Proof. The claim is a special case of [Reference Gille and PoloSGA 3_{III new}, Exposé XXIV, Proposition 3.13, Corollaire 3.14, Proposition 8.4]. The relevant $S'$ is the scheme of Dynkin diagrams of G defined in [Reference Gille and PoloSGA 3_{III new}, Exposé XXIV, Section 3.2 and below].

Proof. Let $T \subset B \subset G$ be a maximal R-torus and an R-Borel subgroup of G (see [Reference Gille and PoloSGA 3_{III new}, Exposé XXIV, Section 3.9]), and let $T' \subset B' \subset (G^{\mathrm {der}})^{\mathrm {sc}}$ be the corresponding maximal R-torus and Borel R-subgroup of $(G^{\mathrm {der}})^{\mathrm {sc}}$ (see [Reference Gille and PoloSGA 3_{III new}, Exposé XXVI, Proposition 1.19]). We have compatible isogenies

$$ \begin{align*} \operatorname{\mathrm{rad}}(G) \times (G^{\mathrm{der}})^{\mathrm{sc}} \rightarrow G \quad\text{and}\quad \operatorname{\mathrm{rad}}(G) \times T' \rightarrow T \end{align*} $$

that have a common finite kernel Z of multiplicative type, and Lemma 2.1 ensures that $H^1(R, T') = 0$. We will use the resulting commutative diagram of long exact cohomology sequences

as well as its analogue over K. Thus, to proceed, we fix an $\alpha \in \operatorname {\mathrm {Ker}}(H^1(R, G) \rightarrow H^1(K, G))$.

By [Reference Colliot-Thélène and SansucCTS87, Theorem 4.3] (which is where we use the assumption on the strict Henselizations of R), the map $H^2(R, Z) \rightarrow H^2(K, Z)$ is injective. Chasing the diagram above and its analogue over K, we therefore find that $\alpha $ comes from some pair

$$ \begin{align*} (\beta, \gamma) \in H^1(R, \operatorname{\mathrm{rad}}(G)) \times H^1(R, (G^{\mathrm{der}})^{\mathrm{sc}}) \end{align*} $$

and that, by the nature of the second vertical arrow there, $\gamma |_K$ is trivial. The assumption on $(G^{\mathrm {der}})^{\mathrm {sc}}$ then implies that $\gamma $ is trivial so that $\alpha $ lifts to a $\beta \in H^1(R, \operatorname {\mathrm {rad}}(G))$ for which $\beta |_K$ lifts to $H^1(K, Z)$. The image of this $\beta $ in $H^1(R, T)$ lifts $\alpha $ and is trivial over K. By [Reference Colliot-Thélène and SansucCTS87, Theorem 4.1] (the Grothendieck–Serre Conjecture 1.1 for T), this image of $\beta $ is trivial, so $\alpha $ is also trivial.

Proof. The claim is a special case of [Reference Gabber, Liu and LorenziniGLL15, Theorem 5.1] (with definitions reviewed in [Reference Gabber, Liu and LorenziniGLL15, page 1207]).

Proof. The pure dimension hypothesis means that all the irreducible components of X have the same dimension, so [Reference Grothendieck and DieudonnéEGA IV₂, Proposition 5.2.1] ensures that X is biequidimensional in the sense that the saturated chains of specializations of its points all have the same length. Similarly to [Reference ČesnavičiusČes21, Section 4.1], part (i) then ensures that each $X \cap H_1 \cap \dotsc \cap H_i$ inherits biequidimensionality, so is also of pure dimension. This reduces us to $t = 1$: by applying this case iteratively and at each step adjoining to the $Y_j$’s all their possible intersections with some of the already chosen $H_i$’s (to ensure (iii)), we will obtain the general case. In the case $t = 1$, we fix closed points $y_1, \dotsc , y_{n} \in X\setminus Z$ that jointly meet every irreducible component of X and of every $Y_j \setminus Z$. Both (iii) and the dimension aspect of (i) will hold as soon as $H_1$ contains no $y_{j}$, so at the cost of requiring this we may forget about the $Y_j$.

For the rest of the argument, we begin with the case when $\operatorname {\mathrm {char}} k = 0$, in which we will use the ‘classical’ Bertini theorem (one could also choose to skip this case because we will only use Lemma 3.2 for finite k). For this, we first claim that for every large $N> 0$ there are global sections $h_i$ of $\mathscr {O}_X(N)$ whose common zero locus contains Z and set-theoretically equals to it. Indeed, by repeatedly applying [Reference Grothendieck and DieudonnéEGA III₁, Corollaire 2.2.4] to shrink the base locus, we first build such $h_{i^{\prime }}'$ (resp., $h_{i^{\prime \prime }}^{\prime \prime }$) for some $N'$ (resp., $N^{\prime \prime }$) that is a power of $2$ (resp., of $3$), then express every large N as $aN' + bN^{\prime \prime }$ with $a, b> 0$, and, finally, let $h_i$ be the collection of all the $h_{i^{\prime }}^{\prime a}h_{i^{\prime \prime }}^{\prime \prime b}$. By [Reference Grothendieck and DieudonnéEGA III₁, Corollaire 2.2.4] and [Reference Grothendieck and DieudonnéEGA IV₄, Corollaire 17.15.9] (which uses the separable residue field assumption), granted that N is sufficiently large, we may build another global section $h_0$ of $\mathscr {O}_X(N)$ whose associated hypersurface contains Z and is smooth at every point of $Z_0$. We adjoin this $h_0$ to the $h_i$ and then discard linear dependencies to assume that the $h_i$ are k-linearly independent. By [Reference Grothendieck and DieudonnéEGA II, Proposition 4.2.3], the $h_i$ determine a morphism

