ESSENTIAL DIMENSION IN MIXED CHARACTERISTIC

PATRICK BROSNAN, ZINOVY REICHSTEIN, AND ANGELO VISTOLI

Abstract. Let R be a discrete valuation ring with residue field k and fraction field K. We say that a finite group G is weakly tame at a prime p if it has no non-trivial normal p-subgroups. By convention, every finite group is weakly tame at 0. We show that if G is weakly tame at char(k), then ed_{K}(G) ⩾ ed_{k}(G). Here ed_{F} (G) denotes the essential dimension of G over the field F. We also prove a more general statement of this type, for a class of étale gerbes 𝒳 over R.

As a corollary, we show that if G is weakly tame at p, then ed_{K} G ⩾ ed_{k} G for any field K of characteristic 0 and any field k of characteristic p, provided that k contains \overline{𝔽}_{p}. We also show that a conjecture of A. Ledet, asserting that ed_{k}(ℤ/p^{n}ℤ) = n for a field k of characteristic p > 0 implies that ed_{ℂ}(G) ⩾ n for any finite group G which is weakly tame at p and contains an element of order p^{n}. To the best of our knowledge, an unconditional proof of the last inequality is out of the reach of all presently known techniques.

1. Introduction

Let R be a discrete valuation ring with residue field k and fraction field K, and let G be a finite group. In this paper we will compare ed_{K}(G) and ed_{k}(G). More generally, we will compare ed_{K}(𝒳 ) to ed_{k}(𝒳 ) for an étale gerbe 𝒳 over R. For an overview of the theory of essential dimension, we refer the reader to [BRV11, Mer13, Rei10].

2010 Mathematics Subject Classification. 14A20, 13A18, 13A50. Key words and phrases. Essential dimension, Ledet’s conjecture, genericity the-

orem, gerbe, mixed characteristic.

Patrick Brosnan was partially supported by NSF grant DMS 1361159. Zinovy Reichstein was partially supported by National Sciences and Engineering

Research Council of Canada Discovery grant 253424-2017.

Angelo Vistoli was partially supported by research funds from the Scuola Normale Superiore.

The authors are grateful to the Collaborative Research Group in Geometric and Cohomological Methods in Algebra at the Pacific Institute for the Mathematical Sciences for their support of this project.

To state our main result, we will need some definitions.

Suppose S is a scheme. By an étale gerbe 𝒳 → S we mean an algebraic stack that is a gerbe in the étale topology on S. Furthermore, we will always assume that there exists an étale covering {S_{i} → S}, such that the pullback 𝒳_{S_{i}} is of the form ℬ_{S_{i}}G_{i}, where G_{i} → S_{i} is a finite étale group scheme.

We say that a finite group G is tame (resp. weakly tame) at a prime number p if p ∤ |G| (resp. G contains no non-trivial normal p-subgroup). Equivalently, G is tame at p if the trivial group is the (unique) p-Sylow subgroup of G, and G is weakly tame at p if the intersection of all p-Sylow subgroups of G is trivial. By convention we say that every finite group is both tame and weakly tame at 0.

By a geometric point of S, we mean a morphism Spec Ω → S with Ω an algebraically closed field. We say that a finite étale group scheme G over S is tame (resp. weakly tame) if, for every geometric point Spec Ω → S, the group G(Ω) is tame (resp. weakly tame) at char Ω. Similarly, we say that an étale gerbe 𝒳 → S is tame (resp. weakly tame) if, for every object ξ over a geometric point Spec Ω → S, the automorphism group Aut_{Ω} ξ is tame (resp. weakly tame) at char Ω.

A key result of [BRV11] is the so called Genericity Theorem for tame Deligne–Mumford stacks, [BRV11, Theorem 6.1]. The proof of this result in [BRV11] was based on the following.

Theorem 1.1 ([BRV11, Theorem 5.11]). Let R be a discrete valuation ring (DVR) with residue field k and fraction field K, and let

𝒳 ⟶ Spec R

be a tame étale gerbe. Then ed_{K} 𝒳_{K} ⩾ ed_{k} 𝒳_{k}.

Here 𝒳_{K} and 𝒳_{k} are respectively the generic fiber and the special fiber of 𝒳 → Spec R.

