On the Structure of Witt-Burnside Rings

Attached to Pro-p Groups

Lance Edward Miller

Received: April 1, 2013

Communicated by Lars Hesselholt

Abstract. The p-typical Witt vectors are a ubiquitous object in algebra and number theory. They arise as a functorial construction that takes perfect fields k of prime characteristic p > 0 to p-adically complete discrete valuation rings of characteristic 0 with residue field k and are universal in that sense. A. Dress and C. Siebeneicher generalized this construction by producing a functor W_{G} attached to any profinite group G. The p-typical Witt vectors arise as those attached to the p-adic integers. Here we examine the ring structure of W_{G}(k) for several examples of pro-p groups G and fields k of characteristic p. We will show that the structure is surprisingly more complicated than the p-typical case.

1 Introduction

The purpose of this article is to explore what kinds of rings lie in the image of the functors W_{G} introduced by A. Dress and C. Siebeneicher [DS88] in the case that G is a pro-p group for prime integer p. These functors were originally defined for all profinite groups and are now called Witt-Burnside functors due to the fact that they generalize both the p-typical (recovered when G = Z_{p} as an additive group) and ‘big’ Witt vector construction (recovered when G = Ẑ), as well as Burnside functors (recovered as W_{G}(Z)). We call rings lying in the image of a Witt-Burnside functor Witt-Burnside rings. These functors are of significant interest, and yet to date the types of rings which are produced from this construction lack description. To illustrate some of their recent importance, we remark that Witt-Burnside functors have been used extensively to study equivariant ring spectra as they arise as a left adjoint for Tambara functors [Bru05, Bru07, Str]. Many constructions of Witt-Burnside functors have been given. Specifically, J. Graham’s construction utilizes ring valued G-sets [Gra93]. J. Elliott gave a unified construction relating the Graham and Dress and Siebeneicher approaches [Ell06]. Y. Oh has studied some decomposition and q-deformation questions [Oh09, Oh07, Oh12]. Beyond the classical G = Z_{p} case, not much about the ring structure of the images of Witt-Burnside functors is known. Motivated by the extensive applications enjoyed by the p-typical and big Witt vectors, this paper addresses structural questions about Witt-Burnside rings under the assumptions that G is an infinite pro-p group and k is a field of characteristic p > 0. The construction of the classical (both p-typical and big) Witt vectors uses the Witt polynomials to define certain addition and multiplication polynomials with rational coefficients which do not obviously have integral coefficients [Wit36]. A. Dress and C. Siebeneicher generalized the Witt polynomials to a family of multivariable polynomials associated to any profinite group G. Like classical Witt polynomials, these polynomials obviously have Q-coefficients and a significant theorem of Dress and Siebeneicher shows that they in fact have integral coefficients. Thus one can use these polynomials, in an analogous way to the construction of classical Witt vectors, to define a functor W_{G} on the category of commutative rings for each profinite group G. For infinite pro-p groups G, the functor W_{G} retains the surprising property of taking rings of characteristic p to rings of characteristic 0. For perfect fields k of characteristic p, the ring W_{Z_{p}}(k) is the ring of classical ptypical Witt vectors. These are p-adically complete discrete valuation domains with maximal ideal (p) and residue field k. From [DS88, Thm. 2] it is known that W_{G}(k) is a ring of characteristic zero when G ≅ Z̸_{p} is an infinite pro-p group and k is a field. We show as expected that W_{G}(k) shares some properties with the p-typical case when G is pro-p and k has characteristic p.

Theorem (Cf., Theorem 2.16) For G a pro-p group and A a local ring of characteristic p > 0, W_{G}(A) is a local ring.

However, the similarities do not run deep!

Theorem (Cf., Theorem 4.5) For G = Z_{p}^{d}, k a field of characteristic p > 0 and d ≥ 2, the maximal ideal of W_{G}(k) is not finitely generated, so W_{G}(k) is not Noetherian.

Witt-Burnside Rings Attached to Pro-p Groups 1293

The core reason for the complication in the case G = Z_{p}^{d} when d ≥ 2 versus d = 1 is the existence of more than one maximal subgroup. In general, when G is any pro-p group that is not pro-cyclic there is more than one maximal open subgroup H, necessarily normal. These subgroups describe certain coordinates in the Witt vectors attached to G for which sums and products have repeated values, or redundencies. In particular, G = Z_{p}^{d} has a fairly homogeneous subgroup structure as every open subgroup is (non-canonically) isomorphic to G. This means the redundancy behavior when d ≥ 2 propagates and manifests as a much smaller square of the maximal ideal than expected. Another consequence of there being more than one maximal subgroup of an infinite pro-p group G when G ≅ Z̸_{p} is the ability to construct zero divisors. This was essentially known in [DS88]. Our second main goal is to give some control on the zero divisors of W_{G}(k) for G = Z_{p}^{2} and k a field of characteristic p.

Theorem (Cf., Theorem 5.17) For G = Z_{p}^{2} and any field k of characteristic p, the ring W_{G}(k) is reduced.

The methods used here fail for d > 2 due to the reliance on a certain property of the subgroup structure of Z_{p}^{2} which is not satisfied more generally. The rest of the paper is organized as follows. Section 2 develops the preliminary definitions, constructions, and facts about Witt-Burnside rings. We also prove necessary lemmas for the paper and close with the proof that W_{G}(A) is local when G is pro-p and A is a local ring of characteristic p. Section 3 discusses in detail the frame of Z_{p}^{d} for d ≥ 2; these groups form the basic examples of the paper. Section 4 discusses the failure of finite generation in the maximal ideal W_{Z_{p}^{d}}(k) for d ≥ 2 when the characteristic of k is p. Finally, Section 5 concerns nilpotent elements in W_{G}(k). Unless otherwise stated, G will always be a profinite group and p denotes a prime integer.

2 Preliminaries

Witt-Burnside rings are constructed utilizing generalized Witt polynomials associated to a profinite group G. The index set of these generalized polynomials is the set of isomorphism classes of discrete finite transitive G-sets, called the frame of G and denoted F(G). For example, F(Z_{p}) = N by the correspondence Z_{p}/p^{n}Z_{p} ↔ n. There is a natural partial ordering on F(G). For T and U in F(G) we say U ≤ T if there is a G-map from T to U. Denote the set of all G-maps from T to U as Map_{G}(T, U) and the number of G-maps #Map_{G}(T, U) by φ_{T} (U). Thus φ_{T} (U) ≠ 0 if and only if T ≤ U. We summarize some facts about F(G)

1. If T and U in F(G) with U ≤ T, then #U divides #T and #T/#U represents the size of any of the fibers of any element of Map_{G}(T, U).

2. If the stabilizer subgroups of the points in T are all equal (we will say in this case that T has normal stabilizers or that T is a normal G-set), then φ_{T} (U) = #U for U ≤ T.

3. For each T in F(G), there are only finitely many U in F(G) with U ≤ T.

The elements of F(G) have a concrete description. Every finite transitive Gset T is isomorphic to some coset space G/H with left G-action, where H is an open subgroup of G that can be chosen as the stabilizer subgroup of any point in T. The partial order ≤ on coset spaces (considered as G-sets up to isomorphism) can be described concretely by G/K ≤ G/H if and only if H is conjugate to a subgroup of K (or equivalently, H is a subgroup of a conjugate of K). For T ∈ F(G), define the T -th Witt polynomial to be

W_{T} ({X_{U}}_{U∈F(G)}) = ∑_{U≤T} φ_{T} (U)X_{U}^{#T/#U} = X_{0}^{#T} + … + φ_{T} (T)X_{T} , (1)

where 0 denotes the trivial G-set G/G. Trivially φ_{T} (0) = 1 for all T in F(G). This is a finite sum since there are only finitely many U ≤ T. For instance, if G = Z_{p} then the finite transitive G-sets up to isomorphism are Z_{p}/p^{n}Z_{p} for n ≥ 0 and the Witt polynomial associated to Z_{p}/p^{n}Z_{p} is the classical n-th p-typical Witt polynomial. One may gauge the complexity of a given frame by looking at the covering relations. In any partially ordered set, one says that U covers T if T ≤ U and there are no elements in between. In the frame of Z_{p} each element has exactly one cover and the frame is linearly ordered. Figure 1 displays the frame of Z_{2}^{2}. When G = Z_{p}^{2}, all G-sets have p + 1 covers. Other than the trivial G-set, each G-set below the horizontal line in Figure 1 has p covers also below the horizontal line. This line is not part of the frame and depicts a property about the stabilizers of various G-sets described in Definition 5.1.

\begin{matrix}Z_{2}^{2}/Z_{2}^{2} \\ Z_{2}^{2}/2Z_{2}^{2} \end{matrix}

Figure 1: The frame F(Z_{2}^{2}).

Remark 2.1. The picture in Figure 1 is reminiscent of the tree of Z_{2}-lattices in Q_{2}^{2} up to scaling, on which PGL_{2}(Q_{2}) acts [Ser77, p. 71]. However, it is different since the G-sets Z_{2}^{2}/2^{r}Z_{2}^{2} appear as separate vertices in Figure 1, while the subgroups 2^{r}Z_{2}^{2} all correspond to the same vertex in the tree for PGL_{2}(Q_{2}).

