Tensorial schemes

Abstract: Jacob Lurie (arXiv:math/0412266) has shown that for geometric stacks X,Y every cocontinuous tensor functor F : Qcoh(X) -> Qcoh(Y) is the pullback of a morphism Y -> X under the additional assumption that F is tame. In this note we get rid of this assumption if X is a projective scheme. In general, we call a scheme X tensorial if every cocontinuous tensor functor Qcoh(X) -> Qcoh(Y) is induced by a unique morphism Y -> X, show that projective schemes are tensorial and tensorial schemes are closed under various operations.