A result on formal linearity

A.J. de Jong*

1 Statement of the result

Let Λ be a complete discrete valuation ring of mixed characteristic (0, p). Let τ : Λ → Λ be an automorphism of Λ. Consider the formal multiplicative group 𝔾̂_{m} over Spf(Λ). Put X = 𝔾̂_{m}^{n} with coordinates q_{1}, …, q_{n}. Let Φ : X → X be the τ–linear morphism such that Φ^{∗}(q_{i}) = (q_{i})^{q}, where q = p^{t} is some positive power of p.

Consider a closed formal subscheme Z ⊂ X. We have Z = Spf(A) for some complete local ring A. We define Φ(Z) ⊂ X to be the smallest closed subscheme such that the morphism Z → X⟶^{Φ}X factors through Φ(Z). Similarly, we define Φ^{−1}(Z) to be the largest closed subscheme such that Φ^{−1}(Z) → X⟶^{Φ}X factors through Z. If Φ(Z) = Spf(B) and Φ^{−1}(Z) = Spf(C), then A → C is finite flat and B → A is finite and injective.

1.1 Definition. We say that Z is a formal subtorus of X or that Z is linear in X if Z is a closed formal subgroup scheme of X isomorphic to 𝔾̂_{m}^{r} for some r ≥ 0. We say that Z is quasi–linear in X if there exists a finite extension Λ ⊂ Λ′ of discrete valuation rings, torsion points t_{1}, …, t_{m} ∈ X(Λ′) and formal subtori T_{1}, …, T_{m} ⊂ X such that

Z_{Λ′} = Z ×_{Spf(Λ)} Spf(Λ′) = ⋃_{i=1}^{m} t_{i} · T_{i} ⊂ X_{Λ′}

as closed formal subschemes of X_{Λ′}.

The object of this note is to prove a characterization of quasi–linear formal subschemes in terms of Φ, see below. This extends the characterization of toroidal formal groups given in the appendix by Katz to [DI]. This characterization may be used to study the reduction of Shimura varieties at ordinary points; see forthcoming work of B.J.J. Moonen [Mo].

1.2 Proposition. Suppose that Z is equidimensional, reduced and flat over Spf(Λ) (this means that A is an equidimensional, flat Λ–algebra without nilpotents).

(1) The following assertions are equivalent (a) the closed formal subscheme Z is quasi–linear X,

(b) for some a > b we have Φ^{a}(Z) ⊂ Φ^{b}(Z).

(2) The following are equivalent (a) Z is a formal subtorus of X,

(b) Z is irreducible and Φ^{a}(Z) ⊂ Z for some a ≥ 1.

Proof. The implication (a) ⇒ (b) is easy in both cases, see also Section 2. Part (2) is restated and proved in Section 2. Remark that if Z ⊂ X is quasi–linear, then so is Φ^{−1}(Z). By our assumption on dimensions, we have that Z is a union of components of

* The research of Dr. A.J. de Jong has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.

Φ^{−b}(Φ^{b}(Z)). Thus to prove (b) ⇒ (a) in the first part we may assume b = 0. Decomposing Z into irreducible compontents Z_{i} we see that a number of Z_{j}’s will satisfy (b) of (2) (with possibly a replaced by a multiple of a) and the others will be mapped into these by some power of Φ. Again the dimension assumption gives the result.

2 Proof of the result

We consider the complete local ring

R = Λ[[x_{1}, …, x_{n}]]

and the elements q_{i} = 1 + x_{i} ∈ R^{∗}. For a fixed integer t ≥ 1, let σ : R → R be the endomorphism given by λ ↦ τ(λ), λ ∈ Λ and q_{i} ↦ (q_{i})^{p^{t}}, i = 1, …, n.

2.1 Lemma. Let 𝔭 ⊂ R be a prime ideal, p ∉ 𝔭 and σ(𝔭) ⊂ 𝔭.

(1) 𝔭 ⊂ (x_{1}, …, x_{n}).

(2) Let K = Q(A) be the fraction field of A := R/𝔭 and let σ : K → K be induced by σ, then

K^{σ} = Q(Λ)^{τ} ,

where Q(Λ) denotes the fraction field of Λ.