To simplify notation, write a tuple of variables X_{T} indexed by all T in F(G) as \underline{X}, e.g., W_{T} (\{X_{U}\}_{U∈F(G)}) = W_{T} (\underline{X}), Z[\{X_{T} \}_{T ∈F(G)}] = Z[\underline{X}], and Z[\{X_{T} , Y_{T} \}_{T ∈F(G)}] = Z[\underline{X}, \underline{Y}]. This underline notation of course depends on G. For any commutative ring A, a polynomial f(\underline{X}) ∈ Z[\underline{X}] defines a function from ∏_{T ∈F(G)} A to A, and for a tuple a = (a_{T})_{T ∈F(G)} with coordinates in A we write f(a) = f({a_{T} }_{T ∈F(G)}) ∈ A. A similar meaning is applied to f(a, b) for a polynomial f(\underline{X}, \underline{Y}) ∈ Z[\underline{X}, \underline{Y}]. Generally, we write sequences indexed by F(G) as bold letters (e.g., a, b, x, y, v) and their T -th coordinate is in italics (e.g., a_{T} , b_{T} , x_{T} , y_{T} , v_{T}). Because X_{T} appears on the right side of (1) just in the linear term φ_{T} (T)X_{T}, and all variables which appear in other terms are X_{U} for U < T, we get the following uniqueness criterion for all the Witt polynomial values together which is equivalent to Lemma 2.1 in [Ell06, p. 331].

Theorem 2.2. If A is a commutative ring which has no φ_{T} (T)-torsion for any T ∈ F(G), then the function ∏_{T ∈F(G)} A → ∏_{T ∈F(G)} A given by a ↦ (W_{T} (a))_{T ∈F(G)} is injective. This function is bijective provided each φ_{T} (T) is a unit in A.

Example 2.3. If G is a pro-p group and T ≅ G/H, then φ_{T} (T) = [N_{G}(H) : H] is a power of p, so if p is invertible in A then every a ∈ ∏_{T ∈F(G)} A has the form (W_{T} (b))_{T ∈F(G)} for a unique b ∈ ∏_{T ∈F(G)} A.

The most important application of Theorem 2.2 is to the ring A = Q[\underline{X}, \underline{Y}] and the vectors (W_{T} (\underline{X})+W_{T} (\underline{Y}))_{T ∈F(G)} and (W_{T} (\underline{X})W_{T} (\underline{Y}))_{T ∈F(G)}. It tells us there are unique families of polynomials \{S_{T} (\underline{X}, \underline{Y})\} and \{M_{T} (\underline{X}, \underline{Y})\} in Q[\underline{X}, \underline{Y}] satisfying

W_{T} (\underline{X}) + W_{T} (\underline{Y}) = W_{T} (\underline{S}) for all T ∈ F(G)

and

W_{T} (\underline{X})W_{T} (\underline{Y}) = W_{T} (\underline{M}) for all T ∈ F(G).

More explicitly, this says

∑_{U≤T} φ_{T} (U)X_{U}^{#T/#U} + ∑_{U≤T} φ_{T} (U)Y_{U}^{#T/#U} = ∑_{U≤T} φ_{T} (U)S_{U}^{#T/#U} (2)

and

( ∑_{U≤T} φ_{T} (U)X_{U}^{#T/#U} ) ( ∑_{U≤T} φ_{T} (U)Y_{U}^{#T/#U} ) = ∑_{U≤T} φ_{T} (U)M_{U}^{#T/#U} (3)

for all T. The polynomials S_{T} and M_{T} each only depend on the variables X_{U} and Y_{U} for U ≤ T. A significant theorem of Dress and Siebeneicher [DS88, p. 107], which generalizes Witt’s theorem (G = Z_{p}), says that the polynomials S_{T} and M_{T} have coefficients in Z. We call the S_{T} ’s and M_{T} ’s the Witt addition and multiplication polynomials, respectively. (Obviously they depend on G, but that dependence will not be part of the notation).

Example 2.4. Taking T = 0, one has

S_{0}(\underline{X}, \underline{Y}) = X_{0} + Y_{0} and M_{0}(\underline{X}, \underline{Y}) = X_{0}Y_{0}.

If T ≅ G/H where H is a maximal open subgroup, so {U ∈ F(G): U ≤ T } is just {0, T } solving for S_{T} and M_{T} in (2) and (3) yields

S_{T} = X_{T} + Y_{T} + \frac{(X_{0} + Y_{0})^{#T} − X_{0}^{#T} − Y_{0}^{#T}}{φ_{T} (T)} ,

M_{T} = X_{0}^{#T} Y_{T} + X_{T} Y_{0}^{#T} + φ_{T} (T)X_{T} Y_{T} .

Compare with the first two classical Witt addition and multiplication polynomials in [Ser79, p. 42]. Further addition and multiplication polynomials could be very complicated to write out explicitly, as is already apparent for the classical Witt vectors if you try to go past the first two polynomials.

Since S_{T} and M_{T} have integral coefficients, they can be evaluated on any ring, including rings where the hypotheses of Theorem 2.2 break down, like a ring of characteristic p when G is a pro-p group.

Definition 2.5. Let G be a profinite group. For any commutative ring A, define the Witt–Burnside ring W_{G}(A) to be the product space ∏_{T ∈F(G)} A as a set, with elements written as a = (a_{T})_{T ∈F(G)}. The ring operations on W_{G}(A) are defined using the Witt addition and multiplication polynomials:

a + b = (S_{T} (a, b))_{T ∈F(G)}

and

a · b = (M_{T} (a, b))_{T ∈F(G)}.

The additive (resp. multiplicative) identity is (0, 0, 0, …) (resp. (1, 0, 0, …)).

We remark that W_{G}(A) = \underleftarrow{lim}_{N} W_{G/N}(A) where the inverse limit runs over open normal subgroups of G and, that giving each W_{G/N}(A) the discrete topology induces a natural profinite topology on W_{G}(A) in which W_{G}(A) is complete, which follows from the proof of [DS88, Thm 3.3.2], see also [Ell06, pg. 357]. Even when G is not abelian, W_{G}(A) is a commutative ring. For G = Z_{p}, the addition and multiplication polynomials are the classical p-typical Witt addition and multiplication polynomials and W_{Z_{p}}(A) is the p-typical Witt vectors. For any ring homomorphism f: A → B define W_{G}(f): W_{G}(A) → W_{G}(B) by applying f to the coordinates:

W_{G}(f)(a) = (f(a_{T}))_{T ∈F(G)} ∈ W_{G}(B).

This is a ring homomorphism and makes W_{G} a covariant functor from commutative rings to commutative rings.

Packaging all the Witt polynomials together, we get a ring homomorphism W : W_{G}(A) → ∏_{T ∈F(G)} A which is W_{T} in the T -th coordinate:

W (a) = (W_{T} (a))_{T ∈F(G)} = ( ∑_{U≤T} φ_{T} (U)a_{U}^{#T/#U} )_{T ∈F(G)} .

This homomorphism is called the ghost map and its coordinates W_{T} (a) are called the ghost components of a. In some cases it is quite useless: if G is pro-p and A has characteristic p then W (a) = (a_{0}^{#T})_{T ∈F(G)}, whose dependence on a only involves a_{0}. If A fits the hypothesis of Theorem 2.2 then the ghost map is injective (i.e., the ghost components of a determine a). Also from Theorem 2.2 the ghost map is bijective if every integer φ_{T} (T) is invertible in A so W_{G}(A) ≅ ∏_{T ∈F(G)} A by the ghost map. That means W_{G}(A) is a new kind of ring only if some φ_{T} (T) is not invertible in A, and especially if A has φ_{T} (T)- torsion for some T (e.g., G is a nontrivial pro-p group and A has characteristic p). The coordinates on which a Witt vector is nonzero is called its support. While the ring operations in W_{G}(A) are generally not componentwise, addition in W_{G}(A) is componentwise on two Witt vectors with disjoint support.

Theorem 2.6. Let {R, S} be a partition of F(G), i.e., R ∪ S = F(G) and R ∩ S = ∅. For every ring A and any a ∈ W_{G}(A), define r(a) and s(a) to be the Witt vectors derived from a with support in R and S:

r(a) = \begin{cases}a_{T} & if T ∈ R, \\ 0 & if T ∈ S, \end{cases} and s(a) = \begin{cases}0 & if T ∈ R, \\ a_{T} & if T ∈ S. \end{cases}

Then a = r(a) + s(a) in W_{G}(A).

Proof. First we will show the result in W_{G}(Z[\underline{X}]) for the particular Witt vector x = (X_{T})_{T ∈F(G)}: x = r(x) + s(x) in W_{G}(Z[\underline{X}]). If we prove this then given any ring A and a ∈ W_{G}(A), there is a ring homomorphism f: Z[\underline{X}] → A such that f(X_{T}) = a_{T} for all T, and applying the ring homomorphism W_{G}(f): W_{G}(Z[\underline{X}]) → W_{G}(A) to the identity x = r(x) + s(x) turns it into a = r(a) + s(a). Since Z[\underline{X}] is a domain of characteristic 0, the ghost map

W: W_{G}(Z[\underline{X}]) → ∏_{T ∈F(G)} Z[\underline{X}]

is an injective ring homomorphism, so it suffices to prove

W (x) = W (r(x) + s(x)).

