Local triviality for G-torsors

Abstract

Let \(C \rightarrow \mathop {{\mathrm{Spec}}}\nolimits (R)\) be a relative proper flat curve over a henselian base. Let G be a reductive C-group scheme. Under mild technical assumptions, we show that a G-torsor over C which is trivial on the closed fiber of C is locally trivial for the Zariski topology.

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Notes

  1. 1.

    With the convention that a set defines a groupoid [47, Tag 001A].

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Acknowledgements

We thank Laurent Moret-Bailly for the extension of the toral case beyond curves, a strengthened version of Lemma  6.2 and several suggestions. We thank Jean-Louis Colliot-Thélène for communicating to us his method to deal with tori. We thank Olivier Benoist for raising a question answered by Theorem 7.2. Finally we thank Lie Fu, Ofer Gabber, Jochen Heinloth and Anastasia Stavrova for useful discussions. The first author is supported by the project ANR Geolie, ANR-15-CE 40-0012, (The French National Research Agency). The second and third authors are partially supported by National Science Foundation grants DMS-1463882 and DMS-1801951.

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Communicated by Vasudevan Srinivas.

Appendices

Appendices

The purpose of this appendix is to provide proofs to statements for algebraic spaces and stacks which are well-known among experts.

Jacobian criterion for stacks

Let S be a scheme and let \({\mathcal {X}}\), \({\mathcal {Y}}\) be quasi-separated algebraic S-stacks of finite presentation. Let \(g: {\mathcal {X}}\rightarrow {\mathcal {Y}}\) be a 1-morphism over S. We have a 1-morphism \(Tg: T({\mathcal {X}}) \rightarrow T({\mathcal {Y}})\) of algebraic stacks [36, 17.14, 17.16].

Let \(s \in S\) and denote by K the residue field of s. Let \(x: \mathop {{\mathrm{Spec}}}\nolimits (K) \rightarrow {\mathcal {X}}\) be a 1-morphism mapping to s. We put \(T({\mathcal {X}})_x= T({\mathcal {X}}/S) \times _{{\mathcal {X}}} \mathop {{\mathrm{Spec}}}\nolimits (K)\) and denote by \(\mathrm {Tan}_x({\mathcal {X}})\) the category \(T({\mathcal {X}})_x(K)\). We denote by \(y= g \circ x: \mathop {{\mathrm{Spec}}}\nolimits (K) \rightarrow {\mathcal {Y}}\) and get the tangent morphism \((Tg)_x: \mathrm {Tan}_x({\mathcal {X}}) \rightarrow \mathrm {Tan}_y({\mathcal {Y}})\).

Proposition 8.1

We assume that \({\mathcal {X}}\) is smooth at x over S. Then the following assertions are equivalent:

  1. (i)

    The morphism g is smooth at x;

  2. (ii)

    The tangent morphism \((Tg)_x: \mathrm {Tan}_x({\mathcal {X}}) \rightarrow \mathrm {Tan}_y({\mathcal {Y}})\) is essentially surjective.

Furthermore, under those conditions, \({\mathcal {Y}}\) is smooth at y over S.

Proof

In the case of a morphism \(g: X \rightarrow Y\) of S-schemes locally of finite presentation such that \(g(x)=y\) and X is smooth at x over S, we have that \(K=\kappa (x)=\kappa (y)\) so that the statement is a special case of [20, 17.11.1]. We proceed now to the stack case.

\((i) \Longrightarrow (ii).\) Up to shrinking, we can assume that \({\mathcal {X}}\) is smooth over S and that g is smooth. We denote by \(i: \mathop {{\mathrm{Spec}}}\nolimits (K) \rightarrow \mathop {{\mathrm{Spec}}}\nolimits (K[\epsilon ])\).

We are given an object of \(\mathrm {Tan}_y({\mathcal {Y}})\), that is a morphism couple \((y', \eta )\) where \(y': \mathop {{\mathrm{Spec}}}\nolimits (K[\epsilon ]) \rightarrow {\mathcal {Y}}\) together with a 2-morphism \(\eta : y' \circ i \rightarrow y\). We remind that g is formally smooth [47, Tag 0DP0] that is, if it is formally smooth on objects as a 1-morphism in categories fibered in groupoids [47, Tag 0DNV]. A special case is for the following commutative diagram

where \(\eta : y' \circ i \rightarrow y= g \circ x=y\) is a 2-morphism witnessing the commutativity of the diagram; there exists a triple \((x', \alpha , \beta )\) where:

  1. (i)

    \(x': \mathop {{\mathrm{Spec}}}\nolimits (K[\epsilon ] ) \rightarrow \mathfrak X\) is a morphism;

  2. (ii)

    \(\alpha : x' \circ i \rightarrow x\), \(\beta : y' \rightarrow g \circ x'\) are 2-arrows such that \(\eta = ( id_g \star \alpha ) \circ ( \beta \star id_i )\).