$$ \begin{align*} X\setminus Z \rightarrow \mathbb{P}^{N'}_k \end{align*} $$

such that the pullback of $\mathscr {O}_{\mathbb {P}^{N'}_k}(1)$ is our $\mathscr {O}_{X\setminus Z}(N)$. The hyperplanes in $\mathbb {P}^{N'}_k$ and, compatibly, the nonzero k-linear combinations of the $h_i$ up to scaling are parametrized by the dual projective space. Due to the existence of a k-linear combination of the $h_i$ whose associated hypersurface does not contain a fixed $y_j$, a generic such hypersurface contains no $y_j$. Likewise, due to the openness of the smooth locus, the existence of a k-linear combination of the $h_i$ whose associated hypersurface is smooth at all the points in $Z_0$, and [Reference Grothendieck and DieudonnéEGA IV₃, Théorème 11.3.8 b) c)], a generic such hypersurface is smooth at all the points in $Z_0$. Finally, by the Bertini theorem [Reference JouanolouJou83, Corollaire 6.11 2)], the hypersurface H associated to a generic k-linear combination of the $h_i$ is such that $(X^{\mathrm {sm}} \setminus Z) \cap H$ is k-smooth. In conclusion, since k is infinite, we may choose our desired $H_1$ to be a generic such H.

The remaining case when $\operatorname {\mathrm {char}} k = p> 0$ is a very minor sharpening of [Reference GabberGab01, Corollary 1.6] that is proved as there. Namely, we use the pure dimension hypothesis to apply [Reference GabberGab01, Theorem 1.1]Footnote ² with

• ○ U there being our $X^{\mathrm {sm}} \setminus (Z \cup \{y_1, \dotsc , y_{n} \})$ and $\mathcal {E}$ there being $\Omega ^1_U$;

• ○ Z there being our $Z_1 \cup \{y_1, \dotsc , y_{n}\} \cup \bigcup _{z \in Z_0} \underline {\operatorname {\mathrm {Spec}}}_{\mathscr {O}_X}(\mathscr {O}_X/\mathscr {I}_{z}^2)$;

• ○ $\Sigma $ there being our $Z_0 \cup \{y_1, \dotsc , y_{n}\}$;

• ○ $m_0$ there being $0$; and

• ○ $\sigma _0$ there being $0$ on our $Z_1$, a unit on each of our $y_1, \dotsc , y_{n}$ and a nonzero cotangent vector at $z \in Z_0$ on each of our $\underline {\operatorname {\mathrm {Spec}}}_{\mathscr {O}_X}(\mathscr {O}_X/\mathscr {I}_{z}^2)$;

to find a finite set of closed points

$$ \begin{align*} F \subset X^{\mathrm{sm}} \setminus (Z \cup \{ y_1, \dotsc, y_{n} \}) \end{align*} $$

and, for every large N divisible by p, a global section h of $\mathscr {O}_X(N)$ whose associated hypersurface contains Z, has a k-smooth intersection with $X^{\mathrm {sm}} \setminus (F \cup Z \cup \{y_1, \dotsc , y_{n} \})$, passes through every $z \in Z_0$ and is k-smooth there (for this we use [Reference Grothendieck and DieudonnéEGA IV₄, Corollaire 17.15.9] and the separable residue field assumption as in the characteristic $0$ case), and does not pass through any $y_{j}$. By [Reference Grothendieck and DieudonnéEGA III₁, Corollaire 2.2.4], if this N is sufficiently large, then there is a global section $h'$ of $\mathscr {O}_X(N/p)$ that vanishes on $Z \cup \{y_1, \dotsc , y_{n}\}$ and is such that $h + h^{\prime p}$ does not vanish at any point of F. We may then let $H_1$ be the hypersurface associated to $h + h^{\prime p}$.

Proof. We decompose $\operatorname {\mathrm {Spec}}(\mathcal {O})$ and $W := \operatorname {\mathrm {Spec}}(A)$ into connected components to assume that $\mathcal {O}$ and A are domains so that A is of pure relative dimension $d> 0$ over $\mathcal {O}$. We embed W into some affine space over $\mathcal {O}$ and then take the schematic image in the corresponding projective space to build an open immersion $W \hookrightarrow X$ into a projective, flat $\mathcal {O}$-scheme X. On the K-fiber with $K := \operatorname {\mathrm {Frac}}(\mathcal {O})$ this immersion has dense image, so $X_K$ is of pure dimension d. It then follows from [Reference de JongSP, Lemmas 0D4J and 02FZ] that X is of pure relative dimension d over $\mathcal {O}$. We will obtain our U and $\pi $ from X via Variant 3.7 (vii). To apply the latter, all we need to do is to check that the schematic image $\overline {Y}$ of Y in X is such that $Y' := \overline {Y} \setminus Y$ is $\mathcal {O}$-fiberwise of codimension $> 1$ in X.

