The behavior of essential dimension under specialization

Let $A$ be a discrete valuation ring with generic point $\eta$ and closed point $s$. We show that in a family of torsors over $\operatorname{Spec}(A)$, the essential dimension of the torsor above $s$ is less than or equal to the essential dimension of the torsor above $\eta$. We give two applications of this result, one in mixed characteristic, the other in equal characteristic.


Introduction
Let k be a field, G be a linear algebraic group over k, K/k be a field extension and τ : T → Spec(K) be a G-torsor.We say that τ (or T ) descends to an intermediate subfield k ⊂ K 0 ⊂ K if there exists a G-torsor τ 0 : T 0 → Spec(K 0 ) such that τ 0 is obtained from τ by a pull-back diagram (1.1) / / Spec(K 0 ).
The essential dimension ed k (τ) is the minimal transcendence degree trdeg k (K 0 ) such that τ descends to K 0 .Essential dimension of torsors has been much studied; for an overview see [Rei10] or [Mer13].In this paper we will investigate how ed k (τ) behaves as we deform τ.Our main theorem shows that under relatively mild assumptions, ed k (τ) does not increase under specialization.We will sometimes write ed k (T ) or ed k ([τ]) in place of ed k (τ), where [τ] is the class of τ in H 1 (K, G).
Following [BRV18], we will refer to a finite group S as being "tame" at a prime p if |S| is not divisible by p and "weakly tame" at p if S does not have a non-trivial normal p-subgroup.By definition every finite group is tame at p = 0. Theorem 1.2.Let A be a complete discrete valuation ring with maximal ideal m, fraction field k and residue field k 0 .Set p := char(k 0 ) 0 and let G be a smooth affine group scheme over A, satisfying one of the conditions (i), (ii) or (iii) below.Let R ⊃ A be a complete discrete valuation ring with fraction field K ⊃ k and residue field K 0 ⊃ k 0 , and assume that m is contained in the maximal ideal of R. Then for every α ∈ H 1 (R, G) we have ] and G A is defined over k 0 , then the above inequality is an equality.
(i) p = 0, and there exist a section σ : k 0 → A of the projection A → k 0 , and a k 0 -group H such that G σ * H. (ii) The neutral component G • is reductive, G/G • is A-finite, and there exists a finite subgroup S ⊂ G(A) such that S is tame at p and for every field L containing k the natural map H 1 (L, S) → H 1 (L, G) is surjective.(iii) G = S A , where S is an abstract finite group which is weakly tame at p.
The assumption that A and R are complete may be dropped; see Theorem 6.4.In the case, where G is a finite group and p = 0, the inequality ed k 0 (α K 0 ) ed k (α K ) was noted in [FS21,Remark 6.3].A version of Theorem 1.2 for essential dimension at a prime will be proved in the Appendix; see Theorem A.1.
It is natural to ask whether or not conditions (ii) and (iii) can be replaced by a single (weaker) assumption.
Question 1.3.Does Theorem 1.2 remain valid if the finite group S in part (ii) is only assumed to be weakly tame, rather than tame?
We have not been able to answer this question; our proof of Theorem 1.2 uses an entirely different argument in case (iii), compared to cases (i) and (ii).Note however, that Theorem 1.2(ii) and (iii) both fail in the case where p > 0 and G is a finite discrete p-group; see Lemma 9.1.Some assumptions on char(k 0 ) are thus necessary.Note also that Question 1.3 has a positive answer if essential dimension is replaced by essential dimension at a prime q, different from p; see Theorem A.1.
We will give two applications of Theorem 1.2.Additional applications of Theorem 1.2 can be found in the companion paper [RS21a].
For our first application, recall that the essential dimension ed k (G) of an algebraic group G defined over k is the supremum of ed k (α), as K ranges over field extensions of k and α ranges over H 1 (K, G).It is shown in [BRV18] that if G is an abstract finite group which is weakly tame at a prime p > 0, then where k is any field of characteristic 0 and k 0 is any field of characteristic p containing the algebraic closure of the prime field F p .Theorem 1.5 below partially extends this inequality to split reductive groups. (1)  Theorem 1.5.Let G be a split reductive group scheme (not necessarily connected) of rank r 0 defined over Z.
Denote the Weyl group of G by W . Then ed k (G k ) ed k 0 (G k 0 ) for any field k of characteristic zero and any field k 0 of characteristic p > 0, as long p does not divide 2 r |W | and k 0 contains the algebraic closure of F p .
Note that in the case where G is a finite constant group (and thus r = 0 and W = G), we only recover Inequality (1.4) in the case where G is tame at p.A positive answer to Question 1.3 would imply that Theorem 1.5 remains valid when W is only assumed to be weakly tame at p = char(k 0 ), as long as p is odd.
For our second application of Theorem 1.2, we briefly recall the definition of essential dimension of a G-variety.Once again, let G be an algebraic group defined over k.By a G-variety we shall mean a separated reduced k-scheme of finite type endowed with a G-action over k.We will say that the G-variety Y is primitive if Y ∅ and G(k) transitively permutes the irreducible components of Y k := Y × k k.We will say that the G-variety Y (or equivalently, the G-action on Y ) is generically free if there exists a dense open subscheme U ⊂ Y such that for every u ∈ U the scheme-theoretic stabilizer G u of u is trivial.
(1) For a recent generalization of the inequality (1.4) in a different (stack-theoretic) direction, see [BV22].Now suppose that Y is a generically free primitive G-variety defined over k.By a G-compression of Y we will mean a dominant G-equivariant rational map Y X, defined over k, where the G-action on X is again generically free and primitive.The essential dimension of Y , denoted by ed k (Y ; G), is defined as the minimal value of dim(X) − dim(G), where the minimum is taken over all G-compressions Y X.This notion is closely related to that of essential dimension of a torsor; see Section 2.2.If the reference to G is clear from the context, we will often write ed k (Y ) in place of ed k (Y ; G).Theorem 1.6.Let k be a field of characteristic p 0, G be a linear algebraic group defined over k, satisfying one of the conditions (1) -(4) below.Let X, Y be primitive generically free G-varieties defined over k.Assume that X is smooth.If there exists a G-equivariant rational map f : Y X, then ed k (X) ed k (Y ).(1) p = 0.
(2) p > 0, G is split reductive, and k contains a primitive root of unity of degree 2n 2 .
(3) p > 0, G is connected reductive, and there exists a maximal torus T of G such that T [n 2 ] is discrete.(4) G is a finite discrete group, weakly tame at p.

Here in parts (2) and (3), n = |W | denotes the order of the Weyl group
If f is dominant, then Theorem 1.6 is obvious from the definition, since any compression of X can be composed with f .The key point here is that f is allowed to be arbitrary.In particular, we do not assume that the G-action on the image of f is generically free.The idea of the proof is to use Theorem 1.2 to deform f .The assumption that X be smooth may not be dropped; see Remark 9.5.

