1 Introduction
1.1 Aim and scope
Let
$S$
be a Dedekind scheme of dimension 1 and
$\unicode[STIX]{x1D702}=\operatorname{Spec}(K)$
its generic point; let
$X$
be a scheme,
$f:X\rightarrow S$
a faithfully flat morphism of finite type and
$f_{\unicode[STIX]{x1D702}}:X_{\unicode[STIX]{x1D702}}\rightarrow \unicode[STIX]{x1D702}$
its generic fibre. Assume that we are given a finite
$K$
-group scheme
$G$
and a
$G$
-torsor
$Y\rightarrow X_{\unicode[STIX]{x1D702}}$
. So far, the problem of extending the
$G$
-torsor
$Y\rightarrow X_{\unicode[STIX]{x1D702}}$
has consisted in finding a finite and flat
$S$
-group scheme
$G^{\prime }$
whose generic fibre is isomorphic to
$G$
and a
$G^{\prime }$
-torsor
$T\rightarrow X$
whose generic fibre is isomorphic to
$Y\rightarrow X_{\unicode[STIX]{x1D702}}$
as a
$G$
-torsor. Some solutions, from Grothendieck’s first ideas until nowadays, are known in some particular relevant cases and are the object of many classical and well-known results and more recent papers; see, for instance, [Reference Grothendieck10, Exposé X], [Reference Raynaud13, §3], [Reference Saïdi14, §2.4], [Reference Tossici15, Corollary 4.2.8], [Reference Antei2] and [Reference Antei3]. However, a general solution does not exist. Moreover, it is known that it can even happen that
$G$
does not admit a finite and flat model (see, for instance, [Reference Milne12]). What is always true is that
$G$
admits at least an affine, quasi-finite, flat
$R$
-group scheme model as an easy consequence of [Reference Waterhouse16, §3.4]. In this paper we study the problem of extending torsors under the action of very general
$G$
, that is, when
$G$
is only affine and of finite type. This approach has been already used in [Reference Antei and Emsalem5], where it has been proved that (at least when
$X$
has dimension 2, the case
$\operatorname{dim}(X)>2$
having a different formulation for which we refer the reader to [Reference Antei and Emsalem5]) every torsor over
$X_{\unicode[STIX]{x1D702}}$
under the action of an affine and flat group scheme can be extended to a torsor over
$X$
up to a finite number of Néron blow ups of
$X$
at a closed subscheme of its special fibre. In this paper we focus essentially, but not only, on a precise example, the case when
$X$
is the affine space
$\mathbb{A}_{R}^{n}$
, i.e., the
$n$
-dimensional affine space defined over a discrete valuation ring
$R$
. In this setting we are able to prove the following result (cf. Theorem 3.3 and Corollary 3.5).
Theorem 1.1. Let
$X=\mathbb{A}_{R}^{n}$
be the
$n$
-dimensional affine space and
$x=(0,\ldots ,0)$
its origin. Let
$G$
be an affine
$K$
-group scheme of finite type and
$f:Y\rightarrow X_{\unicode[STIX]{x1D702}}$
a
$G$
-torsor pointed in
$y\in Y_{x_{\unicode[STIX]{x1D702}}}(K)$
. Then, possibly after pulling back
$Y$
over an automorphism of
$\mathbb{A}_{K}^{n}$
, there exists a
$G^{\prime }$
-torsor
$f^{\prime }:Y^{\prime }\rightarrow \mathbb{A}_{R}^{n}$
, pointed in
$y^{\prime }\in Y_{x}(R)$
, extending the given
$G$
-torsor
$Y$
.
This led us to formulate the following conjecture, which we are not able to prove at the momentFootnote 1 .
Conjecture 1.2. Let
$\unicode[STIX]{x1D70B}^{\text{qf}}(\mathbb{A}_{R}^{n},0)$
denote the quasi-finite fundamental group scheme of
$\mathbb{A}_{R}^{n}$
at its origin as defined in [Reference Antei, Emsalem and Gasbarri6]; then the following faithfully flat morphism:
$$\begin{eqnarray}\unicode[STIX]{x1D70B}(\mathbb{A}_{K}^{n},0)\rightarrow \unicode[STIX]{x1D70B}^{\text{qf}}(\mathbb{A}_{R}^{n},0)\times _{R}K\end{eqnarray}$$
is an isomorphism.
This conjecture is known to be true if we replace
$\mathbb{A}_{R}^{n}$
by an abelian scheme or, more generally, for smooth projective schemes
$X$
(with some extra assumptions) over
$R$
provided we consider the abelianization
$\unicode[STIX]{x1D70B}^{\text{ab}}(X,x)$
of the fundamental group scheme
$\unicode[STIX]{x1D70B}(X,x)$
(cf. [Reference Antei2]).
Similar techniques show that an analog of Theorem 1.1 may indeed be stated for other interesting cases, for instance when
$X=\unicode[STIX]{x1D6FC}_{p,R}$
. This has a particular interest because reduced schemes are often not studied in this context. And this is nothing but a single not reduced point. This leads to a conjecture similar to 1.2 stated in terms of the pseudo-fundamental group scheme, as defined in [Reference Antei and Dey4].
