Problems About Torsors over Regular Rings

Abstract

We overview a web of conjectures about torsors under reductive groups over regular rings and survey some techniques that have been used for making progress on such problems.

Appendix. Resolutions of reductive groups

Yifei Zhao¹⁰

This appendix is an exposition on the construction of flasque and coflasque resolutions of a reductive group G over a general base scheme S, subject only to the condition that rad(G) be isotrivial.

The notions of flasque and coflasque tori are due to Colliot-Thélène–Sansuc [30]. The existence of a coflasque resolution strengthens that of a z-extension of Langlands and Kottwitz [74, Section 1], which is often stated over a field of characteristic zero ([35, Chapter V, Section 3], [13], for example). When the base is a field of arbitrary characteristic, both resolutions are constructed by Colliot-Thélène in [24], but, as observed by González-Avilés [52], the same proof yields the existence of flasque resolutions over locally Noetherian, geometrically unibranch schemes (e.g., a normal scheme). Our proof follows Colliot-Thélène’s strategy, but we replace the hypotheses on S by the isotriviality of rad(G), which holds whenever S is locally Noetherian and geometrically unibranch but could remain valid in other contexts.

Sections A.2–A.3 are preparatory and the main construction appears as Theorem A.4.1. As an application, we explain a simple reduction of the Grothendieck–Serre conjecture 3.1.1 to the case when the derived subgroup is simply connected, see Proposition A.5.1. The author thanks K. Česnavičius for many helpful conversations and comments.

1.1 A.1 Group schemes of multiplicative type

In this section, we review group schemes of multiplicative type. The most important notion for us is the isotriviality of such group schemes.

1.1.1 A.1.1

Let S be a scheme. For an fppf sheaf of abelian groups ${\mathscr{F}}$ over S, one may consider the fppf sheaf $\mathbb D_{S}({\mathscr{F}})$ whose value at an affine S-scheme $S^{\prime }$ is $\text {Hom}({\mathscr{F}}_{S^{\prime }}, \mathbb G_{m, S^{\prime }})$. Here, Hom is viewed in the category of fppf sheaves of abelian groups over $S^{\prime }$. The fppf sheaf $\mathbb D_{S}({\mathscr{F}})$ again takes values in abelian groups, the group structure being inherited from $\mathbb G_{m}$.

1.1.2 A.1.2

An S-group scheme G is diagonalizable if there exist a finitely generated abelian group M and an isomorphism between G and the group scheme $\mathbb D_{S}(M_{S})$, where M_S denotes the constant sheaf with values in M. An S-group scheme G is of multiplicative type if it is diagonalizable fppf locally on S.¹¹ In fact, every S-group scheme of multiplicative type is diagonalizable étale locally on S ([120, Exposé X, Corollaire 4.5] or [20, Proposition B.3.4]).

If an S-group scheme G of multiplicative type becomes diagonalizable after base change along $\widetilde S\rightarrow S$, then G is said to be split by $\widetilde S$. If S is connected, then any S-group scheme of multiplicative type is split by some fppf (equivalently, étale) surjection $\widetilde S\rightarrow S$ ([120, Exposé IX, Remarque 1.4.1]). By fppf descent, any S-group scheme G of multiplicative type is S-affine.

An S-group G of multiplicative type is a torus if fppf (equivalently, étale) locally on S it is of the form $\mathbb D_{S}(M_{S})$ for some finitely generated free abelian group M.

1.1.3 A.1.3

Groups of multiplicative type enjoy certain closure properties:

(1) an S-flat, finitely presented closed subgroup of an S-group scheme of multiplicative type is again of multiplicative type ([120, Exposé X, Corollaire 4.7 b)] or [20, Corollary B.3.3]);

(2) a commutative extension of group schemes of multiplicative type is again of multiplicative type ([120, Exposé XVII, Proposition 7.1.1] or [20, Corollary B.4.2]).

Furthermore, an S-group scheme G of multiplicative type is reflexive in the sense that the natural transformation $G \rightarrow \mathbb D_{S}(\mathbb D_{S}(G))$ is an isomorphism. Indeed, this statement may be verified fppf locally on S, where it follows from [120, Exposé VIII, Théorème 1.2].

