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Localizations of tensor categories and fiber products of schemes

Main Result

Theorem A. Let 𝕂\mathbb{K} be a commutative ring and let XX and YY be two quasi-compact quasi-separated 𝕂\mathbb{K}-schemes over some quasi-compact quasi-separated 𝕂\mathbb{K}-scheme SS. Then Qcoh(XΓ— 𝕂Y)\mathsf{Qcoh}(X \times_\mathbb{K} Y) is the bicategorical pullback of Qcoh(X)\mathsf{Qcoh}(X) and Qcoh(Y)\mathsf{Qcoh}(Y).

Idals

The term ideals is introduced. The name is meant to imply that its like ideals but without β€œembedding”.

Definition 2.1 Let π’ž\mathcal{C} be a tensor category with unit object π’ͺ π’ž\mathcal{O}_\mathcal{C}. An idal is a morphism e:Iβ†’π’ͺ π’že \colon I \to \mathcal{O}_\mathcal{C} such that λ∘(eβŠ—I)=ρ∘(IβŠ—e)\lambda \circ (e \otimes I) = \rho \circ (I \otimes e).

This condition is better expressed as a commuting square or as a string diagram.