ON THE INTEGRALITY OF THE WITT POLYNOMIALS^{1}

BY

Andreas DRESS AND Christian SIEBENEICHER^{2}

Consider, for example, the following covariant functors defined on the category rings of commutative rings with a unit element^{3} and with values in rings :

A ↦ F (A) := A[X]
A ↦ F (A) := A[X]/(X^{2})
A ↦ F (A) := A × A
A ↦ F (A) := A ⊗_{Z} A

These functors share the following property:

If p is a prime number and if p · A = 0, then p · F (A) = 0, that is, char A = p ⟹ char F (A) = p.

Question: Do all functors from rings to rings share this property?

Answer: No.

The simplest counterexample known to us is based on the well known

fact that every prime number p divides the binomial coefficient ( \begin{matrix}p \\ j \end{matrix} ) for all

integers j ∈ {1, …, p − 1}.

Indeed, consider for an arbitrary ring A the subset

A_{p}^{(2)} := {r_{p}(a, b) := (a, a^{p} + p · b) | a, b ∈ A} ⊂ A × A.

128. Séminaire Lotharingien de Combinatoire, 15–18 März 1992, Saint-Nabor, Ottrott.

2Universität Bielefeld, Fakultät für Mathematik, Postfach 10 01 31, D 33 501 Bielefeld.

e–mail: dress@mathematik.uni-bielefeld.de sieben@mathematik.uni-bielefeld.de

3In the following all rings will be assumed to be commutative and to have a unit element, denoted by 1.

of the cartesian product A × A and observe that with

( \begin{matrix}p \\ j \end{matrix} )′ := \frac{1}{p} · ( \begin{matrix}p \\ j \end{matrix} ) (j ∈ \{1, …, p − 1\})

one has

r_{p}(0, 0) = (0, 0) ∈ A_{p}^{(2)},
r_{p}(1, 0) = (1, 1) ∈ A_{p}^{(2)},

as well as

r_{p}(a_{1}, b_{1}) ± r_{p}(a_{2}, b_{2}) =

= (a_{1} ± a_{2}, (a_{1} ± a_{2})^{p} + p(b_{1} ± b_{2} − ∑_{j=1}^{p−1}(±1)^{j} ( \begin{matrix}p \\ j \end{matrix} )′ · a_{1}^{p−j} · a_{2}^{j}))

= r_{p}(a_{1} ± a_{2}, b_{1} ± b_{2} − ∑_{j=1}^{p−1}(±1)^{j} ( \begin{matrix}p \\ j \end{matrix} )′ · a_{1}^{p−j} · a_{2}^{j})

and

r_{p}(a_{1}, b_{1}) · r_{p}(a_{2}, b_{2}) =
= (a_{1} · a_{2}, (a_{1} · a_{2})^{p} + p · (a_{1}^{p} · b_{2} + b_{1} · a_{2}^{p} + p · b_{1} · b_{2}))
= r_{p}(a_{1} · a_{2}, a_{1}^{p} · b_{2} + b_{1} · a_{2}^{p} + p · b_{1} · b_{2})

for all a_{1}, b_{1}, a_{2}, b_{2} ∈ A. So the subset A_{p}^{(2)} is a sub–ring of the product ring A × A and the above formulae suggest to define quite formally a new addition and multiplication, say +^{p} and ^{p∘}, on the set A × A by

(a_{1}, b_{1}) +^{p} (a_{2}, b_{2}) := (a_{1} + a_{2}, b_{1} + b_{2} − ∑_{i=j}^{p−1} ( \begin{matrix}p \\ j \end{matrix} )′ · a_{1}^{p−j} · a_{2}^{j})

and

(a_{1}, b_{1}) ^{p∘} (a_{2}, b_{2}) := (a_{1} · a_{2}, a_{1}^{p} · b_{2} + b_{1} · a_{2}^{p} + p · b_{1} · b_{2}),

so that the map

r_{p} : A × A → A × A (a, b) ↦ r_{p}(a, b)

becomes a homomorphism from (A × A, +^{p}, ^{p∘}) into the product–ring A × A.

Obviously, if A has no p–torsion, the homomorphism r_{p} maps (A × A, +^{p}, ^{p∘}) isomorpically onto A_{p}^{(2)}, which establishes in particular that (A×A, +^{p}, ^{p∘}) is indeed a ring for such A. But even if A has p–torsion, in which case the map r_{p} is no more injective, (A × A, +^{p}, ^{p∘}) is a ring. This can be verified either by direct computation or by using a surjective homomorphism from some appropriate p–torsion free ring, e.g. some polynomial ring over Z, onto the ring A.