Dependence on the base field
Lemma 2.1.Let G be a linear algebraic group over k, K/k be a field extension, and τ : (a) Composing with the natural projection T K → T , we obtain a Cartesian diagram of G-torsors Let K 1 be the subfield of K generated by l and K 0 .Then τ also descends to K 1 .Thus

Essential dimension of a G-variety
Let G be an algebraic group defined over k and let Y be a generically free primitive G-variety.These terms are defined in the paragraph preceding Theorem 1.6, where one can also find the definition of a compression Y X and of ed k (Y ).
Lemma 2.2.Let Y be a generically free primitive G-variety defined over k, and k /k be a field extension.Then Note that if Y is a generically free primitive G-variety, then Y k is a generically free primitive G k -variety. Proof.
Then there exists an intermediate field k ⊂ l ⊂ k such that l is finitely generated over k and f descends to a G l -compression f : Y l → X, where X is a generically free G l -variety and X is a generically free primitive G l .Therefore as desired.
(c) In view of part (b), we may assume without loss of generality that k is finitely generated over k.By part (a), it suffices to show that Then f is actually defined over some k-variety U whose function field is k : In other words, there exists a surjective G-equivariant morphism π : X → U whose generic fiber is X, and a G-equivariant dominant rational map as desired.
Recall that in the Introduction we defined the essential dimension for both G-torsors and G-varieties.These two notions are closely related in the following way.Let Y be a generically free primitive G-variety.After passing to a G-invariant open subvariety of Y , we may assume that Y is the total space of a G-torsor τ : Y → B, where B is irreducible with function field and a direct product of fields in general; however, K is always a field, as long as Y is primitive.Pulling back to the generic point Spec(K) → B of B, we obtain a G-torsor τ Y : T Y → Spec(K).Now an easy spreading out argument shows that Recall from the Introduction that ed k (G) is defined as the maximal value of ed k (τ), where the maximum is taken over all field extensions of K/k and all G-torsors τ : T → Spec(K).
Lemma 2.4.Let G be a linear algebraic group defined over a field k and k /k be a field extension.
(a) Let V be a generically free linear representation of G defined over k.(e) Let V be a generically free linear representation of G defined over k.By Lemma 2.2(b), there exists an intermediate extension k ⊂ l ⊂ k such that l is finitely generated over k and ed l (V l ) = ed k (V k ).By part (a), the left hand side of this equality is ed l (G l ) and the right hand side is ed k (G k ).

Essential dimension at a prime
Let G be a linear algebraic group defined over a field k and let q be a prime integer.Essential dimension at q for G-torsors, G-varieties, and G itself, is defined in a way that parallels the definitions of essential dimension for these objects.We recall these definitions below.
A field K is called q-closed if every finite field extension L/K is of degree [L : K] = q r for some integer r 0. For every field K, there exists a unique algebraic extension K (q) /K such that K (q) is q-closed and the degree [L : K] of every every finite subextension K ⊂ L ⊂ K (q) is prime to q.The field K (q) is called a q-closure of K.It is the fixed field of a q-Sylow subgroup of the absolute Galois group of K.
Let k be a field, and let G be an algebraic k-group.If K/k is a field extension and τ : T → Spec(K) is a G-torsor, then ed k,q (τ) is the minimal value of ed k (τ L ), where L ranges over the finite field extensions of K whose degree [L : K] is prime to q. Equivalently, (2.5) ed k,q (τ) = ed k (τ K (q) ), where K (q) is the q-closure of K.
Recall that a correspondence X Z between G-varieties X and Z of degree where the vertical map is dominant of degree d.We say that X Z is dominant if the rational map X Z in the above diagram is so.Dominant correspondences may be composed in an evident way.Note that sometimes the term "rational correspondence" is used in place of correspondence, to indicate that the maps in Diagram (2.6) are rational maps.All correspondences used in this paper will be rational in this sense.For notational simplicity we will use the term "correspondence" throughout.Now let X be a generically free primitive G-variety over k.The essential dimension of ed k,q (X) of X at q is the minimal value of dim(Z) − dim(G), where the minimum is taken over all G-equivariant dominant correspondences X Z of degree prime to q. Equivalently, ed k,q (X) = maximal value of ed k,q (X ), where the maximum is taken over all G-equivariant dominant rational covers X X of degree prime to q. (2.7)If τ X is the G-torsor over k(X) G associated to X, as in the previous section, then one readily checks that (2.8) ed k,q (X) = ed k,q (τ X ).
The essential dimension ed k,q (G) of G at q is the maximal value of ed k,q (τ), as K ranges over fields containing k and T → Spec(K) ranges over G-torsors over Spec(K).Equivalently, ed k,q (G) is the maximal value of ed k,q (X), as X ranges over the generically free primitive G-varieties.

Torsors over complete discrete valuation rings
Let R be a complete discrete valuation ring, F its fraction field, F 0 its residue field, and s and η geometric points lying over the closed and generic point of Spec R, respectively.We have a group homomorphism sp : Gal(F) → Gal(F 0 ), called specialization homomorphism.Under the identifications Gal(F) = π 1 (Spec F, η) and Gal(F 0 ) = π 1 (Spec F 0 , s), the homomorphism sp is the composition where the map on the left is induced by the inclusion Spec F → Spec R, and the isomorphism on the right by specialization of finite étale covers; see e.g.[Stacks,Tag 0BUP].The homomorphism sp may also be defined Galois-theoretically; see [Ser03, Section 7.1].
Lemma 3.2.Let K be a field, π ∈ K × , and fix an algebraic closure K of K. Then (a) there exists a system π 1/m m 1 of roots of π in K such that (π 1/m 1 m 2 ) m 2 = π 1/m 1 for any m 1 , m 2 1.
in part (a) can be chosen so that π 1/n = α.
Proof.It suffices to prove part (b).Once (b) is established, we deduce (a) by setting n = 1 and α = π.
To prove (b), choose β ∈ K such that β (n−1)!= α.In particular, β n! = π.For every integer e d, we define π 1/e! ∈ K recursively as follows: π 1/n! := β, and if e > n then π 1/e! is defined as an arbitrary e th root of π 1/(e−1)! .Now we define π 1/m for an arbitrary integer m 1 as follows.Choose e n such that m divides e!, and set π 1/m := (π 1/e! ) e!/m .It is immediate to check that π 1/m does not depend on the choice of e, that (π 1/m 1 m 2 ) m 2 = π 1/m 1 for any m 1 , m 2 1 and that π 1/n = (π 1/n! ) (n−1)! = β (n−1)!= α.Lemma 3.3.Let R be a complete discrete valuation ring with fraction field F and residue field F 0 .Set p = char(F 0 ) 0. Choose a uniformizing parameter π for R and a system π 1/n n 1 of roots of π in F such that (π 1/mn ) m = π 1/n for all m, n 1.Let F(π 1/∞ ) be the subfield of F obtained by adjoining all the π 1/n to F. Then we have a commutative diagram Let It remains to construct the top horizontal exact sequence.The map sp is defined as the restriction of sp.We claim that sp is surjective.Let γ ∈ Gal(F 0 ): we will show that γ belongs to the image of sp .Let L 0 /F 0 be a finite Galois extension such that L 0 is invariant under γ, and γ restricts to an element of Gal(L 0 /F 0 ).Let F ⊂ L ⊂ F, where L/F is the unramified extension corresponding to L 0 /F 0 .By construction L/F induces L 0 /F 0 by passing to residue fields, and the homomorphism sp L/F : Gal(L/F) → Gal(L 0 /F 0 ) induced by sp is an isomorphism.Let L(π 1/∞ ) be the subfield of F generated by L and π 1/n n 1 . We have We would like to show that f is an F(π 1/∞ )-algebra isomorphism.First we note that the image of f contains both L and π 1/n for every n 1.Hence, f is surjective.
It remains to show that f is injective, i.e., that f : To prove this, it suffices to show that 1, π 1/n , . . ., π (n−1)/n are linearly independent over L. Indeed, suppose for some l 0 , l 1 , . . ., l n−1 ∈ L. Since L/F is unramified, the given valuation ν : F * → Z lifts to valuations Z which, by abuse of notation, we will also denote by ν.Now observe that the terms on the left hand side of (3.4) all have different valuations, as ν(l i π i/n ) is of the form i/n plus an integer.Thus the only way the sum in (3.4) can be 0 is if each term is 0. In other words, l 0 = l 1 = . . .l n−1 = 0, as desired.This shows that f is injective and hence, an isomorphism.
As a consequence, we have a group isomorphism Gal(L/F) → Gal L(π 1/∞ )/F(π 1/∞ ) such that the composition is sp L/F .This implies that γ belongs to the image of sp .We conclude that sp is surjective, as claimed.
Let Q be the kernel of sp .Then Q is the intersection of Gal(F nr ) with Gal(F(π 1/∞ )) = Gal(F(π 1/∞ ) ).In other words, Q is the absolute Galois group of the composite field F comp of F nr and F(π 1/∞ ) .The field F comp is obtained from F nr by adjoining all the π 1/n for n not divisible by p, and so F comp = F mod .Therefore Q = Gal(F mod ) = P .This proves the exactness of the top horizontal row, and thus completes the construction of the diagram.
The existence of a section of sp follows from the fact that P is a pro-p-group and that Gal(F 0 ) has p-cohomological dimension equal to 1; see [Ser97, § II.4.3,Exercise 2(c)].
Lemma 3.5.Let R be a complete discrete valuation ring, F and F 0 be the fraction field and residue field of R, respectively, and π ∈ R be a uniformizer.Fix an algebraic closure F of F, choose a system (d) Assume further that at least one of the following conditions holds.
] and G is defined over F 0 .(ii) There exists a finite abstract group S of order invertible in F 0 and an R-subgroup embedding S R → G such that the induced map where G • is a smooth, connected and reductive groups scheme and G 0 is a finite group scheme over R.
Passing to non-Abelian cohomology, we obtain a commutative diagram . This shows that the pullback By the Grothendieck-Serre Conjecture over Henselian discrete valuation rings, due to Nisnevich [Nis84, Théorème 4.5], the pullback map ) also has trivial kernel.An easy diagram chase now shows that the vertical map H 1 (R, G) → H 1 (F, G) has trivial kernel as well.
We will now show that the map We want to show that α = β.Let a be a cocycle representing α, and let a G be the R-group scheme obtained by twisting G by a.The fiber of the map H 1 (R, G) → H 1 (F, G) over α F is naturally identified with the kernel of the map H 1 (R, a G) → H 1 (F, a G).Now the same argument as above, with G replaced by its twist a G, shows that the map H 1 (R, a G) → H 1 (F, a G) has trivial kernel.Consequently, the fiber of the map where the left and right vertical arrows are induced by reduction modulo π and π 1/n , respectively.Both are isomorphisms by part (a).We conclude that α R n = β R n and consequently, α = β, as desired.
(d) In view of part (c), it suffices to show that the map From now on we will assume that condition (ii) of part (d) holds.We will break up the proof into several steps given by the claims below.Recall that the homomorphism sp : Gal(F(π 1/∞ )) → Gal(F 0 ) has a section by Lemma 3.3.Denote a section of sp by σ : Gal(F 0 ) → Gal(F(π 1/∞ )), and let Γ be its image.