1.2 Notation and conventions
Let
$S$
be any scheme,
$X$
an
$S$
-scheme,
$G$
an affine (faithfully) flat
$S$
-group scheme and
$Y$
an
$S$
-scheme endowed with a right action
$\unicode[STIX]{x1D70E}:Y\times G\rightarrow Y$
. An
$S$
-morphism
$p:Y\rightarrow X$
is said to be a
$G$
-torsor if it is affine, faithfully flat,
$G$
-invariant and the canonical morphism
$(\unicode[STIX]{x1D70E},pr_{Y}):Y\times G\rightarrow Y\times _{X}Y$
is an isomorphism. Let
$H$
be a flat
$S$
-group scheme and
$q:Z\rightarrow X$
a
$H$
-torsor; a morphism between two such torsors is a pair
$(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FC}):(Z,H)\rightarrow (Y,G)$
, where
$\unicode[STIX]{x1D6FC}:H\rightarrow G$
is an
$S$
-morphism of group schemes and
$\unicode[STIX]{x1D6FD}:Z\rightarrow Y$
is an
$X$
-morphism of schemes such that the following diagram commutes:
(thus
$Y$
is isomorphic to the contracted product
$Z\times ^{H}G$
through
$\unicode[STIX]{x1D6FC}$
; cf. [Reference Demazure and Gabriel8, III, §§4 and 3.2]). In this case we say that
$Z$
precedes
$Y$
. Assume moreover that
$\unicode[STIX]{x1D6FC}$
is a closed immersion. Then
$t$
is a closed immersion too and we say that
$Z$
is a subtorsor of
$Y$
(or that
$Z$
is contained in
$Y$
or that
$Y$
contains
$Z$
).
Let
$q\in S$
be any point. For any
$S$
-scheme
$T$
we will denote by
$T_{q}$
the fibre
$T\times _{S}\operatorname{Spec}(k(q))$
of
$T$
over
$q$
. In a similar way for any
$S$
-morphism of schemes
$v:T\rightarrow T^{\prime }$
we will denote by
$v_{q}:T_{q}\rightarrow T_{q}^{\prime }$
the reduction of
$v$
over
$\operatorname{Spec}(k(s))$
. When
$S$
is irreducible,
$\unicode[STIX]{x1D702}$
will denote its generic point and
$K$
its function field
$k(\unicode[STIX]{x1D702})$
. Any
$S$
-scheme whose generic fibre is isomorphic to
$T_{\unicode[STIX]{x1D702}}$
will be called a model of
$T_{\unicode[STIX]{x1D702}}$
. Furthermore, when
$v_{\unicode[STIX]{x1D702}}$
is an isomorphism, we will often say that
$v$
is a model map. When
$S$
is the spectrum of a discrete valuation ring, then
$s\in S$
will always denote the special point.
Throughout the whole paper a morphism of schemes
$f:Y\rightarrow X$
will be said to be quasi-finite if it is of finite type and, for every point
$x\in X$
, the fibre
$Y_{x}:=Y\times _{X}\operatorname{Spec}(k(x))$
is a finite set. Let
$S$
be any scheme and
$G$
an affine
$S$
-group scheme. Then we say that
$G$
is a finite (respectively quasi-finite/algebraic)
$S$
-group scheme if the structural morphism
$G\rightarrow S$
is finite (respectively quasi-finite/of finite type).
A
$G$
-torsor
$f:Y\rightarrow X$
is said to be finite (respectively quasi-finite/algebraic) if
$G$
is a flat
$S$
-group scheme which is moreover finite (respectively quasi-finite/of finite type). Of course, when
$S$
is the spectrum of a field, an
$S$
-group scheme is quasi-finite if and only if it is finite.
2 Néron blow ups and applications
2.1 Néron blow ups of torsors
In this section we recall the notions of Néron blow up and its applications in order to Néron blow up torsors. This technique in practice provides a useful tool to build new torsors from old ones. As an application we will use this construction to describe all the torsors (cf. Proposition 2.5) under a particular quasi-finite group scheme with generic fibre of order
$p$
and special fibre of order
$1$
, using the well-known description for some finite torsors of order
$p$
. Unless stated otherwise, from now until the end of §2 we only consider the following situation.
Notation 2.1. We denote by
$S$
the spectrum of a discrete valuation ring
$R$
with uniformizing element
$\unicode[STIX]{x1D70B}$
and with fraction and residue fields respectively denoted by
$K$
and
$k$
. As usual,
$\unicode[STIX]{x1D702}$
and
$s$
will denote the generic and special points of
$S$
, respectively. Finally, we denote by
$X$
a faithfully flat
$S$
-scheme of finite type.
Hereafter we recall a well-known result that will be used later.
Proposition 2.2. Let notation be as in 2.1, let
$C$
be a closed subscheme of the special fibre
$X_{s}$
of
$X$
and let
${\mathcal{I}}$
be the sheaf of ideals of
${\mathcal{O}}_{X}$
defining
$C$
. Let
$X^{\prime }\rightarrow X$
be the blow up of
$X$
at
$C$
and
$u:X^{C}\rightarrow X$
denote its restriction to the open subscheme of
$X^{\prime }$
, where
${\mathcal{I}}\cdot {\mathcal{O}}_{X}$
is generated by
$\unicode[STIX]{x1D70B}$
. Then:
(1)
$X^{C}$
is a flat
$S$
-scheme and
$u$
is an affine model map;(2) for any flat
$S$
-scheme
$Z$
and for any
$S$
-morphism
$v:Z\rightarrow X$
such that
$v_{k}$
factors through
$C$
, there exists a unique
$S$
-morphism
$v^{\prime }:Z\rightarrow X^{C}$
such that
$v=u\circ v^{\prime }$
.
Proof. See [Reference Bosch, Lütkebohmert and Raynaud7, §3.2, Proposition 1] or [Reference Anantharaman1, II, 2.1.2 (A)]. ◻
The morphism
$X^{C}\rightarrow X$
(or simply
$X^{C}$
) as in Proposition 2.2 is called the Néron blow up of
$X$
at
$C$
and property 2 is often referred to as the universal property of the Néron blow up.
Now we are going to explain how to Néron blow up torsors.