1.1.4 A.1.4

An S-group scheme G of multiplicative type is called isotrivial if G is split by a finite étale surjection $\widetilde S\rightarrow S$. When S is connected, an isotrivial S-group scheme of multiplicative type is split by a finite connected étale Galois cover $\widetilde S\rightarrow S$. We discuss some ways to obtain isotrivial S-group schemes of multiplicative type.

Lemma A.1.5

Let S be a connected scheme. Then any finite S-group scheme G of multiplicative type is isotrivial.

Proof

Let ${\mathscr{F}}$ denote the fppf sheaf of abelian groups $\mathbb D_{S}(G)$. Since S is connected, there is an fppf surjection $\widetilde S\rightarrow S$ such that ${\mathscr{F}}_{\widetilde S}$ is isomorphic to the constant sheaf $M_{\widetilde S}$ for a finite abelian group M. The descent data of ${\mathscr{F}}_{\widetilde S}$ allow us to construct an Aut(M)-torsor ${\mathscr{P}}$ over S such that ${\mathscr{F}}$ is the fppf sheaf of abelian groups induced from ${\mathscr{P}}$. Since Aut(M) is finite, ${\mathscr{P}}$ is representable by a finite étale surjection $\widetilde S^{\prime }\rightarrow S$. In particular, ${\mathscr{P}}$ is trivialized by $\widetilde S^{\prime }$. It follows that G is split by $\widetilde S^{\prime }$. □

The same argument proves more generally that an S-group scheme of multiplicative type whose maximal torus has rank ≤ 1 is isotrivial. This fact can be compared with Lemma A.1.6 (ii) below.

Lemma A.1.6

Let S be a connected scheme. Given a short exact sequence of S-group schemes of multiplicative type:

$$ 1 \rightarrow G_{1} \rightarrow G \rightarrow G_{2} \rightarrow 1, $$

(i) if G is diagonalizable (resp. isotrivial), then both G₁ and G₂ are diagonalizable (resp. isotrivial);

(ii) if G₁ is isotrivial and G₂ is finite, then G is isotrivial.

Proof

Statement (i) is established in [120, Exposé IX, Proposition 2.11]. To prove statement (ii), we may assume that both G₁ and G₂ are diagonalizable by replacing S with a connected finite étale cover. Since $\mathbb D_{S}$ is an anti-equivalence on reflexive fppf sheaves of abelian groups [120, Exposé VIII, Proposition 1.0.1], it restricts to an exact functor on the full subcategory of S-group schemes of multiplicative type. In particular, we obtain a short exact sequence of fppf sheaves of abelian groups:

$$ 1 \rightarrow M_{2, S} \rightarrow \mathbb D_{S}(G) \rightarrow M_{1, S} \rightarrow 1. $$

Here, M_{i, S} (for i = 1,2) denotes the constant sheaf associated to a finitely generated $\mathbb Z$-module M_i. The finiteness of G₂ allows us to assume that M₂ is finite.

It remains to show that any class in $\text {Ext}^{1}_{\text {fppf}}(M_{1,S}, M_{2,S})$ comes from $\text {Ext}^{1}_{\mathbb Z}(M_{1}, M_{2})$ after passing to a finite étale cover $\widetilde S\rightarrow S$. For this statement, it suffices to treat the case where M₁ is a cyclic group. For $M_{1} {=} \mathbb Z$, we have $\text {Ext}^{1}_{\text {fppf}}(\mathbb Z_{S}, M_{2, S}) {\cong } H^{1}_{\text {fppf}}(S, M_{2}) {\cong } H^{1}_{\acute {\text {e}}\text {t}}(S, M_{2})$, and because M₂ is finite, any class in $H^{1}_{\acute {\text {e}}\text {t}}(S, M_{2})$ vanishes over a finite étale surjection $\widetilde S\rightarrow S$. For $M_{1} = \mathbb Z/n$ for an integer n ≥ 1, we have an exact sequence:

$$ \text{Hom}(\mathbb Z, M_{2}) \xrightarrow{n} \text{Hom}(\mathbb Z, M_{2}) \rightarrow \text{Ext}^{1}_{\text{fppf}}((\mathbb Z/n)_{S}, M_{2,S}) \rightarrow \text{Ext}^{1}_{\text{fppf}}(\mathbb Z_{S}, M_{2,S}). $$