It follows that \(g(x', \alpha )= (g \circ x', g \circ \alpha ) \in \mathrm {Tan}_y({\mathcal {Y}})\) is isomorphic to \((y', \eta )\). This establishes the essential surjectivity of the tangent morphism.

\((ii) \Longrightarrow (i).\) According to [36, Thm. 6.3], there exists a smooth 1-morphism \(\varphi : Y \rightarrow {\mathcal {Y}}\) and a point \(y_1 \in Y(K)\) mapping to y such that Y is an affine scheme. We note that \(K=\kappa (y_1)\). We consider the fiber product \({\mathcal {X}}'= {\mathcal {X}}\times _{\mathcal {Y}}Y\), it is an algebraic stack and there exists a 1-morphism \(x': \mathop {{\mathrm{Spec}}}\nolimits (K) \rightarrow {\mathcal {X}}'\) lifting x and \(y_1\). There exists a smooth 1-morphism \(\psi : X' \rightarrow {\mathcal {X}}'\) and a point \(x_1 \in X'(K)\) mapping to x such that \(X'\) is an affine scheme. By construction we have again that \(K=\kappa (x_1)\). We have then the commutative diagram

According to [36, Lem. 17.5.1], the square

is 2-cartesian. It follows that the square

is 2-cartesian. Our assumption is that the bottom morphism is essentially surjective, it follows that \((Tg')_{x'}: \mathrm {Tan}_{x_1}({\mathcal {X}}') \rightarrow \mathrm {Tan}_{y_1}(Y)\) is essentially surjective as well. Since \(\psi \) is smooth, the map \((T\psi )_{x_1}: \mathrm {Tan}_{x'}(X') \rightarrow \mathrm {Tan}_{x_1}({\mathcal {X}}')\) is essentially surjective. By composition it follows that \(\mathrm {Tan}_{x_1}(X') \rightarrow \mathrm {Tan}_{y_1}(Y)\) is essentially surjective. Since \(X'\) and Y are locally of finite presentation over S, the case of schemes yields that \(g' \circ \psi : X' \rightarrow Y\) is smooth at \(x'\). By definition of smoothness for morphisms of stacks [42, §8.2], we conclude that g is smooth at x.

We assume (ii) and shall show that \({\mathcal {Y}}\) is smooth at y over S. Using the diagrams of the proof, we have seen that the S-morphism \(X' \rightarrow Y\) of schemes is smooth at \(x'\). Once again the classical Jacobian criterion [20, 17.11.1] applies and shows that Y is smooth at \(y_1\) over S. By definition of smoothness for stacks, we get that \({\mathcal {Y}}\) is smooth at y over S. \(\square \)

Lie algebra of an S-group space

Let S be a scheme. Let \(f: X \rightarrow Y\) be a morphism of S-algebraic spaces. We consider the quasi-coherent sheaf \(\Omega ^1_{ X/Y}\) on X defined in [47, Tag 04CT]. Let T be an S-scheme equipped with a closed subscheme \(T_0\) defined by a quasi-coherent ideal \({\mathcal {I}}\) such that \({\mathcal {I}}^2=0\). According to [42, 7.A page 167] for any commutative diagram of algebraic spaces

if there exists a dotted arrow filling in the diagram then the set of such dotted arrows form a torsor under \({{\,\mathrm{Hom}\,}}_{{\mathcal {O}}_{T_0}}(x_{0}^*\Omega ^1_{X/Y} , {\mathcal {I}})\). We extend to group spaces well-known statements on group schemes [44, II.4.11.3].

Lemma 8.2

Let G be an S-group space. We denote by \(e_G: S \rightarrow G\) the unit point and put \(\omega _{G/S}=e_G^*\bigl ( \Omega ^1_{G/S})\).

  1. (1)

    There is a canonical isomorphism of S-functors \(\mathop {{\mathrm{Lie}}}\nolimits (G) \buildrel \sim \over \longrightarrow {\mathbf {V}}(\omega _{G/S})\) which is compatible with the \({\mathcal {O}}_S\)-structure.

