Witt vectors

Witt vectors

University of British Columbia [email protected]

Aunipotentalgebraic group 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 toWitt vectors.

Contents

1. Introduction

2. Unramified p-adic extensions 3. The Witt scheme

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/p²has 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 structure becomes algebraic? Or, equivalently, can we find polynomials defining on F²_pthe structure of an algebraic group whose set of rational points over Fpis isomorphic to Z/p²? 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/p²to Z/p, and let θ be the splitting of π which takes an element a of Z/p to the image in Z/p²of the unique integer m in [0, p) with π(m) = a. For any x in Z/p²its 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:

x1= π x − θ(x0) p

 .

Suppose x and y to be elements of Z/p²with 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

z1= π x + y − θ(x0+ y0) p



= π x + y − θ(x0) − θ(y0) p



+ π θ(x0) + θ(y0) − θ(x0+ y0) p



= π x − θ(x0) + y − θ(y0) p



+ π θ(x0) + θ(y0) − θ(x0+ y0) p



= x1+ y1+ π θ(x0) + θ(y0) − θ(x0+ y0) p

 .

At first sight the question of algebraicity therefore amounts to the question: does there exist a polynomial f (x0, y0) with coefficients in Fpsuch that

f (x0, y0) = π θ(x0) + θ(y0) − θ(x0+ y0) p



?

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 projection π, and we ask the same question about the group o/p²of q²elements, except that now f (x0, y0) is allowed to have coefficients in Fq.

Since x^q = x in Fq, every polynomial over Fq determines the same function on Fⁿ_q as one which has only monomials of degree < q in any one variable. I call this areducedpolynomial. We then have this result, which provides an answer to the previous question: Any Fq-valued function on Fⁿ_q may be expressed by a unique reduced polynomial over Fqin n variables.

Proving this remark is straightforward. It suffices to follow the method of Lagrange interpolation, and construct for each α in Fqthe reduced indicator polynomial Pα(x) in one variable x, satisfying the conditions

Pα(β) = 0 (β 6= 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:

Pα(x) = Q

β6=α(x − β) Q

β6=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 x0of o/p there exists a unique element x of o/p²with the two properties (a) π(x) = x0; (b) x is a p-th power in o/p².

This element is called the^{Teichm ¨uller}representative τ (x0) of x0. To find it: let x0be given, and let a = x^1/p₀ . Choose α in o/p²with π(α) = a, and let x = α^p. 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

α^p_∗= α^p+ cp²α^p−1+ · · · = α^p (mod p²).

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 x0and X1? Well, x0is (still) the image of x modulo p. Say α^p₀ = x0, and α in o/p²has image α0. Then α^p = τ (x0) and hence

X1= π x − α^p p

 .

Similarly write y = β^p. For the second coordinate of the sum x + y we have

(1.2)

Z1= X1+ Y1+ π α^p+ β^p− (α + β)^p p



= X1+ Y1− π(α^p−1β + · · · + αβ^p−1)

= X1+ Y1− (α^p−1₀ β0+ · · · + α0β₀^p−1)

This is still not quite what we would like, because the last part of this expression is not a polynomial in x0

and y0, but only one in α0 = x^1/p₀ and β0 = y^1/p₀ . There is a simple trick to solve thsi problem. Apply the Frobenius automorphism to both sides of this last equation. We get

Z₁^p= X₁^p+ Y₁^p− (x^p−1₀ y0+ · · · + x0y₀^p−1) .

This suggests choosing the second coordinate of x to be x1 = X₁^prather than X1. Since X1= x^1/p₁ we now have

(1.3) x = τ (x0) + τ (x^1/p₁ )p ,

and (1.2) becomes

z1= x1+ y1− (x^p−1₀ y0+ · · · + x0y^p−1₀ ).

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

(x0, x1) + (y0, y1) = x0+ y0, x1+ y1− (x^p−1₀ y0+ · · · + x0y^p−1₀ )
defines a group law.

Proof. The second coordinate of the sum x + y + z is (formally)

x1+ y1+ z1+ α^p₀+ β^p₀+ γ₀^p− (α0+ β0+ γ0)^p p



from which associativity is transparent.

The algebraic group W2defined by the formula fits into a short exact sequence 1 −→ Ga−→ W2−→ Ga −→ 1

that doesn’t split.

2. Unramifiedp-adic extensions

Suppose that q = p^m, let f (x) be an irreducible polynomial with coefficients in Fp defining the extension F_q, 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) = x^p; (d) x^q = x.

The map τ is called theTeichm ¨ullermap.