By the same argument as above, any class in $\text {Ext}^{1}_{\text {fppf}}((\mathbb Z/n)_{S}, M_{2,S})$ has zero image in $\text {Ext}^{1}_{\text {fppf}}(\mathbb Z_{S}, M_{2, S})$ after passing to a connected finite étale cover $\widetilde S\rightarrow S$. Equivalently, this means that over $\widetilde S$, the class comes from $\text {Ext}^{1}_{\mathbb Z}(\mathbb Z/n, M_{2})$. □

Remark A.1.7

In Lemma A.1.6 (ii), the finiteness hypothesis on G₂ cannot be dropped. Indeed, whenever $H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb Z)\neq 0$, there exist self-extensions of $\mathbb G_{m}$ which are not isotrivial. To see this, we use the isomorphism $\text {Ext}^{1}_{\text {fppf}}(\mathbb G_{m}, \mathbb G_{m}) \cong H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb Z)$ and the fact that any class of $H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb Z)$ which vanishes on a finite étale cover of S must already be zero (because $H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb {Z}) \hookrightarrow H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb {Q})$).

1.1.5 A.1.8

There is a convenient condition on the base scheme S which guarantees that all multiplicative type S-group schemes are isotrivial.

A local ring R is geometrically unibranch if its strict Henselization R^sh has a unique minimal prime, see [123, Definition 0BPZand Lemma 06DM]. A scheme S is geometrically unibranch if so are its local rings. For example, a normal scheme (in the usual sense that its local rings are normal domains) is geometrically unibranch. Every connected component of a locally Noetherian, geometrically unibranch scheme is irreducible (see [68, Exercise 3.16 (a)]).

Let S be a locally Noetherian, geometrically unibranch scheme. By [120, Exposé X, Théorème 5.16], every S-group scheme G of multiplicative type splits over a finite étale surjection $\widetilde S \rightarrow S$. When S is connected, we may further assume that $\widetilde S \rightarrow S$ is a connected Galois cover.

1.2 A.2 Flasque and coflasque tori

In this section, we focus on isotrivial tori. The study of these objects is equivalent to that of Galois modules with integral coefficients. We discuss several conditions on isotrivial tori (quasi-trivial, flasque, and coflasque) which are “of Galois cohomology nature.”

1.2.1 A.2.1

Suppose that S is a connected scheme and let $\widetilde S \rightarrow S$ be a connected finite étale Galois cover. By [120, Exposé X, Proposition 1.1], the construction $\mathbb D_{\widetilde S}$ induces an equivalence of categories between

(1) group schemes $G\rightarrow S$ of multiplicative type split by $\widetilde S$;

(2) finitely generated $\mathbb Z$-modules M equipped with a $\text {Gal}(\widetilde S/S)$-action.

Under this equivalence, tori $T\rightarrow S$ split by $\widetilde S$ correspond to finitely generated free $\mathbb Z$-modules M equipped with a $\text {Gal}(\widetilde S/S)$-action—such an M is called the character lattice of T, and we denote it by $\check {{{\varLambda }}}_{T, \widetilde S}$. Its $\mathbb Z$-linear dual is called the cocharacter lattice of T, which we denote by ${{\varLambda }}_{T, \widetilde S}$.

1.2.2 A.2.2

Let Γ be a finite group. A Γ-lattice Λ, i.e., a finitely generated free $\mathbb Z$-module equipped with a Γ-action, is called quasi-trivial if it has a Γ-stable $\mathbb Z$-basis. Clearly, Λ is quasi-trivial if and only if its $\mathbb Z$-linear dual $\check {{{\varLambda }}} := \text {Hom}_{\mathbb {Z}}({{\varLambda }}, \mathbb {Z})$, equipped with the contragredient Γ-action, is quasi-trivial.

The following lemma describes the conditions that end up defining flasque tori.

Lemma A.2.3

Let Γ be a finite group and let Λ be a lattice equipped with a Γ-action. The following conditions are equivalent:

(i) $H^{1}({{\varGamma }}^{\prime }, {{\varLambda }}) = 0$ for any subgroup ${{\varGamma }}^{\prime } \subset {{\varGamma }}$;

(ii) $\text {Ext}^{1}_{\mathbb Z[{{\varGamma }}]}(P, {{\varLambda }}) = 0$ for any quasi-trivial Γ-lattice P.