Claim 3.7. The pullback map σ
Proof of Claim 3.7.Given that S is a finite discrete group, we may identify H 1 (F(π 1/∞ ), S) with Hom Gal(F(π 1/∞ )), S / ∼ , H 1 F Γ , S with Hom(Γ , S)/ ∼ , and σ * with the map Here the symbol ∼ stands for the equivalence relation given by conjugation by an element of S. Let a, b : Gal F(π 1/∞ ) −→ S be group homomorphisms such that a| Γ ∼ b| Γ .After replacing b by an S-conjugate, we may assume that a| Γ = b| Γ .Our goal is to prove that a = b.Let be the equalizer of a and b.Since a| Γ = b| Γ , we know that Γ ⊂ H.By Lemma 3.3, Gal(F(π 1/∞ )) is a semi-direct product of P and Γ , where P is a pro-p group.Hence, Gal(F(π 1/∞ )) = P • H and consequently, On the other hand, consider the group homomorphism and let H a,b := ker φ = ker a ∩ ker b.Then φ factors through an injective homomorphism of finite groups Since S has order prime to p, the index [Gal(F(π 1/∞ ) : H a,b ] is not divisible by p. Since H a,b ⊂ H, we deduce that (3.9) [Gal(F(π 1/∞ )) : H] is not divisible by p.
Claim 3.10.For every finite discrete group S, the pullback map Proof of Claim 3.10.Consider the composition where η is a geometric point lying above the generic point of Spec R. The pullback map H 1 (R, S) → H 1 (F Γ , S) may be identified with the map induced by precomposition with φ.Here once again, ∼ denotes the equivalence relation given by conjugation by an element of S. By construction, the composition is the restriction of sp to Γ , and so it is an isomorphism, with inverse σ .We conclude that φ is also an isomorphism.
Claim 3.11.The map H 1 (R, S) → H 1 F(π 1/∞ ), S is bijective for every finite discrete group S of order invertible in F 0 .
Proof of Claim 3.11.Injectivity follows from (c), and surjectivity from the commutativity of the triangle

∼
Here the vertical map is injective by Claim 3.7 and the diagonal map is bijective by Claim 3.10 applied to S.

Proof of Theorem 1.2 (i) and (ii)
Lemma 4.1.Let A ⊂ R be an inclusion of discrete valuation rings.Let m A and m R be the maximal ideals of A and R, respectively, and assume that m A ⊂ m R .Then there exists a discrete valuation ring R be a uniformizer, and set Since A is noetherian and B is a localization of a finitely generated A-algebra, B is also noetherian.Thus B is a local noetherian domain whose maximal ideal is principal, and consequently, a discrete valuation ring.
Since π is a generator of m R , it is clear that m B R = m R .The map A/m A → B/m B is surjective because π goes to zero in B/m B , and is injective because A/m A is a field.

Proof of Theorem 1.2 assuming (i) or (ii).
Choose a discrete valuation ring B between A and R as in Lemma 4.1, so that m B R = m R and the induced map k 0 = A/m A → B/m B is an isomorphism.Let B be the completion of B. Since R is complete, the universal property of the completion gives rise to an embedding B → R. Denote the fraction field of B by l.By Lemma 2.1(b), ed k (α K ) ed l (α K ).Thus in order to prove the inequality (4.2) of Theorem 1.2, it suffices to show that ed l (α K 0 ) ed l (α K ).Thus for the purpose of proving Inequality (4.2), we may replace A by B. In other words, we may assume we may assume that mR = m R is the maximal ideal of R.
Once again, let π ∈ m R − m 2 R be a uniformizing parameter.By definition of ed k (α K ), there exists an intermediate field k and α descends to F. Let v : K × → Z be the valuation determined by π.Since F contains k, v| F × is surjective.Let O ⊂ F be the valuation ring of v| F × , and F 0 be the residue field of the restriction of v to F.
Let Ô be the completion of O, and let F be the fraction field of Ô, that is, the completion of F as a valued field.Since K is complete, the universal property of the completion gives a unique field embedding F → K extending the inclusion F ⊂ K.We view F as a subfield of K via this embedding, so that Since π is a uniformizer in R, it is also a uniformizer in O and in Ô.Let K be an algebraic closure of K, and fix a system π 1/m m 1 of roots of π in K such that (π 1/mn ) n = π 1/m for all m, n 1.Such a system of roots exists by Lemma 3.2(a).We then define the following subfields of K:

Thus we have a commutative diagram
Passing to Galois cohomology, we obtain the following commutative diagram where the horizontal maps on the left are isomorphisms by Lemma 3.5(a), and those on the right are isomorphisms by Lemma 3.5(d).(This is where assumptions (i) and (ii) are used.)We deduce that α K 0 descends to F 0 , and thus By (4.3) and (4.4), we conclude that This completes the proof of the Inequality (4.2).
In the case, where ] and G A is defined over k 0 , Lemma 2.1(a) implies that the reverse inequality is also true and thus ed k α K = ed k 0 α K 0 .This completes the proof of Theorem 1.2 under assumptions (i) and (ii).(Note that at the beginning of the proof of (4.2) we replaced A by B. For the reverse inequality we work with A = k 0 [[t]] directly, here we do not need B.) We conclude this section with the following remarks on conditions (i) and (ii) of Theorem 1.2.Remarks 4.5.
(1) By Cohen's Structure Theorem, when char(k 0 ) = 0 there exists an isomorphism A k 0 [[t]] inducing the identity on residue fields.Such an isomorphism is not unique.One may restate (i) as follows: we may identify A with k 0 [[t]] in such a way that G is defined over the subfield of constants ) In (i), the fact that G is constant is not automatic in general; see [GP11b, Chapitre XIX, § 5] for an example, where G k((t)) is an étale form of PGL 2 but the fiber G 0 at t = 0 is solvable with two connected components.However, if char(k 0 ) = 0 and G is reductive over then G is constant.To see this, note that by [GP11b, Chapitre XXIV, Corollaire 1.18], letting G split denote the split form of G over A, there exists an étale Aut(G split )-torsor P such that twisting G split by P yields G.By [GP11b, Chapitre XXIV, Théorème 1.3], Aut(G split ) is smooth over A, hence by [GP11a, Chapitre XXIV, Proposition 8.1] the torsor P is defined over k 0 .Since G split is split, it is defined over Z, hence over k 0 .Thus G, being the twist of a group defined over k 0 by a torsor defined over k 0 , is also defined over k 0 .
(3) In (ii), if we have an A-group embedding S A ⊂ G A , then we also have a k 0 -group embedding S k 0 ⊂ G k 0 .
However, the converse does not necessarily hold: the natural group homomorphism G(A) → G(k 0 ) is surjective (by Hensel's lemma, since G is A-smooth) but not necessarily injective.

