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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.

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    Because I read it in the notes of Geer and Moonen about Abelian varieties, (0.4).2011-11-27

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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 $S$.

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    Yes you are right !2011-11-27