A split affine extension of an abelian variety

The group laws on $A$ and ${\mathbf{G}}_m$ give the group laws on their product. The latter group's affine Hopf formulas are $t\mapsto t\otimes t$, $t\mapsto t^{-1}$, and $t\mapsto 1$, so the group laws are regular. The projection $q(a,t)=a$ is a homomorphism, split by $a\mapsto (a,1)$, and its scheme-theoretic kernel is $\{e_A\}\times {\mathbf{G}}_m$. It is normal because conjugation in the product preserves this factor. It is affine by its displayed spectrum, smooth because it is an open subscheme of the affine line, and geometrically connected because $K[t,t^{-1}]$ is a domain for every field extension $K\mathrm{/}k$.

For every $k$-scheme $T$, $q:G(T)=A(T)\times \mathrm{\Gamma }(T,{\mathcal{O}}_T{)}^{\times }\to A(T)$ is onto, and two elements have the same image exactly when they differ by the action of a unique ${\mathbf{G}}_m(T)$ element. Thus the presheaf quotient is already the represented functor $A(-)$, naturally in $T$; by [F4], its fppf sheafification is the same represented functor. This proves the asserted scheme quotient.

The section $A\times \{1\}\subset G$ is closed since $\{1\}\subset \mathrm{Spec}k[t,t^{-1}]$ is defined by $t-1$. If $G$ were affine, [F3] would make that copy of $A$ affine. For $\dim A>0$ this contradicts [F2]. Hence $G$ is a nonaffine algebraic group in that case. AC is carried through [F2].