By [Reference de JongSP, Lemma 01R8], set-theoretically we have $\overline {Y} = \bigcup _{y} \overline {\{ y\}}$ where y ranges over the generic points of Y and the closures are in X. Each y is of height $\ge 2$ in X, so each $\overline {\{y \}}$ intersects the $\mathcal {O}$-fiber of X containing y in a closed subscheme of dimension $\le d - 1$ (even $\le d - 2$ if the fiber is generic). Thus, since $\overline {\{y\}}$ has a nonempty open $\overline {\{y \}} \cap W$, the contribution of y to its $\mathcal {O}$-fiber of $Y'$ is of dimension $\le d - 2$. The only situation in which $\overline {\{y\}}$ may contribute to other $\mathcal {O}$-fibers of $Y'$ is when y lies in the generic $\mathcal {O}$-fiber of X and $\mathcal {O}$ is not a field. However, since the local rings of X are of dimension $\le d + 1$, then the intersection of $\overline {\{y\}}$ with any closed $\mathcal {O}$-fiber of X is of dimension $\le d - 2$. In conclusion, $Y'$ is $\mathcal {O}$-fiberwise of dimension $\le d - 2$, that is, $\mathcal {O}$-fiberwise of codimension $> 1$ in X.

Proof. We decompose $\operatorname {\mathrm {Spec}}(\mathcal {O})$ and $\operatorname {\mathrm {Spec}}(R)$ into connected components to assume that $\mathcal {O}$ and R are domains and then likewise assume that A is a domain. If A is of relative dimension $0$, then R is a Dedekind domain, so, by [Reference GuoGuo20, Theorem 1], our torsor E is trivial and we may choose $C = \mathbb {A}^1_R$ and $\mathscr {E} := E_{\mathbb {A}^1_R}$, the closed subscheme Z being empty and s being the zero section. Thus, we may assume that A is $\mathcal {O}$-fiberwise of pure dimension $d> 0$. Moreover, we localize A and spread out to assume (abusively, from a notational standpoint) that G, B and E begin life over A and, for the last aspect of the claim, that a fixed extension of $B \subset G$ to a quasi-pinning of G likewise exists over A.

By [Reference Gille and PoloSGA 3_{III new}, Exposé XXVI, Corollaire 3.6, Lemme 3.20], the quotient $E/B$ is representable by a projective A-scheme. Thus, due to the generic triviality of E and the valuative criterion of properness, there is a closed subscheme $Y \subset \operatorname {\mathrm {Spec}}(A)$ of codimension $\ge 2$ such that $(E/B)_{\operatorname {\mathrm {Spec}}(A)\setminus Y}$ has a section that generically lifts to E, in other words, such that $E_{\operatorname {\mathrm {Spec}}(A)\setminus Y}$ reduces to a generically trivial $B_{\operatorname {\mathrm {Spec}}(A)\setminus Y}$-torsor $E^B$. Consider the torus

$$ \begin{align*} T := B/\mathscr{R}_u(B) \quad\text{and the induced}\ T_{\operatorname{\mathrm{Spec}}(A)\setminus Y}\text{-torsor}\quad E^T := E^B/\mathscr{R}_u(B). \end{align*} $$

Since Y is of codimension $\ge 2$ in the regular scheme $\operatorname {\mathrm {Spec}}(A)$, by [Reference Colliot-Thélène and SansucCTS79, Corollaire 6.9], there is a unique T-torsor $\widetilde {E^T}$ that extends $E^T$ to all of $\operatorname {\mathrm {Spec}}(A)$. Since the Grothendieck–Serre conjecture is known for tori [Reference Colliot-Thélène and SansucCTS87, Theorem 4.1 (i)] and $\widetilde {E^T}$ is generically trivial, the base change of $\widetilde {E^T}$ to $\operatorname {\mathrm {Spec}}(R)$ is trivial. Thus, we may localize A further around the maximal ideals of R to assume that $\widetilde {E^T}$ is trivial so that $E^T$ is also trivial and $E_{\operatorname {\mathrm {Spec}}(A)\setminus Y}$ reduces to an $\mathscr {R}_u(B)$-torsor.

We now apply Proposition 4.1 to obtain

• ○ an affine open $U \subset \operatorname {\mathrm {Spec}}(A)$ containing $\operatorname {\mathrm {Spec}}(R)$;

• ○ an affine open $S \subset \mathbb {A}^{d - 1}_{\mathcal {O}}$; and

• ○ a smooth $\mathcal {O}$-morphism $U \rightarrow S$ of pure relative dimension $1$ such that $Y \cap U$ is S-finite.

Since R is a localization of the coordinate ring of U, we then set

$$ \begin{align*} C := U \times_S \operatorname{\mathrm{Spec}}(R) \quad\text{and}\quad Z := (Y \cap U) \times_S \operatorname{\mathrm{Spec}}(R). \end{align*} $$

The R-scheme C comes equipped with an R-point $s \in C(R)$ induced by the diagonal of $\operatorname {\mathrm {Spec}}(R)$ over S, and, by base change, (i)–(iii) hold. Finally, we let $\mathscr {G}$, $\mathscr {B}$ and $\mathscr {E}$ be the base changes to C of the restrictions of G, B and E to U so that, by construction, their s-pullbacks are G, B and E, respectively. Since $U \subset \operatorname {\mathrm {Spec}}(A)$ and, by construction, $E_{\operatorname {\mathrm {Spec}}(A)\setminus Y}$ reduces to an $\mathscr {R}_u(B)$-torsor, the restriction of $\mathscr {E}$ to $C \setminus Z$ reduces to an $\mathscr {R}_u(\mathscr {B})$-torsor. In particular, (iv) and (v) hold. To likewise arrange the final claim about quasi-pinnings, simply take the fixed extension of $B \subset G$ to a quasi-pinning of G and base change it to C.

