A principal $ \mathbb{G}_{a}$-bundle over an affine variety is trivial

Question

I was told that the following theorem is in Demazure, and Gabriel's book Introduction to Algebraic Geometry and Algebraic Groups, but I could not find the theorem. The theorem is that a principal $ \mathbb{G}_{a} $-bundle over an affine variety is trivial. Does someone know a reference?

Answer 1

If by principal $\mathbf{G}_a$-bundle you mean a $\mathbf{G}_a$-torsor in the fppf topology, then this can proved as follows. Since such objects are classified by elements in the fppf cohomology of your scheme with coefficients in $\mathbf{G}_a$, the result follows from the following proposition.

Proposition: Let $S$ be an affine scheme. Then $$H^1_{\text{fppf}}(S, \mathbf{G}_a) = 0.$$

Proof: Since $\mathbf{G}_a$ is smooth (over $S$), by a theorem of Grothendieck from Brauer III we have $$H^1_{\text{fppf}}(S, \mathbf{G}_a) = H^1_{\text{ét}}(S, \mathbf{G}_a).$$ On the other hand, by Theorem 2.1 here on the comparison between Zariski and étale cohomology,
$$H^1_{\text{ét}}(S, \mathbf{G}_a) = H^1(S, \mathcal{O}_S).$$ The group on the right vanishes since $S$ is affine, proving the proposition.