Suppose $\mathcal{A}$ is a quasi coherent sheaf of algebras over a group scheme $\mathcal{G}$. Suppose it is generated by global section. Then , what can we say about the external tensor product $\mathcal{A}\boxtimes\mathcal{A}$? Will this sheaf also be generated by the tensor product of the global section with itself? Or is it bigger than this?



External tensor product of sheaves
What do you mean by $A\boxtimes A$? If this is the sheaf $p\_1^\*A\otimes p\_2^\*A$ on $G\times G$ then certainly it is globally generated by the tensor square of global sections. It follows easily from right-exactness of the pullback and of the tensor product --- let $V$ be the space of global sections, then $V\otimes O\_G \to A$ is surjective, hence $V\otimes O\_{G\times G} \to p\_i^\*A$ is surjective, hence $V\otimes V\otimes O\_{G\times G} \to A\boxtimes A$ is surjective.