Proof

The key observation is as follows. Suppose that P is a quasi-trivial lattice with a basis X that consists of a single Γ-orbit. Fix an x ∈ X and let ${{\varGamma }}^{\prime }\subset {{\varGamma }}$ be the stabilizer of x. Then we have an isomorphism $P \cong \mathbb Z[{{\varGamma }}/{{\varGamma }}^{\prime }]$ of $\mathbb Z[{{\varGamma }}]$-modules, and so an isomorphism $\text {Ext}^{1}_{\mathbb Z[{{\varGamma }}]}(P, {{\varLambda }}) \cong H^{1}({{\varGamma }}^{\prime }, {{\varLambda }})$. □

1.2.3 A.2.4

Let Γ be a finite group. A Γ-lattice Λ is called

(1) coflasque if it satisfies the equivalent conditions of Lemma A.2.3; and

(2) flasque if its $\mathbb Z$-linear dual $\check {{{\varLambda }}}$ satisfies the equivalent conditions of Lemma A.2.3.¹²

By Shapiro lemma, any quasi-trivial Γ-lattice is both flasque and coflasque.

1.2.4 A.2.5

In the setting of Section A.2.1, a torus $T\rightarrow S$ split by $\widetilde S$ is called quasi-trivial (resp. flasque; resp., coflasque) if its character lattice $\check {{{\varLambda }}}_{T, \widetilde S}$ is quasi-trivial (resp. flasque; resp., coflasque). By [30, Lemma 1.1], these notions are independent of the choice of the Galois cover $\widetilde S$.

For any scheme S, a torus $T\rightarrow S$ is called quasi-trivial (resp. flasque; resp., coflasque) if every connected component of S admits a connected finite étale Galois cover $\widetilde S$ such that T is split by $\widetilde S$ and quasi-trivial (resp. flasque; resp., coflasque) with respect to $\widetilde S$ (again, these notions do not depend on $\widetilde {S}$). If a torus $T\rightarrow S$ is quasi-trivial (resp. flasque, coflasque), then so is its base change along any morphism $S^{\prime }\rightarrow S$ with $S^{\prime }$ still connected.

Quasi-trivial tori are both flasque and coflasque, and they can be made more explicit as follows.

Lemma A.2.6

Let S be a connected scheme. A torus $T\rightarrow S$ is quasi-trivial if and only if it is a finite product of Weil restrictions of $\mathbb G_{m}$ along finite étale surjections $S^{\prime } \rightarrow S$.

Proof

Suppose that $T\rightarrow S$ is quasi-trivial. Let $\widetilde S\rightarrow S$ be a connected finite étale Galois cover such that T is split by $\widetilde S$. Without loss of generality, we may assume that ${{\varLambda }}_{T, \widetilde S}$ has a basis X that consists of a single $\text {Gal}(\widetilde S/S)$-orbit. Then the $\text {Gal}(\widetilde S/S)$-set X gives rise to a finite étale cover $S^{\prime } \rightarrow S$, and, by Section A.2.1, we have an isomorphism $T \simeq \text {Res}_{S^{\prime }/S}(\mathbb G_{m})$. The converse is analogous. □

Flasque and coflasque tori enjoy the following pleasant splitting property.

Lemma A.2.7

In the setting of Section A.2.1, a short exact sequence of S-tori split by $\widetilde S$

$$ 1 \rightarrow T_{1} \rightarrow T_{2} \rightarrow T_{3} \rightarrow 1 $$

is split if either of the following conditions holds:

(i) T₁ is quasi-trivial and T₃ is coflasque;

(ii) T₁ is flasque and T₃ is quasi-trivial.

Proof

By considering character lattices, we translate the problem to splitting the exact sequence

$$ 0 \rightarrow \check{{{\varLambda}}}_{T_{3}, \widetilde S} \rightarrow \check{{{\varLambda}}}_{T_{2}, \widetilde S} \rightarrow \check{{{\varLambda}}}_{T_{1}, \widetilde S} \rightarrow 0 $$

(A.2.7.1)

of $\text {Gal}(\widetilde S/S)$-lattices. Suppose that T₁ is quasi-trivial and T₃ is coflasque. Then, by definition,

$$ \text{Ext}^{1}_{\mathbb Z[\text{Gal}(\widetilde S/S)]}(\check{{{\varLambda}}}_{T_{1}, \widetilde S}, \check{{{\varLambda}}}_{T_{3}, \widetilde S}) = 0, $$

so (A.2.7.1) splits. Suppose that T₁ is flasque and T₃ is quasi-trivial. Then the dual of (A.2.7.1) splits for the same reason, so, by dualizing again, (A.2.7.1) splits as well. □

Next, we shall construct “resolutions” of S-group schemes of multiplicative type split by $\widetilde S$ in terms of flasque and coflasque tori. The following Lemma of Colliot-Thélène–Sansuc [30] will be the basis of our construction of resolutions of reductive S-group schemes.