Lemma 2.3. Let
$G$
be an affine, algebraic and flat
$S$
-group scheme and
$H$
a closed subgroup scheme of
$G_{s}$
. Let
$Y$
be a
$G$
-torsor over
$X$
and
$Z$
an
$H$
-torsor over
$X_{s}$
, a subtorsor of
$Y_{s}\rightarrow X_{s}$
. Then there exist a faithfully flat
$S$
-scheme of finite type
$X^{\prime }$
and a model map
$\unicode[STIX]{x1D706}:X^{\prime }\rightarrow X$
such that
$Y^{Z}\rightarrow X^{\prime }$
is a
$G^{H}$
-torsor generically isomorphic to
$Y_{\unicode[STIX]{x1D702}}\rightarrow X_{\unicode[STIX]{x1D702}}$
. If moreover
$G$
is quasi-finite, then
$\unicode[STIX]{x1D706}$
can be obtained from
$X$
after a finite number of Néron blow ups.
Proof. This is [Reference Antei and Emsalem5, Proposition 3.7]. ◻
The importance of the previous construction is that we can build new torsors from old ones. In order to use this construction, we need the special fibre of our given torsor to properly contain some other torsors. This happens, for instance, when the special fibre is trivial, as in the following example.
Example 2.4. Assume that
$R$
has positive characteristic
$p$
. Let
$X:=\operatorname{Spec}(R[x])$
be the affine line over
$R$
. Then
$$\begin{eqnarray}Y:=\operatorname{Spec}(R[x,y]/(y^{p}-y-\unicode[STIX]{x1D70B}x))\end{eqnarray}$$
is a nontrivial
$(\mathbb{Z}/p\mathbb{Z})_{R}$
-torsor [Reference Milne11, III, Proposition 4.12], with special fibre
$$\begin{eqnarray}Y_{s}=\operatorname{Spec}(k[x,y]/(y^{p}-y)),\end{eqnarray}$$
which is a trivial
$(\mathbb{Z}/p\mathbb{Z})_{k}$
-torsor. It is then clear that
$X_{s}$
is a subtorsor of
$Y_{s}$
and we can blow up
$Y$
at
$X_{s}$
following Lemma 2.3, thus getting an
$M$
-torsor, where
$M$
is obtained after Néron blowing up
$(\mathbb{Z}/p\mathbb{Z})_{R}$
at
$\{1\}\text{}_{k}=\operatorname{Spec}(k)$
, a closed subgroup scheme of
$(\mathbb{Z}/p\mathbb{Z})_{k}$
, so that
$M=(\mathbb{Z}/p\mathbb{Z})_{R}^{\{1\}\text{}_{k}}=\operatorname{Spec}(R[y]/(\unicode[STIX]{x1D70B}^{p-1}y^{p}-y))$
; indeed,
$M=\operatorname{Spec}(R[M])$
, where
$R[M]:=R[x,\unicode[STIX]{x1D70B}^{-1}x]/(x^{p}-x)=R[y]/(\unicode[STIX]{x1D70B}^{p-1}y^{p}-y)$
, where we have set
$y=\unicode[STIX]{x1D70B}^{-1}x$
. It is flat as the Néron blowing up is always flat, quasi-finite, but clearly not finite. In a similar way,
$Y^{X_{s}}=\operatorname{Spec}(R[x,y]/(\unicode[STIX]{x1D70B}^{p-1}y^{p}-y-x))$
; then we obtain a quasi-finite
$M$
-torsor.
In a very similar way, we obtain the description of
$M$
-torsors over an affine scheme.
Proposition 2.5. Assume that
$R$
has positive characteristic
$p$
. Let
$X:=\operatorname{Spec}(A)$
be affine over
$R$
with
$X_{s}$
integral. Let
$M:=\operatorname{Spec}(R[x]/(\unicode[STIX]{x1D70B}^{p-1}x^{p}-x))$
be the
$R$
-group scheme defined in Example 2.4. Then any
$M$
-torsor over
$X$
is isomorphic to a torsor of the form
$$\begin{eqnarray}Y:=\operatorname{Spec}(A[y]/(\unicode[STIX]{x1D70B}^{p-1}y^{p}-y+a))\end{eqnarray}$$
for some
$a\in A$
.
Proof. This is as in Example 2.4 if we start from any
$(\mathbb{Z}/p\mathbb{Z})_{R}$
-torsor
$$\begin{eqnarray}\operatorname{Spec}(A[y]/(y^{p}-y+\unicode[STIX]{x1D70B}a))\end{eqnarray}$$
and we Néron blow it up in
$\operatorname{Spec}(A_{k}){\hookrightarrow}Y_{s}$
; we obtain the equation
$\operatorname{Spec}(A[y]/(\unicode[STIX]{x1D70B}^{p-1}y^{p}-y+a))$
, which is an
$M$
-torsor. On the other hand, if we start from an
$M$
-torsor
$Y$
over
$X$
then one can consider the contracted product
$Y\times ^{M}(\mathbb{Z}/p\mathbb{Z})_{R}$
, which is a
$(\mathbb{Z}/p\mathbb{Z})_{R}$
-torsor
$Z$
with trivial special fibre, so in particular
$Y$
is easily seen to be the Néron blowing up of
$Z$
in
$X_{s}$
; hence, as we have just observed, it is isomorphic to
$\operatorname{Spec}(A[y]/(\unicode[STIX]{x1D70B}^{p-1}y^{p}-y+a))$
.◻
3 Extension of torsors
Unless stated otherwise, from now util the end of §3 we only consider the following situation.
Notation 3.1. Let
$S$
be a trait, i.e., the spectrum of a discrete valuation ring
$R$
with uniformizing element
$\unicode[STIX]{x1D70B}$
, with fraction and residue fields denoted by
$K$
and
$k$
, respectively. We denote by
$\unicode[STIX]{x1D702}$
and
$s$
the generic and special points of
$S$
.
