Localizations of tensor categories and fiber products of schemes

Abstract

We prove that the tensor category of quasi-coherent modules Qcoh(X ×S Y ) on a fiber product of quasi-compact quasi-separated schemes is the bicategorical pushout of Qcoh(X) and Qcoh(Y ) over Qcoh(S) in the 2-category of cocomplete linear tensor categories. In particular, Qcoh(X × Y ) is the bicategorical coproduct of Qcoh(X) and Qcoh(Y ). For this we introduce idals, which can be seen as non-embedded ideals, and use them to study localizations of cocomplete tensor categories in general.