Proof. We consider the functor X that parametrizes those isomorphisms between the base changes of G and $G'$ that preserve the (corresponding base changes of the) fixed quasi-pinnings. By [Reference Gille and PoloSGA 3_{III new}, Exposé XXIV, Corollaire 1.10, Section 3.10], this X is representable by a scheme that becomes constant étale locally on A. Thus, by [SGA 3_II, Exposé X, Corollaire 5.14] (with [Reference Grothendieck and DieudonnéEGA I, Corollaire 6.1.9]), the geometrically unibranch assumption ensures that the connected components of X are open subschemes that are finite étale over A. The $A/I$-point $\iota $ of X meets finitely many such components, whose union is then the spectrum of a finite étale A-algebra $\widetilde {A}$. By adjoining further components if needed, we may ensure that the closed A-fibers of $\widetilde {A}$ are nonzero so that $\widetilde {A}$ is faithfully flat over A, as desired.

Proof. We decompose $\operatorname {\mathrm {Spec}}(R)$ into connected components to assume that R is a domain. By Proposition 4.2, there are such C, s, Z and $\mathscr {E}$, except that $\mathscr {E}$ there is a torsor under a quasi-split reductive C-group scheme $\mathscr {G}$ that may not be $G_C$ but that comes equipped with a Borel C-subgroup $\mathscr {B} \subset \mathscr {G}$ whose s-pullback is $B \subset G$ and that extends to a quasi-pinning of $\mathscr {G}$ whose s-pullback is a fixed quasi-pinning of G extending B. We replace C by its connected component containing the image of s to arrange that C be connected. Thus, the geometric C-fibers of $\mathscr {G}$ and $G_C$ are of constant types so that, by the condition on the s-pullback, these types are the same.

To replace $\mathscr {B} \subset \mathscr {G}$ by $B_C \subset G_C$, we first use prime avoidance [Reference de JongSP, Lemma 00DS] to construct the semilocalization $\operatorname {\mathrm {Spec}}(A)$ of C at the union of the closed points of Z and of those of the image of s. Since C is R-smooth, the ring A is regular. The image of s gives rise to a closed subscheme $\operatorname {\mathrm {Spec}}(R) \subset \operatorname {\mathrm {Spec}}(A)$ cut out by an ideal $I \subset A$ and, by assumption, $\mathscr {B}_{A/I} \subset \mathscr {G}_{A/I}$ agrees with $B \subset G$, and even the quasi-pinnings of $\mathscr {G}$ and G are likewise compatible. Thus, by Lemma 5.1, there is a finite étale $\operatorname {\mathrm {Spec}}(A)$-scheme $\operatorname {\mathrm {Spec}}(\widetilde {A})$ equipped with an R-point $\widetilde {s}$ lifting s such that $\mathscr {B}_{\widetilde {A}} \subset \mathscr {G}_{\widetilde {A}}$ is isomorphic to $B_{\widetilde {A}} \subset G_{\widetilde {A}}$ compatibly with the fixed identification of $\widetilde {s}$-pullbacks. We may spread out

$$ \begin{align*} \operatorname{\mathrm{Spec}}(\widetilde{A}) \rightarrow \operatorname{\mathrm{Spec}}(A) \quad\text{to a finite }\acute{\text{e}}\text{tale morphism}\quad \widetilde{C} \rightarrow C' \end{align*} $$

for some affine open $C' \subset C$ that contains Z and the image of s, while preserving an $\widetilde {s} \in \widetilde {C}(R)$ and an isomorphism between $\mathscr {B}_{\widetilde {C}} \subset \mathscr {G}_{\widetilde {C}}$ and $B_{\widetilde {C}} \subset G_{\widetilde {C}}$. To arrive at the desired conclusion, it then remains to replace C, s, Z and $\mathscr {E}$ by $\widetilde {C}$, $\widetilde {s}$, $Z \times _{C} \widetilde {C}$ and $\mathscr {E} \times _C \widetilde {C}$, respectively.

Proof. The lemma is a variant of, for instance, [Reference PaninPan19a, Lemma 6.1] or [Reference FedorovFed21b, Lemma 4.5], and we will prove it by using similar arguments as there due to Panin. Since R is semilocal, the finite R-scheme Z has finitely many closed points, which all lie over maximal ideals of R. Thus, we begin by using Lemma 3.1 to construct the semilocalization S of C at the closed points of Z so that Z is also a closed subscheme of S and $s \in S(R)$. It then suffices to construct a finite étale S-scheme $\widetilde {S}$ such that s lifts to an R-point $\widetilde {s} \in \widetilde {S}(R)$ and the preimage $\widetilde {Z} \subset \widetilde {S}$ of Z satisfies the displayed inequalities: indeed, once this is done, we may first spread $\widetilde {S}$ out to a finite étale scheme over an open neighborhood of S in C and then form its schematic image [Reference de JongSP, Lemma 01R8] in the factorization supplied by the Zariski main theorem [Reference Grothendieck and DieudonnéEGA IV₄, Corollaire 18.12.13] to further extend to a desired finite morphism $\widetilde {C} \rightarrow C$.