The right side is W (r(x)) + W (s(x)), which is a sum in the product ring ∏_{T ∈F(G)} Z[\underline{X}], so its T -th coordinate for any T is

W_{T} (r(x)) + W_{T} (s(x)) = ∑_{U∈R^{U≤T}} φ_{T} (U)X_{U}^{#T/#U} + ∑_{U∈S^{U≤T}} φ_{T} (U)X_{U}^{#T/#U}.

Since R ∪ S = F(G) and R ∩ S = ∅, each U with U ≤ T will lie in exactly one of R or S, so

W_{T} (r(x)) + W_{T} (s(x)) = ∑_{U≤T} φ_{T} (U)X_{U}^{#T/#U} = W_{T} (x).

Therefore W (r(x) + s(x)) and W (x) have the same T -th component for all T, so they are equal, which shows r(x) + s(x) = x.

One typically proves an algebraic identity in W_{G}(A) by reformulating it as an identity in a ring of Witt vectors over a polynomial ring over Z. From now on, we will usually prove the reformulation but may not go through the deduction of the identity we want over A from the identity proved over a polynomial ring; instead simply invoke functoriality.

Definition 2.7. For a ∈ A and T ∈ F(G), denote by ω_{T} (a) ∈ W_{G}(A) the Witt vector with T -coordinate a and all other coordinates 0. We call ω_{T} (a) the T -th Teichmüller lift of a.

We denote the trivial G-set G/G as 0 and by ω_{0}(a) the G/G-th Teichmüller lift. The function ω_{0} : A → W_{G}(A) generalizes the classical Teichmüller lift. Like the classical Teichmüller lift, ω_{0} is multiplicative. For a general formula for ω_{T} (a)ω_{T′}(b), see [Ell06, p. 355]. An easy consequence of Theorem 2.6 is that any Witt vector a of finite support satisfies a = ∑_{U∈Supp(a)} ω_{U}(a_{U}) where Supp(a) is the support of a.

Theorem 2.8. For any a ∈ A and b ∈ W_{G}(A),

ω_{0}(a)b = (a^{#T} b_{T})_{T ∈F(G)}.

In particular, ω_{0}(a)ω_{0}(b) = ω_{0}(ab).

Proof. By functoriality, it suffices to show in W_{G}(Z[\underline{X}, \underline{Y}]) that

(X_{0}, 0, 0, 0, …)(Y_{T})_{T ∈F(G)} = (X_{0}^{#T} Y_{T})_{T ∈F(G)},

and to show this equation it suffices to prove the ghost components (Witt polynomial values) of both sides are equal. Since W_{T} : W_{G}(Z[\underline{X}, \underline{Y}]) → Z[\underline{X}, \underline{Y}] is multiplicative,

W_{T} ((X_{0}, 0, 0, 0, …)(Y_{U})_{U∈F(G)}) = W_{T} (X_{0}, 0, 0, 0, …)W_{T} ((Y_{U})_{U∈F(G)})
= X_{0}^{#T} ∑_{U≤T} φ_{T} (U)Y_{U}^{#T/#U}

and

W_{T} ((X_{0}^{#U}Y_{U})_{U∈F(G)}) = ∑_{U≤T} φ_{T} (U)(X_{0}^{#U}Y_{U})^{#T/#U}
= ∑_{U≤T} φ_{T} (U)X_{0}^{#T} Y_{U}^{#T/#U}.

The Witt polynomial W_{T} (\underline{X}) becomes homogeneous of degree #T if we give X_{U} degree #U (e.g., X_{0} has degree 1, not 0). This grading makes the addition and multiplication polynomials homogeneous as well.

Theorem 2.9. Give the ring Z[\underline{X}, \underline{Y}] the grading in which the degree of X_{U} and Y_{U} is #U.

(a) For all T, the polynomial S_{T} is homogeneous of degree #T and M_{T} is homogeneous of degree 2#T.

(b) For all T, we have S_{T} (\underline{X}, 0) = X_{T}, S_{T} (0, \underline{Y}) = Y_{T}, M_{T} (\underline{X}, 0) = 0, and M_{T} (0, \underline{Y}) = 0.

The second part of the theorem is saying S_{T} equals X_{T} + Y_{T} plus monomials X_{U}^{i}Y_{V}^{j} where U < T and V < T (we cannot have U = T or V = T by homogeneity), while M_{T} contains no monomials that are pure \underline{X}-terms or pure \underline{Y} -terms.

Proof. (a) We will work out the homogeneity for M_{T}; the argument for S_{T} is similar. Clearly M_{0}(X_{0}, Y_{0}) = X_{0}Y_{0}, which is homogeneous of degree 2 and its only monomial term contains the factors X_{0} and Y_{0}. Let n ≥ 2 and assume by induction for all transitive G-sets U with #U < n that M_{U} is homogeneous of degree 2#U. Pick a transitive G-set T with #T = n. (If there are no such G-sets then we are vacuously done.) Solving for M_{T} in (3) in Q[\underline{X}, \underline{Y}],

M_{T} = \frac{1}{φ_{T} (T)} ( ∑_{U_{1}≤T} φ_{T} (U_{1})XU_{1}^{\frac{#T}{#U_{1}}} ∑_{U_{2}≤T} φ_{T} (U_{2})YU_{2}^{\frac{#T}{#U_{2}}} − ∑_{U<T} φ_{T} (U)MU^{\frac{#T}{#U}} )

= \frac{1}{φ_{T} (T)} ( ∑_{U_{1},U_{2}≤T} φ_{T} (U_{1})φ_{T} (U_{2})XU_{1}^{\frac{#T}{#U_{1}}} YU_{2}^{\frac{#T}{#U_{2}}} − ∑_{U<T} φ_{T} (U)MU^{\frac{#T}{#U}} ) .

By the inductive hypothesis, each M_{U} for U < T is homogeneous of degree 2#U, so M_{U}^{#T/#U} is homogeneous of degree 2#T. By the definition of the grading, X_{U_{1}}^{#T/#U_{1}} has degree #T and Y_{U_{2}}^{#T/#U_{2}} has degree #T, so X_{U_{1}}^{#T/#U_{1}}Y_{U_{2}}^{#T/#U_{2}} has degree 2#T.

(b) We will work out the result for multiplication polynomials. Since M_{0}(\underline{X}, 0) is X_{0}0 = 0, we may assume T ≠ 0. Set every Y_{U} to 0 in the recursive formula for M_{T} above. Then the formula tells us M_{T} (\underline{X}, 0) is equal to

\frac{1}{φ_{T} (T)} ( ∑_{U_{1},U_{2}≤T} φ_{T} (U_{1})φ_{T} (U_{2})X_{U_{1}}^{#T/#U_{1}} · 0 − ∑_{U<T} φ_{T} (U)M_{U}(\underline{X}, 0)^{#T/#U} ) .

Using the inductive hypothesis, each term is 0.

So the polynomials S_{T} ({X_{U}^{#U}, Y_{U}^{#U}}_{U∈F(G)}) are genuine homogeneous polynomials of degree #T (replace X_{U} and Y_{U} with their #U-th powers everywhere), and similarly for the multiplication polynomials.

2.1 Units

The present goal is to demonstrate that W_{G}(k) is a local ring when G is prop and k is a field of characteristic p. We state a couple of needed lemmas about divisibility relationships among φ_{T} (U) for U, T ∈ F(G) whose proofs are straightforward and left to the reader.

Lemma 2.10. If G is a pro-p group and T ≥ U in F(G) with U nontrivial, then φ_{T} (U) ≡ 0 mod p.

Lemma 2.11. Let G be a nontrivial pro-p group, U < T in F(G) and s ∈ pZ. Then

\frac{φ_{T} (U)}{φ_{T} (T)}s^{#T/#U} ∈ pZ.

We apply these to study a particularly important family of ideals in W_{G}(A).

Definition 2.12. For n ∈ Z^{+}, set

I_{n}(G, A) = {a ∈ W_{G}(A) : a_{T} = 0 for #T < n}.

These are the Witt vectors a with support in {T : #T ≥ n}.

Lemma 2.13. Each I_{n}(G, A) is an ideal in W_{G}(A).

Proof. Each addition polynomial S_{T} depends only on variables indexed by (isomorphism classes of) finite transitive G-sets U ≤ T and has no constant term by Theorem 2.9. So if two Witt vectors are in I_{n}(G, A) then their sum is also in I_{n}(G, A). It remains to show for any a ∈ W_{G}(A) and b ∈ I_{n}(G, A) that ab ∈ I_{n}(G, A). Set c = ab. By the definition of multiplication in W_{G}(A), c_{T} = M_{T} (a, b) for any T. If #T < n, b_{U} = 0 for U ≤ T by hypothesis. Since M_{T} (\underline{X}, \underline{Y}) only depends on Y_{U} for U ≤ T, c_{T} = M_{T} (a, b) = M_{T} (a, 0) and M_{T} (a, 0) = 0 by Theorem 2.9.

Lemma 2.14. Let G be a pro-p group. For all rings A of characteristic p and nonnegative integers n,

I_{p}(G, A)I_{p^{n}}(G, A) ⊂ I_{p^{n+1}}(G, A)

in W_{G}(A).