Proof. (1) If 𝔭 ⊄ (x_{1}, …, x_{n}) then for some k we have p^{k} + ∑ x_{i}a_{i} ∈ 𝔭 . Since σ(𝔭) ⊂ 𝔭 we get for all N

p^{k} + ∑ σ^{N}(x_{i})σ^{N}(a_{i}) ∈ 𝔭.

Taking the limit for N → ∞ we get p^{k} ∈ 𝔭. Contradiction.

(2) Suppose dim(A) = d + 1. If d = 0 then by (1) we get 𝔭 = (x_{1}, …, x_{n}) and the result follows. If d ≥ 1 we can find a subset I ⊂ {1, …, n}, #I = d such that

Ω_{A} ⊗_{A} K ≅ ⨁_{i∈I} Kdx_{i}.

This can be seen by a consideration using the module of continuous differentials Ω_{A} of A over Λ as follows. This module has rank d over A. If for a subset I ⊂ {1, …, n} as above we have a powerseries f = f(x_{i}, i ∈ I) which maps to zero in A, then we have a nontrivial linear relation ∑_{i∈I} a_{i}dx_{i} = 0 in Ω_{A}. Hence such a relation cannot hold for all these subsets.

Suppose that f ∈ K^{σ}. We have

df = ∑_{i∈I} f_{i}dq_{i}/q_{i},

and f_{i} ∈ K satisfies pσ(f_{i}) = f_{i}. But then it follows that p^{n}σ^{n}(f_{i}) = f_{i}. Let A′ be the normalization of A, and let ℘_{1}, …, ℘_{r} be the height 1 prime ideals of A′ with p ∈ ℘_{i}. We have that σ permutes these prime ideals, as σ defines a finite morphism SpecA′ → SpecA′.

Thus some power σ^{n} of σ fixes the ℘_{j}. The formula p^{n}σ^{n}(f_{i}) = f_{i} implies that the valuation of f_{i} in the discrete valuation ring A′_{℘_{j}} is infinite. Hence f_{i} = 0, i.e., df = 0. From this we will deduce that f ∈ Q(Λ)^{τ}.

More generally, consider a domain A = Λ[[x_{1}, …, x_{n}]]/℘ satisfying (1) of the lemma and f ∈ Q(A), with df = 0 in Ω_{A} ⊗ Q(A). We claim that f ∈ Q(Λ) in this more general situation (dropping the requirement of the existence of σ). We choose a finite injective morphism Λ[[x_{1}, …, x_{d}]] → A. Let f^{m} +a_{1}f^{m−1} +…+a_{m} = 0 be the minimal equation of f over L = Q(Λ[[x_{1}, …, x_{d}]]). We get 0 = f^{m−1}da_{1} +…+da_{m}. But the f^{i}, 0 ≤ i ≤ m−1 are linearly independent over L, hence we get da_{i} = 0. This gives a_{i} ∈ Q(Λ), by explicit computation in the case of A = Λ[[x_{1}, …, x_{d}]]. Multiplying f by a suitable element of Λ, we may assume that a_{i} ∈ Λ for all i. The map ψ : A → Λ given by (1) of the lemma implies that the polynomial X^{m} + a_{1}X^{m−1} + … + a_{m} has the solution ψ(f) in Λ. Since it is irreducible, it follows that it is linear.

For \overline{m} = (m_{1}, …, m_{n}) ∈ ℤ_{p}^{n} we put

q\overline{m} = ∏_{i=1}^{n} q_{i}^{m_{i}} = ∏_{i=1}^{n}(1 + x_{i})^{m_{i}} ∈ R^{∗} ,

where

(1 + x)^{α} = ∑_{k≥0} \binom{α}{k}x^{k} .

Note that \binom{α}{k} ∈ ℤ_{p} for α ∈ ℤ_{p}, and that q\overline{m} ·q\overline{m}′ = q_{\overline{m}+\overline{m}′} and σ(q\overline{m}) = (q\overline{m})^{p^{t}}. If M ⊂ ℤ_{p}^{n} is a primitive sublattice, put

𝔭_{M} = ⟨q\overline{m} − 1, \overline{m} ∈ M⟩ ⊂ R .

This is a prime ideal, the quotient R/𝔭_{M} is formally smooth over Λ (since M primitive) and σ(𝔭_{M}) ⊂ 𝔭_{M}.