Lemma 3.2. Let the notation be as in 3.1, where we assume
$X=\operatorname{Spec}(A)$
to be affine and provided with a section
$x\in X(R)$
. Let
$G$
be an affine
$K$
-group scheme of finite type,
$Y=\operatorname{Spec}(B)$
a
$K$
-scheme and
$f:Y\rightarrow X_{\unicode[STIX]{x1D702}}$
a
$G$
-torsor pointed in
$y\in Y(K)$
lying over
$x_{\unicode[STIX]{x1D702}}$
. We need the following technical assumption:
∙ we fix an embedding
$G{\hookrightarrow}\text{GL}_{d,K}$
and we consider the contracted product
$Z:=Y\times ^{G}\text{GL}_{d,K}$
; we assume that
$Z\rightarrow X_{\unicode[STIX]{x1D702}}$
is a trivial
$\text{GL}_{d,K}$
-torsor (i.e.,
$Z\simeq \text{GL}_{d,X_{\unicode[STIX]{x1D702}}}$
).
Then there exists a
$G^{\prime }$
-torsor
$f^{\prime }:Y^{\prime }\rightarrow X^{\prime }$
extending the given
$G$
-torsor
$Y$
, where
$G^{\prime }$
is the closure of
$G$
in
$\text{GL}_{d,R}$
and
$X^{\prime }$
is obtained by
$X$
after a finite number of Néron blow ups of
$x_{s}\in X_{s}$
.
Proof. By assumption,
$X=\operatorname{Spec}(A)$
is an affine scheme over
$S=\operatorname{Spec}(R)$
and we denote by
$X_{\unicode[STIX]{x1D702}}=\operatorname{Spec}(A_{K})$
its generic fibre. The point
$x$
corresponds to an
$R$
-ring morphism
$\unicode[STIX]{x1D6FC}:A\rightarrow R$
, which, tensoring by
$K$
over
$R$
, gives the
$K$
-morphism
$\unicode[STIX]{x1D6FC}_{K}:A_{K}\rightarrow K$
corresponding to
$x_{\unicode[STIX]{x1D702}}$
. Since we are assuming that
$Y$
has a
$K$
-rational point
$y:\operatorname{Spec}(K)\rightarrow Y$
over
$x_{\unicode[STIX]{x1D702}}:\operatorname{Spec}(K)\rightarrow X_{\unicode[STIX]{x1D702}}$
, then in particular
$Y_{x_{\unicode[STIX]{x1D702}}}=\operatorname{Spec}(B\otimes _{A_{K}}K)\simeq G$
and, if we set
$C:=B\otimes _{A_{K}}K$
, we can assume that
$G=\operatorname{Spec}(C)$
. Hence,
$C$
is a quotient of
$B$
and we have the following commutative diagrams:
where
$\unicode[STIX]{x1D6E5}_{C}$
is the comultiplication of the
$K$
-Hopf algebra
$C$
and
$\unicode[STIX]{x1D70C}_{B}$
is the coaction induced by the (right) action of
$\unicode[STIX]{x1D70E}:Y\times G\rightarrow Y$
, thus giving
$B$
a structure of (left) comodule over
$C$
. Finally,
$q$
is the morphism induced by the closed immersion
$G{\hookrightarrow}Y$
and we will denote by
$\unicode[STIX]{x1D700}_{C}:C\rightarrow K$
and
$S_{C}:C\rightarrow C$
, respectively, the counit and the coinverse morphisms of
$C$
. Now consider the surjective morphism of
$A_{K}$
-algebras induced by the closed immersion of
$Y$
into the trivial
$\text{GL}_{d,X_{\unicode[STIX]{x1D702}}}$
-torsor:
$$\begin{eqnarray}u:A_{K}[y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]\rightarrow B;\end{eqnarray}$$
then, if we identify
$B$
with the quotient
$A_{K}[y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]$
by
$\operatorname{ker}(u)$
and we take, via
$\unicode[STIX]{x1D6FC}_{K}$
, the tensor product over
$K$
, we obtain
$$\begin{eqnarray}\begin{array}{@{}c@{}}\displaystyle B=\frac{A_{K}[y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]}{f_{1},\ldots ,f_{s}},\\ \displaystyle C=\frac{K[x_{11},\ldots ,x_{dd},1/\text{det}[x_{ij}]]}{{\unicode[STIX]{x1D6FC}_{K}}_{\ast }(f_{1}),\ldots ,{\unicode[STIX]{x1D6FC}_{K}}_{\ast }(f_{s})},\qquad q:y_{ij}\mapsto x_{ij}.\end{array}\end{eqnarray}$$
For each
$i=1,\ldots ,s$
, we assume that the polynomials
$f_{i}$
have coefficients in
$A$
. Consequently, the
$\unicode[STIX]{x1D6FC}_{\ast }(f_{i})$
have coefficients in
$R$
.