We view s as a closed subscheme $\operatorname {\mathrm {Spec}}(R) \subset Z$ and we list the closed points of Z (that is, of S):

• ○ the closed points $y_1, \dotsc , y_{m}$ of Z not in s with an infinite residue field;

• ○ the closed points $z_1, \dotsc , z_{n}$ of Z not in s with a finite residue field;

• ○ the closed points $y_1', \dotsc , y^{\prime }_{m'}$ of s with an infinite residue field;

• ○ the closed points $z_1', \dotsc , z^{\prime }_{n'}$ of s with a finite residue field.

For any $N> 1$, we may choose monic polynomials

• ○ $f_{y_i} \in k_{y_i}[t]$ that are products of N distinct linear factors; and

• ○ $f_{z_j} \in k_{z_j}[t]$ that are irreducible of degree N.

Likewise, we may choose a monic polynomial $f_{s} \in tR[t]$ of degree N such that

• ○ the image of $f_{s}$ in each $k_{y_i'}[t]$ is a product of N distinct linear factors; and

• ○ the image of $f_{s}$ in each $k_{z_j'}[t]$ is a product of t and an irreducible polynomial not equal to t.

Finally, since $s \sqcup \bigsqcup _{i = 1}^{m} y_i \sqcup \bigsqcup _{j = 1}^{n} z_j$ is a closed subscheme of S, by lifting coefficients we may choose a monic polynomial $f \in \Gamma (S, \mathscr {O}_S)[t]$ of degree N that restricts to $f_{y_i}$ on each $y_i$, to $f_{z_j}$ on each $z_j$ and to $f_{s}$ on s. This f defines a finite étale S-scheme $\widetilde {S}$, which, by construction, is equipped with an R-point $\widetilde {s} \in \widetilde {S}(R)$ lifting s (cut out by the factor t of $f_{s}$) and is such that the number of closed points with finite residue fields in the preimage $\widetilde {Z} \subset \widetilde {S}$ of Z stays bounded as N grows, but, except for the points in $\widetilde {s}$, the cardinalities of these residue fields grow uniformly. Thus, since, for a finite field $\mathbb {F}$, the number of closed points of $\mathbb {A}^1_{\mathbb {F}}$ with a given residue field grows unboundedly together with the degree of that residue field over $\mathbb {F}$, for large N our $\widetilde {S}$ meets the requirements.

Proof. By [Reference Grothendieck and DieudonnéEGA IV₄, Corollaire 17.11.4], an open neighborhood $U \subset C$ of c has an étale k-morphism

$$ \begin{align*} U \rightarrow \mathbb{A}^1_k. \end{align*} $$

Thus, there is a subextension $\ell /k$ of $k_c/k$ generated by a single element with $k_c/\ell $ separable. By the primitive element theorem, we need to check that this forces $k_c/k$ to only have finitely many subextensions $k'/k$. Since there are finitely many possibilities for $k' \cap \ell $, we replace k by $k' \cap \ell $ to reduce to considering those $k'$ for which $k' \cap \ell = k$. Like any finite separable extension, the separable closure of k in $k_c$ has only finitely many subextensions. Thus, there are finitely many possibilities for the maximal separable subextension $k^{\prime \prime }/k$ of $k'/k$. By replacing k by $k^{\prime \prime }$ and $\ell $ by $k^{\prime \prime }\ell $, we therefore reduce to the case when $k'/k$ is purely inseparable. Then the subextension $k'\ell /\ell $ of the separable extension $k_c/\ell $ is also purely inseparable, to the effect that $k' \subset \ell $. However, $\ell /k$ is generated by a single element, so, by the primitive element theorem, it has only finitely many subextensions.

Proof. Proposition 5.2 supplies C, s, Z and $\mathscr {E}$ that satisfy the present (i)–(iv). We view s as a closed subscheme of C, and we apply Lemma 6.1 to the R-finite closed subscheme $(Z \cup s)^{\mathrm {red}}$ of C to see that we may change C to assume that, in addition, for every maximal ideal $\mathfrak {m} \subset R$,

$$ \begin{align*} \#\{ z \in Z_{k_{\mathfrak{m}}} \, \vert\, [k_z : k_{\mathfrak{m}}] = d \} < \#\{ z \in \mathbb{A}^1_{k_{\mathfrak{m}}}\, \vert\, [k_z : k_{\mathfrak{m}}] = d \} \quad\text{for every}\quad d \ge 1. \end{align*} $$

This allows us to apply Lemma 6.3 with $Y = s$ to shrink C and to arrange (v).

Proof. By [Reference de JongSP, Theorem 06FI], the classifying S-stack $\mathbf {B} G$ is algebraic, and, by descent, its diagonal inherits quasi-affineness from G. Thus, the claim is a special case of [Reference Moret-BaillyMB96, Corollaire 6.5.1 (a)].

1. (a) We let A and $A'$ be the coordinate rings of S and $S'$, respectively. By [Reference de JongSP, Lemmas 0ALZ and 0955],

   $$ \begin{align*} \quad\quad R\Gamma_Z(S, \mathscr{F}) \otimes^{\mathbb{L}}_A A' \overset{\sim}{\longrightarrow} R\Gamma_{Z}(S', \mathscr{F}_{S'}). \end{align*} $$
   Thus, since $A'$ is A-flat, to obtain (a) it remains to note that, by [Reference de JongSP, Lemma 05E9], we have
   $$ \begin{align*} \quad\quad H^i_Z(S, \mathscr{F}) \overset{\sim}{\longrightarrow} H^i_Z(S, \mathscr{F}) \otimes_A A' \quad\text{for all}\quad i \in \mathbb{Z}. \end{align*} $$

2. (b) In the case when F is an S-group, the vanishing of quasi-coherent cohomology of affine schemes and the assumed filtration show that both $H^1(S, F)$ and $H^1(S', F)$ vanish. Thus, the assertion about the kernel simply amounts to the claim that every $F_U$-torsor that trivializes over $U'$ extends to an F-torsor. This, however, is immediate from Lemma 7.1.

