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Torsors under the additive group over an affine scheme are trivial

Statement

Assume the Axiom of Choice. Let $R$ be a commutative ring and let $S = Spec R$ . Let $G_{a} = Spec R [t]$ be the additive group scheme over $R$ , with comultiplication $Δ (t) = t \otimes 1 + 1 \otimes t$ , counit $ε (t) = 0$ and antipode $S (t) = - t$ , so that $G_{a} (T) = O_{T} (T)$ as an additive group for every $S$ -scheme $T$ .

A $G_{a}$ -torsor over $S$ is an $S$ -scheme $π : X \to S$ that is faithfully flat and finitely presented, together with an action $α : G_{a} \times_{S} X \to X$ of $G_{a}$ on $X$ over $S$ such that the shear morphism $σ : G_{a} \times_{S} X \to X \times_{S} X, σ (g, x) = (g \cdot x, x)$ is an isomorphism. Then $X$ is trivial: there is an isomorphism $X ≅ G_{a} \times_{S} S = A_{R}^{1}$ of $S$ -schemes, and in particular $π$ admits a section $S \to X$ .

The Axiom of Choice enters through the fppf descent of scheme morphisms used below.

Facts & Assumptions

Given: The Axiom of Choice, a commutative ring $R$ , the affine base $S = Spec R$ , and a $G_{a}$ -torsor $π : X \to S$ with action $α$ and shear isomorphism $σ$ .

[F1]

The given morphism $X \to S$ is flat, surjective, quasi-compact, and locally of finite presentation. An affine morphism is quasi-compact, and a flat surjective morphism is faithfully flat (Faithfully flat scheme morphism, Locally finite presentation morphisms, Quasi-compact and quasi-separated morphisms).

[F2]

Open immersions are flat and locally of finite presentation; flatness and local finite presentation are preserved by composition; and a finite disjoint union of affine schemes is affine (Open immersions of schemes, Locally finite presentation morphisms, Flatness is transitive under a flat change of rings, The spectrum of a finite product ring is the disjoint union of the factor spectra).

[F3]

Fibre products of affine schemes over $S = Spec R$ are affine with the tensor-product coordinate ring (Affine fibre products are spectra of tensor products).

[F4]

For a faithfully flat ring map $R \to B$ , the Amitsur complex $0 \to R \to B \overset{d^{0}}{\to} B \otimes_{R} B \overset{d^{1}}{\to} B \otimes_{R} B \otimes_{R} B, d^{0} (b) = b \otimes 1 - 1 \otimes b,$ is exact, where $d^{1} (c) = c \otimes 1 - c_{13} + 1 \otimes c$ in the three tensor slots. Thus every $1$ -cocycle in $B \otimes_{R} B$ is $d^{0} (b)$ for some $b \in B$ (Stacks Project, Descent, Lemma 35.3.6, tag 023M; already recorded above as a source).

[F5]

Under AC, a morphism $f^{'} : X^{'} \to Z$ descends uniquely along a faithfully flat, quasi-compact, locally finitely presented map $p : X^{'} \to X$ exactly when its two pullbacks to $X^{'} \times_{X} X^{'}$ agree (Scheme morphisms satisfy fppf descent, The Axiom of Choice).

[F6]

The action satisfies $α (g, α (g^{'}, x)) = α (g + g^{'}, x)$ , and the shear isomorphism gives a unique group element carrying one point of a fibre to another. Also $G_{a} (T) = O_{T} (T)$ by the group-scheme description in the Statement.

Proof

Given: The Axiom of Choice, a commutative ring $R$ , and the $G_{a}$ -torsor $π : X \to S$ with action $α$ and shear isomorphism $σ (g, x) = (g \cdot x, x)$ .

1.1F1F2F3

If $S = \emptyset$ , faithful flatness forces $X = \emptyset$ and the claim is immediate. Otherwise $X$ is quasi-compact because $X \to S$ is of finite presentation, so choose a finite affine open cover ${U_{i}}$ of $X$ . Its finite disjoint union $U = ∐_{i} U_{i} = Spec B$ is affine by [F2]. The map $p : U \to S$ is flat because each $U_{i} \to X$ is an open immersion and $X \to S$ is flat; it is surjective because the $U_{i}$ cover $X$ and $X \to S$ is surjective; and it is locally of finite presentation by composition. It is quasi-compact because it is affine. Thus $p$ is faithfully flat, quasi-compact, and locally of finite presentation. Write $S = Spec R$ , so $R \to B$ is faithfully flat by [F1], and [F3] identifies the affine fibre products of $U$ over $S$ with the corresponding tensor products. Let $u : U \to X$ be the covering morphism.

2.1step 1.1F3F4F6

On $U \times_{S} U$ , let $u_{1}, u_{2}$ be the two pullbacks of $u$ . The shear isomorphism gives a unique morphism $δ : U \times_{S} U \to G_{a}$ with $u_{2} = δ \cdot u_{1}$ . By [F3], $δ$ is an element of $B \otimes_{R} B$ . On $U \times_{S} U \times_{S} U$ , uniqueness and the group law give $δ_{13} = δ_{12} + δ_{23}$ , so $d^{1} (δ) = 0$ in the Amitsur complex of [F4]. Exactness gives $b \in B$ with $δ = b \otimes 1 - 1 \otimes b$ , where the two tensor slots correspond to the first and second copies of $U$ .

3.1F4F5step 2.1F6

Regard $b \in B = O_{U} (U)$ as a morphism $U \to G_{a}$ and set $v = b \cdot u : U \to X$ . On $U \times_{S} U$ , $v_{2} = b_{2} \cdot u_{2} = (b_{2} + δ_{12}) \cdot u_{1} = b_{1} \cdot u_{1} = v_{1}$ , since $δ_{12} = b_{1} - b_{2}$ . Hence the two pullbacks of $v$ agree. By [F5], $v$ descends to a morphism $s : S \to X$ ; since $π \circ v = p$ , uniqueness in [F5] gives $π \circ s = {id}_{S}$ . Thus $s$ is a section.

4.1F6step 3.1∎

Define $Φ : G_{a} \times_{S} S \to X$ by $Φ (g, z) = g \cdot s (z)$ . Applying $σ^{- 1}$ to the morphism $X \to X \times_{S} X$ , $y \mapsto (y, s (π (y)))$ , gives a morphism $y \mapsto (g (y), s (π (y)))$ with $g (y) \in G_{a}$ . The map $y \mapsto (g (y), π (y))$ is inverse to $Φ$ : one composite is the identity by the defining equation $g (y) \cdot s (π (y)) = y$ , and the other by uniqueness in the shear isomorphism. Therefore $X ≅ A_{R}^{1}$ over $S$ , and $s$ is the required section.

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
The Stacks Project, Descent, Lemma 35.3.6 (exactness of the extended Amitsur complex) (standard reference, not scraped)