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?
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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 rightexactness 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. 