Unfortunately, the proof of [BRV11, Theorem 5.11] contains an error in the case when char K = 0 and char k > 0. This was noticed by Amit Hogadi, to whom we are very grateful. (See Remark 4.2 for an explanation of the error.) For the applications in [BRV11] only the equicharacteristic case was needed, so this mistake in the proof of Theorem 1.1 does not affect any other results in [BRV11] (the genericity theorem, in particular). However, the assertion of Theorem 1.1 in the mixed characteristic case remained of interest to us, as a way of relating essential dimension in positive characteristic to essential dimension in characteristic 0. In this paper, our main result is the following strengthened version of Theorem 1.1.

Theorem 1.2. Let R be a DVR with residue field k and fraction field K, and let

𝒳 ⟶ Spec R

be a weakly tame étale gerbe. Then ed_{K} 𝒳_{K} ⩾ ed_{k} 𝒳_{k}.

In particular, [BRV11, Theorem 5.11] is valid as stated. Moreover, our new proof is considerably shorter than the one in [BRV11]. And in Section 3 we will deduce some rather surprising consequences.

We will give two proofs of our main result, one for gerbes of the form where 𝒳 = ℬ_{R}G, where G is a (constant) finite group (Theorem 2.4) and the other for the general case. The ideas in these two proofs are closely related; the proof of Theorem 2.4 allows us to introduce these ideas, initially, in the elementary setting of classical valuation theory. A separate proof of Theorem 2.4 also makes the applications in Section 3 accessible to those readers who are not familiar with, or don’t care for, the language of gerbes.

2. Proof of Theorem 1.2 in the constant case

In this section we will prove a special case of Theorem 1.2, where 𝒳 = ℬ_{R}G for G a finite group (viewed as a constant group scheme over Spec R); see Theorem 2.4.

Throughout this section we will assume that L is a field equipped with a (surjective) discrete valuation ν: L^{∗} → ℤ and K is a subfield of L such that ν(K^{∗}) = ℤ. We will denote the residue fields of L and K by l and k, respectively. Similarly, we will denote the valuation rings by 𝒪_{L} and 𝒪_{K}.

The following lemma is a special case of the Corollary to Theorem 1.20 in [Vaq06]. For the convenience of the reader, we supply a short proof.

Lemma 2.1. trdeg_{k}(l) ⩽ trdeg_{K}(L).

Proof. Let u_{1}, …, u_{m} ∈ l be algebraically independent over k. Lift each u_{i} to v_{i} ∈ 𝒪_{L} ⊆ L. It now suffices to show that v_{1}, …, v_{m} are algebraically independent over K. Assume the contrary: f(v_{1}, …, v_{m}) = 0 for some polynomial 0 ≠ f(x_{1}, …, x_{m}) ∈ K[x_{1}, …, x_{m}]. After clearing denominators we may assume that every coefficient of f lies in 𝒪_{K}, and at least one of the coefficients has valuation 0. If f_{0} is the image of f in k[x_{1}, …, x_{m}] then f_{0} ≠ 0 and f_{0}(u_{1}, …, u_{m}) = 0. This contradicts our assumption that u_{1}, …, u_{m} are algebraically independent over k. ♠

Let L_{m} = ν^{−1}(m) ∪ {0} and L_{⩾m} = ⋃_{j⩾m} L_{j}. Note that, by definition, L_{⩾0} = 𝒪_{L} is the valuation ring of ν, L_{⩾1} is the maximal ideal, and L_{⩾0}/L_{⩾1} = l is the residue field.

Lemma 2.2. Assume that g is an automorphism of L of finite order d ⩾ 1, preserving the valuation ν. Let p = char(l) ⩾ 0. If g induces a trivial automorphism on both L_{⩾0}/L_{⩾1} and L_{⩾1}/L_{⩾2}, then

(a) d = 1 (i.e., g = id is the identity automorphism) if p = 0, and (b) d is a power of p, if p > 0.

Part (a) is proved in [BR97, Lemma 5.1]; a minor variant of the same argument also proves (b). Alternatively, with some additional work, Lemma 2.2 can be deduced from [ZS58, Theorem 25, p. 295]. For the reader’s convenience we will give a short self-contained proof below.

Proof. In case (b), write d = mp^{r}, where m is not divisible by p. After replacing g by g^{p^{r}}, we may assume that d is prime to p. In both parts we need to conclude that g is the identity.