Proof. If n = 0 the result is clear since I_{1}(G, A) = W_{G}(A), so without loss of generality assume n ≥ 1. We will derive a mod p congruence for particular Witt vectors over the ring R = Z[\underline{X}, \underline{Y}], which will be sufficient using functoriality. Define x and y in W_{G}(R) by

x_{T} = \begin{cases}0, & if T = 0, \\ X_{T} , & if T ≠ 0, \end{cases} and y_{T} = \begin{cases}0, & if #T < p^{n}, \\ Y_{T} , & otherwise. \end{cases}

Set z = xy. Our aim is to show

#T < p^{n+1} ⟹ z_{T} ≡ 0 mod pR.  (4)

We argue by induction on #T. Clearly z_{0} = x_{0}y_{0} = 0. Let #T = p^{r} < p^{n+1} with r ≥ 1 and assume by induction that for all #U < p^{r}, z_{U} ≡ 0 mod pR. Since the T -th Witt polynomial is a multiplicative function W_{T} : W_{G}(R) → R, W_{T} (z) = W_{T} (x)W_{T} (y):

∑_{U≤T} φ_{T} (U)z_{U}^{#T/#U} = ∑_{T_{1},T_{2}≤T} φ_{T} (T_{1})φ_{T} (T_{2})x_{T_{1}}^{#T/#T_{1}}y_{T_{2}}^{#T/#T_{2}}.

Solving this equation for z_{T} in Q[\underline{X}, \underline{Y}],

z_{T} = ∑_{T_{1},T_{2}≤T} \frac{φ_{T} (T_{1})φ_{T} (T_{2})}{φ_{T} (T)}x_{T_{1}}^{#T/#T_{1}}y_{T_{2}}^{#T/#T_{2}} − ∑_{U<T} \frac{φ_{T} (U)}{φ_{T} (T)}z_{U}^{#T/#U}. (5)

Since z_{U} ∈ pR for U < T, the second term in (5) is 0 mod pR by Lemma 2.11. In the first term in (5), we can assume T_{1} ≠ 0 since x_{0} = 0. If #T_{2} < p^{n} then y_{T_{2}} = 0, so we only need to consider T_{2} where #T_{2} ≥ p^{n}. Since #T < p^{n+1}, T_{2} = T. Then (5) becomes

z_{T} = ∑_{0<T_{1}≤T} φ_{T} (T_{1})x_{T_{1}}^{#T/#T_{1}}y_{T} − ∑_{U<T} \frac{φ_{T} (U)}{φ_{T} (T)}z_{U}^{#T/#U}.

Since φ_{T} (T_{1}) is an integral multiple of p by Lemma 2.10, z_{T} ≡ 0 mod pR.

An immediate useful corollary follows.

Corollary 2.15. For any pro-p group G and ring A of characteristic p, I_{p}(G, A)^{m} ⊂ I_{p^{m}}(G, A).

Proof. This follows directly from repeated applications of Lemma 2.14.

We are now set to prove the main result of this section.

Theorem 2.16. Let G be a pro-p group and A be a ring of characteristic p. The units in W_{G}(A) are W_{G}(A)^{×} = {a : a_{0} ∈ A^{×}}. Consequently, when (A, 𝔫) is local, W_{G}(A) is a local ring with maximal ideal 𝔪 = {a ∈ W_{G}(A) : a_{0} ∈ 𝔫}.

Proof. Obviously W_{G}(A)^{×} ⊂ {a : a_{0} ∈ A^{×}}. To prove the reverse inclusion, let a ∈ W_{G}(A) have a_{0} ∈ A^{×}. By Theorem 2.8 the Witt vector (a_{0}, 0, 0, …) is a unit, with inverse (a_{0}^{−1}, 0, 0, …), so it suffices to show (a_{0}^{−1}, 0, 0, …)a is a unit. The first coordinate of this product is 1, so we are reduced to showing a Witt vector with first coordinate 1 is a unit. That is, we can assume a_{0} = 1. By Theorem 2.6,

a = (1, 0, 0, …) + (0, {a_{T} }_{T ≠0}) ∈ 1 + I_{p}(G, A).

Since I_{p}(G, A)^{m} ⊂ I_{p^{m}}(G, A) by Corollary 2.15, we can invert a using a geometric series since W_{G}(A) is complete in the profinite topology.

3 The Frame of Z_{p}^{d}

The subgroup structure of Z_{p}^{d} is homogeneous in the sense that every open subgroup is isomorphic to Z_{p}^{d} (although there is not a canonical choice of isomorphism, unlike the case when d = 1). A subgroup is open if and only if it has finite index. If H is an open subgroup there is a Z_{p}-basis {e_{1}, e_{2}, …, e_{d}} for G such that G = Z_{p}e_{1} ⊕ Z_{p}e_{2} ⊕ ⋯ ⊕ Z_{p}e_{d} and H = Z_{p}p^{a_{1}}e_{1} ⊕ Z_{p}p^{a_{2}}e_{2} ⊕ ⋯ ⊕ Z_{p}p^{a_{d}}e_{d} for some a_{1}, …, a_{d} ≥ 0. The G-set G/H has the form

G/H ≅ Z_{p}/p^{a_{1}}Z_{p} × Z_{p}/p^{a_{2}}Z_{p} × ⋯ × Z_{p}/p^{a_{d}}Z_{p}.  (6)

As a group, G/H is usually a product of d nontrivial cyclic p-groups. For some H, a_{i} = 0 for all but one i, making T = G/H a cyclic group.

Definition 3.1. Let G be any profinite group and N be an open normal subgroup. A G-set T ≅ G/N where G/N is a cyclic group is called a cyclic G-set.

An important property about cyclic G-sets T when G is a pro-p group is that {U ∈ F(G): U < T }, which is called the strict downset of T, with its induced order is a chain.

Throughout the rest of this section G = Z_{p}^{d} with d ≥ 2.

From the large number of Z_{p}-bases of G it is reasonable to expect that there are many cyclic G-sets of each size as the size grows. We will use cyclic G-sets later (Lemma 3.4) to find nonisomorphic G-sets of the same size with the same strict downsets, i.e., the G-sets lying below them in the frame of G are the same, which will be important in Section 4. For each T ∈ F(G), there is an open subgroup H ⊂ G with T ≅ G/H, and H is uniquely determined by T since G is abelian: H is the common stabilizer of all points in T. Using a Z_{p}-basis {e_{1}, e_{2}, …, e_{d}} of G we can identify H with p^{a_{1}}Z_{p} ×p^{a_{2}}Z_{p} ×⋯×p^{a_{d}}Z_{p} (this amounts to applying an automorphism of G), so the G-sets below G/H in F(G) are in one-to-one correspondence with the (open) subgroups of G that contain p^{a_{1}}Z_{p} × p^{a_{2}}Z_{p} × ⋯ × p^{a_{d}}Z_{p}. The frame of F(Z_{p}) is linearly ordered so every element has a unique cover (G-set lying directly above it). Since every subgroup of Z_{p}^{d} is isomorphic to Z_{p}^{d} and there are p^{d−1} + … + p + 1 maximal subgroups of Z_{p}^{d} the number of covers of each element of F(G) is the same which we now show.

Theorem 3.2. Let T ∈ F(Z_{p}^{d}). It has p^{d−1} + … + p + 1 covers in F(Z_{p}^{d}) and in the case T ≥ Z_{p}^{d}/pZ_{p}^{d}, it covers p^{d−1} + … + p + 1 elements of F(Z_{p}^{d}).

Proof. Let H be the stabilizer of T. Covers of T correspond to subgroups of H with index p, which correspond to maximal subgroups of H/pH ≅ (Z/pZ)^{d}. By duality the number of such subgroups is the number of subgroups of H/pH with size p, namely the number of linear subspaces of (Z/pZ)^{d}. That is (p^{d} − 1)/(p − 1) = 1 + p + … + p^{d−1}. Similarly, if T covers U then U = Z_{p}^{d}/K where H ⊂ K and [K : H] = p, so the sets which T covers correspond to subgroups of size p in Z_{p}^{d}/H. The number of such subgroups is the same as the number of subgroups of index p. When Z_{p}^{d}/H ≥ Z_{p}^{d}/pZ_{p}^{d}, we have H ⊂ pZ_{p}^{d}. All subgroups of Z_{p}^{d} with index p contain pZ_{p}^{d}, so the number of subgroups of index p in Z_{p}^{d}/H and Z_{p}^{d}/pZ_{p}^{d} is the same.

In some calculations, it was noticed that many pairs of coordinates in sums or products of certain Witt vectors are equal. This is described formally by Lemmas 4.1 and 4.2 below and motivates the following definition.

Definition 3.3. A nonisomorphic pair of G-sets T and T′, whose strict downsets agree, is called linked.

For a linked pair of G-sets T and T′, #T = #T′ since for any U < T of maximal size, #T = p#U. Therefore #T′ = p#U = #T. An example of such a pair T and T′ is labeled in Figure 2. The pair V_{1} and V_{2} in Figure 2 is not linked since the G-set U is below V_{2} and not below V_{1}. Our first task is to show that linked pairs of G-sets of arbitrarily large size exist for G = Z_{p}^{d} for d ≥ 2. Of course, for d = 1, there are no linked Z_{p}-sets.