2.2 Proposition. Let 𝔭 ⊂ R be a prime ideal, p ∉ 𝔭 and σ(𝔭) ⊂ 𝔭. Then 𝔭 = 𝔭_{M} for some M ⊂ ℤ_{p}^{n} primitive.

Proof. Write A = R/𝔭 and let dim(A) = d + 1. The lemma shows that the proposition is true for d = 0, and that for d ≥ 1 we can renumber x_{1}, …, x_{n} in such a way that Ω_{A} ⊗ Q(A) = ⨁_{i=1}^{d} Q(A)dx_{i}. Suppose we can show that the (non–trivial) ideals

𝔮_{i} = 𝔭 ∩ Λ[[x_{1}, …, x_{d}, x_{i}]] i = d + 1, …, n

satisfy the conclusion of the proposition. The result will follow for 𝔭. (Using that 𝔭_{M} is prime if and only if M is primitive.)

Thus we may assume that 𝔭 = (f) is a principal ideal. We get σ(f) = g · f for some g ∈ R. We note that f does not divide g, since the morphism σ : R → R is étale over Spec(R[1/p]). We have dσ(f) = fdg + gdf in Ω_{R}, which we expand to get

(∗)  ∑_{i=1}^{n} p^{t}q_{i}^{p^{t}−1}σ (\frac{∂f}{∂q_{i}} ) dq_{i} = fdg + ∑_{i=1}^{n} g \frac{∂f}{∂q_{i}} dq_{i} .

Remark that at least one of the elements ∂f/∂q_{i} maps to a nonzero elelement of A: if not the rank of the module Ω_{A} is too large. Say ∂f/∂q_{1} has this property. The elements

image of q_{i} \frac{∂f}{∂q_{i}} /q_{1} \frac{∂f}{∂q_{1}} in K

lie in the fixed field K^{σ} = Q(Λ)^{τ}. Here we use that g is not mapped to zero in A. We define

N = \{(λ_{1}, …, λ_{n}) ∈ Λ^{n} | ∑_{i=1}^{n} λ_{i}q_{i} \frac{∂f}{∂q_{i}} = 0 in A\} .

This is a primitive sublattice of Λ^{n} which has rank n − 1 by what has gone before. We take a Λ–basis \overline{λ}_{1}, …, \overline{λ}_{n−1}, with \overline{λ}_{i} = (λ_{i1}, …, λ_{in}), of N. Put

ϑ_{i} = ∑_{j=1}^{n} λ_{ij}q_{j} \frac{∂}{∂q_{j}} ∈ Der(R) .

By construction f = 0 is an integral manifold for the vectorfields ϑ_{i}. This means that they induce maps ¬ ϑ_{i} : Ω_{A} → A. The composition

⨁_{i=1}^{n} Adq_{i} ⟶ Ω_{A} ⟶ ⨁_{i=1}^{n−1} A

is given by the matrix (λ_{ij}) and is therefore surjective. We conclude that Ω_{A} ≅ A^{n−1}⊕A/I for some ideal I ⊂ A, I ≠ (0). The kernel of ⊕Adq_{i} → Ω_{A} is given by Adf, hence I is principal.

Let (μ_{1}, …, μ_{n}) ∈ Λ^{n} be a primitive element such that the relation ∑ λ_{j}μ_{j} = 0 holds for all (λ_{1}, …, λ_{n}) ∈ N. Consider the homomorphism R → Q(Λ)[[τ_{1}, …, τ_{n}]] given by q_{i} ↦ exp(τ_{i}). Our f maps to a nonunit f ∈ Q(Λ)[[τ_{1}, …, τ_{n}]], since f ∈ (x_{1}, …, x_{n}) by the lemma. The vectorfields ϑ_{i} extend to the vectorfields ϑ_{i} = ∑_{j=1}^{n} λ_{ij} \frac{∂}{∂τ_{j}} on Q(Λ)[[τ_{1}, …, τ_{n}]], and the equation f = 0 defines an integral manifold for them. This implies that in Q(Λ)[[τ_{1}, …, τ_{n}]] we can write f as

f = ε(τ) · (μ_{1}τ_{1} + … + μ_{n}τ_{n})^{r}

for some unit ε(τ) of the ring Q(Λ)[[τ_{1}, …, τ_{n}]] and some r ≥ 1. Note that the local ring Q(Λ)[[τ_{1}, …, τ_{n}]]/(f) is the completion of the localization of A at the prime ideal (x_{1}, …, x_{n})A. As A is reduced, we get r = 1.