From the comultiplication on
$Z$
(i.e.,
$\unicode[STIX]{x1D6E5}_{Z}(y_{ij})=\sum _{r=1}^{d}y_{ir}\otimes y_{rj}$
), we deduce that
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{B}(y_{ij})=\mathop{\sum }_{r=1}^{d}x_{ir}\otimes y_{rj}\end{eqnarray}$$
and consequently
$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{C}(x_{ij})=\mathop{\sum }_{r=1}^{d}x_{ir}\otimes x_{rj}.\end{eqnarray}$$
Applying to the latter the equality
$(\unicode[STIX]{x1D700}_{C}\otimes \text{id})\unicode[STIX]{x1D6E5}_{C}=\text{id}$
and comparing coefficients, we get
$$\begin{eqnarray}\unicode[STIX]{x1D700}_{C}(x_{ij})=\unicode[STIX]{x1D6FF}_{ij}.\end{eqnarray}$$
Moreover, recalling that
$\unicode[STIX]{x1D6E5}_{C}(S_{C},\text{id})=\unicode[STIX]{x1D700}_{C}$
, we obtain
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{ij}=\mathop{\sum }_{r=1}^{d}S(x_{ir})x_{rj};\end{eqnarray}$$
thus,
$S_{C}(x_{sr})$
is the
$(s,r)$
th entry (
$s$
th row,
$r$
th column) in the
$d\times d$
matrix
$[x_{ij}]^{-1}$
. In particular,
$\unicode[STIX]{x1D6E5}_{C}(1/(\text{det}[x_{ij}]))=1/(\text{det}[x_{ij}])\otimes 1/(\text{det}[x_{ij}])$
, since
$\unicode[STIX]{x1D6E5}_{C}(\text{det}[x_{ij}])=\text{det}[x_{ij}]\otimes \text{det}[x_{ij}]$
.
The isomorphism given by
$Y\times G\stackrel{{\sim}}{\longrightarrow }Y\times _{X_{\unicode[STIX]{x1D702}}}Y,(y,g)\mapsto (y,yg)$
gives rise to the isomorphism
$$\begin{eqnarray}\unicode[STIX]{x1D6F9}:B\otimes _{A_{K}}B\stackrel{{\sim}}{\longrightarrow }C\otimes B,\qquad y_{ij}\otimes y_{rs}\mapsto \unicode[STIX]{x1D70C}(y_{ij})(1\otimes y_{rs}).\end{eqnarray}$$
We are going to describe
$\unicode[STIX]{x1D6F9}^{-1}$
. Since of course
$\unicode[STIX]{x1D6F9}^{-1}(1\otimes y_{ij})=(1\otimes y_{ij})$
, it only remains to compute
$\unicode[STIX]{x1D6F9}^{-1}(x_{ij}\otimes 1)$
. We claim that
$$\begin{eqnarray}\unicode[STIX]{x1D6F9}^{-1}(x_{ij}\otimes 1)=\mathop{\sum }_{r=1}^{d}y_{ir}\otimes H(y_{rj}),\end{eqnarray}$$
where, for all
$(r,s)\in \{1,\ldots ,d\}\text{}^{2}$
,
$H(y_{rs})$
denotes the
$(s,r)$
th entry (
$s$
th row,
$r$
th column) in the
$d\times d$
matrix
$[y_{ij}]^{-1}$
. Indeed,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}\biggl(\mathop{\sum }_{r=1}^{d}y_{ir}\otimes H(y_{rj})\biggr) & = & \displaystyle \mathop{\sum }_{r=1}^{d}\unicode[STIX]{x1D70C}(y_{ir})(1\otimes H(y_{rj}))\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{r=1}^{d}\biggl(\mathop{\sum }_{s=1}^{d}x_{is}\otimes (y_{sr}H(y_{rj}))\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{s=1}^{d}\biggl(x_{is}\otimes \mathop{\sum }_{r=1}^{d}(y_{sr}H(y_{rj}))\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{s=1}^{d}(x_{is}\otimes \unicode[STIX]{x1D6FF}_{sj})=x_{ij}\otimes 1.\nonumber\end{eqnarray}$$
Now it is important to observe that
$H(y_{rj})=P(y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}])\in \mathbb{Z}[y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]$
, so in particular it has coefficients in
$R$
. So, let us set
$$\begin{eqnarray}B^{\prime }:=\frac{A[y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]}{f_{1},\ldots ,f_{s}}.\end{eqnarray}$$
In order for
$\operatorname{Spec}(B^{\prime })$
to be a torsor over
$\operatorname{Spec}(A)$
, we need indeed
$B^{\prime }$
to be
$A$
-faithfully flat, so we divide the reminder of the proof into two steps: in the first we explain that if
$B^{\prime }$
is
$A$
-faithfully flat, then
$\operatorname{Spec}(B^{\prime })$
is a
$\operatorname{Spec}(C^{\prime })$
-torsor over
$\operatorname{Spec}(A)$
, where
$C^{\prime }:=B^{\prime }\otimes _{A}R$
; in the second we will describe how to always reduce to this situation up to Néron blow up of the scheme
$X$
in
$x_{s}$
, the special fibre of the
$R$
-valued point of
$X$
.