   For the surjectivity assertion, we will induct on n. We begin with the case $n = 1$, in which F itself is the vector group associated to some vector bundle $\mathscr {F}$ on U. By applying (a) to $j_*(\mathscr {F})$, where $j \colon U \hookrightarrow S$ is the indicated open immersion, and again using the vanishing of quasi-coherent cohomology of affine schemes, we find that, for all $i\ge 1$, even

   
   For the inductive step, we assume that $n> 1$ and combine the inductive hypothesis, the preceding display for $F_{n - 1}$, and the nonabelian cohomology sequences [Reference GiraudGir71, Chapitre IV, Remarque 4.2.10] of a central extension to obtain the following commutative diagram with exact rows:
   
   We fix an $\alpha ' \in H^1(U', F)$ that we wish to lift to $H^1(U, F)$ and note that, by a diagram chase, there at least is an $\alpha \in H^1(U, F)$ whose image in $H^1(U', F/F_{n - 1})$ agrees with that of $\alpha '$. Every inner fpqc form of F inherits an analogous filtration, even with the same subquotients $F_i/F_{i + 1}$, so the change of origin bijections [Reference GiraudGir71, Chapitre III, Proposition 2.6.1 (i)] allow us to twist F and reduce to the case when the common image of $\alpha $ and $\alpha '$ in $H^1(U', F/F_{n - 1})$ vanishes. In this case, however, the surjectivity of the left vertical arrow suffices.

Proof. Let $B \subset G$ be a Borel R-subgroup. Proposition 6.5 supplies a quasi-finite, affine, flat R-morphism $\pi \colon C \rightarrow \mathbb {A}^1_R$ whose base change to an R-finite closed subscheme $Z \subset \mathbb {A}^1_R$ (called $Z'$ there) is an isomorphism, as well as an $s \in C(R)$ and a $G_C$-torsor $\widetilde {\mathscr {E}}$ (called $\mathscr {E}$ there) with s-pullback E such that $\widetilde {\mathscr {E}}$ reduces to a $\mathscr {R}_u(B)$-torsor over $C \setminus \pi ^{-1}(Z)$. By Lemma 7.2 (b) and Example 7.3, this $\mathscr {R}_u(B)$-torsor descends to a $\mathscr {R}_u(B)$-torsor over $\mathbb {A}^1_R \setminus Z$, so $\widetilde {\mathscr {E}}_{C \setminus \pi ^{-1}(Z)}$ descends to a $G_{\mathbb {A}^1_R \setminus Z}$-torsor. The patching Lemma 7.1 then ensures that $\widetilde {\mathscr {E}}$ itself descends to a $G_{\mathbb {A}^1_R}$-torsor $\mathscr {E}$ that reduces to a $\mathscr {R}_u(B)$-torsor over $\mathbb {A}^1_R \setminus Z$. By postcomposing with a change of coordinate automorphism of $\mathbb {A}^1_R$ to ensure that s map to the zero section of $\mathbb {A}^1_R$, we make the pullback of $\mathscr {E}$ along the zero section be E. Finally, we apply Lemma 3.1 to $\mathbb {P}^1_R$ to enlarge $Z \subset \mathbb {A}^1_R$ to be a hypersurface in $\mathbb {P}^1_R$. This ensures that $\mathbb {A}^1_R \setminus Z$ is affine so that, due to the filtration of $\mathscr {R}_u(B)$ by vector groups as in Example 7.3 and the vanishing of quasi-coherent cohomology of affine schemes, our $\mathscr {E}_{\mathbb {A}^1_R \setminus Z}$ is even trivial.

Proof. By [Reference GilleGil21, Theorem 1.1 and Corollary 4.3], the assumption on $\operatorname {\mathrm {rad}}(G)$ is equivalent to requiring that G may be embedded as a closed subgroup of some $\operatorname {\mathrm {GL}}_{n,\, R}$ and it holds in the indicated parenthetical cases. For the rest, we first use a limit argument to reduce to Noetherian R and then pass to connected components to also assume that $\operatorname {\mathrm {Spec}}(R)$ is connected. Moreover, we begin with the case $G = \operatorname {\mathrm {GL}}_{n,\, R}$, in which we may regard $\mathscr {E}$ as a vector bundle of rank n.