Lemma 3.4. For each n ≥ 1 and cyclic G-set T of size p^{n−1}, there is a linked pair of cyclic G-sets T_{1} and T_{2} covering T.

Proof. Since G is abelian, G-sets G/H for different (open) subgroups H are nonisomorphic. Since d ≥ 2, there is more than one subgroup of index p. That settles the case n = 1.

\begin{matrix}U \\ T_{1} \\ T_{2} \\ V_{1} \\ V_{2} \end{matrix}

Figure 2: Linked and non-linked Z_{2}^{2}-sets.

Now let n ≥ 2 and write T = G/H. There is a unique maximal chain of subgroups

H = K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ K_{n−1} = G

where [K_{i} : H] = p^{i} for all i. There is a Z_{p}-basis of Z_{p}^{d}, {e_{1}, …, e_{d}}, such that H = ⨁_{i=1}^{d−1} Z_{p}e_{i} ⊕ Z_{p}p^{n−1}e_{d}. Consider the subgroups H_{1} = ⨁_{i=1}^{d−1} Z_{p}e_{i} ⊕ Z_{p}p^{n}e_{d} and H_{2} = ⨁_{i=1}^{d−2} Z_{p}e_{i} ⊕ Z_{p}pe_{d−1} ⊕ Z_{p}(e_{d−1} + p^{n−1}e_{d}). Clearly H_{1} ⊂ H and H_{2} ⊂ H, and both G/H_{1} and G/H_{2} are cyclic. Since #G/H_{1} and #G/H_{2} are both p^{n}, G/H_{1} and G/H_{2} are covers of G/H. Set T_{1} = G/H_{1} and T_{2} = G/H_{2}. Since H_{1} ≠ H_{2}, T_{1} and T_{2} are nonisomorphic. To show their strict downsets agree, we use the cyclic condition. The G-sets U < T_{1} are G/K_{i} for i = 0, …, n − 1 since we the given maximal chain of subgroups is unique. The same argument applies to T_{2}, so the same G-sets lie strictly below T_{1} and T_{2} in the frame of G.

4 W_{Z_{p}^{d}}(k) is not Noetherian for d ≥ 2

When k is a perfect field of characteristic p and d ≥ 2, one might expect the rings W_{Z_{p}^{d}}(k) generalize the classical Witt vectors W_{Z_{p}}(k) in the same way that k[[X_{1}, …, X_{d}]] generalizes k[[X_{1}]]: the power series ring in d variables over a field is a complete local Noetherian domain with dimension d. It essentially follows from [DS88] that W_{Z_{p}^{d}}(k) is not a domain but is a local ring whether or not k is perfect. Since it is also complete in its profinite topology it is plausible to guess, by analogy to k[[X_{1}, …, X_{d}]], that the maximal ideal of W_{Z_{p}^{d}}(k) is generated by the Witt vectors ω_{T} (1) for #T = p, but we will see this is false in a very strong way: W_{Z_{p}^{d}}(k) is not Noetherian, whether or not k is perfect. This is because, as we will see, the square of the maximal ideal of W_{Z_{p}^{d}}(k) is much smaller than intuition suggests. To prove W_{Z_{p}^{d}}(k) is not Noetherian when d ≥ 2, note that Lemma 3.4 guarantees lots of linked paris of Z_{p}^{d}-sets and we next prove two lemmas that describe the behavior of sums and products on these linked pairs of coordinates.. The upshot, Theorem 4.4, is that Witt vectors in the square of the maximal ideal of W_{Z_{p}^{d}}(k) have built-in redundancies in linked coordinates which occur arbitrarily far out into the frame of Z_{p}^{d} when d ≥ 2. We continue to use the convention that G stands for Z_{p}^{d}.

Lemma 4.1. Let A be a ring. Given linked G-sets T and T′ and a finite collection of Witt vectors a_{1}, a_{2}, …, a_{m} ∈ W_{G}(A) such that (a_{i})_{T} = (a_{i})_{T′} for

1 ≤ i ≤ m, set s = ∑_{i=1}^{m} a_{i}. Then s_{T} = s_{T′}.

Proof. By induction it suffices to check the case m = 2. Since T and T′ are linked i.e., the G-sets strictly below T and T′ in F(G) are the same, #T = #T′ and the Witt polynomials W_{T} (\underline{X}) and W_{T′}(\underline{X}) are the same up to the roles of X_{T}, Y_{T} and X_{T′}, Y_{T′} in them. Therefore the two sum polynomials S_{T} and S_{T′} are the same up to the roles of X_{T}, Y_{T}, X_{T′}, and Y_{T′} in them (and we know how X_{T}, Y_{T}, X_{T′} and Y_{T′} appear in S_{T} and S_{T′} by Theorem 2.9). So when Witt vectors a_{1} and a_{2} have the same T and T′ coordinates, since T and T′ are linked we have S_{T} (a_{1}, a_{2}) = S_{T′}(a_{1}, a_{2}). Thus (a_{1} + a_{2})_{T} = (a_{1} + a_{2})_{T′}.

Unlike the previous lemma, the next one is specific to rings of characteristic p.

Lemma 4.2. Let A be a ring of characteristic p. Given a pair of linked G-sets T and T′ and elements a and b of W_{G}(A) such that a_{0} = b_{0} = 0, the product m = ab satisfies m_{T} = m_{T′}.

Proof. Let R = Z[\underline{X}, \underline{Y}] and define x and y in W_{G}(R) by x_{U} = X_{U} and y_{U} = Y_{U} for all U. Set z = xy. We will show

z_{T} − z_{T′} ≡ (x_{T} − x_{T′})y_{0}^{p^{n}} + (y_{T} − y_{T′})x_{0}^{p^{n}} mod pR.

Since A has characteristic p, it would then follow by functoriality that ab has equal T and T′ coordinates in W_{G}(A). Since the T -th Witt polynomial is a multiplicative function W_{T} : W_{G}(R) → R, W_{T} (z) = W_{T} (x)W_{T} (y):

∑_{U≤T} #Uz_{U}^{#T/#U} = ( ∑_{U≤T} #Ux_{U}^{#T/#U} ) ( ∑_{U≤T} #Uy_{U}^{#T/#U} ) .

Isolating the z_{T} term,

#T z_{T} = ( ∑_{U≤T} #Ux_{U}^{#T/#U} ) ( ∑_{U≤T} #Uy_{U}^{#T/#U} ) − ∑_{U<T} #Uz_{U}^{#T/#U}. (7)

Likewise looking at the T′ coordinate we have

#T′z_{T′} = ( ∑_{U≤T′} #Ux_{U}^{#T′/#U} ) ( ∑_{U≤T′} #Uy_{U}^{#T′/#U} ) − ∑_{U<T′} #Uz_{U}^{#T′/#U}.

(8)

Since T and T′ are linked, #T = #T′ and {U < T } = {U < T′} so the z-terms being subtracted on the right side of (7) and (8) are the same. Subtracting (8) from (7) and setting #T = #T′ = p^{n}, we have

p^{n}z_{T} − p^{n}z_{T′} = ( ∑_{U≤T} #UxU^{\frac{p^{n}}{#U}} ) ( ∑_{U≤T} #UyU^{\frac{p^{n}}{#U}} ) − ∑_{U<T} #UzU^{\frac{p^{n}}{#U}}

− ( ∑_{U≤T′} #UxU^{\frac{p^{n}}{#U}} ) ( ∑_{U≤T′} #UyU^{\frac{p^{n}}{#U}} ) + ∑_{U<T′} #UzU^{\frac{p^{n}}{#U}}

= p^{2n}x_{T} y_{T} + p^{n}x_{T} ∑_{U<T} #UyU^{\frac{p^{n}}{#U}} + p^{n}y_{T} ∑_{U<T} #UxU^{\frac{p^{n}}{#U}}

−p^{2n}x_{T′}y_{T′} − p^{n}x_{T′} ∑_{U<T′} #UyU^{\frac{p^{n}}{#U}} − p^{n}y_{T′} ∑_{U<T′} #UxU^{\frac{p^{n}}{#U}}

≡ p^{n}x_{T} y_{0}^{p^{n}} + p^{n}x_{0}^{p^{n}}y_{T} − p^{n}x_{T′}y_{0}^{p^{n}} − p^{n}x_{0}^{p^{n}}y_{T′} mod p^{n+1}R
≡ p^{n}(x_{T} y_{0}^{p^{n}} + x_{0}^{p^{n}}y_{T} − x_{T′}y_{0}^{p^{n}} − x_{0}^{p^{n}}y_{T′}) mod p^{n+1}R
≡ p^{n}((x_{T} − x_{T′})y_{0}^{p^{n}} + (y_{T} − y_{T′})x_{0}^{p^{n}}) mod p^{n+1}R.

So z_{T} − z_{T′} ≡ (x_{T} − x_{T′})y_{0}^{p^{n}} + (y_{T} − y_{T′})x_{0}^{p^{n}} mod pR.

Remark 4.3. The characteristic p hypothesis in Lemma 4.2 is necessary. Consider the case G = Z_{p}^{d} for d ≥ 2 and W_{G}(Z). Any pair of nonisomorphic G-sets of size p is linked. Given two vectors a, b ∈ W_{G}(Z) such that a_{0} = b_{0} = 0, set m = ab. For T ∈ F(G) such that #T = p, the formula for M_{T} in Example 2.4 implies m_{T} = pa_{T} b_{T}, which depends on T for suitable choices of a and b.

