does not depend on the choice of α, and we can therefore define τ(a) to be x. Why doesn’t x depend on α? If α and α∗ are two choices, then α∗ = α + cp for some c in o, and

p p 2 p−1 p 2 α∗ = α + cp α + = α (mod p ).

The map τ is clearly a multiplicative homomorphism. With this choice of splitting, we can do calculations explicitly. Suppose x in o/p with

(1.1) x = τ(x0)+ τ(X1)p .

(We’ll see later why I use X instead of x. Be assured, it is only temporary.) How do we specify x0 and X1? p 2 p Well, x0 is (still) the image of x modulo p. Say α0 = x0, and α in o/p has image α0. Then α = τ(x0) and hence x − αp X = π . 1 p Similarly write y = βp. For the second coordinate of the sum x + y we have

αp + βp − (α + β)p Z = X + Y + π 1 1 1 p (1.2) p−1 p−1 = X1 + Y1 − π(α β + + αβ ) p−1 p−1 = X1 + Y1 − (α0 β0 + + α0β0 ) Witt vectors 3

This is still not quite what we would like, because the last part of this expression is not a polynomial in x0 1/p 1/p and y0, but only one in α0 = x0 and β0 = y0 . There is a simple trick to solve thsi problem. Apply the Frobenius automorphism to both sides of this last equation. We get

p p p p−1 p−1 Z1 = X1 + Y1 − (x0 y0 + + x0y0 ) .

p 1/p This suggests choosing the second coordinate of x to be x1 = X1 rather than X1. Since X1 = x1 we now have

1/p (1.3) x = τ(x0)+ τ(x1 )p , and (1.2) becomes p−1 p−1 z1 = x1 + y1 − (x0 y0 + + x0y0 ). We shall see later that the entire ring structure of o can be recovered similarly. At any rate, we now have a candidate for a commutative algebraic group structure on affine space of dimension two in characteristic p: 1.4. Proposition. The formula

p−1 p−1 (x0, x1) + (y0,y1)= x0 + y0, x1 + y1 − (x0 y0 + + x0y0 ) defines a group law. Proof. The second coordinate of the sum x + y + z is (formally)

αp + βp + γp − (α + β + γ )p x + y + z + 0 0 0 0 0 0 1 1 1 p from which associativity is transparent.

The algebraic group W2 defined by the formula fits into a short exact

1 −→ Ga −→ W2 −→ Ga −→ 1 that doesn’t split.

2. Unramified p-adic extensions

m Suppose that q = p , let f(x) be an irreducible polynomial with coefficients in Fp defining the extension Fq, and let F (x) be any polynomial with coefficients in Zp reducing to f(x) modulo p. Hensel’s Lemma implies that the field extension of Qp obtained by adjoining a root of F (x) is Galois and unramified. The Galois group is cyclic of degree m, generated by the Frobenius F. Let o be its ring of integers, π the canonical projection o → Fq. 2.1. Proposition. There exists a unique multiplicative splitting τ of π. The following are equivalent: (a) the element x lies in the image of τ; (b) the element x is a pn•th power for every n> 0; (c) F(x)= xp; (d) xq = x. The map τ is called the Teichmuller¨ map.

Proof. All these assertions will follow from the simple claim that given a in Fq there exists a unique x in o q such that π(x)= a; (2) x = x. To see this, suppose x0 to be any element of o with π(x)= a. I claim that the sequence q xn+1 = xn Witt vectors 4 converges in o to some x with xq = x. This is immediate from

k k 2.2. Lemma. In any ring, if x ≡ y (mod p)) then xp ≡ yp (mod pk). To prove this lemma, it suffices to show that if x ≡ y (mod pk) then xp ≡ yp (mod pk+1). This is immediate from the binomial theorem. Define τ(a) to be the limit x. Any element x of o may be expressed as a unique series

2 x = τ(x0)+ τ(x1)p + τ(x2)p + and since Fq is perfect it may in fact be expressed as some other series

2 1/p 1/p 2 (2.3) x = τ(x0)+ τ(x1 )p + τ(x2 )p + .