Let G be the cyclic group generated by g; then G is linearly reductive. Since the action of G on ℓ is trivial, the induced action on L_{⩾i}/L_{⩾i+1} is ℓlinear. Furthermore, let t ∈ L_{1} be a uniformizing parameter. By our assumption g(t) = t (mod L_{⩾2}). Thus multiplication by t^{i−1} induces the ℓ-linear G-equivariant isomorphism (L_{⩾1}/L_{⩾2})^{⊗i} ≃ L_{⩾i}/L_{⩾i+1}. Consequently, G acts trivially on L_{⩾i}/L_{⩾i+1} for all i ⩾ 0. Since G is linearly reductive, from the exact sequence

0 ⟶ L_{⩾i}/L_{⩾i+1} ⟶ L/L_{⩾i+1} ⟶ L/L_{⩾i} ⟶ 0

we deduce, by induction on i, that G acts trivially on L/L_{⩾i} for every i ⩾ 1. Since ⋂_{i⩾0} L_{i} = 0, this implies that the action of G on L is trivial. Since G acts faithfully on L, we conclude that G = {1}, and the lemma follows. ♠

Proposition 2.3. Consider a faithful action of a finite group G on L, such that G preserves ν and acts trivially on K. Let Δ be the kernel of the induced G-action on l. Then Δ = {1} if char(k) = 0 and Δ is a p-subgroup if char(k) = p.

Proof. Assume the contrary. Then we can choose an element g ∈ Δ of prime order q, such that q ≠ char(k). Let M be the maximal ideal of the valuation ring 𝒪_{L}. Since we are assuming that ν(K^{∗}) = ν(L^{∗}) = ℤ, we can choose a uniformizing parameter t ∈ K for ν. Since g ∈ Δ, g acts trivially on both l = 𝒪_{L}/M and M/M^{2} = l · t. By Lemma 2.2, g acts trivially on L. This contradicts our assumption that G acts faithfully on L. ♠

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

Theorem 2.4. Let (R, ν) be a discrete valuation ring with residue field k and fraction field K, and G be a finite group. If p = char(k) > 0, assume that G is weakly tame at p. Then ed_{K}(G) ⩾ ed_{k}(G).

Proof. Set d := ed_{K}(G). Let R[G] be the group algebra of G and let V_{R} = (𝔸_{R})^{|G|} denote the corresponding R-scheme equipped with the (left) regular action of G. By definition d is the minimal transcendence degree trdeg_{K}(L), where L ranges over G-invariant intermediate subfields K ⊂ L ⊂ K(V_{K}) such that the G-action on L is faithful; see [BR97]. Choose a G-invariant intermediate subfield L such that trdeg_{K}(L) = d.

We will now construct a G-invariant intermediate subfield k ⊂ l ⊂ k(V_{k}), where V_{k} is the regular representation of G over k, as follows. Lift the given valuation ν: K^{∗} → ℤ to the purely transcendental extension K(V_{K}) of K in the obvious way. That is, ν : K(V_{K})^{∗} → ℤ is the divisorial valuation corresponding to the fiber of V_{R} over the closed point in Spec R. The residue field of K(V_{K}) is then k(V_{k}). By restriction, ν is a valuation on L with ν(L^{∗}) = ℤ. Let l be the residue field of L. Clearly k ⊂ l ⊂ k(V_{k}) and ν is invariant under G. By Proposition 2.3, G acts faithfully on l. Moreover, by Lemma 2.1, trdeg_{k}(l) ⩽ d. Thus ed_{k}(G) ⩽ d = ed_{K}(G), as desired. ♠

3. Applications and examples

We begin with the following easy corollary of Theorem 2.4.

Corollary 3.1. Let p be a prime, G a finite group weakly tame at p. Then

(a) (cf. [Tos17, Corollary 4.2]) ed_{ℚ} G ⩾ ed_{𝔽_{p}} G.

(b) If K is a field of characteristic 0 and k a field of characteristic p containing \overline{𝔽}_{p}, then ed_{K} G ⩾ ed_{k} G.

Proof. (a) follows directly from Theorem 2.4 by taking R to be the localization of the ring of integers ℤ at a prime ideal pℤ.

(b) Let \overline{K} be the algebraic closure of K. Since ed_{K}(G) ⩾ ed\overline{K}(G), we may replace that K by \overline{K} and thus assume that K is algebraically closed. Note that ed_{K} G = ed\overline{ℚ} G and ed_{k} G = ed_{\overline{𝔽}_{p}} G; see [BRV07, Proposition 2.14] or [Tos17, Example 4.10].