  2. (2)

    If \(\omega _{G/S}\) is a locally free coherent sheaf, then \(\mathop {{\mathrm{Lie}}}\nolimits (G) \buildrel \sim \over \longrightarrow {\mathbf {W}}(\omega _{G/S}^\vee )\). In particular we have an isomorphism

    $$\begin{aligned} \mathop {{\mathrm{Lie}}}\nolimits (G)(R) \otimes _R R' \buildrel \sim \over \longrightarrow \mathop {{\mathrm{Lie}}}\nolimits (G)(R') \end{aligned}$$

    for each morphism of S-algebras \(R \rightarrow R'\).

  3. (3)

    Assume that G is smooth and quasi-separated over S. Then \(\omega _{G/S}\) is a finite locally free coherent sheaf and (2) holds.

Under the conditions of (2) or (3), we denote also by \(\mathcal {L}ie(G)=\omega _{G/S}^\vee \) the locally free coherent sheaf.

Proof

  1. (1)

    Let \(T_0\) be an S-scheme and consider \(T=T_0[\epsilon ]\). We apply the above fact to the morphism \(G \rightarrow S\) and the points \(x_0=e_{G_{T_0}}\) and \(y : T \rightarrow S\) the structural morphism. It follows that \(\ker \bigl ( G(T) \rightarrow G(T_0) \bigr )\) is a torsor under \({{\,\mathrm{Hom}\,}}_{{\mathcal {O}}_{T_0}}(e_{G_{T_0}}^* \Omega ^1_{G/S} , \epsilon \, {\mathcal {O}}_{T_0}) \cong {{\,\mathrm{Hom}\,}}_{{\mathcal {O}}_{T_0}}(e_{G_{T_0}}^*\Omega ^1_{G/S},{\mathcal {O}}_{T_0}) = {{\,\mathrm{Hom}\,}}_{{\mathcal {O}}_{T_0}}(\omega ^1_{G/S} \otimes _{{\mathcal {O}}_S} {\mathcal {O}}_{T_0},{\mathcal {O}}_{T_0})\). We have constructed a isomorphism of S-functors \(\mathop {{\mathrm{Lie}}}\nolimits (G) \, \buildrel \sim \over \longrightarrow \, {\mathbf {V}}(\omega _{G/S})\) and the compatibility of \({\mathcal {O}}_S\)-structures is a straightforward checking.

  2. (2)

    If \(\omega _{G/S}\) is a locally free coherent sheaf, then \(\mathop {{\mathrm{Lie}}}\nolimits (G) \buildrel \sim \over \longrightarrow {\mathbf {V}}(\omega _{G/S}) \buildrel \sim \over \longrightarrow {\mathbf {W}}(\omega _{G/S}^\vee )\). The next fact follows from [19, 12.2.3].

  3. (3)

    According to [47, Tag 0CK5], \(\Omega ^1_{G/S}\) is a finite locally free coherent sheaf over G. If follows that \(\omega ^1_{G/S}\) is a finite locally free coherent sheaf over S. \(\square \)

Lemma 8.3

Let G be a smooth S-group space and let T be an S-scheme equipped with a closed subscheme \(T_0\) defined by a quasi-coherent ideal \({\mathcal {I}}\) such that \({\mathcal {I}}^2=0\). We denote by \(t_0: T_0 \rightarrow S\) the structural morphism, \(G_0 = G \times _S T_0\) and assume that \(t_0\) is quasi-compact and quasi-separated.

(1) We have an exact sequence of fppf (resp. étale, Zariski) sheaves on S

$$\begin{aligned} 0 \rightarrow {\mathbf {W}}\Bigl ( (t_0)_*\bigl ( \mathcal {L}ie(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}}\bigr ) \Bigr ) \rightarrow \prod _{T/S} G \rightarrow \prod _{T_0/S} G \rightarrow 1 . \end{aligned}$$

(2) If \(T= \mathop {{\mathrm{Spec}}}\nolimits (A)\) is affine and \(T_0=\mathop {{\mathrm{Spec}}}\nolimits (A/I)\), we have an exact sequence

$$\begin{aligned} 0 \rightarrow \mathop {{\mathrm{Lie}}}\nolimits (G)(A) \otimes _A I \rightarrow G(A) \rightarrow G(A/I) \rightarrow 1. \end{aligned}$$

