Tensor functors between categories of quasi-coherent sheaves

Abstract

For a quasi-compact quasi-separated scheme X and an arbitrary scheme Y we show that the pullback construction f↦f⁎ implements an equivalence between the discrete category of morphisms Y→X and the category of cocontinuous tensor functors Qcoh(X)→Qcoh(Y). This is an improvement of a result by Lurie and may be interpreted as the statement that algebraic geometry is 2-affine. Moreover, we prove the analogous version of this result for Durovʼs notion of generalized schemes.