Finite torsors over strongly $F$-regular singularities

We investigate finite torsors over big opens of spectra of strongly $F$-regular germs that do not extend to torsors over the whole spectrum. Let $(R,\mathfrak{m},k)$ be a strongly $F$-regular $k$-germ where $k$ is an algebraically closed field of characteristic $p>0$. We prove the existence of a finite local cover $R \subset R^{\star}$ so that $R^{\star}$ is a strongly $F$-regular $k$-germ and: for all finite algebraic groups $G/k$ with solvable neutral component, every $G$-torsor over a big open of $\mathrm{Spec} R^{\star}$ extends to a $G$-torsor everywhere. To achieve this, we obtain a generalized transformation rule for the $F$-signature under finite local extensions. Such formula is used to show that that the torsion of $\mathrm{Cl} R$ is bounded by $1/s(R)$. By taking cones, we conclude that the Picard group of globally $F$-regular varieties is torsion-free. Likewise, it shows that canonical covers of $\mathbb{Q}$-Gorenstein strongly $F$-regular singularities are strongly $F$-regular.


Introduction
Inspired by the relationship between strongly F-regular singularities in characteristic p > 0 and KLT singularities in characteristic zero, K. Schwede, K. Tucker, and the author obtained in [C-RST18, Theorem A] an analog of [Xu14, Theorem 1]. Concretely, they proved that the order of the étale fundamental group of a big open of the spectrum of a strongly F-regular germ (R, m, k, K) is at most 1/s(R) and prime-to-p (where s(R) is the F-signature of R). It is worth noting that B. Bhatt, O. Gabber, and M. Olsson introduced a spreading out technique to give a proof of Xu's result from ours [BGO17]. See Subsection 2.1 for more on strong F-regularity and F-signatures.
The aforementioned result gives a satisfactory answer to what extent there are nontrivial finite étale torsors over big opens U ⊂ X Spec R (with R as above). We use the word torsor as in [Mil80,III,§4] (i.e. principal homogeneous spaces in the fppf topology). Let us refer to a finite torsor over a big open as a quasitorsor. Indeed, [C-RST18, Theorem A] establishes the existence of a generically Galois finite extension (R, m, k, K) ⊂ (Ré t , mé t , k, Ké t ) that restricts to a Gal(Ké t /K)-torsor over X reg and: • (Ré t , mé t , k, Ké t ) is a strongly F-regular germ, • every étale quasitorsor over Xé t Spec Ré t extends to a torsor over Xé t , and • its generic degree (i.e. the order of Gal(Ké t /K)) is at most 1/s(R) and prime-to-p.
Cartier's theorem establishes that, in characteristic zero, all affine (resp. finite) algebraic groups are smooth (resp. étale); see [Mil17,3.23,3.24], [DG70, II, §6.1]. This result fails in positive characteristics, thereby showing that the local étale fundamental group is not suited to classify quasitorsors over a singularity (as it does in characteristic zero). This had been observed in the failure in positive characteristics of the celebrated Mumford-Flenner theorem characterizing the regularity of normal surface singularities by the triviality of their local fundamental group [Mum61,Fle75]. Remarkably, H. Esnault and E. Viehweg gave an explanation for this failure via their local Nori fundamental group-scheme. Their central idea was incorporating general quasitorsors into the picture; see [EV10]. This raises the question of whether there is a maximal cover as above for general finite torsors.
In the rest of this introduction, we explain the main steps in achieving Theorem A, whereby outlining the paper and pinpointing some important results obtained in its afterglow.
In studying Question 1.1, it is natural to stick as much as possible to the strategy that was so successful in [C-RST18] for the étale case. That strategy was based on bounding above the degree of a connected element ofȞ 1 (U fl , G) (for any étale G/k). This was done via a novel transformation rule for the F-signature under local étale quasitorsors; see [C-RST18, Theorem B]. Thus, one may hope to get an analogous transformation rule for general local quasitorsors. Under closer inspection, one sees that such transformation was built upon the trace map of such local étale quasitorsors and exploiting its most basic properties. Thus, we are left facing two questions. Do such trace maps exist for general quasitorsors? If so, do they have the same or analogous properties to those of the trace map in the étale case? We dedicate Section 3 entirely to these questions. Fortunately, the former question will turn out to have a quite satisfactory answer whereas the latter will end up making things very interesting. In Section 3, we adapt to our purposes the theory of integrals and traces found in the Hopf algebras literature. We further prove the properties we need from these traces but could not find a reference for. The following theorem summarizes the section. We denote by o(G) the order of a finite algebraic group G/k. Theorem A (Subsection 3.1, Theorem 3.9, Corollary 3.13). Let q : Y → X be a quotient by a finite algebraic group G/k; see Remark 2.3. There exists Tr Y /X ∈ Hom X (q * O Y , O X ) such that: q is G-torsor at x ∈ X if and only if (q * O Y ) x is a free O X,x -module of rank o(G) and Tr Y /X ⊗O X,x freely generates Hom O X,x ((q * O Y ) x , O X,x ) as a (q * O Y ) x -module. In particular, assuming that X, Y are S 2 schemes and q is a G-quasitorsor, the O Y -module q ! O X = Hom X (q * O Y , O X ) is freely generated by its global section Tr Y /X . Using this, we are able to obtain the following transformation rule.
This formula can be used along the same lines of [C-RST18] to investigate Question 1.1 modulo these two hypotheses on Tr S/R . Just as in [C-RST18, §2.3], the surjectivity of the trace (i.e. cohomological tameness [KS10,CEPT96]) can be granted by R being a splinter using Theorem A and so this is not an issue for us either. However, the condition Tr S/R (n) ⊂ m is rather troublesome; see Example 4.10. Recall such condition is automatic in the étale case; see [C-RST18, Lemma 2.10]. This work is largely concerned with understanding such property in the infinitesimal case. Such analysis occupies most of Section 4. To see how we do it, consider the commutative infinitesimal case. There are two diametrically opposite classes of such groups: the linearly reductive groups (e.g. µ p ) and the unipotent ones (e.g. α p ). We can make Tr S/R (n) ⊂ m work in the most critical class of linearly reductive quasitorsors. Namely, Veronese-type cyclic covers (yet those of Kummer-type may still be problematic); see Terminology 4.19. This suffices to solve a longstanding question raised by K. Watanabe [Wat91] on whether canonical covers of strongly F-regular singularities are strongly F-regular: Theorem C (Proposition 4.20, Example 4.22). Let (R, m, k, K) be a strongly F-regular (resp. F-pure) local F p -domain. Let D be a Q-Cartier divisor on Spec R of index n ∈ N and write div a + nD = 0 for some a ∈ K × . Then, the corresponding cyclic cover R ⊂ C = n−1 i=0 R(iD) is strongly F-regular (resp. F-pure) and s(C) = n · s(R). In particular, canonical covers (also known as index-1 covers) of Q-Gorenstein strongly F-regular (resp. F-pure) singularities are strongly F-regular (resp. F-pure) even if p divides the Gorenstein index.
Despite the success we could have in the linearly reductive case, the unipotent case stands firmly as a potential truly troublesome case. We deal with it in a rather radical way: Theorem D (Theorem 4.12). Work in Setup 1.2. Then, 1 X,U (G) :Ȟ 1 (X fl , G) →Ȟ 1 (U fl , G) is surjective for all unipotent (not necessarily commutative) finite algebraic groups G/k. This result should be thought of as a vast generalization of [C-RST18, Corollary 2.11], for the p-groups are the étale unipotent algebraic groups. Thus, it should be thought of as a generalization of the tameness of π ét 1 (U ) in [C-RST18, Theorem A]. Putting together our observations from the linearly reductive as well as the unipotent cases, we get the following result which is the essential step in obtaining our Main Theorem inductively.
Theorem E (Theorem 4.24). Work in Setup 1.2. Suppose that 1 X,U (G) is not surjective for some finite algebraic group G/k such that G • is commutative. There exist a nontrivial linearly reductive finite algebraic group G /k and (R, m, k, K) ⊂ (S, n, k, L) a G -torsor over U but not everywhere such that S is a strongly F-regular k-germ with s(S) = o(G ) · s(R).
Another instance of étale quasitorsors missing part of the picture in characteristic p > 0 is that we were able to bound only the prime-to-p torsion of Cl R in [C-RST18]. Here, we complete the picture by bounding all torsion. By taking cones, we get that globally F-regular varieties have torsion-free divisor class group. We discuss these applications in Section 5.