Proof of Theorem 1.5
Recall that a commutative ring with identity is said to be semilocal if it has finitely many maximal ideals.Proof.Let T ⊂ G be a split maximal torus over Z, let N ⊂ G be the normalizer of T , and let W := N /T be the Weyl group scheme.Since W is étale over Z, it is a disjoint union of copies of Spec Z. Since Pic(Z) = 0 and T is split, we have H 1 (Z, T ) = 0.It follows that we have a short exact sequence Claim 5.2.The Z-subgroup Γ is reduced and finite flat of degree 2 r n over Z.
Proof of Claim 5.2.Since Γ is defined as the scheme-theoretic closure of the constant subscheme otherwise Γ would contain a Z-subgroup isomorphic to G m and hence Γ (Q) would be infinite, a contradiction.The quotient T /(T ∩ Γ ) is also finite over Z because it is a closed subgroup of W . Since Γ is an extension of T /(T ∩ Γ ) by T ∩ Γ , it is finite over Z as well.In particular, the degree of Γ over Z is equal to the degree of Claim 5.4.The group scheme Γ is the union of the 2 r n irreducible components given by sections of N → Spec Z.Moreover, the restriction of Γ to Spec Z[1/2 r ] is constant.
Proof of Claim 5.4.Let Γ be the closed subscheme of N given by the union of the images of the 2 r n sections of N → Spec Z. (Note that N is separated, hence every section of N → Spec(Z) is a closed immersion.Therefore Γ is closed.)We have Γ ⊂ Γ by Claim 5.3.By Claim 5.2 there exist a Z-algebra R, finite and free of rank 2 r n, and an ideal I ⊂ R such that Γ = Spec(R) and Γ = Spec(R/I).Since Γ is the union of 2 r n copies of Spec(Z) over Z, the rank of R/I is also 2 r n.It follows that I is torsion.Since R is free, this implies that I = 0, hence Γ = Γ .In order to prove the second part of Claim 5.4, we consider the commutative diagram of short exact sequences where F p is the field of p elements and p is prime to 2 r .The vertical map on the left is the inclusion {±1} r → (F × p ) r , and in particular is injective.On the other hand, the map is a disjoint union of 2 r n copies of Spec F p , which means that the irreducible components of Γ do not intersect above p.Therefore the restrictions to Spec Z[1/2 r ] of the 2 r n irreducible components of Γ are pairwise disjoint.This concludes the proof of Claim 5.4.
For each positive integer m 1, let φ m : T → T be the m th power map given by t → t m .As usual, we will denote the kernel of φ m by T [m].For each m 1, T [m] is a flat finite normal subgroup of N of degree m r .Claim 5.5.We have Proof of Claim 5.5.We have T (Z) = T [2](Z) = {±1} r and T [2] (Spec Z[t]/(t 2 − 1)) r .It follows that the union of the 2 r sections of T → Spec Z is equal to T [2] and that T [2] becomes constant over Spec Z[1/2 r ].
By Claim 5.4, Γ → Spec Z is the union of all sections of N → Spec Z.Let s : Spec Z → N be a section.The composition of s with the projection N → W is a section w : Spec Z → W .If w is trivial, then the image of s is contained in T , and if w is non-trivial, then the image of s is disjoint from T .This shows that T ∩ Γ is the union of the sections of N → Spec Z which map to the identity section of W , i.e. the sections of T → Spec Z. Therefore T ∩ Γ = T [2].This proves Claim 5.5.
We are now ready to finish the proof of Proposition 5.1.Let S be the subgroup of N generated by Γ and φ −1 n (Γ ∩ T ).By construction, S is finite flat over Z.This concludes the proof of Proposition 5.1.
Corollary 5.6.Let G be a split reductive group over Spec(Z).Then there exists a G-torsor τ : X → B, where B is a smooth irreducible Z-scheme of finite type and for every field k we have ed Proof.By [GP11a, Exposé VI B , Proposition 13.2] there exists an embedding G → GL n,Z over Z.Since GL n,Z and G are reductive, by [Alp14, Theorem 9.4.1] the sheaf-theoretic quotient GL n,Z /G is represented by an affine Z-scheme.It is clear that GL n,Z /G is of finite type over Z and irreducible.It is flat over Z by [Stacks, Tag 02JZ] and it has smooth fibers, hence it is smooth over Z.We can now take X = GL n,Z , B = GL n,Z /G and τ the quotient map.For any field k, the G k -torsor X k → B k will then be GL n,k → GL n,k /G k .Here G k acts on X k = GL n,k by right translations.Since this action is birationally isomorphic to the linear action of G k on the affine space of n × n matrices over k, we conclude that Proof of Theorem 1.5.By Lemma 2.4(c), ed k (G) ed k (G) where k is the algebraic closure of k.Hence, we may, without loss of generality, replace k by k and thus assume that k is algebraically closed.By Lemma 2.4(d), ed k (G) = ed Q (G) for any algebraically closed field k of characteristic 0, where Q is the field of algebraic numbers.Thus we may assume without loss of generality that (5.7) k = Q.
By Lemma 2.4(e) there exists a number field L ⊂ Q such that We may assume that L contains ζ, where ζ is a primitive root of unity of degree 2 r n.Otherwise we can simply replace L by L(ζ).Indeed, by Lemma 2.4, and (5.8) forces ed Let O L be the ring of algebraic integers in L and p ⊂ O L be a prime ideal lying over p.Note that ζ, being an algebraic integer, lies in O L .Let A be the completion of the localization of O L at p. Then A is a complete local ring whose fraction field F contains L and whose residue field f is a finite field of characteristic p.

Recall that ζ ∈ O
Now let X → B be the G-torsor constructed in Corollary 5.6.Consider the Cartesian diagram where Spec(F) is the generic point of Spec(A) and Spec(f ) is the closed point.By Theorem 1.2(ii), we have (5.9)ed F (X F ) ed f (X f ).
Note that condition (ii) of Theorem 1.2 is satisfied here: a finite subgroup S ⊂ G A such that the morphism H 1 (L, S) → H 1 (L, G) is surjective for every field L containing F is constructed in Proposition 5.1.Our assumption that p does not divide 2 r n allows us to factor the inclusion Z → A through Z[1/2 r , ζ].Thus S A is constant by Proposition 5.1(c), as required.
By Corollary 5.6, we infer We finally get as desired.Here (a) and (b) follow from our assumptions (5.7) and (5.8) on k and L, (c) from Lemma 2.4(c) using the inclusion L ⊂ F, (d) from (5.9) and (5.10), and (e) again from Lemma 2.4(c) using the inclusions f → F p → k 0 .This completes the proof of Theorem 1.5.