Let us look at the morphism

Spf(R ˆ⊗_{Λ}R) = Spf(R) ×_{Spf(Λ)} Spf(R) ⟶ Spf(R)

given by q_{i} ↦ q_{i} ⊗ q_{i}. This makes Spf(R) into a formal group over Spf(Λ). The formula above and the fact that r = 1 show that the completion of the local ring of A ˆ⊗_{Λ}A at the prime ideal (x_{1} ⊗ 1, …, x_{n} ⊗ 1, 1 ⊗ x_{1}, …, 1 ⊗ x_{n})A ˆ⊗A maps into Spec(A) ⊂ Spec(R) under the morphism Spec(R ˆ⊗_{Λ}R) → Spec(R).

2.3 Lemma. Spec(A ˆ⊗_{Λ}A) is irreducible and reduced.

From this and the above it follows that Spf(A) ⊂ Spf(R) is a formal closed subgroup scheme of Spf(R). Since Spf(R) ≅ 𝔾̂_{m}^{n}, and by the classification of closed subgroups of 𝔾̂_{m}^{n} we get Spf(A) ≅ 𝔾̂_{m}^{d} and everything (but the lemma) is proven.

Proof of 2.3. Recall that we defined an ideal I ⊂ A giving the singular locus of A. Remark that p^{t}σ(I) ⊂ I by formula (∗). This gives σ(I′) ⊂ I′, where I′ = {a ∈ A | p^{N}a ∈ I some N}. Arguing as in the proof of (1) of the first lemma, we get that I′ ≠ A implies I′ ⊂ (x_{1}, …, x_{n})A. This would imply that I ⊂ (x_{1}, …, x_{n})A, which is a contradiction with the fact that the local ring of A at (x_{1}, …, x_{n})A is regular (by the result r = 1 above). Thus we have I′ = A, and p^{N} ∈ I for some N. It follows that A[1/p] is a regular ring.

We consider the morphism

p_{1} : Spec(A ˆ⊗_{Λ}A[1/p]) ⟶ Spec(A[1/p])

given by a ↦ a ⊗ 1. A closed point of Spec(A ˆ⊗_{Λ}A[1/p]) has a residue field which is a finite extension of Q(Λ), hence it maps to a closed point of Spec(A[1/p]). If A[1/p] → Q′ is a Q(Λ)-homomorphism onto a finite extension of Q(Λ), then the fibre of p_{1} over this (closed) point is

Spec(Q′ ⊗_{Q(Λ)} A[1/p]) .

This is a regular irreducible scheme: the scheme Spec(A[1/p]) is regular irreducible and has a Q(Λ)-rational point. Thus all fibres of p_{1} over closed points are irreducible and meet the irreducible closed subscheme Δ(Spec(A[1/p])). The lemma follows if we show that Spec(A ˆ⊗_{Λ}A[1/p]) is a regular scheme. Let 𝔪 be a maximal ideal of A ˆ⊗_{Λ}A[1/p]. By the above the ideals 𝔪_{i} ⊂ A[1/p] corresponding to the projections of the point 𝔪 are maximal. The completion of A ˆ⊗_{Λ}A[1/p] in the maximal ideal 𝔪 is a factor of the semi-local ring

A[1/p]_{𝔪_{1}}^{∧} ˆ⊗_{Q(Λ)}A[1/p]_{𝔪_{2}}^{∧} .

The local rings A[1/p]_{𝔪_{i}}^{∧} are isomorphic to powerseries rings over finite extensions of Q(Λ) by what was said before. We have won.

Acknowledgements. I would like to thank B.J.J. Moonen for several discussions in which the question studied in this report turned up. I thank Prof. T. Zink for pointing out a mistake in the original proof of Lemma 2.1. Thanks also to G. Weckermann for typing quicker than I could have done.

References

[DI] P. Deligne and L. Illusie, Cristaux ordinaires et coordonnées canoniques (with an appendix by N. Katz), in: Surfaces Algébriques, J. Giraud, L. Illusie and M. Raynaud, eds., Lecture Notes in Mathematics 900, Springer Verlag, Berlin, 1982, pp. 101-228.

[Mo] B.J.J. Moonen, Special points and linearity properties of Shimura varieties, preliminary version April 1995.