Step 1. Let us assume that
$B^{\prime }$
is
$A$
-faithfully flat: thus,
$$\begin{eqnarray}C^{\prime }=\frac{R[x_{11},\ldots ,x_{dd},1/\text{det}[x_{ij}]]}{\unicode[STIX]{x1D6FC}_{\ast }(f_{1}),\ldots ,\unicode[STIX]{x1D6FC}_{\ast }(f_{s})}\end{eqnarray}$$
is
$R$
-flat and it becomes a Hopf algebra over
$R$
when provided with the comultiplication given by the restriction of
$\unicode[STIX]{x1D6E5}$
to
$C^{\prime }$
:
$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{C^{\prime }}:C^{\prime }\rightarrow C^{\prime }\otimes C^{\prime },\qquad x_{ij}\mapsto \mathop{\sum }_{r=1}^{d}x_{ir}\otimes x_{rj},\end{eqnarray}$$
the coinverse given by
$$\begin{eqnarray}S_{C^{\prime }}:C^{\prime }\rightarrow C^{\prime },\qquad x_{ij}\mapsto S_{C^{\prime }}(x_{ij}),\end{eqnarray}$$
where
$S_{C^{\prime }}(x_{rs})$
denotes the
$(s,r)$
th entry in the matrix
$[x_{ij}]^{-1}$
, and finally the counity given by
$$\begin{eqnarray}\unicode[STIX]{x1D700}_{C^{\prime }}:C^{\prime }\rightarrow R,\qquad x_{ij}\mapsto \unicode[STIX]{x1D6FF}_{ij}.\end{eqnarray}$$
Moreover,
$B^{\prime }$
acquires a structure of (left) comodule over
$C^{\prime }$
when provided with the coaction given by
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{B^{\prime }}:B^{\prime }\rightarrow C^{\prime }\otimes _{R}B^{\prime },\qquad y_{ij}\mapsto \mathop{\sum }_{r=1}^{d}x_{ir}\otimes y_{rj}.\end{eqnarray}$$
Furthermore, the natural morphism
$$\begin{eqnarray}\unicode[STIX]{x1D6F9}^{\prime }:B^{\prime }\otimes _{A}B^{\prime }\longrightarrow C^{\prime }\otimes B^{\prime },\qquad y_{ij}\otimes y_{rs}\mapsto \unicode[STIX]{x1D70C}_{B^{\prime }}(y_{ij})(1\otimes y_{rs})\end{eqnarray}$$
has an inverse given by
$$\begin{eqnarray}\unicode[STIX]{x1D6F9}^{\prime -1}:C^{\prime }\otimes B^{\prime }\stackrel{{\sim}}{\longrightarrow }B^{\prime }\otimes _{A}B^{\prime },\qquad x_{ij}\otimes y_{uv}\mapsto \mathop{\sum }_{r=1}^{d}y_{ir}\otimes (H(y_{rj})y_{uv})\end{eqnarray}$$
and it is thus an isomorphism. Setting
$G^{\prime }:=\operatorname{Spec}(C^{\prime })$
and
$Y^{\prime }:=\operatorname{Spec}(B^{\prime })$
, then
$G^{\prime }$
is an
$R$
-flat group scheme of finite type acting on
$Y^{\prime }$
such that
$Y^{\prime }\rightarrow X$
is a
$G^{\prime }$
-invariant morphism. Finally, inverting arrows in (8) and (9), we obtain the desired isomorphism
$$\begin{eqnarray}Y^{\prime }\times G^{\prime }\stackrel{{\sim}}{\longrightarrow }Y^{\prime }\times _{X}Y^{\prime },\end{eqnarray}$$
so, by definition,
$Y^{\prime }\rightarrow X$
is a
$G^{\prime }$
-torsor.
Step 2. When
$B^{\prime }$
is not
$A$
-faithfully flat, we Néron blow up
$X$
: first,
$A$
being of finite type over
$R$
, we can write
$$\begin{eqnarray}A=R[t_{1},\ldots ,t_{r}]/u_{1}(t_{1},\ldots ,t_{r}),\ldots ,u_{m}(t_{1},\ldots ,t_{r}),\end{eqnarray}$$
so we rewrite in a useful way equations (2):
$$\begin{eqnarray}B=\frac{K[t_{1},\ldots ,t_{r},y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]}{u_{1},\ldots ,u_{m},f_{1},\ldots ,f_{s}},\end{eqnarray}$$
where the
$u_{i}=u_{i}(t_{1},\ldots ,t_{r}),i=1,\ldots ,m$
and
$f_{n}=f_{n}(t_{1},\ldots ,t_{r},y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]),n=1,\ldots ,s$
are polynomials with coefficients in
$K$
. Chasing denominators if necessary, we can assume that these polynomials have coefficients in
$R$
with at least one coefficient with valuation equal to zero. Since
$X$
is affine, we can also assume, up to a translation, that the point
$x\in X(R)$
is the origin, so that for
$C$
we obtain the following description:
$$\begin{eqnarray}C=\frac{K[x_{11},\ldots ,x_{dd},1/\text{det}[x_{ij}]]}{\unicode[STIX]{x1D6FC}_{\ast }(f_{1}),\ldots ,\unicode[STIX]{x1D6FC}_{\ast }(f_{s})}\end{eqnarray}$$
and moreover for every
$n=1,\ldots ,s$
,
$f_{n}(t_{1},\ldots ,t_{r},y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}])$
can be rewritten as
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6FC}_{\ast }(f_{n})(y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}])\nonumber\\ \displaystyle & & \displaystyle \quad +\,\mathop{\sum }_{l=1}^{L_{n}}v_{nl}(y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}])g_{nl}(t_{1},\ldots ,t_{r})\end{eqnarray}$$
for
$L_{n}\in \mathbb{N}$
, where
$v_{nl}$
and
$g_{nl}$
are polynomials with coefficients in
$R$
, by the above assumption, and
$g_{nl}(0,\ldots ,0)=0$
. Hence, we write
$B^{\prime }$
as follows:
$$\begin{eqnarray}B^{\prime }=\frac{R[t_{1},\ldots ,t_{r},y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]}{u_{1},\ldots ,u_{m},f_{1},\ldots ,f_{s}};\end{eqnarray}$$
we can assume that
$B^{\prime }$
is
$R$
-flat, otherwise we can add other polynomials
$f_{s+1},\ldots ,f_{s^{\prime }}$
in
$R[t_{1},\ldots ,t_{r},y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]$