Notations, conventions, and preliminaries
Convention 2.1. All schemes are defined over an algebraically closed field k of characteristic p > 0. We let F : X → X denote the Frobenius endomorphism of a scheme X. For instance, if R is a ring then F e * R denotes the R-module obtained by restriction of scalars under R → R, r → r p e . Let us recall that a scheme X is said to be F-finite if its Frobenius endomorphism is finite. We assume all our schemes to be F-finite and noetherian, and so excellent. Fibered/tensor products are defined over k unless otherwise explicitly stated.

F-signatures and strong F-regularity
In view of Kunz's theorem [Kun69], it is natural to use Frobenius splittings to measure singularities in positive characteristic. Strongly F-regular singularities were introduced by M. Hochster and C. Huneke [HH89]. Let (R, m, k, K) be a local ring. The e th F-splitting number of R is defined by where λ R (M) denotes the length of an R-module M. This number turns out to be the maximum of the set {a | ∃ F e * R → R ⊕a surjective} ⊂ Z; see [BST12]. In particular, setting δ dim R + log p [k 1/p : k], we have 0 ≤ a e (R) ≤ p eδ where p eδ is the (generic) rank of F e * R and at the same time the free rank of F e * R if R were regular. For details, see [BST12,PT18].
In this way, we may reinterpret Kunz's theorem by saying that: R is regular if and only if a e (R) = p eδ for some (equivalently all) e ≥ 1. Moreover, one says that R is F-pure if a e 0 for some (equivalently all) e ≥ 0. To define strong F-regularity, one considers the asymptotic behavior of the splitting numbers as e → ∞. To be precise, after [HL02,SdB97], one defines the F-signature of R as the limit  [SZ15,ST12]. See [PT18] for more on F-signatures.

Linear and bilinear forms
Let M be a module over a commutative ring R. We denote its dual Hom R (M, R) by M ∨ . A bilinear form on M over R is the same as an element of (M ⊗ R M) ∨ M ∨2 . By ⊗-Hom adjointness, there are two natural isomorphisms υ i : M ∨2 → Hom R (M, M ∨ ). A bilinear form Θ is symmetric if υ 1 (Θ) = υ 2 (Θ), and we write υ(Θ) for either one. It is said to be nondegenerate (resp. nonsingular) if υ(Θ) is injective (resp. an isomorphism). If M is free of finite rank, there is a determinant function det : M ∨2 → R. Note that Θ is nondegenerate (resp. nonsingular) if and only if det Θ is a nonzerodivisor (resp. a unit). If M is locally free of finite rank, we associate to Θ a locally principal ideal det Θ of R, for naturally gives a canonical way to define a bilinear form from a linear one. We refer to θ ∈ S ∨ as nondegenerate or nonsingular if so is ∆ ∨ S/R (θ). If S/R is finite and flat, one defines the discriminant of θ as disc θ det ∆ ∨ S/R (θ).

Group-schemes and their actions on schemes
For the most part, we follow [Mil17,Tat97,Mon93]. Let G/k be an affine group-scheme. We denote by O(G) its corresponding (commutative) Hopf algebra. Given a Hopf algebra H/k, we denote by u : k → H and ∆ : H ⊗ H → H the algebra morphisms of unit and product whereas the coalgebra morphisms counit and coproduct are denoted by e : H → k and ∇ : H → H ⊗ H. For the antipode, we use ι : [Mil17,11.c]. The dual of a finite Hopf algebra H is denoted by which splits as k is perfect [Mil17,11.3]. In other words, G = G • π 0 (G). In fact, G red π 0 (G) and the splitting is (uniquely) given by the canonical closed embedding G red → G.

Examples of affine algebraic groups.
Finite constant group-schemes are examples of finite étale algebraic groups and (since k is separably closed) these are the only ones [Mil17,2.b]. If G is a finite discrete group, the Hopf algebra of the constant group-scheme it defines (also denoted by G) is O(G) = Hom Set (G, k). The coproduct is defined by The counit is the evaluation-at-1 map and the antipode is given by: From a group Γ , one constructs another Hopf algebra as follows. The underlying algebra is k[Γ ]-the group k-algebra of Γ . The coproduct, counit, and antipode are given by ∇(g) = g ⊗ g, e(g) = 1, ι(g) = g −1 , for all g ∈ Γ . If Γ is abelian and finitely generated, we denote the corresponding algebraic group by D(Γ ) and refer to them as the diagonalizable algebraic groups [Mil17,Chapter 12]. For example, D(Z) is the multiplicative group G m , with O(G m ) = k[ζ, ζ −1 ]. Also, D(Z/n) is the group of n-th roots of unity µ n , with O(µ n ) = k[ζ]/(ζ n − 1). Likewise, D(Z ⊕n ) = D n is the group of invertible diagonal n × n matrices. In fact, D is an exact contravariant functor [Mil17,12.9]. Applying it to 0 → Z ·n − → Z → Z/n → 0 yields the Kummer sequence: The additive group G a (see [Mil17,2.1]) is represented by the Hopf algebra k[ξ] where ∇(ξ) = ξ ⊗ 1 + 1 ⊗ ξ, e(ξ) = 0 and ι(ξ) = −ξ. Its e th k-linear Frobenius homomorphism O(G a ) → O(G a ) is given by ξ → ξ p e .
It is an injective homomorphism of Hopf algebras and so it is faithfully flat [Mil17,3.i]. The kernel of the corresponding quotient homomorphism F e : G a → G a is the Cartier self-dual infinitesimal group-scheme α p e [Mil17,2.5], which is represented by O(α p e ) = k[ξ]/ξ p e . Thus, we have the following short exact sequence Likewise, we have the Artin-Schreier exact sequence Let T n ⊂ GL n be the subgroup of upper triangular matrices and U n ⊂ T n be the one of upper triangular matrices with 1 along the diagonal [Mil17,2.9]. In fact, U n is a normal subgroup of T n whose quotient is D n and T n = U n D n .