In other words, the above construction defines a functor

W_{C_{p}} : rings → rings

A ↦ W_{C_{p}}(A) := (A × A, +^{p}, ^{p∘})

(h : A → A′) ↦ (W_{C_{p}}(h) : A × A → A′ × A′ (a, b) ↦ (h(a), h(b)))

for which there exists a canonical natural transformation

Φ : W_{C_{p}} → id × id
Φ(A) : W_{C_{p}}(A) → A × A : (a, b) ↦ r_{p}(a, b).

This functor provides a counter–example for the assumption made above, i.e. if A is a ring for which p · A = 0, then p · W_{C_{p}}(A) ≠ 0:

Indeed the calculation

r_{p}(p^{∘}(a, b)) = p · r_{p}(a, b)
= (pa, pa^{p} + p^{2}b)
= r_{p}(pa, (1 − p^{p−1})a^{p} + pb)

shows that

p^{∘}(1, 0) = (p, 1 − p^{p−1})

holds at least if A has no p–torsion, and therefore, as above, this identity must hold for all rings A.

Hence if char A = p, then for the unit element (1, 0) of W_{C_{p}}(A) one has

p^{∘}(1, 0) = (0, 1) ≠ (0, 0).

More generally, E. Witt observed that for every ring A the subset

{(a_{1}, a_{1}^{2} + 2a_{2}, …, ∑_{d|n} d · a_{d}^{n/d}, …) | a_{1}, a_{2}, … ∈ A}

of the infinite product ring A^{N}, N = {1, 2, 3, …} constitutes a sub–ring of A^{N} and that, as above, this allows to construct a functor

W : rings → rings

which is uniquely determined by the following properties:

• W(A) = AN

• W(h : A → A′) = hN : (a_{1}, a_{2}, …) ↦ (h(a_{1}), h(a_{2}), …)

• for every n ∈ N one has a natural transformation

Φ_{n} : W ⟶ id
Φ_{n}(A) : W(A) → A : (a_{1}, a_{2}, …) ↦ ∑_{d|n} d · a_{d}^{n/d}

To understand these constructions from a structural rather than a purely computational point of view, consider even more generally a pro–finite group G and let O(G) denote the set of open subgroups of G. For every ring A, one considers the ring

A^{O(G)/∼} := {f : O(G) → A | f(U) = f(V) if U ∼^{G} V }

of all functions f : O(G) → A which are constant on G–conjugacy classes. Then the subset of all those maps g : O(G) → A for which there exists some f ∈ A^{O(G)/∼} such that

g(U) = ∑_{W ∈O(G)}′#Fix_{U}(G/W) · f(W)^{(W:U)}

(where the symbol ∑′ is meant to indicate that for each conjugacy class of open subgroups W of G exactly one summand has to be taken and with (W : U) := (G : U)/(G : W)^{4}) can be shown to be a sub–ring of A^{O(G)/∼}. As above, this allows to construct an associated functor W_{G} from rings to rings described in

Theorem 1: Let G be a pro–finite group and let O(G) denote the set of open sub–groups of G. Then there exists a unique functor W_{G} : rings → rings with the following properties:

• W_{G}(A) := A^{O(G)/∼},

• for every ring homomorphism h : A → A′ one has

W_{G}(h) : W_{G}(A) → W_{G}(A′) : f ↦ h^{∘}f,

• for every open subgroup U ∈ O(G) one has a natural transformation

Φ_{U} : W_{G} ⟶ id,

defined by

Φ_{U}(A) : W_{G}(A) → A : f ↦ ∑_{V ∈O(G)}′#Fix_{U}(G/V) · f(V)^{(V:U)}.

4which is an integer whenever Fix_{U}(G/W) is non empty

Remarks:

(1) Witt’s theorem presents the special case where G is the pro–finite completion Ĉ of the infinite cyclic group C.

(2) The functor W_{C_{p}} considered in our first example is precisely the functor W_{C_{p}} for G the cyclic group C_{p} with p elements.

Further results concerning this construction are:

Theorem 2: With F_{p} the finite field with p elements, one has p^{n} · W_{G}(F_{p}) = 0 if and only if p · #G_{p} divides p^{n}, where G_{p} denotes a p–Sylow subgroup of G. In particular, if G_{p} is infinite, one has p^{n} · W_{G}(F_{p}) ≠ 0 for all n ∈ N.