In this vector bundle case,

$$ \begin{align*} \mathscr{V} := \mathscr{H} om_{\mathscr{O}_{\mathbb{P}^1_R}}(\mathscr{O}_{\mathbb{P}^1_R}^{\oplus n}, \mathscr{E}) \cong \mathscr{E}^{\oplus n} \end{align*} $$

is also a vector bundle on $\mathbb {P}^1_R$. By [Reference Grothendieck and DieudonnéEGA III₁, Théorème 3.2.1], the R-module $V := \Gamma (\mathbb {P}^1_R, \mathscr {V})$ is finite. By assumption, $\mathscr {E}|_{\mathbb {P}^1_{k_{\mathfrak {m}}}}$ is trivial for every maximal ideal $\mathfrak {m} \subset R$, so for such $\mathfrak {m}$ we choose an isomorphism

$$ \begin{align*} \mathscr{O}_{\mathbb{P}^1_{k_{\mathfrak{m}}}}^{\oplus n} \overset{\sim}{\longrightarrow} \mathscr{E}|_{\mathbb{P}^1_{k_{\mathfrak{m}}}}, \quad\text{which corresponds to some}\quad v_{\mathfrak{m}} \in \Gamma(\mathbb{P}^1_{k_{\mathfrak{m}}}, \mathscr{V}|_{\mathbb{P}^1_{k_{\mathfrak{m}}}}). \end{align*} $$

Likewise, each vector bundle $\mathscr {V}|_{\mathbb {P}^1_{k_{\mathfrak {m}}}}$ is trivial, so $H^1(\mathbb {P}^1_{k_{\mathfrak {m}}}, \mathscr {V}|_{\mathbb {P}^1_{k_{\mathfrak {m}}}}) \cong 0$. Thus, by cohomology and base change [Reference Grothendieck and DieudonnéEGA III₁, Proposition 4.6.1], there is a $\widetilde {v}_{\mathfrak {m}} \in V/\mathfrak {m} V$ that maps to $v_{\mathfrak {m}}$. Since R is semilocal and $\mathfrak {m}$ ranges over its maximal ideals, we may then find a $v \in V$ that maps to all the $\widetilde {v}_{\mathfrak {m}}$, so also to all the $v_{\mathfrak {m}}$. By construction and the Nakayama lemma [Reference de JongSP, Lemma 00DV], the $\mathscr {O}_{\mathbb {P}^1_R}$-module homomorphism

$$ \begin{align*} \mathscr{O}_{\mathbb{P}^1_R}^{\oplus n} \rightarrow \mathscr{E} \end{align*} $$

corresponding to v is surjective at every closed point, so it is surjective. Cayley–Hamilton [Reference de JongSP, Lemma 05G8] then ensures that this homomorphism is an isomorphism, so $\mathscr {E}$ is trivial, as desired.

To deduce the general case, we use our closed embedding $G \hookrightarrow \operatorname {\mathrm {GL}}_{n,\, R}$. Namely, the settled case of $\operatorname {\mathrm {GL}}_{n,\, R}$ and the nonabelian cohomology sequence [Reference GiraudGir71, Chapitre III, Proposition 3.2.2] show that our $G_{\mathbb {P}^1_R}$-torsor $\mathscr {E}$ comes from a some $\mathbb {P}^1_R$-point of $\operatorname {\mathrm {GL}}_{n,\, R}/G$. However, G is reductive, so, by [Reference AlperAlp14, Theorem 9.4.1 and Theorem 9.7.5], this quotient is affine. By [Reference Mumford, Fogarty and KirwanMFK94, Proposition 6.1] (to reduce to an R-fiber), this means that the only R-morphisms from $\mathbb {P}^1_R$ to $\operatorname {\mathrm {GL}}_{n,\, R}/G$ are constant, in particular, that our $\mathbb {P}^1_R$-point comes from an R-point. This then implies that our $G_{\mathbb {P}^1_R}$-torsor $\mathscr {E}$ is the base change of a G-torsor, as desired.

Proof. Due to the canonical decomposition [Reference Gille and PoloSGA 3_{III new}, Exposé XXIV, Sections 5.2--5.3, Proposition 5.10 (i)], we may assume that

$$ \begin{align*} G \cong \operatorname{\mathrm{Res}}_{R'/R} G' \end{align*} $$

for a finite étale R-algebra $R'$ and a semisimple, simply connected $R'$-group $G'$ whose geometric $R'$-fibers are simple (in the sense that the Dynkin diagrams of these geometric fibers are connected). By [Reference Gille and PoloSGA 3_{III new}, Exposé XXIV, Proposition 8.4], we may then replace R and G by $R'$ and $G'$, respectively, to assume that the geometric R-fibers of G are simple.

We let t be the inverse of the coordinate on $\mathbb {A}^1_R$ and consider as the completion of $\mathbb {P}^1_R$ along infinity. Due to its R-finiteness, Z is closed in $\mathbb {P}^1_R$, so its pullback to is also closed and hence is even empty because it does not meet the locus $\{ t = 0\}$. Thus, we may use formal glueing supplied by, for instance, [Reference Bouthier and ČesnavičiusBČ22, Lemma 2.2.11 (b)] (or Lemma 7.1 when R is Noetherian) to extend $\mathscr {E}$ to a $G_{\mathbb {P}^1_R}$-torsor $\overline {\mathscr {E}}$ by glueing $\mathscr {E}$ with the trivial -torsor. It suffices to argue that we can glue like this so that $\overline {\mathscr {E}}_{\mathbb {P}^1_{k_{\mathfrak {m}}}}$ be trivial for every maximal ideal $\mathfrak {m} \subset R$: Lemma 8.3 will then imply that $\overline {\mathscr {E}}$ is the base change of its pullback by the section at infinity, and hence that $\overline {\mathscr {E}}$ and $\mathscr {E}$ are trivial.

