Tensorial schemes

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.

💡 Research Summary

The paper introduces the notion of a “tensorial scheme” and proves that projective schemes belong to this class, thereby extending Jacob Lurie’s reconstruction theorem for geometric stacks without the need for the tameness hypothesis. A tensorial scheme X is defined by the property that every cocontinuous tensor functor Qcoh(X) → Qcoh(Y) arises uniquely as the pull‑back functor f* for a morphism f : Y → X. The author first reviews the language of R‑linear tensor categories, emphasizing that the tensor product must preserve colimits in each variable, i.e. the category is a “cocomplete tensor category” or, in higher‑categorical terms, an R‑2‑algebra.

The core technical tool is the universal cocompletion of a small tensor category. Given a small tensor category C, its presheaf category (\widehat{C}= \operatorname{Hom}(C^{op},\operatorname{Mod}(R))) can be equipped with a unique tensor structure extending that of C, making (\widehat{C}) the free cocomplete tensor category generated by C. Two fundamental examples are worked out in detail.

1. For an R‑algebra S, the one‑object tensor category ({S}) has cocompletion (\widehat{{S}} \simeq \operatorname{Mod}(S)). Consequently, cocontinuous tensor functors (\operatorname{Mod}(S) \to \mathcal{C}) correspond bijectively to R‑algebra homomorphisms (S \to \operatorname{End}( \mathbf{1}_{\mathcal{C}} )). This recovers the classical affine case: morphisms (\operatorname{Spec} S \to X) are in bijection with tensor functors (\operatorname{Mod}(S) \to \operatorname{Qcoh}(X)).

2. For a G‑graded R‑algebra (S = \bigoplus_{g\in G} S_g), the small tensor category (C