Choose a number field E ⊆ \overline{ℚ} such that ed_{E} G = ed\overline{ℚ} G and let 𝔭 ⊆ 𝒪_{E} a prime in the ring 𝒪_{E} of algebraic integers in E lying over p. Set E_{0} =^{def} 𝒪_{E}/𝔭. Since k contains \overline{𝔽}_{p}, there is an embedding E_{0} ⊆ k. By Theorem 2.4, ed_{E}(G) ⩾ ed_{E_{0}}(G) and since E_{0} ⊂ k, ed_{E_{0}} G ⩾ ed_{k} G. ♠ Remark 3.2. One could reasonably conjecture that, under the hypotheses of Corollary 3.1(b), ed_{K} G = ed_{k} G, provided that K is algebraically closed.

For our second application, recall the following conjecture of A. Ledet [Led04].

Conjecture 3.3. If k is a field of characteristic p > 0, n is a natural number, and C_{p^{n}} is a cyclic group of order p^{n}, then ed_{k}(C_{p^{n}}) = n.

It is known that in characteristic p, ed(C_{p^{n}}) ⩽ n for every n ⩾ 1 (see [Led04]) and ed(C_{p^{n}}) ⩾ 2 if n ⩾ 2 ([Led07, Theorems 5 and 7]). Thus the conjecture holds for n = 1 and n = 2; it remains open for every n ⩾ 3.

Conjecture 3.3 has the following surprising consequence.

Corollary 3.4. Assume that a finite group G is weakly tame at a prime p and contains an element of order p^{n}. Let K be a field of characteristic 0. If Conjecture 3.3 holds for C_{p^{n}}, then ed_{K}(G) ⩾ n.

Proof. By Corollary 3.1(b), with k = \overline{𝔽}_{p}, we have ed_{K}(G) ⩾ ed_{k}(G). Since G contains C_{p^{n}}, ed_{k}(G) ⩾ ed_{k}(C_{p^{n}}), and by Conjecture 3.3, ed_{k}(C_{p^{n}}) = n. ♠

Example 3.5. Let p be a prime and n a positive integer. Choose a positive integer m such that q =^{def} mp^{n} + 1 is a prime. (By Dirichlet’s theorem on primes in arithmetic progressions, there are infinitely many such m.) Let C_{q} be a cyclic group of order q. Then Aut C_{q} = (ℤ/qℤ)^{∗} is cyclic of order mp^{n}; let C_{p^{n}} ⊆ (ℤ/qℤ)^{∗} denote the subgroup of order p^{n}. Set G =^{def} C_{p^{n}} ⋉ C_{q}. Then

(a) G is weakly tame at p, and

(b) if Conjecture 3.3 holds, then ed_{K}(G) ⩾ n for any field K of characteristic 0. To prove (a), suppose S ⊆ G is a normal p-subgroup. Then S lies in every Sylow p-subgroup of G, in particular, in C_{p^{n}}. Our goal is to show that S = {1}. The cyclic group C_{q} of prime order q acts on S by conjugation. Since q > p^{n} ⩾ |S|, this action is trivial. In other words, S is a central subgroup of G. In particular, S acts trivially on C_{q} by conjugation. On the other hand, by the definition of G, C_{p^{n}} acts faithfully on C_{q} by conjugation. We conclude that S = {1}, as desired. Part (b) follows from Corollary 3.4. ♠

Note that the inequality of part (b) is equivalent to

(3.1)  ed_{ℂ}(C_{p^{n}} ⋉ C_{q}) ⩾ n ,

where ℂ is the field of complex numbers (once again, see [BRV07, Proposition 2.14] or [Tos17, Example 4.10]). The essential dimension of C_{p^{n}} ⋉ C_{q} at a prime r is given by

ed_{ℂ}(G; r) = \begin{cases}ed_{ℂ}(C_{p^{n}}; p) = 1, if r = p \\ ed_{ℂ}(C_{q}; q) = 1, if r = q, and \\ ed_{ℂ}(\{1\}; r) = 0, if r ≠ p or q. \end{cases}

In this case, the inequality (3.1) can be proved unconditionally (i.e., without assuming Conjecture 3.3) for n = 2 and 3 by appealing to the classifications of finite groups of essential dimension 1 and 2 over ℂ in [BR97, Theorem 6.2] and [Dun13, Theorem 1.1] respectively. So, for n ⩾ 2, ed_{ℂ}(G; r) is smaller than the (absolute) essential dimension ed_{ℂ}(C_{p^{n}} ⋉ C_{q}). Thus, in the language of [Rei10, Section 5], proving the inequality (3.1) is a “Type 2 problem”. As is explained in [Rei10], such problems are only accessible by existing techniques in low dimensions. To the best of our knowledge, an unconditional proof of (3.1), or even of the inequality ed_{ℂ}(C_{p^{n}} ⋉ C_{q}) ⩾ 4 for any n ⩾ 4, is out of the reach of all currently available techniques.