In this expression, the xi are called the Teichmuller¨ coordinates of x. The Frobenius automorphism acts simply in terms of Teichmuller¨ coordinates, since F(x) = xp for every x in the image of τ, and F(p)= p. If x is expressed as in (2.3), then for each n> 0

2 n 1/p 1/p 2 F (x)= F(τ(x0)) + F(τ(x1 ))p + F(τ(x2 ))p + . . . − − pn pn 1 pn 2 2 = τ(x0) + τ(x1) p + τ(x2) p + − − pn pn 1 pn 2 2 n n+1 ≡ τ(x0) + τ(x1) p + τ(x2) p + + τ(xn)p (mod p ) according to Proposition 2.1.(b). Define for each n the Witt polynomials

n n−1 n−2 p p p 2 n Wn(X)= Wn(X0,X1,...,Xn)= X0 + X1 p + X2 p + . . . + Xnp so that the congruence above can be translated to the assertion that if

−1 −1 2 x = α0 + F (α1)p + F (α2)p + with all the αi in τ(Fq ) then n n+1 F (x) ≡ Wn(α0, α1,...,αn) (mod p ) In fact, a converse is also true: 2.4. Lemma. If n n+1 F (x) ≡ Wn(α0, α1,...,αn) (mod p ) for elements αi in o then the xi = π(αi) are the first n +1 Teichmuller¨ coordinates of x. pn Proof. It suffices to show that the Teichmuller¨ coordinates of F(x) are the xi . By hypothesis

n n−1 n p p n+1 F (x)= α0 + pα1 + (mod p )

n−i n−i p i i n+1 p so it need only be shown that each αi p is of the form τ(y)p modulo p , or equivalently that αi is in the image of τ modulo pn−i+1. But this is an application of (b).

2.5. Theorem. For any polynomial R(X, Y ) with coefficients in Z there exist polynomials r0(X0, Y0), r1(X0,X1, Y0, Y1), . . . with coefficients in Fp such that if

2 1/p 1/p 2 x = τ(x0)+ τ(x1 )p + τ(x2 )p + ,

2 1/p 1/p 2 y = τ(y0)+ τ(y1 )p + τ(y2 )p + Witt vectors 5 and zi = ri(x0,...,xi,y0,... ,yi)

then 2 1/p 1/p 2 R(x, y)= τ(z0)+ τ(z1 )p + τ(z2 )p +

In other words, the Teichmuller¨ coordinates of R(x, y) are polynomials in the coordinates of x and y. The polynomials ri will be determined explicitly by induction. Proof. The basic idea is simple—that the Frobenius map on F is an automorphism, and deals well with Teichmuller¨ coordinates, is a very strong property. Strong enough, in fact, to have as consequence the formulas for sums and products. n+1 In order to verify that the n•th coordinate of R(x, y) is rn(x, y) it suffices to verify it modulo r . But since F is an automorphism FnR(x, y)= R Fn(x), Fn(y) n+1 ≡ RWn(x), Wn(y) (mod p ). By Lemma 2.4 it therefore suffices to prove:

2.6. Proposition. For any polynomial R(X, Y ) with coefficients in Z there exist polynomials ri(X, Y ) with coefficients in Z such that for every n

R Wn(X), Wn(Y ) = Wn(r0, r1,... ,rn).

Proof. For n = 0 we choose r0 to be the reduction of R modulo p. From here we proceed by induction. (I follow the exposition of George Bergman in [Mumford:1966].) Start by observing that

p p p n+1 Wn+1(X0,X1,...Xn)= Wn(X0 ,X1 ,...Xn)+ p Xn+1 , (a) and that n+1 if X ≡ Y (mod p) then Wn(X) ≡ Wn(Y ) (mod p ) . (b) We can solve R(Wn+1(X), Wn+1(Y )) = Wn+1(r0, r1,...rn+1) p p n+1 = Wn(r0 ,...rn)+ p rn+1 for rn+1. This gives the inductive definition

n+1 p p rn+1(x, y)=(1/p )[R(Wn+1(X), Wn+1(Y )) − Wn(r0 ,...rn] which is allowable if: 2.7. Lemma. For every n p p R(Wn+1(X), Wn+1(Y )) − Wn(r0 ,...rn) has coefficients divisible by pn+1.