Proof. All these assertions will follow from the simple claim that given a in Fq there exists a unique x in o such that π(x) = a; (2) x^q = x. To see this, suppose x0to be any element of o with π(x) = a. I claim that the sequence

xn+1= x^q_n

converges in o to some x with x^q = x. This is immediate from 2.2. Lemma.In any ring, if x ≡ y (mod p)) then x^pk≡ y^pk(mod p^k).

To prove this lemma, it suffices to show that if x ≡ y (mod p^k) then x^p≡ y^p(mod p^k+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

x = τ (x0) + τ (x1)p + τ (x2)p²+ · · · and since Fqis perfect it may in fact be expressed as some other series (2.3) x = τ (x0) + τ (x^1/p₁ )p + τ (x^1/p₂ ²)p²+ · · · . In this expression, the xiare called the^{Teichm ¨}uller coordinatesof x.

The Frobenius automorphism acts simply in terms of Teichm ¨uller coordinates, since F(x) = x^pfor every x in the image of τ , and F(p) = p. If x is expressed as in (2.3), then for each n > 0

Fⁿ(x) = F(τ (x0)) + F(τ (x^1/p₁ ))p + F(τ (x^1/p₂ ²))p²+ . . .

= τ (x0)^pn+ τ (x1)^pn−1p + τ (x2)^pn−2p²+ · · ·

≡ τ (x0)^pn+ τ (x1)^pn−1p + τ (x2)^pn−2p²+ · · · + τ (xn)pⁿ(mod pⁿ⁺¹) according to Proposition 2.1.(b). Define for each n theWitt polynomials

Wn(X) = Wn(X0, X1, . . . , Xn) = X₀^pn+ X₁^pn−1p + X₂^pn−2p²+ . . . + Xnpⁿ so that the congruence above can be translated to the assertion that if

x = α0+ F⁻¹(α1)p + F⁻¹(α2)p²+ · · · with all the αiin τ (Fq) then

Fⁿ(x) ≡ Wn(α0, α1, . . . , αn) (mod pⁿ⁺¹) In fact, a converse is also true:

2.4. Lemma.If

Fⁿ(x) ≡ Wn(α0, α1, . . . , αn) (mod pⁿ⁺¹)

for elements αiin o then the xi = π(αi) are the first n + 1Teichm ¨uller coordinatesof x.

Proof. It suffices to show that the Teichm ¨uller coordinates of F(x) are the x^p_iⁿ. By hypothesis Fⁿ(x) = α^p₀ⁿ+ pα^p₁ⁿ⁻¹+ · · · (mod pⁿ⁺¹)

so it need only be shown that each α^p_iⁿ⁻ⁱpⁱis of the form τ (y)pⁱmodulo pⁿ⁺¹, or equivalently that α^p_iⁿ⁻ⁱis 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 Fpsuch that if

x = τ (x0) + τ (x^1/p₁ )p + τ (x^1/p₂ ²)p²+ · · · , y = τ (y0) + τ (y₁^1/p)p + τ (y^1/p₂ ²)p²+ · · ·

and

zi= ri(x0, . . . , xi, y0, . . . , yi) then

R(x, y) = τ (z0) + τ (z₁^1/p)p + τ (z₂^1/p2)p²+ · · ·

In other words, the Teichm ¨uller coordinates of R(x, y) are polynomials in the coordinates of x and y. The polynomials riwill be determined explicitly by induction.

Proof. The basic idea is simple—that the Frobenius map on F is an automorphism, and deals well with Teichm ¨uller coordinates, is a very strong property. Strong enough, in fact, to have as consequence the formulas for sums and products.

In order to verify that the n-th coordinate of R(x, y) is rn(x, y) it suffices to verify it modulo rⁿ⁺¹. But since Fis an automorphism

FⁿR(x, y) = R Fⁿ(x), Fⁿ(y)

≡ R Wn(x), Wn(y)
(mod pⁿ⁺¹).

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 r0to 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

Wn+1(X0, X1, . . . Xn) = Wn(X₀^p, X₁^p, . . . X_n^p) + pⁿ⁺¹Xn+1, (a) and that

if X ≡ Y (mod p) then Wn(X) ≡ Wn(Y ) (mod pⁿ⁺¹) . (b) We can solve

R(Wn+1(X), Wn+1(Y )) = Wn+1(r0, r1, . . . rn+1)

= Wn(r^p₀, . . . r_n^p) + pⁿ⁺¹rn+1

for rn+1. This gives the inductive definition

rn+1(x, y) = (1/pⁿ⁺¹)[R(Wn+1(X), Wn+1(Y )) − Wn(r^p₀, . . . r^p_n] which is allowable if:

2.7. Lemma.For every n

R(Wn+1(X), Wn+1(Y )) − Wn(r₀^p, . . . r^p_n) has coefficients divisible by pⁿ⁺¹.