cutting the
$R$
-torsion (thus making it the only
$R$
-flat quotient of
$A[y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]$
which is isomorphic to
$B$
after tensoring with
$K$
over
$R$
[Reference Grothendieck9, Lemme 2.8.1.1]); finally,
$C^{\prime }:=B^{\prime }\otimes _{A}R$
is as follows:
$$\begin{eqnarray}C^{\prime }=\frac{R[x_{11},\ldots ,x_{dd},1/\text{det}[x_{ij}]]}{\unicode[STIX]{x1D6FC}_{\ast }(f_{1}),\ldots ,\unicode[STIX]{x1D6FC}_{\ast }(f_{s})}.\end{eqnarray}$$
Now let
$e\in \mathbb{N}$
be a positive integer; we Néron blow up
$e$
times
$X$
in
$x_{s}$
, the special fibre of the point
$x\in X(R)$
that we are assuming to be the origin. This is equivalent to the following construction: we set
$$\begin{eqnarray}t_{\unicode[STIX]{x1D6FE}}^{\prime }:=\unicode[STIX]{x1D70B}^{-e}t_{\unicode[STIX]{x1D6FE}},\quad \unicode[STIX]{x1D6FE}=1,\ldots ,r\end{eqnarray}$$
and
$$\begin{eqnarray}A^{\prime }:=\frac{R[t_{1}^{\prime },\ldots ,t_{r}^{\prime }]}{u_{1}^{\prime }(t_{1}^{\prime },\ldots ,t_{r}^{\prime }),\ldots ,u_{m}^{\prime }(t_{1}^{\prime },\ldots ,t_{r}^{\prime })},\end{eqnarray}$$
where
$u_{i}^{\prime }$
is obtained from
$u_{i}$
by replacing
$t_{\unicode[STIX]{x1D6FE}}$
with
$\unicode[STIX]{x1D70B}^{e}t_{\unicode[STIX]{x1D6FE}}^{\prime }$
and dividing it by a suitable power of
$\unicode[STIX]{x1D70B}$
so that the resulting polynomial has coefficients in
$R$
with at least one with valuation zero. If we call
$X^{\prime }:=\operatorname{Spec}(A^{\prime })$
, then
$X^{\prime }$
is the desired Néron blow up of
$X$
in
$x_{s}\in X_{s}$
$e$
times. In a similar way, from
$B^{\prime }$
we obtain the
$R$
-flat algebra
$B^{\prime \prime }$
$$\begin{eqnarray}\frac{R[t_{1}^{\prime },\ldots ,t_{r}^{\prime },y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]}{u_{1}^{\prime },\ldots ,u_{m}^{\prime },\{\unicode[STIX]{x1D6FC}_{\ast }(f_{n})+\mathop{\sum }_{l=1}^{L_{n}}v_{nl}g_{nl}^{\prime }\}\text{}_{n=1,\ldots ,s}^{\prime }},\end{eqnarray}$$
where we have first obtained
$g_{nl}^{\prime }$
from
$g_{nl}$
by replacing
$t_{\unicode[STIX]{x1D6FE}}$
with
$\unicode[STIX]{x1D70B}^{e}t_{\unicode[STIX]{x1D6FE}}^{\prime }$
and then we have divided by a suitable power of
$\unicode[STIX]{x1D70B}$
the polynomials
$\unicode[STIX]{x1D6FC}_{\ast }(f_{n})+\sum _{l=1}^{L_{n}}v_{nl}g_{nl}^{\prime }$
, thus obtaining
$\{\unicode[STIX]{x1D6FC}_{\ast }(f_{n})+\sum _{l=1}^{L_{n}}v_{nl}g_{nl}^{\prime }\}\text{}^{\prime }$
, which now has coefficients in
$R$
with at least one with valuation zero. We set
$Y^{\prime \prime }:=\operatorname{Spec}(B^{\prime \prime })$
(it thus coincides, by construction, with the only closed subscheme of
$Y^{\prime }\times _{X}X^{\prime }$
which is
$R$
-flat and generically isomorphic to
$Y$
) and
$G^{\prime \prime }:=Y_{x}^{\prime \prime }=\operatorname{Spec}(C^{\prime \prime })$
, where
$$\begin{eqnarray}C^{\prime \prime }=\frac{R[x_{11},\ldots ,x_{dd},1/\text{det}[x_{ij}]]}{\{\unicode[STIX]{x1D6FC}_{\ast }(f_{1})\}\text{}^{\prime },\ldots ,\{\unicode[STIX]{x1D6FC}_{\ast }(f_{s})\}\text{}^{\prime }}.\end{eqnarray}$$
For a sufficiently big
$e$
, the exponent of
$\unicode[STIX]{x1D70B}$
in the equations
$t_{\unicode[STIX]{x1D6FE}}^{\prime }=\unicode[STIX]{x1D70B}^{-e}t_{\unicode[STIX]{x1D6FE}}$
, we have
$$\begin{eqnarray}Y_{s}^{\prime \prime }=\operatorname{Spec}\biggl(\frac{k[t_{1}^{\prime },\ldots ,t_{r}^{\prime },y_{11},\ldots ,y_{dd},1/\text{det}[y_{ij}]]}{u_{1}^{\prime },\ldots ,u_{m}^{\prime },\{\unicode[STIX]{x1D6FC}_{\ast }(f_{1})\}\text{}^{\prime },\ldots ,\{\unicode[STIX]{x1D6FC}_{\ast }(f_{s})\}\text{}^{\prime }}\biggr),\end{eqnarray}$$
which is isomorphic to
${G_{s}^{\prime \prime }}_{X_{s}^{\prime }}:=G_{s}^{\prime \prime }\times _{k}X_{s}^{\prime }$
and thus faithfully flat over
$X_{s}^{\prime }$
(note that
$G^{\prime \prime }$
being contained in
$\text{GL}_{d,R}$
always contains a section and it is thus surjective over
$\operatorname{Spec}(R)$
). By the so called critère de platitude par fibres (fiberwise criterion for flatness), it follows that
$Y^{\prime \prime }\rightarrow X^{\prime }$
is faithfully flat too (and consequently
$G^{\prime \prime }\rightarrow \operatorname{Spec}(R)$
is flat) and this concludes the proof.◻
The proof is made in such a way that we always find a model, for the given torsor, which is trivial on the special fibre. However, this has only been made for computational purposes. It is clear that in the proof we may Néron blow up too much; so in order to obtain a less trivial model we need to blow up a smaller amount of time. Indeed, we can choose the first
$X^{\prime }$
where the given torsor has a model. This is in general not caught by the proof.