The point of these last two lemmas is that for linked G-sets T and T′, the T and T′ coordinates of a sum are the same if we make an assumption about the T and T′ coordinates of the summands, while the T and T′ coordinates of a product are the same if we make an assumption about the coordinates of the factors at the trivial G-set. Let k be a field of characteristic p. For n ≥ 0, recall the notation

I_{p^{n}} = I_{p^{n}}(G, k) = I_{p^{n}}(Z_{p}^{d}, k) = {a ∈ W_{Z_{p}^{d}}(k); a_{T} = 0 if #T < p^{n}}.

The unique maximal ideal of W_{G}(k) is 𝔪 = I_{p}(G, k).

Theorem 4.4. Let T and T′ be linked G-sets. Any element of 𝔪^{2} has equal T and T′ coordinates.

Proof. Any element of 𝔪^{2} is a_{1}b_{1} +⋯+a_{r}b_{r} for some a_{i} and b_{i} in 𝔪. Lemma 4.2 shows that a_{i}b_{i} has the same T and T′ coordinates for all i. Applying Lemma 4.1 shows that this equality of coordinates is preserved when passing to the sum of these products over all i.

Since Lemma 3.4 shows that there are linked G-sets of arbitrarily large size, Theorem 4.4 puts infinitely many constraints on elements of 𝔪^{2}. Theorem 4.4 was first observed in examples, where many redundancies were noticed in different coordinates of a product of elements of W_{Z_{2}^{2}}(F_{2}[\underline{X}, \underline{Y}]) that each have first coordinate 0. This is also how the importance of linked G-sets was discovered. We are now set to use them to prove our first main structural theorem.

Theorem 4.5. The maximal ideal of W_{G}(k) is not finitely generated, so W_{G}(k) is not Noetherian.

Proof. Using Lemma 3.4, for each n ≥ 1 there are linked G-sets T_{n} and T′_{n} in F(G) of size p^{n}. In particular, {U : U < T_{n}} = {U : U < T′_{n}}. The Witt vector ω_{T_{n}}(1) lies in 𝔪. For each r ≥ 1 we will show the r Witt vectors ω_{T_{1}}(1), …, ω_{T_{r}}(1) are linearly independent over k in 𝔪/𝔪^{2}. In 𝔪/𝔪^{2}, suppose we have a k-linear relation

α_{1}ω_{T_{1}}(1) + ⋯ + α_{r}ω_{T_{r}}(1) ≡ 0 mod 𝔪^{2},

for some α_{i} ∈ k. The product α_{i}ω_{T_{i}}(1) in 𝔪/𝔪^{2} really means, as a Witt vector, ω_{0}(α_{i})ω_{T_{i}}(1) mod 𝔪^{2}. By Theorem 2.8, ω_{0}(α_{i})ω_{T_{i}}(1) = ω_{T_{i}}(α_{i}^{p^{i}}). Therefore

ω_{T_{1}}(α_{1}^{p}) + ⋯ + ω_{T_{r}}(α_{r}^{p^{r}}) ≡ 0 mod 𝔪^{2}.  (9)

Since the supports of ω_{T_{i}}(α_{i}^{p^{i}}) for 1 ≤ i ≤ r are disjoint, by Theorem 2.6 these Witt vectors can be added coordinatewise: the left side of (9) is the Witt vector with T_{i}-coordinate α_{i}^{p^{i}} and other coordinates equal to 0. Since this sum is in 𝔪^{2}, its T_{i}- and T′_{i}-coordinates are the same by Theorem 4.4. The T′_{i}-coordinate is 0 for all i, so α_{i}^{p^{i}} = 0 for all i. Thus every α_{i} is 0 in k. Since we have found r linearly independent elements of 𝔪/𝔪^{2} for any r ≥ 1, its k-dimension is infinite. Therefore 𝔪 is not finitely generated.

Remark 4.6. For an infinite pro-p group G with arbitrarily large pairs of linked normal G-sets, the results of this section go through. For such pro-p groups, W_{G}(k) is not Noetherian when k is a field (or ring) of characteristic p.

5 Reducedness of W_{G}(k)

Although W_{Z_{p}^{d}}(k) with d > 2 is not a domain and is not Noetherian, this is not a pathological state of affairs among p-adic rings. For instance, the ring C(Z_{p}, Q_{p}) of continuous functions from Z_{p} to Q_{p} is not a domain and is not Noetherian. This ring is reduced. So we ask: is W_{Z_{p}^{d}}(k) reduced? We answer this in the case d = 2 and for any p. To do this, we keep track of a new partial ordering on F(G). This new partial ordering makes sense with no extra effort for G = Z_{p}^{d} with any d ≥ 1 so we state it in this generality. Consider the descending subgroup filtration

G ⫌ pG ⫌ p^{2}G ⫌ ⋯ ⫌ p^{n}G ⫌ p^{n+1}G ⫌ ⋯,

which leads to the rising family of G-sets

0 = G/G < G/pG < G/p^{2}G < ⋯ < G/p^{n}G < G/p^{n+1}G < ⋯

in F(G).

Definition 5.1. For G = Z_{p}^{d}, the level of T ∈ F(G) is the largest n ≥ 0 such that T ≥ G/p^{n}G. We write Lev(T) = n.

This definition makes sense, since if T ≥ G/p^{n}G then #T ≥ p^{nd}, so n is bounded above, and T ≥ 0 so we have somewhere to begin. To get a feel for this concept, we describe it on coset spaces. Write T ≅ G/H for a unique open subgroup H of G. We have T ≥ G/p^{n}G if and only if H ⊂ p^{n}G. Therefore Lev(G/H) is the largest n ≥ 0 such that H ⊂ p^{n}G. See Figure 3.

Level 0 Level 1 Level 2

Figure 3: Initial Z_{2}^{2}-sets of level 0, 1, 2.

If G = Z_{p} then Lev(T) and #T are basically the same concept, since #T = p^{Lev(T)}. The level is something genuinely new when G = Z_{p}^{d} for d ≥ 2. In this case neither #T nor Lev(T) determines the other; all we can say in general is that #T ≥ p^{dLev(T)}. Writing the cyclic decomposition of G/H as Z/p^{a_{1}}Z × Z/p^{a_{2}}Z × ⋯ × Z/p^{a_{d}}Z, its level is min{a_{1}, a_{2}, …, a_{d}}. This makes it easy to produce examples.

Example 5.2. For a ≥ 1, the G-set Z_{p}^{d}/(Z_{p} × p^{a}Z_{p}^{d−1}) of size p^{a(d−1)} has level 0 since Z_{p} × p^{a}Z_{p}^{d−1} is contained in G but not pG. So when d ≥ 2, arbitrarily large G-sets can have level 0 (but not when d = 1).

Example 5.3. There are arbitrarily large G-sets of any chosen level n when d ≥ 2: use Z_{p}^{d}/(p^{n}Z_{p} × p^{a+n}Z_{p}^{d−1}) with a → ∞.

Theorem 5.4. For d ≥ 2, each Z_{p}^{d}-set of level n is covered by more than one G-set of level n.

Proof. Let T be a G-set with Lev(T) = n. Pick a Z_{p}-basis {e_{1}, …, e_{d}} of Z_{p}^{d} so that T ≅ G/H with H = Z_{p}p^{a_{1}}e_{1} + … + Z_{p}p^{a_{d}}e_{d}, where n = a_{1} ≤ a_{2} ≤ … ≤ a_{d}. When d ≥ 3, ∑_{i=1}^{d−1} Z_{p}p^{a_{i}}e_{i} + Z_{p}p^{a_{d}+1}e_{d} and ∑_{i≠d−1} Z_{p}p^{a_{i}}e_{i} + Z_{p}p^{a_{d−1}+1}e_{d−1} are subgroups of H with index p and they are the stabilizers of two distinct G-sets covering T both with level n. If d = 2, on the other hand, Z_{p}p^{a_{1}}e_{1} + Z_{p}p^{a_{2}+1}e_{2} and Z_{p}(p^{a_{1}+1}e_{1} + p^{a_{1}}e_{2}) + Z_{p}p^{a_{2}}e_{2} are subgroups of H with index p and they are the stabilizers of two distinct Z_{p}^{2}-sets covering T both with level n.

For any T ∈ F(G), {U : #U ≤ #T } and {U : U ≤ T } are finite (the latter is a subset of the former), but {U : Lev(U) ≤ Lev(T)} is infinite. This is an important distinction to remember.

Remark 5.5. When d ≥ 2, any nontrivial cyclic G-set looks like Z_{p}^{d}/(Z_{p}^{d−1} × p^{a}Z_{p}) with a ≥ 1 after a suitable choice of basis for Z_{p}^{d}, and Z_{p}^{d−1} × p^{a}Z_{p} is not contained in pZ_{p}^{d} (this is false for d = 1), so all cyclic G-sets have level 0. When d = 2, a G-set of level 0 is isomorphic to Z_{p}^{2}/(Z_{p} × p^{a}Z_{p}), so having level 0 and cyclic in F(Z_{p}^{2}) mean the same thing. For d ≥ 3, some G-sets of level 0 are not cyclic, such as Z_{p}^{d}/(Z_{p} × pZ_{p} × p^{a}Z_{p}^{d−2}) with a ≥ 1.