Remark 3.6. It is shown in [RV17] that if G is a finite group and k is a field of characteristic p, then

(3.2)  ed_{k}(G; p) = \begin{cases}1, if p divides |G|, and \\ 0, otherwise. \end{cases}

In particular, Conjecture 3.3 is also a Type 2 problem. Note also that in view of (3.2), Corollary 3.1(b) continues to hold if we replace essential dimension by essential dimension at p, for trivial reasons. Moreover, under the assumptions of Corollary 3.1, (a′) ed_{ℚ}(G; p) ⩾ ed_{𝔽_{p}}(G; p) and (b′) ed_{K}(G; p) ⩾ ed_{k}(G; p), for any finite group G, not necessarily weakly tame. In (b′) we can also drop the requirement that k should contain \overline{𝔽}_{p}. Note however that our proof of Theorem 2.4 breaks down if we replace essential dimension by essential dimension at p.

Example 3.7. The following example shows that Theorem 2.4 fails if we do not assume that G is weakly tame. Choose R so that char K = 0, char k = p > 0, and K contains a p^{2}-th root of 1. Let G = C_{p^{2}} be the cyclic group of order p^{2}. Since K contains a primitive p^{2}-th root of 1, ed_{K}(G) = ed_{K}(C_{p^{2}}) = 1. On the other hand, as we pointed out above, Conjecture 3.3 is known for n = 2, and thus ed_{k}(G) = ed_{k}(C_{p^{2}}) = 2.

Example 3.8. Here is an example showing that Theorem 2.4 fails if we do not assume that R is a DVR. Let R ⊆ ℂ[[t]] be the subring consisting of power series in t with constant coefficient, whose constant term is real. Then R is a one-dimensional complete Noetherian local ring with quotient field K = ℂ((t)) and residue field k = ℝ, but not a DVR. In this situation ed_{K}(G) = ed_{ℂ((t))}(C_{4}) = 1, while ed_{k}(G) = ed_{ℝ}(C_{4}) = 2; see [BF03, Theorem 7.6].

Example 3.9. (cf. [Tos17, Remark 4.5(ii)]) This example shows that essential dimension does not satisfy any reasonable semicontinuity, even in characteristic 0. Consider the scheme

S =^{def} Spec ℚ[u, x]/(x^{2} − u) .

The embedding ℚ[u] ⊆ ℚ[u, x]/(x^{2} − u) gives a finite map S → 𝔸_{ℚ}^{1}. If p is an odd prime, the inverse image of the prime (u − p) ⊆ ℚ[u] in S consists of a point s_{p} with residue field k(p) = ℚ[x](x^{2} − p). Then ed_{k(p)}(C_{4}) = 1 if −1 is a square modulo p, and ed_{k(p)} C_{4} = 2 if −1 is not a square modulo p; once again, see [BF03, Theorem 7.6]. Equivalently, ed_{k(p)}(C_{4}) = 1 if p ≡ 1 (mod 4), and ed_{k(p)} C_{4} = 2 is p ≡ 3 (mod 4). We conclude that the set of points s ∈ S with ed_{k(s)} C_{4} = 1 is dense in S, and likewise for the set of points s ∈ S with ed_{k(s)} C_{4} = 2 is also dense in S.

4. Proof of Theorem 1.2

We begin by remarking that an étale gerbe 𝒳 → S is weakly tame if and only if there exists an étale cover {S_{i} → S} such that each 𝒳_{S_{i}} → S_{i} is equivalent to ℬ_{S_{i}}G_{i} → S_{i} with G_{i} weakly tame étale group schemes over S_{i}.

Our proof of Theorem 1.2 will rely on the following fact.

