Localizations of tensor categories and fiber products of schemes

Abstract

We prove that the tensor category of quasi-coherent modules $\mathsf{Qcoh}(X{\times }_{S}Y)$ on a fiber product of quasi-compact quasi-separated schemes is the bicategorical pushout of $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$ over $\mathsf{Qcoh}(S)$ in the $2$-category of cocomplete linear tensor categories. In particular, $\mathsf{Qcoh}(X\times Y)$ is the bicategorical coproduct of $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$. 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.

32 pages

References in corpus (4)

• Tannaka Duality for Geometric Stacks
• Tensor categorical foundations of algebraic geometry
• A characterization of categories of coherent sheaves of certain algebraic stacks
• Bicategorical colimits of tensor categories

Cited by in corpus (2)

• Sheaf representation of monoidal categories
• Exponentiable Grothendieck categories in flat Algebraic Geometry