Proof of Theorem 1.2(iii)
We have not been able to adapt the Galois-theoretic approach of Section 4 to prove Theorem 1.2 in case (iii).The proof below is based on valuation-theoretic methods in the spirit of [BRV18]; see also [FS21, Lemma 6.2].Proposition 6.1.Let L be a field and v : L × → Z be a discrete valuation.Assume that a finite group G acts faithfully on L and that v is invariant under this action.Assume further that k is a subfield of K = L G such that v(k × ) = Z.Denote the residue fields of L, K, and k by L 0 , K 0 and k 0 , respectively.Finally, assume that G is weakly tame at p = char(k 0 ) 0. Then By part (a), the G-action on L 0 is faithful.We claim that (L 0 ) G = K 0 .Clearly K 0 ⊂ (L 0 ) G ⊂ L 0 and, since G acts faithfully on L 0 , we have [L 0 : (L 0 ) G ] = |G|.On the other hand, by [Lan02, Corollary XII.6.3] the degree [L 0 : K 0 ] divides [L 0 : K 0 ] = |G|, hence K 0 = (L 0 ) G .This proves the claim.
We thus obtain the following diagram of field extensions: where the right side are obtained from the left side by passing to residue fields.In other words, the G-Galois extension L 0 /K 0 descends to K 0 .By [BRV18, Lemma 2.1], trdeg k 0 (K 0 ) trdeg k (K ).We conclude that as desired.
We are now ready to complete the proof of Theorem 1.2 assuming (iii).The class α ∈ H 1 (R, G) is represented by a finite étale R-algebra E such that G acts faithfully on E over R and transitively permutes the connected components of Spec(E).Recall that K is the field of fractions of R. Since R is complete, it is Henselian, hence every connected component of Spec E is local, i.e. is of the form Spec(E 1 ), where E 1 is a discrete valuation ring with valuation ν : E 1 \ {0} → Z extending the valuation on R. Denote the fraction field of E 1 by L and the residue field of E 1 by L 0 .Let us now consider two cases.
Case 1. E is integral, i.e.E = E 1 .In this case the class α K ∈ H 1 (K, G) is represented by the field extension L/K, where L is the fraction field of E. By Proposition 6.1(a) the class α K 0 is represented by the field extension L 0 /K 0 .The inequality (6.2) of Theorem 1.2 now translates to the inequality of Proposition 6.1(b).
Case 2. E 1 E. To handle this case we will need the following.
Lemma 6.3.Let G be a finite discrete group, H be a subgroup of G, K/k be a field extension and T → Spec(K) be an Proof.A proof is implicit in [BRV07, Proposition 2.17] and is worked out in detail in [Bre21, Lemma 2.4].
We can now complete the proof of Theorem 1.2 (iii) in Case 2. Denote the stabilizer of E 1 in G by H. Then Spec(E) = Spec(E 1 ) × H G. In other words, the G-torsor Spec(E) → Spec(R) is induced by the H-torsor Spec(E 1 ) → Spec(R) is an H-torsor.Replacing E by E 1 and G by H, and using Lemma 6.3, we reduce Case 2 to Case 1.This completes the proof of Theorem 1.2.
We conclude this section with a variant of Theorem 1.2, where A and R are not required to be complete.Theorem 6.4.Let A be a discrete valuation ring with maximal ideal m, fraction field k, residue field k 0 and m-adic completion Â. Set p := char(k 0 ) 0 and let G be a smooth affine group scheme over A, satisfying one of the conditions (i), (ii) or (iii) below.Let R ⊃ A be a discrete valuation ring with fraction field K ⊃ k and residue field K 0 ⊃ k 0 , and assume that m is contained in the maximal ideal of R. Then for every α ∈ H 1 (R, G) we have ] and G Â is defined over k 0 , then the above inequality is an equality.
(i) p = 0, and there exist a section σ : and there exists a finite subgroup S ⊂ G( Â) such S is tame at p and for every field L containing k the natural map where S is an abstract finite group which is weakly tame at p. Proof.We have where R is the completion of R, and k, K are the fraction fields of Â, R, respectively.Here (a) follows from Theorem 1.2, (b) from Lemma 2.1(b), and (c) from Lemma 2.1(a).

Proof of Theorem 1.6
Throughout this section, (7.1) G will denote a linear algebraic group and X a G-variety, both defined over a field k.We begin with a brief review of the concepts that will be used in the proof.

Twisting
For any field K/k and G-torsor τ : T → Spec(K), we can define the algebraic space τ X over K as the quotient of T × k X K by the diagonal action of G.Note that in this situation G acts freely on T × K X K , and τ X can be defined by descent so that the quotient morphism T × K X K → τ X is a G-torsor.If X is quasi-projective, then τ X is a K-variety and not just an algebraic space.The algebraic space τ X is separated and of finite type over K.This may be checked after splitting τ; when τ is split, we have τ X X K , where denotes isomorphism over K.In general, τ X does not inherit a G-action from X.There is a natural action of the twisted group τ G on τ X, but we will not need it in the sequel.For a more detailed discussion of the twisting construction and further references, see [DR15, Section 3].

Large fields
A field K is called large if the following property holds for every irreducible K-variety Z: if Z has a smooth K-point, then K-points are dense in Z.This notion is due to F. Pop.Note that if Z is irreducible and has a smooth K-point, then it is absolutely irreducible.A variant of this definition assumes that Z is an irreducible K-curve, rather than an irreducible K-variety of arbitrary dimension.This a priori weaker property of K turns out to be equivalent.The most important examples of large fields for this paper are fields of Laurent series K((t)), where K is an arbitrary field (see [Pop14, Section 1A(2)]) and p-closed fields (see [CT00,p. 360] or [Pop14, Theorem 1.3]).See Section 2.3 for the definition of a p-closed field and [Pop14] for a detailed discussion of large fields, including further examples.Lemma 7.2.Let K be a large field and Z be an irreducible separated algebraic space of finite type over K.If Z has a smooth K-point, then K-points are dense in Z.
Proof.Let z ∈ Z(K) be a smooth point, and let Z := Proj Z (⊕ i 0 I i ) → Z be the blow-up of Z at z, where I is the ideal sheaf of z; see [Stacks,Tag 085Q].It follows that the exceptional divisor E of Z → Z is given by Proj z (⊕ i 0 I i /I i+1 ) = P(T Z,z ), where T Z,z denotes the tangent space at z. Since K is large, it is infinite, hence K-points are dense in E. Since z is smooth, any K-point in E is smooth.By [Stacks, Tag 0ADD], there exists an open embedding U → Z such that U is a scheme and U ∩ E ∅.Thus U contains smooth K-points and so, since K is large and U is irreducible, K-points are dense in U .Since the blow-up map Z → Z is surjective, we conclude that K-points are dense in Z, as desired.