Explicitly, the elements of give rise to all the possible glueings of $\mathscr {E}$ and the trivial -torsor to a $G_{\mathbb {P}^1_R}$-torsor and likewise over the residue fields $k_{\mathfrak {m}}$ of R. We will first build a trivial $G_{\mathbb {P}^1_{k_{\mathfrak {m}}}}$-bundle $\overline {\mathscr {E}}_{\mathbb {P}^1_{k_{\mathfrak {m}}}}$ from $\mathscr {E}_{\mathbb {P}^1_{k_{\mathfrak {m}}}}$ by such a glueing for every maximal ideal $\mathfrak {m} \subset R$ and then argue that these glueings come from a single glueing over R. These steps reduce, respectively, to the following:

1. (1) for every maximal ideal $\mathfrak {m} \subset R$, the $G_{\mathbb {A}^1_{k_{\mathfrak {m}}}}$-torsor $\mathscr {E}_{\mathbb {A}^1_{k_{\mathfrak {m}}}}$ is trivial;

2. (2) letting $\mathfrak {m} \subset R$ range over all the maximal ideals, the following map is surjective:

   

For (1), since $\mathscr {E}_{\mathbb {A}^1_{k_{\mathfrak {m}}}}$ is trivial away from $Z_{k_{\mathfrak {m}}}$, we may glue it arbitrarily with the trivial -torsor to obtain a $G_{\mathbb {P}^1_{k_{\mathfrak {m}}}}$-torsor whose pullback along the infinity section is trivial. By [Reference GilleGil02, Lemme 3.12] (see also [Reference GilleGil05]), such torsors are trivial over $\mathbb {A}^1_{k_{\mathfrak {m}}}$, so $\mathscr {E}_{\mathbb {A}^1_{k_{\mathfrak {m}}}}$ is trivial, that is, (1) holds.

For the claim (2), we first note that the isotropicity assumption implies that G has a proper parabolic subgroup $P \subset G$ (see Definition 8.1). By [Reference GilleGil09, Fait 4.3, Lemme 4.5] (this is where we use the assumptions on G), the Whitehead group of the base changes of G is of unramified nature, that is,

where $G(k_{\mathfrak {m}}(\!( t )\!))^+ \subset G(k_{\mathfrak {m}}(\!( t )\!))$ is the subgroup generated by $(\mathscr {R}_u(P))(k_{\mathfrak {m}}(\!( t )\!))$ and $(\mathscr {R}_u(P^-))(k_{\mathfrak {m}}(\!( t )\!))$ with $P^- \subset G$ being a parabolic opposite to P in the sense of [Reference Gille and PoloSGA 3_{III new}, Exposé XXVI, Définition 4.3.3, Corollaire 4.3.5 (i)]. To conclude (2), it suffices to show that the following pullback maps are surjective:

$$ \begin{align*} \textstyle (\mathscr{R}_u(P))(R(\!( t )\!)) \twoheadrightarrow \prod_{\mathfrak{m}} (\mathscr{R}_u(P))(k_{\mathfrak{m}}(\!( t )\!)) \quad\text{and}\quad (\mathscr{R}_u(P^-))(R(\!( t )\!)) \twoheadrightarrow \prod_{\mathfrak{m}} (\mathscr{R}_u(P^-))(k_{\mathfrak{m}}(\!( t )\!)). \end{align*} $$

For this, we combine the surjectivity of the map $R(\!( t)\!) \twoheadrightarrow \prod _{\mathfrak {m}} k(\!( t )\!)$ with [Reference Gille and PoloSGA 3_{III new}, Exposé XXVI, Corollaire 2.5], according to which both $\mathscr {R}_u(P)$ and $\mathscr {R}_u(P^-)$ are isomorphic to affine spaces $\mathbb {A}^d_R$.

Proof. We pass to connected components to assume that $\operatorname {\mathrm {Spec}}(R)$ is connected so that R is a domain and, in particular, $R \neq 0$. We use Proposition 2.2 to replace G by $(G^{\mathrm {der}})^{\mathrm {sc}}$ and, hence, reduce to the case when our quasi-split G is also semisimple and simply connected.

Let E be a G-torsor that trivializes over K, so also over $R[\frac {1}{r}]$ for some $r \in R \setminus \{ 0\}$. By Popescu theorem [Reference de JongSP, Theorem 07GC], the ring R is a filtered direct limit of smooth $\mathcal {O}$-algebras. Thus, a limit argument allows us to assume that R is the semilocalization of a smooth $\mathcal {O}$-algebra at finitely many primes. In this case, Proposition 7.4 gives a $G_{\mathbb {A}^1_R}$-torsor $\mathscr {E}$ whose pullback along the zero section is E such that $\mathscr {E}$ is trivial away from an R-finite closed subscheme $Z \subset \mathbb {A}^1_R$. By Proposition 8.4 (with Example 8.2), this $\mathscr {E}$ is trivial, so E is also trivial, as desired.

Proof. We pass to connected components to assume that $\operatorname {\mathrm {Spec}}(R)$ is connected so that R is a domain, and we set $K := \operatorname {\mathrm {Frac}}(R)$. Only the ‘if’ part requires an argument, so we assume that $G_{K}$ is split. The geometric fibers of G have a constant type (see [Reference Gille and PoloSGA 3_{III new}, Exposé XXII, Définition 1.13]), and we let $\mathbf {G}$ be a split reductive R-group of this type so that G is a form of $\mathbf {G}$ that corresponds to some $x \in H^1(R, \underline {\operatorname {\mathrm {Aut}}}(\mathbf {G}))$ whose pullback to $H^1(K, \underline {\operatorname {\mathrm {Aut}}}(\mathbf {G}))$ is trivial. We wish to show that x is trivial.