Lemma A.2.8

In the setting of Section A.2.1, let $G\rightarrow S$ be a group scheme of multiplicative type split by $\widetilde S$. There exist S-tori T₁ and T₂ split by $\widetilde S$ that fit into a short exact sequence of S-group schemes

$$ 1 \rightarrow G \rightarrow T_{1} \rightarrow T_{2} \rightarrow 1. $$

(A.2.8.1)

Furthermore, we may arrange (A.2.8.1) so that either of the following conditions is satisfied:

(i) T₁ is flasque and T₂ is quasi-trivial;

(ii) T₁ is quasi-trivial and T₂ is coflasque.

Proof

By Section A.2.1, the problem translates into one concerning finitely generated $\mathbb Z$-modules equipped with a $\text {Gal}(\widetilde S/S)$-action, which is addressed in [30, Lemma 0.6]. □

Remark A.2.9

In the setting of Section A.2.1, let $T \rightarrow S$ be a torus split by $\widetilde S$. In the same vein as Lemma A.2.8, [30, Lemma 0.6] implies the existence of resolutions by tori split by $\widetilde S$:

$$ 1 \rightarrow T_{1} \rightarrow T_{2} \rightarrow T \rightarrow 1, $$

such that either

(i) T₁ is flasque and T₂ is quasi-trivial; or

(ii) T₁ is quasi-trivial and T₂ is coflasque.

These resolutions are the flasque, respectively coflasque resolutions of the torus T. The main result we shall prove (Theorem A.4.1) can be viewed as its generalization where T is replaced by a reductive S-group with isotrivial radical. In its proof, however, we will only need a special case of the result for T: the existence of a surjection $P \twoheadrightarrow T$ from a quasi-trivial torus P split by $\widetilde S$.

1.3 A.3 Central isogenies and the simply connected cover

Before proceeding to construct the promised resolutions of reductive groups in Section 7, we review the notion of central isogenies that plays an important role there. Recall the notion of the center Z_G of a reductive group scheme $G\rightarrow S$ as defined in Section 1.3.3.

1.3.1 A.3.1

For a scheme S, a morphism $f \colon G^{\prime } \rightarrow G$ of reductive S-group schemes is called a central isogeny if

(1) f is finite, flat, and surjective;

(2) $\ker (f)$ lies in the center of $G^{\prime }$.

We only define the notion of central isogenies for reductive S-group schemes, as is done in [121, Exposé XXII, Définition 4.2.9]. One may generalize this notion to other S-group schemes, but it may become pathological: for example, the composition of two central isogenies may fail to be central, see [20, Exercise 3.4.4 (ii)]. We now show that such phenomena do not occur for reductive group schemes and then we use central isogenies to define the simply connected cover of a semisimple group scheme in Proposition A.3.4.

Lemma A.3.2

Let $f \colon G^{\prime } \rightarrow G$ be a central isogeny of reductive S-group schemes.

(i) The induced map $Z_{G^{\prime }} \rightarrow f^{-1}(Z_{G})$ is an isomorphism.

(ii) For any other central isogeny $g \colon G^{\prime \prime } \rightarrow G^{\prime }$ of reductive group schemes over S, the composition $f \circ g \colon G^{\prime \prime } \rightarrow G$ is also a central isogeny.

Proof

In (i), the problem is étale local on S, so we may assume that G contains a split maximal torus T, whose inverse image $T^{\prime } := f^{-1}(T)$ is then a split maximal torus of $G^{\prime }$. Since f is a central isogeny, the induced map on character lattices $\check {{{\varLambda }}}_{T} \rightarrow \check {{{\varLambda }}}_{T^{\prime }}$ restricts to a bijection between the roots of (G, T) and $(G^{\prime }, T^{\prime })$, see [20, Example 6.1.9]. The result then follows from the characterization of Z_G as the kernel of the adjoint action $T \rightarrow \text {GL}(\text {Lie}(G))$ (see Section 1.3.3), that is, as the intersection of the $\ker (\alpha )$ over all the roots $\alpha \colon T \rightarrow \mathbb G_{m}$ of (G, T).