To prove that W_{Z_{p}^{2}}(k) is reduced when k has characteristic p, we seek the right coordinate to look at in a power of a nonzero Witt vector to know that the power is also not 0. Say x ∈ W_{G}(k) and x ≠ 0. It is natural to consider how x sits in the descending ideal filtration {I_{p^{n}}}: there is some I_{p^{n}} for which x ∈ I_{p^{n}} and n is as large as possible, so x_{T} = 0 for #T < p^{n} and some x_{T} is nonzero where #T = p^{n}. In this case it is not hard to check that x^{2} ∈ I_{p^{2n}} (Corollary 2.15), and we can anticipate (if W_{G}(k) is reduced) that x^{2} has a nonzero coordinate at some G-set of size p^{2n}. Which one? We need a way to predict a nonzero coordinate in x^{2} when x ≠ 0.

Lemma 5.6. Let G = Z_{p}^{d}. If U ≤ T in F(G) then Lev(U) ≤ Lev(T).

Proof. Let n = Lev(U), so U ≥ G/p^{n}G. Since T ≥ U, we have T ≥ G/p^{n}G, so Lev(T) ≥ n.

Note that if U < T and d ≥ 2 it need not follow that Lev(U) < Lev(T): we might have Lev(U) = Lev(T). For example, if U = G/H and T = G/K where H = pZ_{p}^{d} = pZ_{p}^{d−1} × pZ_{p} and K = pZ_{p}^{d−1} × p^{2}Z_{p} then K ⊂ H so U < T and Lev(U) = Lev(T) = 1.

Lemma 5.7. Let G = Z_{p}^{d} and T ∈ F(G) have level n. Write T ≅ G/H with H ⊂ p^{n}Z_{p}^{d} and H ⊄ p^{n+1}Z_{p}^{d} so H = p^{n}Ĥ for a unique open subgroup Ĥ of Z_{p}^{d}. Set T̂ = G/Ĥ. Then T̂ has level 0. Moreover, if U ∈ F(G) and Lev(U) = n then U ≤ T if and only if Û ≤ T̂.

Proof. By choosing a suitable basis of Z_{p}^{d}, we may assume H = p^{a_{1}}Z_{p} × … × p^{a_{d}}Z_{p}. The statement that T has level n means n = min{a_{1}, …, a_{d}}, so H = p^{n}(p^{b_{1}}Z_{p} × … × p^{b_{d}}Z_{p}) with some b_{i} = 0. Then Ĥ = p^{b_{1}}Z_{p} × … × p^{b_{d}}Z_{p} and T̂ has level 0. To prove the second claim, write U ≅ G/K and K = p^{n}K̂. We have U ≤ T if and only if H ⊂ K and Û ≤ T̂ if and only if Ĥ ⊂ K̂. The conditions H ⊂ K and Ĥ ⊂ K̂ are the same.

Lemma 5.8. For G = Z_{p}^{d} with d ≥ 2 and T ∈ F(G), set #T = p^{n}. For each m ≥ n, there exist T′ such that #T′ = p^{m}, T ≤ T′ and Lev(T) = Lev(T′).

Proof. Write T = G/H. Choose a Z_{p}-basis {e_{1}, e_{2}, …, e_{d}} of Z_{p}^{d} such that H = ∑_{i=1}^{d} Z_{p}p^{a_{i}}e_{i}. Without loss of generality, assume a_{1} ≤ a_{2} ≤ … ≤ a_{d} and set K = ∑_{i=1}^{d−1} Z_{p}p^{a_{i}}e_{i}+Z_{p}p^{a_{d}+m−n}e_{d}. Then K ⊂ H and [H : K] = p^{m−n} so G/K is a cover of G/H. Since a_{1} ≤ a_{2} ≤ … ≤ a_{d} and d ≥ 2, min{a_{1}, a_{2}, …, a_{d}} = min{a_{1}, a_{2}, …, a_{d} + m − n}, so Lev(G/K) = Lev(G/H).

Consider Figure 3. The two horizontal lines divide the diagram into regions of Z_{2}^{2}-sets with the same level. Lemma 5.8 just says that these levels go infinitely far out in the frame. Returning to the assumption G = Z_{p}^{2} we have the following lemma.

Lemma 5.9. When G = Z_{p}^{2} and T and T′ in F(G) satisfy T ≤ T′ and Lev(T) = Lev(T′),

{U ∈ F(G) : U ≤ T′ and Lev(U) = Lev(T) and #U = #T } = {T }.

Proof. The set of G-sets of level zero is a tree (see Remark 2.1), so the property is obviously true when Lev(T) = Lev(T′) = 0. Using Lemma 5.7 one has the result for any level.

Lemma 5.10. Let G = Z_{p}^{2}, A be a nonzero commutative ring, and choose any nonzero a ∈ W_{G}(A). There is T_{0} ∈ F(G) such that a_{T_{0}} ≠ 0 and a_{U} = 0 under either of the following conditions:

• Lev(U) < Lev(T_{0}),

• Lev(U) = Lev(T_{0}) and #U < #T_{0},

Proof. Given a ≠ 0 in W_{G}(A), among {T : a_{T} ≠ 0} first select all T with minimal level, and then among the T of that minimal level, choose one T of minimal size. Call that T_{0}. If Lev(U) < Lev(T_{0}) then a_{U} = 0 since T_{0} is a nonzero coordinate with minimal level. If Lev(U) = Lev(T_{0}) and #U < #T_{0} then a_{U} = 0 since otherwise T_{0} is not a nonzero coordinate of minimal size among nonzero coordinates of minimal level.

Remark 5.11. Lemma 5.10 is expressed in a form convenient for the applications we have in mind, but it’s not a result about nonzero Witt vectors so much as a property of F(G): for any nonempty subset S of F(G), there is a T_{0} ∈ S such that U ∉ S if Lev(U) < Lev(T_{0}), or if Lev(U) = Lev(T_{0}) and #U < #T_{0}.

Remark 5.12. In our application of Lemma 5.10 it is important to note the order in which the concepts are minimized. Here we are first choosing nonzero G-sets of minimal level, then among those we choose one of minimal size. These two minimizations do not commute. Figure 4 depicts a nonzero element of W_{Z_{2}^{2}}(k), where all coordinates are zero except for the circled ones which are non-zero. The T coordinate is the one of minimal size first and then level, whereas the U coordinate is the one of minimal level first and then size.

Level 0 Level 1 T U

Figure 4: Nonzero element in W_{Z_{2}^{2}}

Now we will use the concept of level to prove something about multiplication in W_{G}(k). The end of the next lemma identifies a formula for a specific coordinate in the product of two Witt vectors if all the “smaller” coordinates are 0. It is analogous to something simple when G = Z_{p}: if a = p^{n}(a_{0} + pa′) and b = p^{n}(b_{0} + pb′) then ab = p^{2n}(a_{0}b_{0} + pc). (It is not assumed that a_{0} and b_{0} are nonzero.) There is a similar formula even if the p-powers in a and b are not equal, but for G = Z_{p}^{d} with d ≥ 2 we don’t have a formula that broad. It seems to be the price we pay for W_{G}(k) not being a domain.

Lemma 5.13. Let G be an abelian pro-p group and A be a ring of characteristic p. Let V ∈ F(G) with #V ≤ p^{m+n} for positive integers m and n. Consider Witt vectors a and b in W_{G}(A) such that a_{U} = 0 for all U < V such that #U < p^{m} and b_{U} = 0 for all U < V such that #U < p^{n}. Set c = ab. Then c_{V} = 0.

Proof. This will follow from functoriality by proving the following mod p congruence for particular Witt vectors over the ring R = Z[\underline{X}, \underline{Y}]. Define x, y ∈ W_{G}(R) by

x_{T} = \begin{cases}0, & #T < p^{m} and T < V, \\ X_{T} , & otherwise, \end{cases} and y_{T} = \begin{cases}0, & #T < p^{n} and T < V, \\ Y_{T} , & otherwise. \end{cases}

Set z = xy. We will show that

T ≤ V ⟹ z_{T} ≡ 0 mod pR.  (10)

Returning to the proof of (10), we argue by induction on #T. If #T = 1 then T = 0 and z_{0} = x_{0}y_{0} = 0. Let p^{r} ≤ p^{m+n} with r ≥ 1 and assume by induction that for all U < V such that #U < p^{r}, z_{U} ≡ 0 mod pR. Pick T ≤ V with #T = p^{r}. Since the T -th Witt polynomial is a multiplicative function W_{T} : W_{G}(R) → R, W_{T} (z) = W_{T} (x)W_{T} (y):

∑_{U≤T} φ_{T} (U)z_{U}^{#T/#U} = ∑_{T_{1},T_{2}≤T} φ_{T} (T_{1})φ_{T} (T_{2})x_{T_{1}}^{#T/#T_{1}}y_{T_{2}}^{#T/#T_{2}}.