Trigonalizable groups.
The coradical C 0 of a coalgebra C is defined as the sum of its simple subcoalgebras [Mon93,5.1.5]. This definition applies to a Hopf algebra as well under the caveat that its coradical may not be a Hopf subalgebra. However, we have: Since k is separably closed, we may refer to diagonalizable group-schemes as group-schemes of multiplicative type; see [Mil17,12.8]. These are the commutative linearly reductive groups [Mil17,12.54]. See [Mil17,12.18] for more. Nagata's theorem [Nag62] establishes that G is linearly reductive if and only if p o(π 0 (G)) and G • is of multiplicative type; see [Mon93,Theorem 5.7.4], [Mil17,12.56]. Thus, G is linearly reductive if and only if π 0 (G) has prime-to-p order and G • = D(Γ ) for some abelian group Γ whose torsion is p-torsion.
Unipotent groups are, up to isomorphism, the subgroups of U n [Mil17,14.4]. Now, U n has a central normal series ([Mil17, 6.f]) whose intermediate quotients are canonically isomorphic to G a . Hence, every unipotent group admits a central normal series whose quotients are (isomorphic to) subgroups of G a [Mil17,14.21]. Thus, G a and its subgroups (e.g. α p and Z/p) are the building blocks for unipotent groups. Likewise, the trigonalizable groups are the subgroups of T n [Mil17,16.2]. A trigonalizable group G admits a factorization G u G mt where G u is unipotent and G mt is of multiplicative type; see [Mil17,16.6,16.26].

Actions, quotients, and torsors.
Let Y /k be a scheme and G/k be an affine group-scheme. A G-torsor is a principal G-bundle in the fppf-topology. That is, it is an fppf morphism q : Y → X together with an action a : Y ×G → Y such that q is G-invariant and the induced morphism a×pr 1 : Y ×G → Y × X Y is an isomorphism; see [Mil80, III, Proposition 4.1]. Torsors are quotients as fppf maps are strict epimorphisms [Mil80, I, Theorem 2.17]. Since G/k is affine, the isomorphism classes of G-torsors over X are naturally classified by the pointed-setȞ 1 (X fl , G); see [Mil80,III,§4], [Gir71]. The distinguished point is the class of the trivial G-torsor pr 1 : X × G → X. If φ : G → H is a homomorphism, one defines the map of pointed-setš The right action of H on Y ∧ G H is given by the rule (y, h 1 ) · h 2 = (y, which can be continued using Giraud's second nonabelian cohomology [Gir71,Deb77] · · · →Ȟ 1 (X fl , G ) In the commutative case, this agrees with the long exact sequence from derived-functor abelian cohomology with respect to the fppf site [Gir71, III, §3.4]. Thus, when G is commutative, we shall often write H i (X fl , G) = H i (X fl , G) for i = 0, 1, 2.
Remark 2.2. Let G → G be a faithfully flat homomorphism of finite group-schemes over k with kernel Thus, q : Y → X is locally given by spectra of rings of invariants. By a finite G-quotient q : Y → X, we always mean a morphism as above. If the action of G on Y is free (e.g. q is a G-torsor) then q : Y → X is finite and flat. Likewise, if Y , X are S 2 and G acts freely on a big open of Y (e.g. q is a quasitorsor), then q is finite.
Let us restate the axioms for actions in a way that will be useful later. We may base change (G, ∇, e, ι) by Y /k to get (G Y , ∇ Y , e Y , ι Y )-a group-scheme over Y . The two axioms for right actions translate into the commutative diagrams satisfying the following two commutative diagrams: The ring of coinvariants is ) and the identity e S is given by id ⊗e. Thus, the coaction axioms are the commutative diagrams: In particular, α is injective by the second axiom. If H/k is finite, an action (resp. coaction) of H is the same as a coaction (resp. action) of H ∨ in such a way that rings of invariants and coinvariants coincide. Indeed, if H coacts on S by α : S → S ⊗ H, then H ∨ acts on S via η · s = (id ⊗η)(α(s)) for all η ∈ H ∨ , s ∈ S. It is clear that coinvariant elements are invariant. The converse is true yet it relies on H/k being finite: to check two elements (e.g. α(s) and s ⊗ 1) in the finite rank free S-module S ⊗ H are the same, it suffices to show that their images under id ⊗η agree for all η ∈ H ∨ as these maps generate the S-dual module of S ⊗ H. Thus, it is a priori easier to show that an element is an invariant than a coinvariant although these are a posteriori equivalent due to finiteness. Finally, given an action H ∨ ⊗ S → S, its associated coaction is only defined after choosing a k-basis for H. If h 1 , . . . , h d is a basis with corresponding dual basis η 1 , . . . , η d , then the coaction is given by the rule

Traces of quotients by finite group-schemes
Let q : X → Y be a finite G-quotient as in Remark 2.3. In this section, we construct an O X -linear map Tr Y /X : q * O Y → O X generalizing the classic trace map for étale G/k. Further, we show its properties necessary to study Question 1.1 in the same vein of [C-RST18].

Construction of the trace
We start with the affine case and obtain the general one by gluing on affine charts. If such trace maps Tr S/S G : S → S G exist, they must exist for the action of G on itself, where O(G) G = k. We discuss this core case first and construct Tr S/S G from Tr G/k and the given action. Our first goal then is to explain the existence of a special k-linear map We want Tr G ∈ O(G) ∨ to yield invariants when acting on elements via this action. That is, we need Tr G ·γ to be an invariant for all γ ∈ O(G), i.e.: g · (Tr G ·γ) = g(1)(Tr G ·γ), for all g ∈ O(G) ∨ . This gives the desired property: Following Hopf algebras nomenclature, we are requiring Tr G to be a left integral of the Hopf algebra O(G) ∨ .
To the best of the author's knowledge, this concept was introduced by R. Larson and M. Sweedler in [LS69]. We summarize their definition, existence, and uniqueness in the following theorem. See [Mon93, Ch. 2] for further details.

