External tensor product of sheaves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:33:47Z http://mathoverflow.net/feeds/question/41254 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41254/external-tensor-product-of-sheaves External tensor product of sheaves Neha 2010-10-06T11:08:22Z 2010-10-07T14:51:51Z <p>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?</p> http://mathoverflow.net/questions/41254/external-tensor-product-of-sheaves/41269#41269 Answer by Sasha for External tensor product of sheaves Sasha 2010-10-06T13:25:33Z 2010-10-06T13:25:33Z <p>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.</p> http://mathoverflow.net/questions/41254/external-tensor-product-of-sheaves/41423#41423 Answer by Neha for External tensor product of sheaves Neha 2010-10-07T14:51:51Z 2010-10-07T14:51:51Z <p>Thanks Sasha for the answer.</p> <p>Yes, $A\boxtimes A$ is the sheaf given by $p_1^∗A⊗p_2^∗A$ on $G\times G$. Can you please tell me if $V\otimes O_G\to A$ is surjective, then how and why should $V\otimes O_{G\times G}\to p_{i}^{∗} A$ be surjective? And finally how will we get the last step i.e. why would this map become $V\otimes V\otimes O_{G\times G}\to A\boxtimes A$ is surjective? I am sure it is trivial for you, but I am not able to figure out the correct reason, so please help.</p>