It is only by the induction hypothesis that r0, . . . rnare defined and known to have integral coefficients.

Proof. This expression is equal to

R(Wn(X^p) + pⁿ⁺¹Xn+1,Wn(Y^p) + pⁿ⁺¹Yn+1) − Wn(r^p)

≡ R(Wn(X^p), Wn(Y^p)) − Wn(r^p)

But since

r_n^p(X, Y ) ≡ rn(X^p, Y^p) (mod p) we have also

Wn(r^p(X, Y )) ≡ Wn(r(X^p, Y^p)) (mod pⁿ⁺¹) so this in turn is equal to

R(Wn(X^p), Wn(Y^p)) − Wn(R(X^p, Y^p))) = 0.

by induction.

If R(X, Y )) is taken to be respectively X + Y , XY then we get sequences S0(X, Y ) = X0+ Y0

S1(X, Y ) = X1+ Y1+ X₀^p+ Y₀^p− (X0+ Y0)^p p



. . . = . . . M0(X, Y ) = X0Y0

M1(X, Y ) = X1Y₀^p+ X₀^pY1+ pX1Y1

. . . = . . .

3. The Witt scheme

Suppose now R to be any ring. We can use the polynomials Sn, Mnconstructed 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, theWitt scheme, over Z.

The polynomials Wn(x) define a map from R^Nto itself:

(x0, x1, . . . ) 7→ (W0(x), W1(x), . . . )

By definition of the Snand Mnthis amounts to a ring homomorphism from W (R) to R^Nendowed with the usual direct product ring structure. Immediately from the definition of the Wnone 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.

TheTeichm ¨uller mapfrom R to W (R) takes x to (x, 0, 0, . . . ). It is multiplicative and functorial. In addition one can define morphisms F and V :

F(x0, x1, x2, . . . ) = (x^p₀, x^p₁, x^p₂, . . . ), V (x0, x1, x2, . . . ) = (0, x0, x1, . . . ).

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

V F(x0, x1, x2, . . . ) = (0, x^p₀, x^p₁, . . . ).

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.

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))

= pWn(X0, X1, . . . , Xn) − Wn(0, X₀^p, X₁^p, . . . )

= pX₀^pn+ · · · + pⁿ⁺¹Xn− pX₀^pn− · · · − pⁿX_n−1^p

= pⁿ⁺¹Xn.

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

pⁿyn = pⁿ⁺¹xn− (y₀^pn+ . . . pⁿ⁻¹y_n−1^p )

≡ 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).

The Witt scheme W has finite-dimensional quotients Wn with Wn(R) isomorphic to Rⁿ 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 Wmas an additive subscheme of Wn+m. For n ≥ m we have also the restriction maps

Rn,m: (x0, x1, . . . xn−1) 7→ (x0, . . . xm−1).

The sequence

0 −→ Wm Vⁿ

−→Wn+m Rn+m,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 formal power series F [[t]]. For each n > 0 let Rnbe the ring F [[t]]/(tⁿ). This is isomorphic to the ring of n × n upper triangular matrices

a0+ a1T + a2T²+ · · · + an−1Tⁿ⁻¹ where

T =















0 1 0 . . . 0 0 0 0 1 . . . 0 0 0 0 0 . . . 0 0

. . . 0 0 0 . . . 0 1 0 0 0 . . . 0 0













 .

The additive group of Rnis 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 6= 0 is a scalar diagonal matrix and v is a unipotent matrix, which I’ll call a unipotent unit. The group V = V1of unipotent matrices may be filtered by the subgroups Vkof units v ≡ I modulo T^k. The quotient Vk/Vk+1is isomorphic to the additive group

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

exp(ν) = I + ν +ν² 2 +ν³

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 Rnis therefore isomorphic to the additive group of nilpotent matrices in Rn, the F -vector space spanned by T , T², . . . , Tⁿ⁻¹. More generally, every power series in T can be expressed uniquely as a product of exponentials

F (T ) = Y

m≥1

exp(cmT^m) ,

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 M ¨obius function

µ(n) =

 0 n has a square factor

(−1)^m if n is the product of m distinct primes

4.1. Lemma.We have an identity

exp(−x) =

∞

X

n=0

(−x)ⁿ

n! =Y

n

(1 − xⁿ)^µ(n)/n.

Proof. I follow [Demazure:1972].