We now state and prove the main consequence of the previous lemma.
Theorem 3.3. Let
$X=\mathbb{A}_{R}^{n}=\operatorname{Spec}(R[x_{1},\ldots ,x_{n}])$
be the
$n$
-dimensional affine space and
$x=(0,\ldots ,0)$
its origin. Let
$G$
be an affine
$K$
-group scheme of finite type and
$f:Y\rightarrow X_{\unicode[STIX]{x1D702}}$
a
$G$
-torsor pointed in
$y\in Y_{x_{\unicode[STIX]{x1D702}}}(K)$
. Then there exists an
$R$
-affine and flat group scheme
$G^{\prime }$
, a
$G^{\prime }$
-torsor
$f^{\prime }:Y^{\prime }\rightarrow X^{\prime }$
, pointed in
$y^{\prime }\in Y_{x}(R)$
, extending the given
$G$
-torsor
$Y$
, where
$X^{\prime }$
is obtained by
$X$
after a finite number of Néron blow ups of
$x_{s}\in X_{s}$
.
Proof. By the Quillen–Suslin theorem, it is known that every
$\text{GL}_{n,\mathbb{A}_{K}^{n}}$
-torsor is trivial; then we apply Lemma 3.2.◻
Lemma 3.4. Let
$X=\mathbb{A}_{R}^{n}$
be the
$n$
-dimensional affine space and
$x=(0,\ldots ,0)$
its origin as before. Let again
$X^{\prime }$
be the
$R$
-scheme obtained by
$X$
after a finite number of Néron blow ups of
$x_{s}\in X_{s}$
. Then there exists an
$R$
-isomorphism
$\mathbb{A}_{R}^{n}\rightarrow X^{\prime }$
.
Proof. This is well known, at least for
$n=1$
(see, for instance, [Reference Waterhouse and Weisfeiler17, proof of Theorem 2.2]), but for any
$n$
the proof is very similar: we compute
$X^{\prime }$
after
$m\in \mathbb{N}$
Néron blow ups: if
$m=1$
, then
$$\begin{eqnarray}X^{\prime }=\operatorname{Spec}(R[x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}]/(x_{1}-\unicode[STIX]{x1D70B}y_{1},\ldots ,x_{n}-\unicode[STIX]{x1D70B}y_{n})).\end{eqnarray}$$
Hence, if
$m$
is any natural integer,
$$\begin{eqnarray}X^{\prime }=\operatorname{Spec}(R[x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}]/(x_{1}-\unicode[STIX]{x1D70B}^{m}y_{1},\ldots ,x_{n}-\unicode[STIX]{x1D70B}^{m}y_{n})).\end{eqnarray}$$
The morphism
$$\begin{eqnarray}X^{\prime }\rightarrow \mathbb{A}_{R}^{n}:=\operatorname{Spec}(R[t_{1},\ldots ,t_{n}]),\qquad t_{i}\mapsto y_{i}\end{eqnarray}$$
gives the desired isomorphism. ◻
This allows us to improve Theorem 3.3 as follows.
Corollary 3.5. Let
$X=\mathbb{A}_{R}^{n}$
be the
$n$
-dimensional affine space and
$x=(0,\ldots ,0)$
its origin as before. Let
$G$
be an affine
$K$
-group scheme of finite type and
$f:Y\rightarrow X_{\unicode[STIX]{x1D702}}$
a
$G$
-torsor pointed in
$y\in Y_{x_{\unicode[STIX]{x1D702}}}(K)$
. Then, possibly after pulling back
$Y$
over an automorphism of
$\mathbb{A}_{K}^{n}$
, there exists a
$G^{\prime }$
-torsor
$f^{\prime }:Y^{\prime }\rightarrow \mathbb{A}_{R}^{n}$
, pointed in
$y^{\prime }\in Y_{x}(R)$
, extending the given
$G$
-torsor
$Y$
.
Remark 3.6. It is worth observing that one can choose any
$R$
-point of
$\mathbb{A}_{R}^{n}$
and move it to the origin with a simple translation.
Of course Lemma 3.2 has other interesting applications not yet stated; for instance, it can be applied to the spectrum of local rings. As an example (among many), we mention
$X=\operatorname{Spec}(R[x]/x^{n})$
, which becomes interesting when
$n=p=\operatorname{char}(R)$
and
$X$
is nothing but
$\unicode[STIX]{x1D6FC}_{p,R}$
where we forget the group structure. We state in this case an analog of Corollary 3.5.
Corollary 3.7. Let
$X=\unicode[STIX]{x1D6FC}_{p,R}$
and
$x$
its identity element. Let
$G$
be an affine
$K$
-group scheme of finite type and
$f:Y\rightarrow X_{\unicode[STIX]{x1D702}}$
a
$G$
-torsor pointed in
$y\in Y_{x_{\unicode[STIX]{x1D702}}}(K)$
. Then, possibly after pulling back
$Y$
over an automorphism (as a scheme) of
$\unicode[STIX]{x1D6FC}_{p,K}$
, there exists a
$G^{\prime }$
-torsor
$f^{\prime }:Y^{\prime }\rightarrow \unicode[STIX]{x1D6FC}_{p,R}$
, pointed in
$y^{\prime }\in Y_{x}(R)$
, extending the given
$G$
-torsor
$Y$
.
Acknowledgements
Marco Antei would like to thank Michel Emsalem and Arijit Dey for interesting discussions on the subject.