It is only by the induction hypothesis that r0, ...rn are defined and known to have integral coefficients. Proof. This expression is equal to

p n+1 p n+1 p R(Wn(X )+ p Xn+1,Wn(Y )+ p Yn+1) − Wn(r ) p p p ≡ R(Wn(X ), Wn(Y )) − Wn(r ) Witt vectors 6

But since p p p rn(X, Y ) ≡ rn(X , Y ) (mod p) we have also p p p n+1 Wn(r (X, Y )) ≡ Wn(r(X , Y )) (mod p ) so this in turn is equal to p p p p R(Wn(X ), Wn(Y )) − Wn(R(X , Y ))) = 0. by induction. If R(X, Y )) is taken to be respectively X + Y , XY then we get

S0(X, Y )= X0 + Y0 Xp + Y p − (X + Y )p S (X, Y )= X + Y + 0 0 0 0 1 1 1 p . . . = . . .

M0(X, Y )= X0Y0 p p M1(X, Y )= X1Y0 + X0 Y1 + pX1Y1 . . . = . . .

3. The Witt scheme

Suppose now R to be any ring. We can use the polynomials Sn, Mn constructed in the last section to define a ring structure on vectors (x0, x1,...) with coefficients in R:

x + y = (S0(x, y),S1(x, y),... )

xy = (M0(x, y),M1(x, y),... )

This construction is functorial in R, and defines therefore a ring scheme W, the Witt scheme, over Z. N The polynomials Wn(x) define a map from R to itself:

(x0, x1,... ) → (W0(x), W1(x),... )

N By definition of the Sn and Mn this amounts to a ring homomorphism from W (R) to R endowed with the usual direct product ring structure. Immediately from the definition of the Wn one can see that it is injective if p is not a zero•divisor in R, and an isomorphism if p is invertible in R. In practice one is interested in the case where R has characteristic p, but several proofs rely on the fact that W (R) can be constructed for arbitrary R. The Teichmuller¨ map from R to W (R) takes x to (x, 0, 0,... ). It is multiplicative and functorial. In addition one can define morphisms F and V :

p p p F(x0, x1, x2,... ) = (x0, x1, x2,... ), V (x0, x1, x2,... )=(0, x0, x1,... ).

The map F is called the Frobenius, and it is a ring endomorphism. The map V is the Verschiebung (or shift), and is only an additive homomorphism. Note that

p p V F(x0, x1, x2,... )=(0, x0, x1,... ).

3.1. Proposition. For any x in R V F(x) − px ≡ 0 (mod p) in the sense that all its components are divisible by p. Witt vectors 7

Proof. It suffices to prove this when R = Z[X0,X1,... ] and x = (X0,X1,... ). If

y = FV (x) − px in W (R) then for each n

Wn(y)= pWn(x) − Wn(V F(x)) p p = pWn(X0,X1,...,Xn) − Wn(0,X0 ,X1 ,... ) n n p n+1 p n p = pX0 + + p Xn − pX0 −− p Xn−1 n+1 = p Xn.

Hence, for example, y0 = px0. Proceed by induction:

n n n+1 p n−1 p p yn = p xn − (y0 + ...p yn−1) ≡ 0 (mod pn+1). since each yi ≡ 0 (mod p) and p ≥ 2. 3.2. Corollary. If R has characteristic p, V F = p in W (R). n The Witt scheme W has finite•dimensional quotients Wn with Wn(R) isomorphic to R as an R•module. If R has characteristic p, then by Corollary 3.2 these may be identified with W (R)/pnW (R). The map V m identifies Wm as an additive subscheme of Wn+m. For n ≥ m we have also the restriction maps

Rn,m: (x0, x1,...xn−1) → (x0,...xm−1).