A straightforward consequence of the definition is that X

d|n

µ(d) = 0

unless n = 1. But then

−t =X

n≥1

−tⁿ n

X

d|n

µ(d)

=X

d≥1

µ(d) d

X

m

−t^dm

m (n = md)

=X

d≥1

µ(d)

d log(1 − t^d) .

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 the unipotent unit group of F2[[T ]]/(Tⁿ). For n = 3, the elements of the group are

1, 1 + T, 1 + T + T², 1 + T², which is isomorphic to Z/4 since

(1 + T )²= 1 + T², (1 + T²)²= 1 .

Similarly, in 1 + (T )/1 + (T⁵) (still with F = F2) the element 1 + T has order 8:

(1 + T )⁰= 1 (1 + T )¹= 1 + T (1 + T )²= 1 + T²

(1 + T )³= 1 + T + T²+ T³ (1 + T )⁴= 1 + T⁴

(1 + T )⁵= 1 + T + T⁴ (1 + T )⁶= 1 + T²+ T⁴

(1 + T )⁷= 1 + T + T²+ T³+ T⁴ (1 + T )⁸= 1 + T⁸= 1 .

The element 1 + T³has order 2; the group 1 + (T )/1 + (T⁵) 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 < n relatively prime to p let nmbe the smallest integer with p^nm ≥ n/m. The unipotent quotient group 1 + (T )/1 + (Tⁿ) in F [[T ]]/(Tⁿ) is isomorphic to

Y

(m,p)=1 1≤m<n

Wnm(F ).

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 nmfor p = 2

n m: 1 3 5 7 9

2 1

3 2

4 2 1

5 3 1

6 3 1 1

7 3 2 1

8 3 2 1 1

9 4 2 1 1

One might conjecture that the group 1 + (T )/1 + (Tⁿ) has as basis the elements 1 + T^mwith m odd. This fits in with the assertion in the Proposition, since its order is 2^kwhere k is the least such that km ≥ n.

The proof starts by defining an exponential function.

5.2. Lemma.We have

exp(−T − T^p/p − T^p2/p²− · · · ) = Y

(n,p)=1

(1 − T )^µ(n)/n.

Here µ is the M ¨obius function.

Proof. We know that

exp(−T ) = Y

n≥1

(1 − T )^µ(n)/n, so

F (T ) = Y

(n,p)=1

(1 − T )^µ(n)/n

= Y

n≥1

(1 − T )^µ(n)/n Y

n≥1

(1 − T )^µ(np)/np

= exp(−T )/Y

n≥1

(1 − T )^µ(np)/np.

Since µ(np) = 0 if p divides n and µ(np) = −µ(n) otherwise, this leads to F (T ) = exp(−T )

Q

(n,p)=1

(1 − T )^−µ(n)/n
1/p

F (T ) = exp(−T )F (T^p)^1/p

= exp(−T ) exp(−T^p/p)F (T^p2)^1/p2. . .

= exp(−T − T^p/p − T^p2/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 ]]:

E(T, x) = E(T, x0, x1, . . .) = Y

n≥0

F (xnT^pn) .

5.4. Proposition.If x = (xi) is a Witt vector then E(T, x) = exp −X

m≥0

Wn(x) T^pm p^m

! .

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 Q

(m,p)=1wm to Q E(T^m, wm) is an isomorphism of the product of the Wm(m relatively prime to p) is an isomorphism with 1 + (T ).

In [Dieudonn´e:1957] it is shown that any homomorphism W (k) → 1 + T k[[T ]] (k a perfect field) is of the form E^ℓ, with ℓ a p-adic integer.

6. References

1. E. Artin and H. Hasse, ‘Die beiden Erg¨anzungs¨atze zum Reziprozit¨atgesetz der lⁿ-ten Potenzreste im K¨orper der lⁿ-ten Einheitswurzeln’,Abhandlungen aus dem Mathematischen Seminar der Universit¨at Ham- burg⁶(1928), 146–162.

2. M. Demazure,Lectures on p-divisible groups,Lecture Notes in Mathematics³³², 1972.

3. J. Dieudonn´e, ‘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 inCollected Papersof Ernst Witt, Springer, 1996.

5. H. Hasse, ‘Die Gruppe der pⁿ-prim¨aren Zahlen f ¨ur einen Primteiler p von p’,Journal f ¨ur die Reine und Angewandte Mathematik¹⁷⁶(1936), 174–183.

6. D. Mumford,Lectures on curves on an algebraic surface,Annals of Mathematics Series⁵⁹, Princeton, 1966. Chapter 26 is an introduction to Witt schemes by George Bergman.

7. J-P. Serre,Groupes alg ´ebriques et corps de classes, Hermann, 1964.

8. ——,Corps Locaux, Hermann, 1968.