Theorem 3: There exists a canonical isomorphism from W_{G}(Z) onto the (completed) Burnside ring^{5} Ω̂(G). It has the following property: If for every positive integer q ∈ N and for every U ∈ O(G) one denotes by C^{0}(U, q)) the U–set of all continuous maps from U into the discrete set {1, …, q} ^{6} and if ind_{U}^{G}(C^{0}(U, q)) denotes the almost finite G–set induced from it,^{7} then the canonical isomorphism maps every f ∈ W_{G}(Z) with f(U) ≥ 0 for all U ∈ O(G) onto the disjoint union

[f] := ˙ ⋃_{U∈O(G)}′ ind_{U}^{G}(C^{0}(U, f(U))),

taken over all conjugacy classes in O(G).

Remark: Using this isomorphism the above formula in Theorem 1 for the natural transformation Φ_{U}(A) has a rather natural interpretation: for any f ∈ W_{G}(Z) as in Theorem 3 the number of U–invariant elements in the almost finite G–set [f] is precisely ∑′_{V ∈O(G)} #Fix_{U}(G/V) · f(V)^{(V:U)}. In other words, using the identification W_{G}(Z) = Ω̂(G), the homomorphism Φ_{U}(Z) : W_{G}(Z) → Z coincides with the homomorphism φ : Ω̂(G) → Z, induced by associating to each almost finite G–set the number of its U–invariant elements.

5that is the Grothendieck ring of those discrete G–spaces—called almost finite G–sets— where for every open subgroup U ∈ O(G) there are only finitely many points which are invariant under U.

6C^{0}(U, q)) is easily seen to be an almost finite U–set.

7For an almost finite U–set X we denote by ind_{U}^{G}(X) the almost finite G–set induced by X. It is the by definition the set of U–orbits \overline{(g, x)} in the cartesian product G × X relative to the (free) U–action U × (G × X) → G × X defined by (u, (g, x)) ↦ (gu^{−1}, ux) where of course g_{1} · \overline{(g_{2}, x)} := \overline{(g_{1}g_{2}, x)}.

Theorem 4:

1. For every open subgroup U ∈ O(G) there are natural transformations

• F_{U} : W_{G} → W_{U}

• V_{U} : W_{U} → W_{G}

where for every ring A

• the map F_{U}(A) : W_{G}(A) → W_{U}(A) is a ring homomorphism,

• the map V_{U}(A) : W_{U}(A) → W_{G}(A) is an additive homomorphism.

2. Using the identification from Theorem 3 F_{U}(Z) : W_{G}(Z) → W_{U}(Z) coincides with the restriction map res_{U}^{G} : Ω̂(G) → Ω̂(U) and V_{U}(Z) : W_{U}(Z) → W_{G}(Z) coincides with the induction map ind_{U}^{G} : Ω̂(U) → Ω̂(G).

3. The standard identities relating restriction and induction hold more generally for F and V, e.g. for any ring A and any x ∈ W_{G}(A) and y ∈ W_{U}(A) one has x · V_{U}(A)(y) = V_{U}(A)(F_{U}(A)(x) · y) (Frobenius reciprocity) and for U_{1}, U_{2} ∈ O(G) and x ∈ W_{U_{1}}(A) one can compute F_{U_{2}}(A)(V_{U_{1}}(A)(x)) ∈ W_{U_{2}}(A) according to an appropriate variant of the Mackey sub–group formula.

Remark: In case G = Ĉ, the natural transformations F and V specialize to the well known Frobenius and Verschiebung maps defined for universal Witt vectors. Moreover, the well known identities relating the Frobenius and Verschiebung maps follow from the third assertion of Theorem 4 in this particular case.

To prove Witt’s theorem as well as Theorems 1 to 4 one needs to show that

certain rational numbers—like e.g. \frac{1}{p} ( \begin{matrix}p \\ j \end{matrix} )—are indeed integers. In the case

\frac{1}{p} ( \begin{matrix}p \\ i \end{matrix} ) this, of course, can be shown by direct computation, but it can also be

shown without any computation by realizing that \frac{1}{p} ( \begin{matrix}p \\ j \end{matrix} ) is the number of orbits

of the action of the cyclic group C_{p} of order p on the set ( \begin{matrix}C & p \\ j \end{matrix} ) of its subsets

of cardinality j.

It is this way of using group actions to prove integrality results of this type which is fundamental for the proof of our theorems and which—first of all—suggested that a rather general variant of Witt’s construction should exist, based on the equivariant combinatorics of arbitrary rather than of cyclic pro–finite groups, only.

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