The sequence n V Rn+m,m 0 −→ Wm−→Wn+m −→ Wn −→ 0 is exact. If k is a perfect field, then W (k) is a complete unramified discrete valuation ring with W (k)/pW (k) = k, and we have seen in the previous section that if o is any such ring then there exists an isomorphism of W (k) with o.

4. The ring of power series

For the moment let F be an arbitrary field, R the ring of F [[t]]. For each n> 0 let Rn be the ring F [[t]]/(tn). This is isomorphic to the ring of n × n upper triangular matrices

2 n−1 a0 + a1T + a2T + + an−1T where 0 1 0 . . . 0 0 0 0 1 . . . 0 0  0 0 0 . . . 0 0  T = .  . . .     0 0 0 . . . 0 1     0 0 0 . . . 0 0    The additive group of Rn is just a vector space of dimension n over F , but the structure of its multiplicative units R× is more subtle. Any unit can be expressed as u v where u =0 is a scalar diagonal matrix and v is a unipotent matrix, which I’ll call a unipotent unit. The group V = V1 of unipotent matrices may be filtered k by the subgroups Vk of units v ≡ I modulo T . The quotient Vk/Vk+1 is isomorphic to the additive group Witt vectors 8 of F . But the detailed structure of V depends strongly on F . In this section, I’ll recall what happens if F has characteristic 0, and in the next I’ll explain what modifications have to be made in characteristic p> 0. Suppose F has characteristic 0. If ν is any nilpotent upper triangular matrix then νn =0 and

ν2 ν3 exp(ν)= I + ν + + + 2 3! is a finite series and is equal to a unipotent matrix. If ν is in the ring Rn then exp(ν) is a unipotent unit in Rn. The inverse map is the Taylor series for log(1 + x) and the group of unipotent units in Rn is therefore 2 isomorphic to the additive group of nilpotent matrices in Rn, the F •vector space spanned by T , T , . . . , T n−1. More generally, every power series in T can be expressed uniquely as a product of exponentials

m F (T )= exp(cmT ) , m≥1 and something similar holds for Rn The definition of an exponential map in characteristic p > 0 in the next section will be motivated by an apparently little known result about the classical exponential function. First recall the Mobius¨ function

0 n has a square factor (n)= (−1)m if n is the product of m distinct primes

4.1. Lemma. We have an identity

∞ (−x)n exp(−x)= = (1 − xn)(n)/n . n! n=0 n

Proof. I follow [Demazure:1972]. A straightforward consequence of the definition is that

(d)=0 d|n unless n =1. But then tn −t = − (d) n n≥1 d|n (d) tdm = − (n = md) d m m d≥1 (d) = log(1 − td) . d d≥1 Witt vectors 9

5. The Artin-Hasse exponential

In characteristic p one also uses an exponential map to determine the group of unipotent units in Rn. The starting point is that the group of units is not equal to a product of copies of F . For example, consider n the unipotent unit group of F2[[T ]]/(T ). For n =3, the elements of the group are

1, 1+ T, 1+ T + T 2, 1+ T 2 , which is isomorphic to Z/4 since

(1 + T )2 =1+ T 2, (1 + T 2)2 =1 .

5 Similarly, in 1 + (T )/1 + (T ) (still with F = F2) the element 1+ T has order 8:

(1 + T )0 =1 (1 + T )1 =1+ T (1 + T )2 =1+ T 2 (1 + T )3 =1+ T + T 2 + T 3 (1 + T )4 =1+ T 4 (1 + T )5 =1+ T + T 4 (1 + T )6 =1+ T 2 + T 4 (1 + T )7 =1+ T + T 2 + T 3 + T 4 (1 + T )8 =1+ T 8 =1 .