Remark 3.2 (Geometric description of integrals). Let
where id ⊗τ ⊗ id is the isomorphism swapping the two middle tensor factors of (F The canonical module ω G = u ! ω k is a major example of a Hopf module. Indeed, ω G (or rather its global sections) is a Hopf k-algebra, for it is the dual Hopf algebra of O(G). In particular, there is a product morphism ω G ⊗ ω G → ω G which can be conveniently rewritten as When the Fundamental Theorem of Hopf modules is applied to the Hopf module ω G , we obtain a canonical isomorphism u * (ω G ) G → ω G . On the other hand, it is an immediate consequence of the Remark 3.4. To be consistent with the forthcoming discussion, we ought to define Tr G as the k-linear map γ → t · γ (for a choice 0 t ∈ O(G) ∨ ) rather than γ → t(γ). Nonetheless, we get the same result either way as In other words, the following diagram is commutative For example, t = g∈G g is a left integral and by uniqueness k[G] = k · t. Thus, for the constant group-scheme G, a trace For instance t = ξ p e −1 , and so α p e = k · ξ p e −1 . Hence, a trace for α p e is obtained by projecting onto the direct k-summand generated by ξ p e −1 . In particular, Tr α p e (1) = 0.
We work out the affine case next. Let S be a k-algebra endowed with an action α : is an R-linear map as it is a ring homomorphism and R = S G . We then get an R-linear map Tr G S •α : S → S, which is none other than s → Tr G ·s (using the induced action of O(G) ∨ on S). Definition-Proposition 3.6. The R-linear map Tr G S •α : S → S has image in R. One then defines Tr S/R : S → R to be the restriction of the codomain.
Proof. Recall that Tr G S •α : s → Tr G ·s. One readily verifies that Tr G ·s is an invariant under the action of O(G) ∨ on S and so it is a coinvariant under the coaction of O(G); see Remark 2.4. However, we present a direct, conceptual proof. Consider the following diagram: The bottom sequence is exact by definition. The top sequence, although not necessarily exact, satisfies α(S) ⊂ ker(α ⊗ id, ∇ S ) according to first axiom of α being a coaction. Thus, it suffices to prove that the following two squares are commutative The commutativity of the first square is fairly straightforward. The commutativity of the second one follows from base changing by S the commutative square in Remark 3.4.
Remark 3.7. The trace map Tr S/R : S → R depends on the action of G on S as well as on the choice of Tr G : O(G) → k. Our notation is misleading as it does not reflect that. We hope this ambiguity will not lead to confusion but would rather improve the readability.
For a general finite G-quotient q : Y → X, we may define an O X -linear map Tr Y /X : q * O Y → O X by gluing the trace maps previously constructed on affine charts; see Remark 2.3. To show that trace maps can be glued together, we have: Proposition 3.8. Let R ⊂ S be a finite G-quotient and W ⊂ R be a multiplicatively closed set. Then, Further, the corresponding ring of coinvariants is W −1 R. It follows that

Initial properties of the trace
Let L/k be a finite field extension and G be a finite group acting (on the left) by k-endomorphisms on L. Letting K ⊂ L denote the fixed subfield, then L/K is Galois with Galois group G if and only if G acts faithfully on L. This is equivalent to Tr L/K being nonsingular and having [L : K] = o(G). We obtain the following analog of this principle as the main result of this section. Proof. The statements are local on X and so we may work on an affine chart R S G ⊂ S. Consider the fibered coproduct diagram By definition, R ⊂ S is a G-torsor if and only if it is a fppf map and ϕ u S ⊗ α is an isomorphism. In this case, R ⊂ S is locally free of rank o o(G). Thus, we may assume this and prove that ϕ is an isomorphism if and only if Tr S/R is nonsingular. Note that both statements are local on R and that for all p ∈ Spec R, we have ϕ p = u S p ⊗ α p , and Tr S p /R p = Tr S/R ⊗ R R p by Theorem 3.6. Then, we may assume that R is local and so that R ⊂ S is a semi-local free extension.
To  , 1 ⊗ s 1 , . . . , 1 ⊗ s o is a basis for p 1 : S → S ⊗ R S and 1 ⊗ γ 1 , . . . , 1 ⊗ γ o is a basis for u S : S → O(G S ). Next, we set: That is, M (a m i ) m,i is the S-matrix representation of ϕ in these bases. Thus, ϕ is an isomorphism if and only if M is nonsingular. We describe next the symmetric R-matrix associated to Tr S/R as an R-bilinear form on S, in terms of M. Let T (Tr G (γ m γ n )) m,n be the k-matrix representing Tr G as a k-bilinear form on O(G), in the k-basis γ 1 , . . . , γ o . Recall that T is nonsingular (as well as symmetric) by Remark 3.3. We then have: Proof of the claim. This amounts to the following computation: Since M and T are injective, it remains to explain why if M · v = 0 for a column vector v ∈ R ⊕o then v = 0. This is just another way to say that α is injective: if v = (r 1 , . . . , r d ) and M · v = (t 1 , . . . , t o ) , then α(r 1 s 1 + · · · + r d s d ) = t 1 ⊗ γ 1 + · · · + t o ⊗ γ o . However, α is injective for id S = e S • α-the second action axiom.
In conclusion, the determinant of M T M is a nonzerodivisor on R and so Tr S/R is nondegenerate. If S is further a domain, the determinant of M T M is also a nonzerodivisor on S. Therefore, M T M defines an injective S-linear operator S ⊕o → S ⊕o forcing M to be injective and so ϕ to be an isomorphism. Proof. Since q is flat, the sheaf of principal ideals disc Tr Y /X ⊂ O X cuts out the locus of points x ∈ X where q x : Y × X Spec O X,x → Spec O X,x is not a G-torsor under the induced action. Thus, if q is not a G-torsor then it fails to be so in pure codimension 1.
There is an open neighborhood W x such that q x is faithfully flat for all x ∈ W .

Cohomological tameness and total integrals
We are ready to formulate the notion of tameness our covers will have. In [KS10], several notions of tameness are considered yet cohomological tameness resulted to be the strongest one. Such condition is going to be imposed by splinters, which are those rings splitting off from any finite extension. As a matter of fact, strongly F-regular rings are splinters. See [Ma88,Hoc73].
Remark 3.16. In the Hopf algebras literature, the surjectivity of Tr S/S G is referred to as the existence of total integrals for the right O(G)-comodule S. To the best of the author's knowledge, total integrals were introduced by Y. Doi [Doi85] yet they were formulated in slightly different terms. The equivalence between the existence of Doi's total integrals and the surjectivity of the trace was established in [CF92]. See [Mon93, §4.3].

Proposition 3.17. Let R ⊂ S be a finite G-quotient. Suppose that R is a splinter (and so normal) and that S satisfies S 2 . If R ⊂ S is a G-quasitorsor then it is tame.
Proof. By Corollary 3.13, Tr S/R ·S Hom R (S, R) as S-modules. Since R is a splinter, there is a splitting Tr S/R ·s : S → R of R ⊂ S. That is, Tr S/R ·s : 1 → 1 and so Tr S/R : s → 1.

On the existence of a maximal cover
We come now to Question 1.1. We work in Setup 1.2. Suppose that the restriction of torsors pointed map 1 X,U (G) :Ȟ 1 (X fl , G) →Ȟ 1 (U fl , G) is not surjective for some finite algebraic group G/k. This means that there is a G-torsor V → U that cannot be extended to a G-torsor over X. Our first task is to convert this into a local algebra setting so that Theorem 4.8 can be applied.

