Group structure on geometric vector bundles Let $S$ be a scheme and $\mathcal A$  a quasicoherent $\mathcal O_S$-Algebra.
One knows that then one can associate the affine $S-$scheme $Spec(\mathcal A)$ over $S$.
In particular I can consider $Spec(Sym(\mathcal E))$ for a quasicoherent sheaf $\mathcal E$ on $S$, where $Sym$ denotes the symmetric algebra of the sheaf.
My question is:
Is there a natural structure of a $S-$group scheme on $Spec(Sym(\mathcal E))$?
At least if $\mathcal E$ is locally free of finite rank, this should be true.
One could argue that one just glues the local addition maps as locally on $S$ the bundle is just affine $n-$space.
But I would be interested in what group functor it represents.
 A: We have to show that for any $S$-scheme $T$, the set of $S$-morphisms 
$$ T\to \mathrm{Spec}({\mathcal Sym}(\mathcal E))$$
has a natural structure of group. 
Start with the affine case. Let $M$ be a module over a ring $A$ and let $B$ be an $A$-algebra, then the canonical map 
$$ \mathrm{Hom}_{A-algebras}(\mathrm{Sym}(M), B) \to \mathrm{Hom}_{A-modules}(M, B)$$ 
which takes $\phi : \mathrm{Sym}(M)\to B$ to its restriction to $M$ (elements of degree $1$ in $\mathrm{Sym}(M)$) is bijective. Therefore the canonical map
$$ \mathrm{Mor}_{A-schemes}(\mathrm{Spec}B, \mathrm{Spec}(\mathrm{Sym}(M)) 
\to \mathrm{Hom}_{A-modules}(M, B)$$ 
is bijective. 
Now for any $S$-scheme $T$, we have a canonical bijection 
$$ \mathrm{Mor}_{S-schemes}(T, \mathrm{Spec}(\mathcal{Sym}(\mathcal E)) 
\to \mathrm{Hom}_{O_S-modules}(\mathcal E, \pi_*O_T) $$ 
where $\pi: T\to S$ is the structural morphism. As the right hand side has a canonical group structure, this shows that $\mathrm{Spec}(\mathcal{Sym}(\mathcal E))$ is a group scheme over $S$. 
When $\mathcal E$ is free of rank $n$, we get the additive group $\mathbb G_{a}^n$ over $S$. 