Solving this equation for z_{T} in Q[\underline{X}, \underline{Y}],

z_{T} = ∑_{T_{1},T_{2}≤T} \frac{φ_{T} (T_{1})φ_{T} (T_{2})}{φ_{T} (T)}x_{T_{1}}^{#T/#T_{1}}y_{T_{2}}^{#T/#T_{2}} − ∑_{U<T} \frac{φ_{T} (U)}{φ_{T} (T)}z_{U}^{#T/#U}. (11)

Since z_{U} ∈ pR for U < T, the second term in (11) is 0 mod pR by Lemma 2.11. In the first term in (11), if T_{1} = V then T_{1} = T = V and φ_{T} (T_{1})φ_{T} (T_{2})/φ_{T} (T) ≡ φ_{T} (T_{2}) ≡ 0 mod pR by Lemma 2.10 provided T_{2} ≠ 0, while if T_{2} = 0 then y_{T_{2}} = 0. A similar argument holds if T_{2} = V so we can assume T_{1} < V and T_{2} < V. If either #T_{1} < p^{m} or #T_{2} < p^{n} then x_{T_{1}} = 0 or y_{T_{2}} = 0 respectively. The remaining terms in the first sum in (11) have T_{1} < V with #T_{1} ≥ p^{m} and T_{2} < V with #T_{2} ≥ p^{n}. In this case #T_{1}#T_{2} ≥ p^{m+n} > p^{r} = #T. Since G is abelian, φ_{T} (U) = #U and so the coefficient in the first sum in (11) is an integral multiple of p. Thus z_{T} ≡ 0 mod pR.

Remark 5.14. The only place where G being abelian played a role in the proof of Lemma 5.13 was at the last step calculating the p-divisibility of the coefficients in the first sum in (11). For general pro-p groups, this ratio need not even be integral, however the conclusion of Lemma 5.13 holds true not just for abelian pro-p groups, but any pro-p which satisfies

\frac{φ_{T} (T_{1})φ_{T} (T_{2})}{φ_{T} (T)} ∈ pZ

for any T, T_{1}, T_{2} ∈ F(G).

Lemma 5.15. For G = Z_{p}^{2} and k a field of characteristic p. Let a and b be in W_{G}(k) such that there is a T_{0} ∈ F(G) with a_{U} = b_{U} = 0 if Lev(U) < Lev(T_{0}), or if Lev(U) = Lev(T_{0}) and #U < #T_{0}. Set #T_{0} = p^{n}. For any T ∈ F(G) with size p^{2n} such that T_{0} ≤ T and Lev(T_{0}) = Lev(T), the product ab has T -coordinate (a_{T_{0}}b_{T_{0}})^{p^{n}}.

Proof. Let R = Z[\underline{X}, \underline{Y}]. Define x, y in W_{G}(R) by

x_{U} = \begin{cases}0 & if Lev(U) < Lev(T_{0}), \\ 0 & if Lev(U) = Lev(T_{0}) and #U < p^{n}, \\ X_{U} & otherwise, \end{cases}

and

y_{U} = \begin{cases}0 & if Lev(U) < Lev(T_{0}), \\ 0 & if Lev(U) = Lev(T_{0}) and #U < p^{n}, \\ Y_{U} & otherwise. \end{cases}

and set z = xy. For any G-set T of size p^{2n} with T_{0} ≤ T and Lev(T_{0}) = Lev(T) we will show z_{T} ≡ (X_{T_{0}}Y_{T_{0}})^{p^{n}} mod pR. By functoriality the lemma would then follow. For any G-set T, W_{T} (z) = W_{T} (x)W_{T} (y):

∑_{U≤T} #Uz_{U}^{#T/#U} = ( ∑_{U≤T} #Ux_{U}^{#T/#U} ) ( ∑_{U≤T} #Uy_{U}^{#T/#U} ) . (12)

By Lemma 5.8 we can choose T ≥ T_{0} such that #T = p^{2n} and Lev(T) = Lev(T_{0}). First we look at the left side of (12). Let V < T, so #V < p^{2n}. By Lemma 5.6 Lev(V) ≤ Lev(T) = Lev(T_{0}). So either Lev(V) < Lev(T) or Lev(V) = Lev(T). For all U < V with #U < p^{n}, both x_{U} and y_{U} are zero by hypothesis. The proof of Lemma 5.13 tells us that z_{V} ≡ 0 mod pR when V < T and #V < p^{2n}. So #V z_{V}^{#T/#V} ≡ 0 mod p^{2n+1}R for V < T and #V < p^{2n}, so the left side of (12) is p^{2n}z_{T} mod p^{2n+1}R. Now we turn to the right side of (12). If U ≤ T then Lev(U) ≤ Lev(T) (Lemma 5.6) and since Lev(T) = Lev(T_{0}) our hypotheses tells us x_{U} = 0 and y_{U} = 0 if Lev(U) < Lev(T), or if Lev(U) = Lev(T) and #U < p^{n}. So the only Uterms in each sum on the right side of (12) that are not automatically 0 have Lev(U) = Lev(T) and #U ≥ p^{n}.

Dropping the terms in (12) where x_{U} = 0 and y_{U} = 0 and reducing modulo p^{2n+1}R, one has that p^{2n}z_{T} is equivalent to

( ∑_{#U≥p^{n}^{{U≤T \atop Lev(U)=Lev(T)}}} #UX_{U}^{#T/#U} ) ( ∑_{#U≥p^{n}^{{U≤T \atop Lev(U)=Lev(T)}}} #UY_{U}^{#T/#U} ) mod p^{2n+1}R.

Each summand has a coefficient divisible at least by p^{n}, so any product of a term from each sum is divisible by p^{2n}. A term divisible by p^{n+1} in one sum has a product with any term in the other sum that is 0 mod p^{2n+1}R, so

p^{2n}z_{T} ≡ ( ∑_{#U=p^{n}^{{U≤T \atop Lev(U)=Lev(T)}}} p^{n}X_{U}^{p^{n}} ) ( ∑_{#U=p^{n}^{{U≤T \atop Lev(U)=Lev(T)}}} p^{n}Y_{U}^{p^{n}} ) mod p^{2n+1}R.

Dividing through by p^{2n} and using additivity of the pth power map in R/pR,

z_{T} ≡ ( ∑_{#U=p^{n}^{{U≤T \atop Lev(U)=Lev(T)}}} X_{U} · ∑_{#U=p^{n}^{{U≤T \atop Lev(U)=Lev(T)}}} Y_{U} )^{p^{n}} mod pR.  (13)

What are the U lying below T in F(G) with the same level as T and of size p^{n}? One example is U = T_{0}. By Lemma 5.9 this is the only example, so z_{T} ≡ (X_{T_{0}}Y_{T_{0}})^{p^{n}} mod pR. The argument did not depend on the choice of T and so holds for any T with T ≥ T_{0}, #T = p^{2n}, and Lev(T) = Lev(T_{0}).

We will only apply Lemma 5.15 when the two Witt vectors are equal, i.e., to the square of a nonzero element of W_{G}(k).

Remark 5.16. Most of the proof of Lemma 5.15 goes through for G = Z_{p}^{d} for d ≥ 3 and not just d = 2. In fact (13) is true for G = Z_{p}^{d} for d ≥ 2 and it was only at the last step where we used the fact that d = 2, which hinged on Lemma 5.9. Lemma 5.9 is not true for G = Z_{p}^{d} with d ≥ 3 since the level 0 part of F(Z_{p}^{d}) is not a tree. This is well-known after one realizes the level 0 part of F(Z_{p}^{d}) is in order bijection with the Z_{p}-lattices in Q_{p}^{d} up to scaling (see Remark 2.1), which forms the Bruhat-Tits building for SL_{d}(Q_{p}) which is not a tree for d ≥ 3 (see [Bro08, pg. 137].) More concretely, when d ≥ 3, Z_{p}^{d}/(Z_{p} × pZ_{p} × p^{a}Z_{p}^{p−2}) with a ≥ 1 is not a cyclic group and so its strict downset is not a chain.

Theorem 5.17. For any field k of characteristic p, the ring W_{G}(k) is reduced for G = Z_{p}^{2}.

Proof. Let v be nonzero in W_{G}(k). If v_{0} ≠ 0 then v ∈ W_{G}(k)^{×} by Theorem 2.16. Otherwise v ∈ 𝔪 and it suffices to show for all n ∈ Z^{+} that v^{2^{n}} is nonzero. Thus it suffices to show if v ∈ 𝔪 and v ≠ 0 then v^{2} ≠ 0. By Lemma 5.10 there is a T_{0} ∈ F(G) such that v_{T_{0}} is nonzero but v_{U} is zero for all U ∈ F(G) where Lev(U) < Lev(T_{0}) or where Lev(U) = Lev(T_{0}) and #U < #T_{0}. Lemma 5.15 with a = b = v tells us that v^{2} has a coordinate equal to v_{T_{0}}^{2#T_{0}} ≠ 0, so v^{2} ≠ 0.

Acknowledgments

The author is forever indebted to Keith Conrad and would also like to thank Sarah Glaz, Álvaro Lozano-Robledo, Jeremy Teitelbaum, Anurag Singh, Jesse Elliott, and Karl Schwede for their readings and enlightening discussions as well as the reviewer and to Lars Hesselholt for his guidance.

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Lance Edward Miller University of Arkansas Department of Mathematical Sciences 301 SCEN Fayetteville, AR 72701. lmiller@math.utah.edu