In (ii), f ∘ g is finite, flat, and surjective, so we need to verify that $\ker (f \circ g) \subset Z_{G^{\prime \prime }}$. Indeed, we have

$$ \ker(f \circ g) \cong g^{-1}(\ker(f)) \subset g^{-1}(Z_{G^{\prime}}) \cong Z_{G^{\prime\prime}}, $$

where the last isomorphism comes from (i). □

Remark A.3.3

Suppose that $f : G^{\prime } \rightarrow G$ is a central isogeny of reductive S-group schemes. Then $\ker (f)$ is an S-group scheme of multiplicative type. Indeed, $Z_{G^{\prime }}$ is of multiplicative type (see Section 1.3.3) so this assertion follows from the closure property in Section A.1.3.

Proposition A.3.4

Let S be a scheme and let G be a semisimple S-group scheme. Consider the category of pairs $(G^{\prime }, f)$ consisting of a semisimple S-group scheme $G^{\prime }$ and a central isogeny $f \colon G^{\prime } \rightarrow G$, with morphisms $(G^{\prime }_{1}, f_{1}) \rightarrow (G^{\prime }_{2}, f_{2})$ being given by central isogenies $\alpha \colon G_{1}^{\prime } \rightarrow G_{2}^{\prime }$ such that f₁ = f₂ ∘ α. This category has an initial object (G^sc,f), the simply connected cover of G.

Proof

The proof relies on the classification of pinned reductive groups by root data ([121, Exposé XXV, Théorème 1.1] or [20, Theorem 6.1.16]). The universal property allows us to work étale locally on S, so we may assume that G is split with respect to a split maximal torus T ⊂ G.

The split maximal torus T allows us to extract the root data $({{\varLambda }}_{T}, {{\varPhi }}, \check {{{\varLambda }}}_{T}, \check {{{\varPhi }}})$. Let ${{{\varLambda }}_{T}^{r}}\subset {{\varLambda }}_{T}$ denote the sublattice generated by the coroots Φ. There is a morphism of root data:

$$ ({{{\varLambda}}_{T}^{r}}, {{\varPhi}}, \check{{{\varLambda}}}_{T}^{r}, \check{{{\varPhi}}}) \rightarrow ({{\varLambda}}_{T}, {{\varPhi}}, \check{{{\varLambda}}}_{T}, \check{{{\varPhi}}}) $$

(A.3.4.1)

which induces the identity maps on Φ and $\check {{{\varPhi }}}$. The root data $({{{\varLambda }}_{T}^{r}}, {{\varPhi }}, \check {{{\varLambda }}}_{T}^{r}, \check {{{\varPhi }}})$ define a pinned reductive S-group G^sc with split maximal torus T^sc and (A.3.4.1) comes from a central isogeny $f \colon G^{\text {sc}} \rightarrow G$ compatible with the splitting (i.e., mapping T^sc to T), but f is only unique up to conjugation by (T/Z_G)(S) ([20, Theorem 6.1.16 (1)]). The pair (G^sc,f), however, is canonically defined thanks to the isomorphism $T^{\text {sc}}/Z_{G^{\text {sc}}} \cong T/Z_{G}$ induced by f. Next, we argue that (G^sc,f) is canonically independent of the choice of the split maximal torus T ⊂ G. Indeed, conjugation defines an isomorphism between G/N_G(T) and the scheme parametrizing maximal tori of G ([20, Theorem 3.1.6]) so the claim follows from the isomorphism $G^{\text {sc}}/N_{G^{\text {sc}}}(T^{\text {sc}}) \cong G/N_{G}(T)$ induced by f.

