Tensorial schemes

TL;DR

This paper proves that projective schemes are tensorial, meaning every cocontinuous tensor functor between their categories of quasi-coherent sheaves is induced by a unique scheme morphism, extending previous results without the tameness assumption.

Contribution

It removes the tameness assumption for projective schemes, establishing that they are tensorial and showing tensorial schemes are stable under certain operations.

Findings

Projective schemes are tensorial.

Tensorial schemes are closed under various operations.

Every cocontinuous tensor functor from a projective scheme is induced by a unique morphism.

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.