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?



A principal $ \mathbb{G}_{a}$-bundle over an affine variety is trivial
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](http://virtualmath1.stanford.edu/%7Econrad/Weil2seminar/Notes/L4.pdf) 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.