The element 1+ T 3 has order 2; the group 1 + (T )/1 + (T 5) has order 16; and it is therefore isomorphic to (Z/8) × (Z/2). The main result of this section is:

5.1. Proposition. Let F be a field of characteristic p, n > 1. For each m

Wnm (F ). (m,p)=1 1≤m

The proof will provide an explicit map from the product to the group of units. In order to make things more concrete, I include a table for p =2:

Table of nm for p =2 n m:1 3 5 7 9 2 1 3 2 4 21 5 31 6 311 7 321 8 3211 9 4211 One might conjecture that the group 1 + (T )/1 + (T n) has as basis the elements 1+ T m with m odd. This fits in with the assertion in the Proposition, since its order is 2k where k is the least such that km ≥ n. Witt vectors 10

The proof starts by defining an exponential function. 5.2. Lemma. We have 2 exp(−T − T p/p − T p /p2 − )= (1 − T )(n)/n . (n,p)=1

Here is the Mobius¨ function. Proof. We know that exp(−T )= (1 − T )(n)/n , n≥1 so F (T )= (1 − T )(n)/n (n,p)=1 = (1 − T )(n)/n (1 − T )(np)/np n≥1 n≥1 = exp(−T )/ (1 − T )(np)/np . n≥1 Since (np)=0 if p divides n and (np)= −(n) otherwise, this leads to exp(−T ) F (T )= 1/p (1 − T )−(n)/n (n,p)=1 F (T ) = exp(−T )F (T p)1/p 2 2 = exp(−T ) exp(−T p/p)F (T p )1/p . . . 2 = exp(−T − T p/p − T p /p2 + ) .

5.3. Corollary. The coefficients of the power series F (T ) lie in

Z(p) = Q ∩ Zp = {m/n| (n,p)=1} .

Now define an exponential map from the Witt ring W(F ) to the unipotent units 1 + (T ) in F [[T ]]:

pn E(T, x)= E(T, x0, x1,...)= F (xnT ) . n≥0

5.4. Proposition. If x = (xi) is a Witt vector then

m T p E(T, x) = exp − Wn(x) m . p m≥0

5.5. Corollary. For two Witt vectors x and y E(T, x + y)= E(T, x)E(T,y) .

As a consequence, the map taking an array of Witt vectors (wm) for (m,p) = 1 to (m,p)=1 wm to E(T m, w ) is an isomorphism of the product of the W (m relatively prime to p) is an isomorphism m m with 1 + (T ). In [Dieudonne:1957]´ it is shown that any homomorphism W (k) → 1+ T k[[T ]] (k a perfect field) is of the form Eℓ, with ℓ a p•adic integer. Witt vectors 11

6. References

1. E. Artin and H. Hasse, ‘Die beiden Erganzungs¨ atze¨ zum Reziprozitatgesetz¨ der ln•ten Potenzreste im Korper¨ der ln•ten Einheitswurzeln’, Abhandlungen aus dem Mathematischen Seminar der Universitat¨ Ham• burg 6 (1928), 146–162. 2. M. Demazure, Lectures on p-divisible groups, Lecture Notes in Mathematics 332, 1972. 3. J. Dieudonne,´ ‘On the Artin•Hasse exponential series’, Proceedings of the American Mathematical Society 8 (1957), 210–214. 4. G. Harder, ‘An essay on Witt vectors’, pp. 165–193 in Collected Papers of Ernst Witt, Springer, 1996. 5. H. Hasse, ‘Die Gruppe der pn•primaren¨ Zahlen fur¨ einen Primteiler p von p’, Journal fur¨ die Reine und Angewandte Mathematik 176 (1936), 174–183. 6. D. Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Series 59, Princeton, 1966. Chapter 26 is an introduction to Witt schemes by George Bergman. 7. J•P. Serre, Groupes algebriques´ et corps de classes, Hermann, 1964. 8. ——, Corps Locaux, Hermann, 1968.