Lemma 4.1. Let 𝒳_{F} → Spec F be a finite étale gerbe over a field F. Suppose that A is a non-zero finite F -algebra, and that the morphism Spec A → Spec F has a lifting ϕ: Spec A → 𝒳_{F}. Consider the locally free sheaf ϕ_{∗}𝒪_{Spec A} on 𝒳_{F}; call 𝒱 → 𝒳_{F} the corresponding vector bundle on 𝒳_{F}. Then 𝒱 has a non-empty open subscheme U ⊆ 𝒱 . Furthermore, if k(U) is the field of rational functions on U, the composite Spec k(U) → U ⊆ 𝒱 → 𝒳_{F} gives a versal object of 𝒳_{F} (k(U)).

Proof. Let us show that 𝒱 is generically a scheme. We can extend the base field F, so that it is algebraically closed; in this case 𝒳_{F} is the classifying space ℬ_{F} G of a finite group G, and there exists a homomorphism of F -algebras A → F. The vector bundle 𝒱 → 𝒮_{F} corresponds to a representation V of G; by the semicontinuity of the degree of the stabilizer for finite group actions, it is enough to show that V has a point with trivial stabilizer. The homomorphism A → F gives a morphism Spec F → Spec A, and the composite Spec F → Spec A → ℬ_{F} G corresponds to the trivial G-torsor on Spec F. If we call 𝒲 the pushforward of 𝒪_{Spec F} to ℬ_{F} G, then 𝒲 ⊆ 𝒱 . On the other hand 𝒲 corresponds to the regular representation of G, and so the generic stabilizer is trivial, which proves what we want.

Let us show that the composite Spec k(U) → U ⊆ 𝒱 → 𝒳_{F} is versal; the argument is standard. Suppose that K is an extension of F that is an infinite field, and consider a morphism Spec K → 𝒳_{F}. It is enough to prove that for any open subscheme U → 𝒱 , the morphism Spec K → 𝒳_{F} factors through u ⊆ 𝒱 → 𝒳_{F}. The pullback V_{K} → Spec K of 𝒱 → 𝒳_{F} is a vector space on K, and the inverse image U_{K} ⊆ V_{K} of U ⊆ 𝒱 is a non-empty open subscheme; hence U_{K}(K) ≠ ∅, which ends the proof. ♠

Proof of Theorem 1.2. Let R̂ be the completion of R and K̂ be the fraction field of R̂. Then clearly K ⊂ K̂ and thus ed_{K}(𝒳_{K}) ⩾ edK̂(𝒳K̂. Thus for the purpose of proving Theorem 1.2, we may replace R by R̂. In other words, we may (and will) assume that R is complete.

Let R → A be an étale faithfully flat algebra such that 𝒳 (A) ≠ ∅; since R is henselian, by passing to a component of Spec A we can assume that R → A is finite. Consider the lifting ϕ: Spec A → 𝒳 ; this is flat and finite. Let 𝒱 → 𝒳 be the vector bundle corresponding to ϕ_{∗}𝒪_{Spec A}. If U → 𝒱 is the largest open subscheme of 𝒱 , the Lemma above implies that U → Spec R is surjective. Denote by U_{K} and U_{k} respectively the generic and special fiber of U → Spec R; call E and E_{0} the fields of rational functions on U_{K} and U_{k} respectively. Again because of the Lemma, the objects ξ: Spec E → 𝒳_{K} and ξ_{0} : Spec E_{0} → 𝒳_{k} are versal.

Consider the local ring 𝒪_{E} of U at the generic point of U_{k}, which is a DVR. The residue field of 𝒪_{E} is E_{0}, and we have a morphism Ξ: Spec 𝒪_{E} → 𝒳 whose restrictions to Spec K and Spec k are isomorphic to ξ and ξ_{0} respectively.

Set m =^{def} ed_{K} 𝒳_{K}; we need to show that ξ_{0} has a compression of transcendence degree at most m.

There exists a field of definition K ⊆ L ⊆ E for ξ such that trdeg_{K} L = m; call θ: Spec L → 𝒳 the corresponding morphism, so that we have a factorization Spec E → Spec L ⟶^{θ} 𝒳 for ξ. Consider the intersection 𝒪_{L} =^{def} 𝒪_{E} ∩ L ⊆ E; then 𝒪_{L} is a DVR with quotient field L. Call L_{0} it residue field; we have L_{0} ⊆ E_{0}. By Lemma 2.1, trdeg_{k} L_{0} ⩽ trdeg_{K} L.