Proof

(1) We have

$$\begin{aligned} {{\,\mathrm{Hom}\,}}_{{\mathcal {O}}_{T_0}}(\omega _{G_0/T_0},{\mathcal {I}}) = H^0(T_0, \mathcal {L}ie(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}}) = H^0\Bigl (T, t_{0,*}( \mathcal {L}ie(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}}) \Bigr ), \end{aligned}$$

whence an exact sequence

$$\begin{aligned} 0 \rightarrow H^0\Bigl (T, t_{0,*}( \mathcal {L}ie(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}}) \Bigr ) \rightarrow G(T) \rightarrow G(T_0). \end{aligned}$$

Now let \(h: S' \rightarrow S\) be a flat morphism locally of finite presentation and denote by \(G'=G \times _S S'\), \(h_T: T' \rightarrow T\), \(\dots \) the relevant base change to \(S'\). Since \(t_0\) is quasi-compact and quasi-separated, the flatness of h yields an isomorphism [47, Tag 02KH]

$$\begin{aligned} h^*\bigl ( t_{0,*}( \mathcal {L}ie(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}})\bigr ) \buildrel \sim \over \longrightarrow t'_{0,*}\Bigl ( h_{T_0}^*\bigl ( \mathcal {L}ie(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}}\bigr ) \Bigr ) = t'_{0,*}( \mathcal {L}ie(G'_0) \otimes _{{\mathcal {O}}_{T'_0}} {\mathcal {I}}'). \end{aligned}$$

The similar sequence for \(T'_0\) reads then

$$\begin{aligned} 0 \rightarrow H^0\Bigl (T', t_{0,*}( \mathcal {Lie}(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}}) \Bigr ) \rightarrow G(T') \rightarrow G(T'_0). \end{aligned}$$

We have then an exact sequence of fppf sheaves

$$\begin{aligned} 0 \rightarrow {\mathbf {W}}\Bigl ( (t_0)_*\bigl ( \mathcal {Lie}(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}}\bigr ) \Bigr ) \rightarrow \prod _{T/S} G \rightarrow \prod _{T_0/S} G . \end{aligned}$$

For T an affine scheme, the map \(G(T) \rightarrow G(T_0)\) is onto since the smooth S-group space G is formally smooth [47, Tag 04AM], that is, it satisfies the infinitesimal lifting criterion [47, Tag 049S, 060G].

It implies the exactness for the the Zariski, étale and fppf topologies.

(2) We can assume that \(S=T=\mathop {{\mathrm{Spec}}}\nolimits (A)\). In this case, we have

$$\begin{aligned} H^0\Bigl (S, (t_0)_*\bigl ( \mathcal {Lie}(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}}\bigr )\Bigr ) = H^0(T_0, \mathcal {Lie}(G_0) \otimes _{{\mathcal {O}}_{T_0}} {\mathcal {I}}) = \mathop {{\mathrm{Lie}}}\nolimits (G_0)(A/I) \otimes _{A/I} I. \end{aligned}$$

We have \(\mathop {{\mathrm{Lie}}}\nolimits (G_0)(A/I) = \mathop {{\mathrm{Lie}}}\nolimits (G)(A) \otimes _{A} A/I\) in view of Lemma 8.3.(2) whence the identification \(\mathop {{\mathrm{Lie}}}\nolimits (G_0)(A/I) \otimes _{A/I} I \cong \mathop {{\mathrm{Lie}}}\nolimits (G)(A) \otimes _{A} I\). Then (1) provides the exact sequence

$$\begin{aligned} 0 \rightarrow \mathop {{\mathrm{Lie}}}\nolimits (G)(A) \otimes _A I \rightarrow G(A) \rightarrow G(A/I) \end{aligned}$$

and the right map is onto since G is smooth. \(\square \)

Remark 8.4

  1. (a)

    A special case of (1) is \(T=S[\epsilon ]\) and \(T_0=S\). We get an exact sequence of fppf (resp. étale, Zariski) sheaves on S

    $$\begin{aligned} 0 \rightarrow {\mathbf {W}}( \mathcal {Lie}(G)) \rightarrow \prod _{S[\epsilon ]/S} G \rightarrow G \rightarrow 1 . \end{aligned}$$
  2. (b)

    In the group scheme case, (2) is established in [15, proof of II.5.2.8].

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Gille, P., Parimala, R. & Suresh, V. Local triviality for G-torsors. Math. Ann. (2021). https://doi.org/10.1007/s00208-020-02138-7

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