To show that the pair (G^sc,f) satisfies the universal property of an initial object, we suppose being given another central isogeny $f^{\prime } \colon G^{\prime }\rightarrow G$. For a split maximal torus T ⊂ G as above, we write $T^{\prime } \subset G^{\prime }$ for the induced maximal torus. Arguing with root data as above, we find a central isogeny $\alpha _{1} \colon G^{\text {sc}} \rightarrow G^{\prime }$ such that f and $f^{\prime }\circ \alpha _{1}$ differ by conjugation by an element of (T/Z_G)(S). The isomorphism $T^{\prime }/Z_{G^{\prime }} \cong T/Z_{G}$ then allows us to construct the unique central isogeny $\alpha \colon G^{\text {sc}} \rightarrow G^{\prime }$ which satisfies $f = f^{\prime }\circ \alpha$. □

Remark A.3.5

Another definition of the simply connected cover is given in [20, Exercise 6.5.2 (i)], which characterizes the central isogeny $f \colon G^{\text {sc}} \rightarrow G$ by the fact that the geometric S-fibers of G^sc are simply connected, i.e., they admit no nontrivial central isogenies from semisimple groups. It is easy to see that the two definitions agree. In particular, the formation of the simply connected cover G^sc of G commutes with arbitrary base change $S^{\prime } \rightarrow S$.

1.4 A.4 Existence of resolutions

In this section, we construct flasque and coflasque resolutions of reductive group schemes with isotrivial radical tori. Recall that to a reductive S-group scheme G, we have associated several other reductive S-group schemes in the main text: the derived subgroup G^der, which is semisimple, and the tori rad(G) and corad(G) := G/G^der (see Section 1.3.3).

Theorem A.4.1

Let S be a connected scheme and let G be a reductive S-group scheme such that rad(G) is isotrivial. Fix a central isogeny $f \colon G^{\prime {\text {der}}} \rightarrow G^{{\text {der}}}$. There exists a central extension

$$ 1 \rightarrow T_{1} \rightarrow G^{\prime} \rightarrow G \rightarrow 1 $$

(A.4.1.1)

of reductive S-group schemes such that $\text {rad}(G^{\prime })$ is isotrivial and $G^{\prime } \rightarrow G$ induces f on derived subgroups. Furthermore, setting $T_{2} := \text {corad}(G^{\prime })$, we may arrange (A.4.1.1) so that one of the following conditions is satisfied:

(a) T₁ is a flasque torus and T₂ is a quasi-trivial torus;

(b) T₁ is a quasi-trivial torus and T₂ is a coflasque torus.

Remark A.4.2

The most typical application of Theorem A.4.1 is with $f \colon G^{\prime {\text {der}}} \rightarrow G^{{\text {der}}}$ being the simply connected cover reviewed in Lemma A.3.4. In this case, we obtain a resolution (A.4.1.1) where $G^{\prime }$ has a simply connected derived subgroup and the tori T₁, T₂ satisfy the conditions above. These are called flasque, respectively coflasque resolutions of G.

Note that if an S-torus T admits a flasque (or coflasque) resolution, then it must be isotrivial (Lemma A.1.6 (i)). Thus the hypothesis that rad(G) be isotrivial cannot be dropped.

Finally, we remark that if S is locally Noetherian and geometrically unibranch (such as a normal scheme), then the isotriviality condition on rad(G) is automatically satisfied (see Section A.1.8).

Proof of Theorem A.4.1

By composing the canonical central isogeny $G^{{\text {der}}} \times \text {rad}(G) \rightarrow G$ of (1.3.3.1) with f ×id_rad(G), we obtain a central isogeny of reductive S-group schemes:

$$ 1 \rightarrow H_{2} \rightarrow G^{\prime{\text{der}}} \times \text{rad}(G) \rightarrow G \rightarrow 1. $$

(A.4.2.1)

In particular, H₂ is a finite S-group scheme of multiplicative type (Remark A.3.3).

Let us denote by $\widetilde S \rightarrow S$ a connected finite étale Galois cover which splits rad(G). By Remark A.2.9, we may choose a short exact sequence of tori split by $\widetilde S$:

$$ 1 \rightarrow H_{1} \rightarrow P \rightarrow \text{rad}(G) \rightarrow 1, $$

(A.4.2.2)

where P is quasi-trivial. Compose the central isogeny (A.4.2.1) with the surjection $P\rightarrow \text {rad}(G)$, we obtain a central extension of reductive S-groups:

$$ 1 \rightarrow M \rightarrow G^{\prime{\text{der}}} \times P \rightarrow G \rightarrow 1. $$

(A.4.2.3)