Local finite torsors
In this section, we explain why, if 1 X,U (G) is not surjective, there exist a finite algebraic group G /k with (G ) • = G • and a local finite G -quotient (R, m, k, K) ⊂ (S, n, k, L) that is a G -torsor over U but not everywhere and S satisfies S 2 . This is done by taking integral closures. Let h : V → U be a finite morphism. Its integral closure is the finite morphismh : Y → X where Y = Spec S and S H 0 (O V , V ). Taking integral closures is functorial on finite U -schemes and the pullback ofh to U recovers h. Also, if h : V → U is the restriction of a G-torsor Y → X, then Y → X must be the integral closure of h.
Proof. We show that the coaction of a on global sections givesã. Note that Remark 4.2. Let R = S G ⊂ S be a G-quasitorsor. As in Corollary 3.12, let W ⊂ X be the big open over which Y /X is a torsor. Since R is S 2 , then R = H 0 (W , O W ). However, S → H 0 (Y W , O Y W ) may not be an isomorphism unless S was S 2 to start with.
In Lemma 4.1, we say that h extends across its integral closure ifh is a G-torsor. Our problem then reduces to study the extent to which finite G-torsors h : V → U extend across their integral closure. Next, we may assume V connected and so S local (as R is henselian):  G -torsor, where G G × π 0 (G) Gal(W /U ), as X is integral and x ∈ X(k). 7 Note that (G ) • G • as G = G • × π 0 (G) as schemes. It remains to explain why V is connected.
Claim 4.4. If X 2 → X 1 is a finite G 0 -torsor with X 1 and G 0 connected, then X 2 is connected.
Proof of the claim. The following argument was kindly provided to the author by one of the anonymous referees. It is well-known that X 2 → X 1 is a (universal) geometric quotient; see [EvdGM18,Theorem 4.16] for details. Further, since G 0 is homeomorphic to a point, it follows that X 2 → X 1 is a homeomorphism. The claim then follows.
The lemma follows by noting that V → W is a G • -torsor. 6 Equivariant means the existence of a homomorphism ϕ : G → G so that a • (f × ϕ) = f • a , where a, a are the corresponding actions.
7 By [Nor82], if V i → U are finite G i -torsors (i = 0, 1, 2) and f i : V i → V 0 (i = 1, 2) are equivariant maps, then V 1 × V 0 V 2 is a G 1 × G 0 G 2 -torsor over U provided that U is integral and U (k) ∅. In our case, U is integral but U (k) = ∅. This is remedied in [EV10] by using that U ⊂ X and X(k) ∅. Proposition 4.5. Let h : V → U be a G-torsor that is not the restriction of a G-torsor over X. Then, there exists a local finite G -quotient Y = Spec(S, n, k) → X that restricts to a torsor over U but not over X. Moreover, S is an S 2 ring and (G ) • G • .
Remark 4.6. The residue fields are the same because k is algebraically closed. This is to ensure that every cover is endowed with a k-rational point lying over x.

The generalized transformation rule for the F-signature
By the transformation rule in [C-RST18], the size of s(R) ∈ [0, 1]-the F-signature of R-imposes constraints on the existence and size of local extensions R ⊂ S in Proposition 4.5 if G /k is étale. We abstract here the properties we require for this transformation rule to exist in further generality.

holds. In particular, B is a strongly F-regular domain if (and only if) so is A.
Proof. If A is not a domain, then neither is B. Hence, both of them fail to be strongly F-regular as they are local and strongly F-regular algebras are products of strongly F-regular domains; see [HH89]. Therefore Assuming A is a domain, set q : Spec B → Spec A and Then, where λ − is used to denote lengths. Our first hypothesis is q ! A = Hom A (B, A) = B · T . Then, by Grothendieck duality for q, composing-with-T yields an A-isomorphism τ : q * Hom B (F e * B, B) → Hom A (q * F e * B, A) under which τ(q * Hom B (F e * B, b)) = Hom A (q * F e * B, a). Indeed, the inclusion "⊂" follows at once from T (b) ⊂ a. The converse containment; rather its contrapositive, follows from the surjectivity of T , for τ(ϕ) = T • ϕ is surjective if so is ϕ. Our analysis resumes as follows See [Tuc12,Theorem 4.11] for the last equality (cf. [BST12, Proposition 3.5, Lemma 3.6]).
The following result is implicit in the proof of Theorem 4.8.
In order to apply Theorem 4.8 to a finite G-quasitorsor (R, m, k, K) ⊂ (S, n, k, L), we must check that Tr S/R satisfies the three hypotheses in Theorem 4.8. The first one follows from Corollary 3.13 and S being S 2 in Proposition 4.5. The second one follows from R being a splinter; see Proposition 3.17. The third hypothesis, however, will occupy us for the rest of this section. In case it holds, s(S) = o(G) · s(R). It is satisfied in the étale case; see [C-RST18, Lemma 2.10], [Spe20, Lemma 9]. The following example warns us to be careful in general.

Example 4.10. Let S = R[t]/(t p −r)
and consider an arbitrary field k. Note that S is local for all r. However, what its maximal ideal n is depends on r. Let y be the closed point of Spec S and x be the one of Spec R. If r ∈ m, then n = m ⊕ R · t ⊕ · · · ⊕ R · t p−1 and so y is a k-rational point lying over x. If r m, there are two cases depending on whether or not r has a p-th root residually. If r = u p + z for some u ∈ R × and z ∈ m, then n = mS + (t − u), so y is a k-rational point too. If r has no p-th roots even residually, then n = mS. However, that would be impossible if we demand y to be a k-rational point as we have k k(r 1/p ) at the residue fields level. Now, (R, m) ⊂ (S, n) is an α p -torsor for all r ∈ R via the coaction α : t → t ⊗ 1 + 1 ⊗ ξ. If r ∈ m, t p−1 ∈ n as we saw above. However, see Example 3.5. Hence, Tr S/R (n) m. This can also happen even for µ p -torsors. Indeed, if r is a unit, then (R, m) ⊂ (S, n) is a µ p -torsor under the coaction t → t ⊗ ζ. If r = u p + z as above, then t − u ∈ n but Tr S/R (t − u) = u by a similar computation to the one before. Interestingly, if r has no p-th roots even residually, Tr S/R (n) ⊂ m and the transformation rule takes the form p · s(S) = p · s(R), so s(S) = s(R). In view of this, one may ask whether, if r = u p + z with z 0, there is any chance that s(S) ≥ s(R). The following example discards this. Take p = 3, R = k s, z 3 , and r = 1 + z 6 . Then, S = k s, z 2 , z 3 , which is not even normal. The extra variable "s" is for the sake of having a 2-dimensional example.
In Example 4.10, R ⊂ S was a torsor everywhere. This motivates the following question.
Question 4.11. Work in Setup 1.2. Let (R, m, k, K) ⊂ (S, n, k, L) be a finite G-quotient that restricts to a G-torsor over U but not everywhere. Is Tr S/R (n) contained in m?
We investigate Question 4.11 for unipotent and linearly reductive group-schemes separately.