τ-versality
Suppose K/k is a field extension and τ : T → Spec(K) is a G-torsor.We say that X is τ-weakly versal if there exists a G-equivariant morphism T → X K defined over K.We say that X is τ-versal if every dense G-invariant open subvariety of X is τ-weakly versal.Note that if X is primitive, then every non-empty G-invariant open subvariety is automatically dense.
With these preliminaries out of the way, we can get started on the proof of Theorem 1.6.
Lemma 7.3.Let G and X be as in (7.1), K/k be a field extension, and τ : T → Spec(K) be a G-torsor.
(a) X is τ-weakly versal if and only if the twisted variety τ X has a K-point.
(b) Assume further that X is smooth, and K is a large field.Then X is weakly τ-versal if and only if X is τ-versal.
Proof.Part (a) is proved at the beginning of Section 4 in [DR15].Note that the proof relies on [DR15, Proposition 3.2], where X is not assumed to be irreducible.
(b) If X is τ-versal, then X is obviously weakly τ-versal.Conversely, suppose X is weakly τ-versal.We want to show that X is τ-versal.Assume the contrary: there exists a Zariski dense open G-invariant subvariety U of X which is not weakly τ-versal.Then τ U is a dense open algebraic subspace of τ X (see [DR15, Corollary 3.4]).By part (a), τ U has no K-points but τ X has a K-point.Denote this point by x.Since X is smooth, so is τ X.Hence, x lies on a unique irreducible component of τ X. Denote this irreducible component by X 1 .Since X 1 is smooth and irreducible and K is large, K-points are dense in τ X 1 .In particular, there is a K-point in τ U ∩ X 1 , contradicting our assumption that τ U has no K-points.
Lemma 7.4.Let G and X be as in (7.1).Assume that X is generically free and primitive, K/k is a field extension and τ : Proof.Let f : X Z be a G-compression (i.e. a G-equivariant dominant rational map defined over k) to a generically free G-variety Z of minimal possible dimension, dim(Z) = ed k (X) + dim(G).As we explained in Section 2.2, after replacing Z by a dense open G-invariant subvariety we may assume that Z is the total space of a G-torsor Z → B, over some k-variety B of dimension Let X 0 ⊂ X be the domain of f .Since X is τ-versal, there exists a G-equivariant map T → (X 0 ) K = X 0 × Spec(k) Spec(K) defined over K. Projecting to the first component and composing with f , we obtain a G-equivariant morphism T → Z defined over k.This morphism induces a pull-back diagram of G-torsors.Restricting to the generic point of the image of Spec(K) in B (whose residue field we will denote by K 0 ), we obtain a pull-back diagram of G-torsors Note that Lemma 7.4 is in the same general spirit as Theorem 1.6.Indeed, let τ = τ Y : T → Spec(K) be the G-torsor over K = k(Y ) G constructed from Y , as in Section 2.2.Then ed k (τ Y ) = ed k (Y ), and the existence of a G-equivariant rational map f : Y X is equivalent to saying that X is weakly τ-versal.Lemma 7.4 requires X to be τ-versal, which is a stronger assumption in general.To bridge the gap between "weakly τ-versal" and "τ-versal", we will use the fact that these two notions are equivalent when K is a large field; see Lemma 7.3(b).Our strategy is thus to replace k by k((t)), K by K((t)) and τ by τ((t)) = τ ⊗ K K((t)).The following lemma tells us that under assumptions (1) -(4) on G of Theorem 1.6, passing from τ to τ((t)) does not change the essential dimension.

Lemma 7.5. Let G be a linear algebraic group over k, satisfying one of the conditions (1) -(4) of Theorem 1.6. Then for A = k[[t]] the group scheme G A satisfies one of the conditions (i) -(iii) of Theorem 1.2.
Proof.In the proof (i), (ii), (iii) will always refer to conditions of Theorem 1.2 and (1), (2), (3), (4) to conditions of Theorem 1.6.The following implications are obvious: We are now ready to complete the proof of Theorem 1.6.
) viewed as a G-variety over k((t)).(Here G acts on the first factor.)The G-equivariant map f : T → X defined over k naturally extends to a G-equivariant map T K((t)) → X K(t)) = (X 1 ) K((t)) defined over K((t)).This shows that X 1 is weakly τ((t))-versal.Since K((t)) is a large field, Lemma 7.3(b) tells us that X 1 is τ((t))-versal.Now observe that (7.7) Here (a) follows from Lemma 2.2(a), (b) follows from Lemma 7.4, and (d) is a restatement of (2.3).Finally, (c) follows from Theorem 1.2.Note that Theorem 1.2 can be applied in this situation under any of the assumptions (1) -(4) of Theorem 1.6 by Lemma 7.5.

Some consequences of Theorem 1.6
Throughout this section k, G and X will be as in (7.1).Recall that X is called weakly versal (respectively, versal) if it is τ-weakly versal (respectively, τ-versal) for every field K/k and every G-torsor τ : T → Spec(K); see [DR15,p. 499].
Corollary 8.1.Assume that G satisfies one of the conditions (1) -(4) of Theorem 1.6 and the G-variety X is generically free and primitive.
(a) If X is smooth and weakly versal, then ed k (X) = ed k (G).
Proof.(a) Let V be a generically free linear representation of G and τ V : T → Spec(K) be the G-torsor over K = k(V ) G constructed from V , as in Section 2.2.Since X is τ V -versal, there exists a G-equivariant rational map V X defined over k.By Theorem 1.6, ed(X) ed(V ).On the other hand, by Lemma 2.4(a), ed(V ) = ed(G) ed(X) and part (a) follows.
(b) After replacing X by its smooth locus, we may assume that X is smooth.The constant map T → X K sending T to the G-fixed point in X is G-equivariant for every G-torsor τ : T → Spec(K).Thus X is weakly versal and part (a) applies.
Example 8.2.Let k be a field of characteristic p 0 and G be a subgroup of the symmetric group S m .Assume that G is weakly versal at p.For any absolutely irreducible variety Z defined over k, we may view Z m as a G-variety, where G acts on Z m by permuting the factors.Then Indeed, if z is a smooth point of Z, then (z, . . ., z) is a smooth G-fixed point of Z m , and Corollary 8.1(b) applies.
Example 8.3.Let G be a linear algebraic group satisfying one of the conditions (1) -(4) of Theorem 1.6, and let X be a smooth connected algebraic group defined over a field k (not necessarily linear).Suppose G acts generically freely on X by group automorphisms.Then (8.4) Indeed, the identity element of X is a smooth fixed point of G, and Corollary 8.1(b) applies.
In the case, where X is affine, this was previously known.In this case X is versal as a G-variety (see [DR15, Example 7.4]), and (8.4) follows.If X is not affine, it is no longer versal as a G-variety (only weakly versal) but equality (8.4) holds nevertheless.
Corollary 8.5.Let G be a linear algebraic group satisfying one of the conditions (1) -( 4) of Theorem 1.6, and let X and Y be G-varieties.Assume that Y is generically free and primitive, and that X is smooth and absolutely irreducible.If there exists a G-equivariant rational map f : Y X, then Proof.Consider the G-equivariant maps (i) f × k id : Y X × k Y and (ii) pr 2 : X × k Y → Y .By Theorem 1.6, (i) implies that ed k (X × k Y ) ed k (Y ).Similarly, (ii) implies that ed k (Y ) ed k (X × k Y ), and Equality (8.6) follows.
Note that the G-action on X in the statement of Corollary 8.5 is not assumed to be generically free.As a special case, (8.6) holds if G acts trivially on X and X(k) ∅.This was previously known only in the case where k-points are dense in X.