Let us study the commutative S-group scheme M. By construction, it is an extension of H₂ by H₁. Since H₁ is an isotrivial torus and H₂ is a finite S-group scheme of multiplicative type, M is of multiplicative type (Section A.1.3) and even isotrivial (Lemma A.1.6 (ii)). Thus, we may take another connected finite étale Galois cover $\widetilde S^{\prime }\rightarrow \widetilde S$ and assume that M is split by $\widetilde S^{\prime }$. Using Lemma A.2.8, we find a resolution of M by S-tori which are also split by $\widetilde S^{\prime }$:

$$ 1 \rightarrow M \rightarrow T_{1} \rightarrow Q \rightarrow 1, $$

(A.4.2.4)

where either

(1) T₁ is flasque and Q is quasi-trivial; or

(2) T₁ is quasi-trivial and Q is coflasque.

Let us form the pushout of the extension (A.4.2.3) along the map $M \rightarrow T_{1}$. This gives rise to a central extension of G by T₁ that fits into a commutative diagram

By construction, the map α induces an isomorphism on derived subgroups. Hence, the morphism $G^{\prime } \rightarrow G$ induces the given central isogeny $f \colon G^{\prime {\text {der}}} \rightarrow G^{{\text {der}}}$ on derived subgroups. Recall that the formation of radicals is preserved under quotient maps. (This statement may be verified over geometric points, where it is [121, Exposé XIX, Section 1.7].) Hence $\text {rad}(G^{\prime })$ is a quotient of the torus T₁ × P. Since the latter is split by $\widetilde S^{\prime }$, so is $\text {rad}(G^{\prime })$ (Lemma A.1.6 (i)).

Finally, we show that the two types of resolutions (A.4.2.4) give rise to the two conditions in the statement of Theorem A.4.1. Indeed, write $T_{2} := \text {corad}(G^{\prime })$. Since T₂ is a quotient of $\text {rad}(G^{\prime })$, it is also split by $\widetilde S^{\prime }$. We have a short exact sequence of S-tori split by $\widetilde S^{\prime }$:

$$ 1 \rightarrow P \rightarrow T_{2} \rightarrow Q \rightarrow 1. $$

(A.4.2.5)

Since P is quasi-trivial and Q is at least coflasque, Lemma A.2.7 shows that (A.4.2.5) splits. In particular, T₂ is quasi-trivial (resp. coflasque) whenever Q is. □

1.5 A.5 An application to the Grothendieck–Serre conjecture

We use Theorem A.4.1 to reduce the Grothendieck–Serre conjecture 3.1.1 to the case when the group G has a simply connected derived subgroup. This argument is suggested to me by K. Česnavičius.

Proposition A.5.1

Let R be a regular local ring, let K := Frac(R), let G be a reductive R-group scheme, and consider the pullback map

$$ H^{1}_{\acute{\text{e}}\text{t}}(R, G) \rightarrow H^{1}_{\acute{\text{e}}\text{t}}(K, G). $$

(A.5.1.1)

If this map has trivial kernel whenever G is replaced by some central extension $G^{\prime }$ of G whose derived subgroup $G^{\prime {\text {der}}}$ is simply connected, then it has trivial kernel for G itself.

Proof

By Theorem A.4.1, we may find a central extension of reductive R-group schemes

$$ 1 \rightarrow T_{1} \rightarrow G^{\prime} \rightarrow G \rightarrow 1, $$

such that the map $G^{\prime } \rightarrow G$ induces the simply connected cover $G^{\prime {\text {der}}} \cong (G^{{\text {der}}})^{\text {sc}} \rightarrow G^{{\text {der}}}$ on the derived subgroups and T₁ is a quasi-trivial torus.¹³ By (1.2.2.1), this extension gives rise to the following map of exact sequences of pointed sets:

Since T₁ is a quasi-trivial torus, by Lemma A.2.6, it is isomorphic to a finite product of tori of the form $\text {Res}_{R^{\prime }/R}(\mathbb G_{m})$ for some finite étale maps $R \rightarrow R^{\prime }$. In particular, Hilbert 90 implies that α₁ is an isomorphism between singletons. Grothendieck’s theorem on the Brauer group [64, Corollaire 1.8] implies that α₂ is injective. Therefore, if $\beta ^{\prime }$ has trivial kernel, then so does β, as desired. □