The unipotent case
We show that Question 4.11 is empty if G is unipotent.
We will provide two proofs. The first one is an application of the work in [CEPT96] and hence shorter looking. The second proof was our original approach and is quite direct. We consider the techniques involved in our proof to be quite valuable and interesting in their own right. In fact, we reuse them in the proof of our main theorem (Theorem 4.24).
First proof of Theorem 4.12. According to [CEPT96, Proposition 6.2], if R ⊂ S is a tame G-quotient by a unipotent finite algebraic group G/k, then R ⊂ S must be a G-torsor. Therefore, the result follows from Proposition 4.5 and Proposition 3.17.
The second proof relies on unipotent group-schemes admitting a central normal series whose intermediate quotients are (isomorphic to) subgroups of G a and in particular commutative; see [Mil17,14.21]. In view of this, we will show Theorem 4.12 first in the commutative case and obtain the general case by doing induction on the order.

The commutative unipotent case.
If G is commutative and I = m (i.e. U is the punctured spectrum), we have the following exact sequence from [Bou78, III, Corollaire 4.9]: Recall that, since G is commutative, the group H i (X fl , G) (defined via derived-functor cohomology of sheaves of abelian groups) is isomorphic to the groupȞ i (X fl , G) (defined via torsors and gerbes); see subsubsection 2.3.3. Thus, every commutative G-torsor over U extends to a G-torsor over X if and only if Hom(G ∨ , Pic R/k (U )) is trivial. This could be used to simplify our forthcoming arguments. However, to the best of the author's knowledge, it is unknown whether Boutot's theory of the local Picard scheme and (4.1) extend to general I of height ≥ 2. This is rather limiting for us. First, we are interested in obtaining potential global results like those in [BC-RGST19]. Second, the case I = m is most interesting for surfaces singularities yet we are interested in higher dimensions. We take a closer look at Boutot's arguments in [Bou78,III] to see what information on the cokernel of ρ 1 X,U (G) we can still get. We introduce the following notation (4.2) Ob X,U (G) coker(ρ 1 X,U (G) : H 1 (X fl , G) → H 1 (U fl , G)) Lemma 4.13. Let * → G → G → G → * be a short exact sequence of commutative affine group-schemes over k with Ob X,U (G ) = 0 = Ob X,U (G ). Then, Ob X,U (G) = 0. −) is obtained as the left-derived natural transformation of Γ (X, −) → Γ (U , −) and so it is compatible with the δ-structures. We then have the following commutative and horizontally-exact diagram: Our hypothesis is that second and fourth vertical arrows are surjective. Hence, according to the 5-lemma, to get surjectivity of the third one, we need the fifth arrow to be injective. However, Boutot shows in [Bou78, III, Corollaire 4.9] that H 2 (X fl , G ) = 0 for all commutative G . In fact, all cohomologies higher than 1 vanish; see [Bha12, Proposition 3.1].
The following proposition demonstrates Theorem 4.12 in the commutative case.
Proposition 4.14. If G is a commutative unipotent finite algebraic group then Ob X,U (G) = 0.
Proof. By Lemma 4.13, it suffices to treat the simple case, i.e. G = Z/p and G = α p .
Proof of the claim. From the long exact sequence on flat cohomology derived from (2.2), we get This kernel is zero, by definition, for an F-injective X and I = m. For general I, one can use F-purity to , forcing it to be injective. This proves the claim.
The proposition then holds.

Acknowledgement.
The author would like to thank Christian Liedtke who first observed the results of Claim 4.15 and Claim 4.16 and made him aware of them. These results will be treated in an upcoming work by Christian Liedtke and Gebhard Martin [LMM21b].

The general unipotent case.
To handle the general case, we proceed by induction on the order of G along with the fact it admits a central commutative (necessarily) unipotent subgroup whose quotient is (necessarily) unipotent [Mil17,14.21]. However, we shall require the use of nonabelian first and second flat cohomology as treated in [Gir71,Deb77].
where the horizontal sequences are exact sequences of pointed sets. As before,Ȟ 2 (X fl , G ) = H 2 (X fl , G ) = 0, for G is commutative [Bou78, III, Corollaire 4.9]. The first and third vertical arrows are surjective by the inductive hypothesis. Unfortunately, we cannot apply the 5-lemma to get the surjectivity of the middle one as the sets in consideration are no longer groups. Let us chase the diagram to see how to get around this. Let t 0 ∈Ȟ 1 (U fl , G). It maps to t 1 ∈Ȟ 1 (U fl , G ) which extends to t 2 ∈Ȟ 1 (X fl , G ). However, t 2 lifts to t 3 ∈Ȟ 1 (X fl , G) asȞ 2 (X fl , G ) is trivial. Let t 4 ∈Ȟ 1 (U fl , G) be the restriction of t 3 to U . At this point, we wish to substract t 4 from t 0 as t 4 → t 1 . However, such operation does not make sense in this setting. Fortunately, we can make sense of this by changing the origin via twisted forms; see [Gir71, III, §2.6]. Indeed, consider the conjugate representation G → Aut G of G being defined by the action of G on itself by inner automorphisms. This gives a map of pointed setsȞ 1 (U fl , G) →Ȟ 1 (U fl , Aut G), whereȞ 1 (U fl , Aut G) classifies the so-called twisted forms of G [Gir71, III, §2.5]. Let us write t → t G for the map realizing G-torsors as twisted forms of G. Moreover, we have a bijection: Thus, what we need to do is to twist * → G → G → G → * by the G-torsor t 4 . Indeed, G acts by inner automorphism on both G and G . Then, t 4 can be realized as a twisted form of both G and G . Since G is central, the corresponding twisted form is the trivial one, namely G itself. Since t 4 → t 1 , the twisted form of G given by t 4 is t 1 G . Thus, we have a short exact sequence * → G → t 4 G → t 1 G → * (on U ) and a commutative diagram See [Gir71, III, 3.3.5]. Likewise, we twist * → G → G → G → * by t 3 to get the short exact sequence * → G → t 3 G → t 2 G → * on X. Further, we have the following commutative diagram with exact horizontal sequences. By (4.4), θ t 4 (t 0 ) ∈Ȟ 1 (U fl , t 4 G) maps to θ t 1 (t 1 ) ∈Ȟ 1 (U fl , t 1 G ), which is the trivial t 1 G-torsor. Hence, there exists t 5 ∈ H 1 (U fl , G ) mapping to θ t 4 (t 0 ). Nevertheless, by the inductive hypothesis, t 5 extends to a torsor t 6 ∈ H 1 (X fl , G ), which maps to a torsor t 7 ∈Ȟ 1 (X fl , t 3 G). By commutativity, t 7 restricts to θ t 4 (t 0 ). On the other hand, we have the commutative squarě from which it follows that the unique t 8 ∈Ȟ 1 (X fl , G) such that θ t 3 (t 8 ) = t 7 restricts to t 0 , i.e. 1 X (G)(t 8 ) = t 0 ; as desired.
Lemma 4.18. Work in Setup 1.2. Let * → G → G → G → * be an exact sequence of affine k-group-schemes such that X,U (G ) and X,U (G ) are surjective. IfȞ 1 (X fl , G) →Ȟ 1 (X fl , G ) is surjective (e.g. G is solvable or G = G G ), then so is X,U (G).
Proof. We start by mentioning whyȞ 1 (X fl , G) →Ȟ 1 (X fl , G ) is surjective if either G is solvable or G = G G . Consider the diagram (4.3). Note thatȞ 2 (X fl , G ) is trivial if G is solvable. Indeed, this follows from the commutative case ([Bha12, Proposition 3.1]) by induction on the order and taking long exact sequences on nonabelian cohomologies. Finally, if G → G is a split epimorphism then so iš H 1 (X fl , G) →Ȟ 1 (X fl , G ); so it is surjective.
We chase (4.3) and define t 0 , t 1 , t 2 , t 3 , and t 4 as above. Next, we twist the whole diagram by t 3 to obtain a commutative diagramȞ where θ t 4 (t 0 ) ∈Ȟ 1 (U fl , t 4 G) is mapped to the trivial class inȞ 1 (U fl , t 1 G ). Therefore, θ t 4 (t 0 ) has a preimage t 5 ∈ H 1 (U fl , t 4 G ). On the other hand, we consider the commutaive diagram where the horizontal arrows are bijections. Since the left arrow is surjective, then so is the right arrow. Thus, we may consider t 6 ∈Ȟ 1 (X fl , t 3 G ) to be a preimage of t 5 . Let t 7 ∈Ȟ 1 (X fl , t 3 G) be the image of t 6 . Notice that t 7 maps to θ t 4 (t 0 ) inȞ 1 (U fl , t 4 G). Therefore, the untiwst of t 7 , say t 8 θ −1 t 3 (t 7 ) ∈Ȟ 1 (X fl , G), maps to t 0 underȞ 1 (X fl , G) →Ȟ 1 (U fl , G).