Counterexamples
Lemma 9.1.Let G be a linear algebraic group defined over k, K be a large field containing k and τ : T → Spec(K) be a G-torsor.Proof.(a) After replacing X by its smooth locus, we may assume that X is smooth.The constant map T → X K sending T to the fixed point shows that X is weakly τ-versal.By Lemma 7.3(b), X is τ-versal.
The inequality ed k (τ) ed k (X) now follows from Lemma 7.4.The inequality ed k (X) dim(X) − dim(G) follows from the definition of essential dimension.
(b) In this case there exists a smooth irreducible G-curve X defined over k such that G acts generically freely (i.e.faithfully) on X and fixes a k-point; see [RV18,Lemma 3].By part (a), ed k (τ) dim(X) − dim(G) = 1.
(c) By [Led07, Proposition 5], ed k (G) 2. By Lemma 2.4(a), this tells us that (9.2) where V is the regular representation of G over k.Let X be a smooth G-curve with a fixed k-point, as in part (b), and f : V → X be the constant G-equivariant morphism sending V to this G-fixed point.If Theorem 1.6 were true in this case, then we would have 2).This shows that Theorem 1.6 fails for G.
If Corollary 8.1 were valid for X, we would conclude that ed k (X) = ed k (G).This leads to a contradiction because ed k (G) 2 but ed k (X) 1.
Since the G-action on both X and V is generically free, and both have smooth G-fixed points, there exists G-equivariant maps X → V and V → X.If Corollary 8.5 were true in this setting, it would tell us that contradicting (9.2).Thus Corollary 8.5 fails in this setting as well.
Finally, to show that Theorem 1.2 fails, let K = k(V ) G and τ : T → Spec(K) be the G-torsor associated to the G-action on V , as in Section 2.2.Then ed k (τ) = ed k (V ) 2. On the other hand, since K((t)) is a large field, part (b) tells us that ed k((t)) τ((t)) 1.In particular, ed k((t)) τ((t)) < ed k (τ).This shows that Theorem 1.2 fails for ] and α = τ R .(Note that none of the conditions (i), (ii), (iii) of Theorem 1.2 are satisfied here.)Remark 9.3.Lemma 9.1(b) is proved in [RV18] in the case where K is a p-closed field.Recall that p-closed fields are large; see [CT00, p. 360].
Remark 9.4.A conjecture of A. Ledet [Led04] asserts that ed k (Z/p n Z) = n for any infinite base field k of characteristic p and any integer n 1.This conjecture has been proved for n 2 and is open for every n 3.Moreover, Ledet showed that ed k (Z/p n Z) n, so his conjecture is equivalent to the existence of a field extension K/k and a Z/p n Z-torsor τ : T → Spec(K) such that ed k (τ) n.
This conjecture has many interesting consequences [BRV18], [Tos19].It looks even more remarkable in light of Lemma 9.1(b), which asserts that ed k((t)) τ((t)) 1 for every field K containing k and every G-torsor τ over Spec(K).
Remark 9.5.The smoothness assumption on X in the statement of Theorem 1.6 cannot be dropped.
Indeed, let G be a finite group.Then there exists a curve C of genus 2 with a faithful G-action.To construct C, start with a curve C n with a faithful S n -action as in [BR97, Remark 4.5].For large n, C n will automatically be of genus 2, since S n cannot act faithfully on a rational or elliptic curve.After embedding G into S n (again, for a suitably large n), we may view C = C n as a G-curve.Next we embed C into a projective space P(V ) in an S n -equivariant way, where S n acts linearly on V (e.g. by using a pluri-canonical embedding) and we consider X the affine cone over C. The origin 0 in X is fixed by S n (and hence, by G).Note that since the center of S n is trivial for any n 2, we may assume that no element of S n (and hence, of G) acts by scalar multiplication on V .
The natural projection X C tells us that ed k (X) dim(C) − dim(G) = 1.We claim that Theorem 1.6 fails for the constant map f : V → X sending all of V to the origin.Here V is a faithful linear representation of G over k (e.g. the regular representation).Indeed, if Theorem 1.6 were true for f , it would tell us that 1 ed k (X) ed k (V ) = ed k (G) for every irreducible G-variety Y , i.e. 1 ed k (G), which is false for most finite groups; see [BR97, Theorem 6.2] or [Led07].Of course, the reason for this failure of Theorem 1.6 in this example is that X is singular at the origin.Similar (even easier) examples can be constructed for a connected group G as follows.Start with a generically free representation G → GL(V ), let w be a point in general position in P(W ), where W = V ⊕ k, and X be the cone over the Zariski closure of the orbit G • w in P(W ).Clearly ed k (X) = 0. Once again, the "constant" morphism f : V → X taking the whole of V to 0 is G-equivariant.If Theorem 1.6 were true for f , then we would obtain 0 ed k (V ) = ed k (G), which is false for most connected groups G, e.g. for G = SO n (n 3) or PGL n (n 2).Once again the reason Theorem 1.6 fails in this example is that X is singular at the origin.
10.An analogue of Theorem 1.6 for essential dimension at a prime Theorem 10.1.Let k be a field, G be a linear algebraic group defined over k and X, Y be generically free primitive G-varieties defined over k.Assume that X is smooth.If there exists a G-equivariant correspondence f : Y X of degree prime to q, then ed k,q (X) ed k,q (Y ).
The proof of Theorem 10.1 is similar to the proof of Theorem 1.6, but simpler.Since the q-closure K (q) of any field K is large, we do not need to pass to k((t)) or appeal to Theorem 1.2.Note that we do not require G to be smooth or reductive here, and char(k) can be arbitrary.
Proof.Let K/k be a field extension and τ = τ Y : T → Spec(K) be a G-torsor constructed from Y , as in Section 2.2.The correspondence Y X of degree prime to q gives rise to a G-equivariant rational map τ K (q) : T K (q) → X, where K (q) is the q-closure of K.In other words, X is weakly τ K (q) -versal.Since K (q) is a large field, Lemma 7.3 tells us that X is τ K (q) -versal.By Lemma 7.4, where the first equality is (2.5) and the second equality is (2.8).Moreover, suppose π : X X is a Gequivariant dominant rational map of G-varieties of degree prime to q. Viewing the inverse of π as a correspondence π −1 : X X and composing it with f , we obtain a correspondence π −1 • f : Y X of degree prime to q.By (10.2), we get ed k (X ) ed k,q (Y ).
Taking the minimum over all G-equivariant rational covers X X of degree prime to q and using (2.7), we arrive at the desired inequality ed k,q (X) ed k,q (Y ).
Using Theorem 10.1, we can deduce prime-to-q analogues of all of the corollaries and examples in Section 8.Note however, that Corollary 8.1 is less novel in this context.In particular, by [DR15, Theorem 8.3] a weakly q-versal smooth G-variety X is q-versal.If X is generically free and primitive, this implies ed k,q (X) = ed k,q (G).The case where X has a fixed point is considered in [DR15, Corollary 8.6].The equality ed k (X) = ed k (G) of Corollary 8.1 is more intricate, because X may not be versal.
On the other hand, to the best of our knowledge, the following prime-to-q analogue of Corollary 8.5 is new.It is proved by the same argument as Corollary 8.5, with Theorem 10.1 used in place of Theorem 1.6.
Corollary 10.3.Let X be Y be G-varieties.Assume that Y is generically free and primitive, and X is smooth and absolutely irreducible.If there exists a G-equivariant correspondence f : Y X of degree prime to q, then ed k,q (X × k Y ) = ed k,q (Y ).