10 PATRICK BROSNAN, ZINOVY REICHSTEIN, AND ANGELO VISTOLI

Now it suffices to show that ξ_{0} : Spec E_{0} → 𝒳 factors through Spec L_{0}. Assume that we have proved that the morphism θ: Spec L → 𝒳 extends to a morphism Θ: Spec 𝒪_{L} → 𝒳 . The composite Spec E ⊆ Spec 𝒪_{E} ⟶^{Ξ} 𝒳 is isomorphic to the composite Spec E → Spec L ⊆ Spec 𝒪_{L} ⟶^{Θ} 𝒳 ; since 𝒳 is separated, it follows from the valuative criterion of separation that the composite Spec 𝒪_{E} ⟶^{Θ} Spec 𝒪_{L} → 𝒳 is isomorphic to Ξ: Spec 𝒪_{L} → 𝒳 . By restricting to the central fibers we deduce that ξ_{0} : Spec E_{0} → 𝒳 is isomorphic to the composite Spec E_{0} → Spec L_{0} → 𝒳 , and we are done.

To prove the existence of the extension Θ: Spec 𝒪_{L} → 𝒳 , notice that the uniqueness of such extension implies that to prove its existence we can pass to a finite étale extension R ⊆ R′, where R′ is a DVR; it is straightforward to check that formation of 𝒪_{L} and 𝒪_{E} commutes with such a base change. Hence we can assume that 𝒳 has a section, so that 𝒳 = ℬ_{R}G, where G → Spec R is a finite étale weakly tame group scheme. By passing to a further covering we can assume that G → Spec R is constant, that is, the product of Spec R with a finite group Γ. If A is an R-algebra, an action of G on Spec A correspond to action of Γ.

The vector bundle 𝒱 → 𝒳 corresponds to a vector bundle V_{R} → Spec R with an R-linear action of Γ, such that the induced representations of Γ on V_{K} and V_{k} are faithful. Call Ẽ the function field of V_{K} and Ẽ_{0} the function field of V_{k}; then Ẽ^{Γ} = E, and therefore 𝒪Γ ; Ẽ = 𝒪_{E}. The factorization Spec E → Spec L → 𝒳 gives a Γ-torsor Spec L̃ → Spec L whose lift to Spec E is isomorphic to Spec Ẽ → Spec E; then L̃ is a Γ-invariant subfield of Ẽ. Then 𝒪L̃ =^{def} L̃ ∩ 𝒪Ẽ is a Γ-invariant DVR,

and 𝒪Γ ; L̃ = L̃^{Γ} ∩ 𝒪Γ ; Ẽ = 𝒪_{L}.

Call 𝔪L̃ ⊆ 𝒪L̃ the maximal ideal; then 𝒪L̃/𝔪L̃ = L̃_{0}. If t ∈ R is the uniformizing parameter, the image of t in 𝒪L̃, which we denote again by t, is a uniformizing parameter; this is Γ-invariant. The action of Γ on 𝒪L̃ descends to an action of Γ on L̃_{0}. By Proposition 2.3, this action is faithful.

So the action of Γ on Spec L_{0} is free over k; this implies that the action of Γ on Spec 𝒪L̃ → Spec R is free, so Spec 𝒪L̃ → (Spec 𝒪L̃)/G = Spec 𝒪_{L} is a Γ-torsor. This gives the desired morphism Θ: Spec 𝒪_{L} → 𝒳 , and ends the proof of the Theorem. ♠

Remark 4.2. The problem with the proof of [BRV11, Theorem 5.11] was in the last sentence of the second paragraph on page 1094. It is the claim there that we can replace the discrete valuation ring R in the proof with the ring called W (k(s)). Since the essential dimension of the generic point can go up when we make this replacement, this is, in fact, not allowable. (In effect, our mistake boils down to using an inequality in the wrong direction.)

Note also that the proof of the characteristic 0 genericity theorem in [BRV07] does not rely on Theorem 1.1. For that argument, which was different from the proof of [BRV11, Theorem 6.1], see [BRV07, Theorem 4.1].

Acknowledgments

We are grateful to Najmuddin Fakhruddin for helpful comments.

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(Brosnan) Department of Mathematics, 1301 Mathematics Building, University of Maryland, College Park, MD 20742-4015, USA

E-mail address: pbrosnan@umd.edu

(Reichstein) Department of Mathematics, 1984 Mathematics Road, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

E-mail address: reichst@math.ubc.ca

(Vistoli) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

E-mail address: angelo.vistoli@sns.it