The linearly reductive case
We keep working in Setup 1.2. However, most of our discussion in this section can be relaxed to R not being necessarily strictly henselian. Recall that linearly reductive finite algebraic groups are extensions of étale groups with prime-to-p order by D(Γ ) where Γ is an abelian p-group. Hence, we may focus on µ p e . By Kummer Theory [Mil80, Chapter III, §4], µ n -torsors over U are (up to isomorphisms) in one-to-one correspondence with pairs (L, ϕ), where L is an invertible sheaf on U and ϕ is an isomorphism O U → L n . The µ n -torsor, say V → U , associated to the pair (L, ϕ) is the cyclic cover Spec U n−1 i=0 L i . That is, V is the open subscheme of Spec C(L, ϕ) lying over U where C(L, ϕ) is the following finite R-algebra; see Proposition 4.5. As an R-module, C(L, ϕ) equals n−1 i=0 H 0 (U , L i ). Its ring product is given by the R-linear maps: The coaction C(L, ϕ) → C(L, ϕ) ⊗ O(µ n ) is defined by sending f ∈ H 0 (U , L i ) to f ⊗ ζ i . Therefore, the trace Tr C(L,ϕ)/R is the projection onto the zeroth-degree summand in the above direct sum. See Example 3.5. Let us write L = O U (D) for some Cartier divisor D on U . Since U is big, D extends uniquely to a Weil divisor on X. Thus, Defining an isomorphism ϕ : O U → O U (nD) amounts to give a ∈ K × such that div(a) + nD = 0, which implies R(nD) = R · a ⊂ K. Hence, we may present the data of a cyclic cover as C(L, ϕ) = C(D, a; n). In this way, the ring product is performed internally by the pairing if i + j < n. In case m i + j − n ≥ 0, we have: where the first isomorphism " ← −" uses that nD is Cartier. Succinctly, if i + j ≥ n, we have Proposition 4.20. With notation as above, suppose that R ⊂ C(D, a; n) is of Veronese-type. Then, C = C(D, a; n) is a local domain with maximal ideal n C = m ⊕ n−1 i=1 R(iD) and field of fractions K(a 1/n ). In particular, Question 4.11 has an affirmative answer in this case. Therefore, C, n C , k, K(a 1/n ) is a strongly F-regular k-germ with s(C) = n · s(R) (or just F-pure if R were assumed only F-pure). Remark 4.21. Proposition 4.20 works just fine if we do not assume that R is strictly henselian. In that case, C would not be necessarily strictly henselian either.
Example 4.22. Suppose that R is Q-Gorenstein with canonical divisor K R of index n; say div a + nK R = 0. The corresponding Veronese-type cyclic cover is called a canonical (or index-1) cover of R. As a direct application of Proposition 4.20, both strong F-regularity and F-purity transfer to canonical covers. Assuming p n, this was originally demonstrated by K.-i. Watanabe [Wat91] but remained open otherwise. It is worth noting that neither F-rationality nor KLT-ness are transferred to cyclic covers in general [Sin03,Kaw99].
Let us look at Question 4.11 for cyclic covers, i.e. S = C, G = µ n . We learned from Example 4.10 that Kummer-type cyclic covers may be a source of problems for us whereas from Proposition 4.20 that Veronese-type ones are suitable for our purposes. A general cyclic cover is an intricate mixture of both types and so we cannot expect to have an affirmative answer for Question 4.11 in general. At least, we have the following way out.
Proposition 4.23. Work in the setup of Question 4.11 assuming G = µ n . Then, there exist a nontrivial Veronesetype cyclic cover (R, m, k, K) ⊂ (S , n , k, L ). In particular, (S , n , k, L ) is a strongly F-regular k-germ with F-signature at least 2 · s(R).
Proof. Write S C(D, a; n) as above. By hypothesis, the index of D is not 1. Thus, we may take S to be C (D, b, m), where m 1 is the index of D and we choose b ∈ K × so that div b + mD = 0. For the last statement, use Proposition 4.20.

Main results
We are ready to summarize with our main results. We commence by answering to Question 4.11 when G • is trigonalizable.
Theorem 4.24. Work in Setup 1.2. Suppose that 1 X,U (G) is not surjective and G • is trigonalizable. Then, there exist a nontrivial finite linearly reductive group-scheme G /k and a G -quotient (R, m, k, K) ⊂ (S , n , k, L ) restricting to a G -torsor over U but not everywhere such that the answer to Question 4.11 is affirmative in this case. In particular, (S , n , k, L ) is a strongly F-regular k-germ with s(S ) = o(G ) · s(R) and so o(G ) ≤ 1/s(R).
Proof. Using G = G • π 0 (G) as in Remark 4.7 and [C-RST18, Corollary 2.11], we may assume that G/k is infinitesimal and further trigonalizable by hypothesis. Thus, we may write G = G u G mt as in subsubsection 2.3.2. By Lemma 4.18 and Theorem 4.12, we may assume G = G mt i µ p e i . Using Lemma 4.13, we may say G = µ p e and the result then follows directly from Proposition 4.23.
is a G i -quasitorsor and: (1) G i /k is a linearly reductive finite algebraic group for all i, is surjective for all big opens U ⊂ X and all finite algebraic groups G/k for which G • is solvable.
Proof. By iterating Theorem 4.24 until s(R) is exhausted, we obtain the weaker version of this result where in part (c) we assume that G • is trigonalizable. To see part (b), just notice that the transformation rule in Theorem 4.8 yields It remains to show part (c) when we let G • to be solvable. As in the proof of Theorem 4.24, it suffices to prove that 1 X ,U (G) is surjective for all infinitesimal solvable groups G/k. We proceed by induction on o(G). The base case is covered by the commutative case. We may assume G is not commutative and so that there is a nontrivial proper normal subgroup G ⊂ G; which is necessarily solvable, such that G = G/G is commutative. The result follows by applying Lemma 4.18 and the inductive hypothesis.
Remark 4.26. The maximal cover R ⊂ R in Theorem 4.24 can be constructed as R ⊂ Ré t ⊂ R where R ⊂ Ré t is the universal Galois quasi-étale cover from [C-RST18], which is tame of prime-to-p degree, and Ré t ⊂ R is obtained as a finite chain of infinitesimal Veronese-type cyclic covers and so it is purely inseparable of degree p e for some e ≥ 0. It is unclear to the author whether R ⊂ R can be realized as a G -quasitorsor for some linearly reductive algebraic group G /k. It is even unclear to him whether this can be done for Ré t ⊂ R (see [TW92, Example 2.7]). However, this is true in dimension 2 by [LMM21a, Theorem 11.2].