Appendix A. Essential dimension at a prime
In this appendix we will prove a variant of Theorem 1.2, where essential dimension is replaced by essential dimension at a fixed prime q.Note that conditions (i), (ii) and (iii) of Theorem 1.2 simplify in this setting; they are replaced by conditions (i') and (ii') below.
Theorem A.1.Let A be a complete discrete valuation ring with maximal ideal m, fraction field k and residue field k 0 .Set p := char(k 0 ) 0, q p be a prime number, and let G be a smooth affine group scheme over A, satisfying one of the conditions (i') or (ii') below.
Let R ⊃ A be a complete discrete valuation ring with fraction field K ⊃ k and residue field K 0 ⊃ k 0 , and assume that mR is the maximal ideal of R. Then for every α ∈ H 1 (R, G) we have ed k 0 ,q (α K 0 ; G K 0 ) ed k,q (α K ; G K ).Furthermore, if A = k 0 [[t]] and G A is defined over k 0 , then the above inequality is an equality.(i') char(k 0 ) = 0, and there exist a section σ : k 0 → A of the projection A → k 0 and a k 0 -group H such that G σ * H; (ii') G • is reductive, there exists a finite subgroup S ⊂ G(A) of order invertible in k 0 such that for every q-closed field L containing k the natural map H 1 (L, S) → H 1 (L, G) is surjective.
Note that if S is the finite group in (ii') and S q is the Sylow q-subgroup of S, then H 1 (L, S q ) → H 1 (L, S) is an isomorphism for every q-closed field L. Hence, we may replace S by S q in (ii').In other words, if the finite subgroup group S of G(A) in (ii') exists, we may assume that S is a q-group.
Our proof of Theorem A.1 below is analogous to that of Theorem 1.2 in Section 4.However, since the definition of ed k,q (α K ; G K ) allows replacing K by a finite extension L/K of degree prime to q (and similarly for ed k 0 ,q (α K 0 ; G K 0 )) there are some complications involving extensions of valuations from K to L.
Proof of Theorem A.1.Let π ∈ A be a uniformizer.Using Lemma 4.1, as we did at the beginning of the proof of Theorem 1.2 in Section 4, we may assume that mR is the maximal ideal of R, that is, that π is also a uniformizer in A. Let K /K be a finite extension of degree prime to q and k ⊂ F ⊂ K be a field of definition for α K such that (A.2) ed k,q α K = trdeg k F.
Let v : (K ) × → 1 d Z be the unique surjective valuation extending the given valuation on K. Let R ⊂ K be the local ring and K 0 the residue field of v. Since q does not divide [K : K], it does not divide d.Moreover, since F contains k and π ∈ A ⊂ k, we have Let O ⊂ F be the valuation ring and F 0 ⊂ K 0 the residue field of v| F × .The inclusion A ⊂ O of valuation rings induces an inclusion k 0 ⊂ F 0 of residue fields.By [RS21b, Lemma 7.1], we have Let R be the π-adic completion of R, and let F be the fraction field of R, that is, the completion of F as a valued field.Since K is complete, the universal property of the completion gives a unique field embedding F → K extending the inclusion F ⊂ K .We view F as a subfield of K via this embedding, so that F ⊂ F ⊂ K , O ⊂ Ô ⊂ R , and the residue field of F is F 0 .If L is a field containing k and n 1 is an integer invertible in k, we write L n for the étale L-algebra L[x]/(x n − π).Then L n factors as a product of finite-dimensional separable field extensions of L, each of which contains an n th root of π.Moreover, if L is the fraction field of a complete discrete valuation ring with uniformizer π, then by Eisenstein's Criterion L n is a field.Thus k n , Fn , and K n are fields, while K n is not necessarily a field.Since k n is a field, the inclusion k n ⊂ K n factors through a unique field factor K n ⊂ K n .Thus k n , Fn and K n are all contained in K n .
We have a commutative diagram Passing to Galois cohomology, we obtain the following commutative diagram: (A.4) where the horizontal maps on the left are isomorphisms by Lemma 3.5(a).
For the injectivity of the horizontal maps on the right side of (A.4), we argue as follows.Let L be a field containing k, n 1 be an integer invertible in k, L be an algebraic closure of L, and L n be a field factor of L n .Let π 1/n ∈ L be such that (π 1/n ) n = π, and consider the homomorphism φ n : L n → L induced by x → π 1/n .The restriction of φ n to L n is a field embedding L n → L which factors through L(π 1/n ).By Lemma 3.2(b), we can extend π 1/n to a system π 1/m m 1 of roots of π in L such that (π 1/mm ) m = π 1/m for every m, m 1. Letting L(π 1/∞ ) := m 1 L(π 1/m ) ⊂ L, we have the inclusions L ⊂ L n ⊂ L(π 1/∞ ).Letting L be F, K n or K n , we deduce from Lemma 3.5(c) that the horizontal maps on the right are injective.
Since α K is defined over F, it is defined over F. We let β ∈ H 1 ( F, G) be such that β K = α K .We now observe that β F(π 1/∞ ) is defined over Ô.Indeed, if (i') holds then this follows from [Flo06,Proposition 5.4].If (ii') holds then, letting S be a finite subgroup of G(A) as in (ii'), β F(π 1/∞ ) admits reduction of structure to some γ ∈ H 1 ( F(π 1/∞ ), S).Now Claim 3.11 shows that γ is defined over Ô, hence so is β F(π 1/∞ ) .The field F(π 1/∞ ) is the increasing union of the Fn , hence there exists some n 1 such that β Fn is defined over Ô.Since β K = α K , we have β K n = α K n , hence α K n is also defined over Ô.From the commutativity of the right-hand side of (A.4) and the injectivity of the map H 1 (R , G) → H 1 (K n , G), we deduce that α R is defined over Ô.The commutativity of the left-hand side of (A.4) now implies that α K 0 is defined over F 0 .Since [K 0 : K 0 ] divides d, it is not divisible by q.Thus trdeg k 0 F 0 ed k 0 ,q α K 0 .
Combining this with (A.2) and (A.3), we conclude that ed k,q α K = trdeg k F trdeg k 0 F 0 ed k 0 ,q α K 0 .Suppose now that A = k 0 [[t]] and that G A is defined over k 0 .Let K (q) be a q-closure of K, and let K (q) 0 be a q-closure of K 0 contained in K (q) .By (2.5), we have ed k,q α K = ed k α K (q) and ed k 0 ,q α K 0 = ed k 0 α K (q) 0 .Now Lemma 2.1(a) implies that ed k,q α K ed k 0 ,q α K 0 .Therefore ed k,q α K = ed k 0 ,q α K 0 , completing the proof of Theorem A.1.
(a) The inequality ed k (Y ) ed k (Y k ) follows directly from the definition of essential dimension, since every G-compression Y X gives rise to a G k -compression Y k X k by base-change.(b) By (a), it suffices to find l so that ed l Tos17, Lemma 4.8]) There exists an intermediate field k ⊂ l ⊂ k such that l is finitely generated over k and ed l (G l ) = ed k (G k ).Proof.(a) This follows from the fact that the G-action on V is versal, which is, in turn, a consequence of Hilbert's Theorem 90.See [Mer13, Propositions 3.10 and 3.11] for details.(b) For any generically free primitive G-variety Y , ed k (Y ) = ed k (τ Y ) ed k (G).On the other hand, note that since G is a linear algebraic group, there exists a closed embedding G → GL n for some n 1.Let G act on the space V = M n of n × n matrices by left multiplication via this embedding.Now part (a) tells us that ed k (V ) = ed k (G), and part (b) follows.(c-d)By Lemma 2.2, ed(V ; G) ed(V k ; G k ); moreover, equality holds if k is algebraically closed.The desired conclusions now follows from part (a).
Proof.(a) See [GP11a, Chapter XXIV, Proposition 8.1].Only smoothness of G is needed here.(b) First we will show that the map H 1 (R, G) → H 1 (F, G) has trivial kernel.Consider the short exact sequence of group R-schemes 0 then the surjectivity of this map follows from [Flo06, Proposition 5.4].

Proposition 5. 1 .
Let G be a split reductive group over Spec Z of rank r 0. Let n be the order of the Weyl group of G. Then there exists a finite flat Z-subgroup S ⊂ G with the following properties.(a)For every semilocal ring R the natural mapH 1 (R, S) → H 1 (R, G) is surjective.(b) If a prime number p divides the degree of S, then p divides 2 r n.(c) S is constant over R = Z[1/2 r , ζ]and ζ is a primitive root of unity of degree 2 r n.In other words, S R is the constant R-group scheme associated to an abstract finite group.
(a) G acts faithfully on L 0 , and (b) ed k (L/K) ed k 0 (L 0 /K 0 ).Proof.(a) Let ∆ be the kernel of the G-action on L 0 .By our assumption, v(k × ) = Z and thus v(K × ) = Z.If char(k) = 0, then [BRV18, Proposition 2.3] tells us that ∆ = 1 and we are done.If char(k) = p > 0, then [BRV18, Proposition 2.3] tells us that ∆ is a p-group.Since we are assuming that G is weakly tame at p, this implies that ∆ = 1 in this case as well.(b)Suppose ed k (L/K) = d.This means that there exists an intermediate field k ⊂ K ⊂ K and a G-Galois extension L /K such that L = L ⊗ K K and trdeg k (K ) = d.Let K 0 and L 0 be the residue fields for the restriction of v to K and L , respectively.
(a) Let X be a generically free G-variety defined over k.If X has a smooth G-fixed k-point, then ed k (τ) ed k (X) dim(X) − dim(G).(b) If char(k) = pand G is a finite (constant ) p-group, then ed k (τ) 1.In other words, ed k (L/K) 1 for every G-Galois field extension L/K.(c) If char(k) = p and G is a finite p-group which is not elementary Abelian, then Theorem 1.6, Corollary 8.1, Corollary 8.5 and Theorem 1.2 all fail for G.
Proof of Claim 5.3.Clearly sQ : Spec Q → N Q factors through Γ Q .Moreover, since Γ is finite over Z, by the existence part of the valuative criterion for properness s Z (p) : Spec Z (p) → N Z (p) factors through Γ Z (p) for every prime p.By the uniqueness part of the valuative criterion for properness, this factorization is unique for each p.Therefore, these local factorizations glue to a global factorization of s through Γ over Z.
also reduced, and hence Γ is flat over Z; see [Liu02, Lemma 4.3.9].Claim 5.3.Let s : Spec Z → N be a Z-point of N .Then s factors through Γ .