Relation with Esnault-Viehweg's local Nori fundamental group-scheme
It is natural to wonder what is the relationship between our maximal cover R ⊂ R ; see Theorem 4.25, and Esnault-Viehweg's construction of the local Nori fundamental group-scheme in [EV10]. For simplicity, let us consider the commutative case only. Then, the property our cover has is that its abelian local Nori fundamental group-scheme π N,ab 1,loc (U , X , x ) is trivial. Nevertheless, it is not clear to the author whether or not this implies that π N,ab 1,loc (U , X, x) is finite. He is deeply thankful to Christian Liedtke who pointed out to him an example suggesting this should not be always the case. The example is a surface D 4 singularity in characteristic 2 that is a Z/2Z-quotient ofÂ 2 k but admits a nontrivial µ 2 -quasitorsor. The singularity is k x, y, x /(z 2 − xyz − x 2 y − xy 2 ). The divisor defined by (x, z) has index 2. Therefore, its local abelian Nori fundamental group-scheme cannot be Z/2Z as it must take into account µ 2 as well. This singularity (although F-pure) is not strongly F-regular. See [ST14, Example 7.12] for a closer look into this singularity, cf. [Art75].

Beyond the solvable case
The author is grateful to Axel Stäbler for making him aware of the classification of simple infinitesimal rank-1 algebraic groups via the classification of simple restricted Lie algebras by Block-Wilson-Premet-Strade [BW88,SW91,PS08]. See [Viv10] for a nice account. With 1 X,U (G) as above, the following questions are of interest.
Question 4.27. For which simple infinitesimal rank 1 algebraic groups G/k is 1 X,U (G) surjective? Is this the case for those G/k of Cartan type?
Question 4.28. If G/k is a simple infinitesimal rank 1 algebraic group such that 1 X,U (G) is not necessarily surjective, do we have an analog of Proposition 4.20?
Question 4.29. Given a simple infinitesimal rank 1 algebraic group G/k, for which type of (F-)singularity X, if any, is 1 X,X {m} (G) naturally surjective? Question 4.30. Let S be the spectrum of a strictly local ring and G/S be a (not necessarily solvable) finite flat group-scheme. Does the vanishingȞ 2 (S fl , G) = 0 hold?
As we have seen in the proofs of Theorem 4.12 and Theorem 4.25, we need to answer this last question in order to bootstrap the simple case to the general case.

Applications to divisor class groups
The methods in [C-RST18] were able to bound the prime-to-p torsion of Cl R = Pic U from above by 1/s(R). Using Proposition 4.20, we are able to bound all torsion. Recall that Cl R ⊂ Cl R h ⊂ ClR for a normal local domain R; see [Mum76,§1C].
Corollary 5.1. Let R be a local strongly F-regular ring. If D ∈ Cl R is torsion with index n, then n ≤ 1/s(R) (i.e. Cl R is n-torsion with n ≤ 1/s(R) ). Thus, Cl R is torsion-free if s(R) > 1/2.
Remark 5.2. In view of Corollary 5.1, one may wonder whether (Cl R) tor is finite with order at most 1/s(R). After a former version of this preprint was public, this was addressed effectively by T. Polstra and I. Martin [Pol22,Mar21]. We would like to remark next that finiteness of (Cl R) tor follows from our methods as well. However, the author acknowledges that he only thought of this possibility after seeing Polstra's and Martin's work. The argument is as follows. Let R ⊂ R be as in Theorem 4.24. Note that Cl R is torsion-free. Moreover, the kernel of the induced homomorphism Cl R → Cl R is (Cl R) tor . Indeed, R ⊂ R restricts to a finite faithfully flat cover U → U of degree  The author is thankful to Fabio Bernasconi for showing him the following example and for kindly letting him to include it here. The example shows that we cannot replace Pic Y by Cl Y in Corollary 5.4, i.e. there are globally F-regular varieties with torsion in their divisor class group. In fact, it shows that there are (necessarily non-factorial) toric varieties in any characteristic with torsion in their divisor class group.
Example 5.7. Set X = P 1 × P 1 and denote the projection morphisms by π 1 , π 2 : X → P 1 . Here, we may define P 1 over an arbitrary algebraically closed field of any characteristic (including zero). Let us write P 1 = G m ∪ {0, ∞} for the standard toric structure on P 1 . For i = 1, 2, let us define F i π * i 0 ⊂ X. Let X 1 → X be the blowup of X at the intersection point F 1 ∩ F 2 and let E ⊂ X 1 denote the corresponding exceptional curve. By abuse of notation, we shall denote the strict transform of F i in X 1 by F i . Let X 2 → X 1 be the blowup of X 1 at the intersection points F 1 ∩ E and F 2 ∩ E, and let D 1 , D 2 denote the corresponding exceptional curves. Observe that X 1 and X 2 are toric varieties, for we blew up points fixed by the torus action. Note that E is a (−3)-curve on X 2 whereas F 1 and F 2 are (−2)-curves (once again, we have denoted strict transforms by the same letters). By contracting the curves F 1 and F 2 on X 2 (which are fixed by the torus action), we obtain a toric surface Y with two singular points y 1 , y 2 ∈ Y corresponding respectively to the contractions of F 1 , F 2 ⊂ X 2 along X 2 → Y . It is worth noting that these are A 1 points and so non-factorial. In fact, Cl O Y ,y i Z/2. For i = 1, 2, let D i ⊂ Y be the push-forward of D i ⊂ X 2 along the contraction X 2 → Y . Observe that D D 1 − D 2 has Cartier index 2, for it generates the divisor class groups of O Y ,y 1 and O Y ,y 1 . In particular, D is not linearly equivalent to zero. However, by the symmetry in the construction, one sees that D is numerically trivial. Thus, 2D is a Cartier divisor numerically equivalent to zero. Since