ScalingStacks

1.2. Tensors and functors

For X,X′X,X^{\prime} ordinary schemes over a ring kk, a theorem of Toën [To1] identifies the dg category of kk-linear continuous (that is, colimit preserving) functors with the dg category of integral kernels

Funk⁡(QC⁡(X),QC⁡(X′))≃QC⁡(X×kX′).\Fun_{k}(\qc(X),\qc(X^{\prime}))\simeq\qc(X\times_{k}X^{\prime}).

For dg categories of perfect (equivalently, bounded coherent) complexes on smooth projective varieties, an analogous result was proved by Bondal, Larsen and Lunts [BLL] as well as by Toën [To1] (generalizing Orlov’s theorem [O] characterizing equivalences as Fourier-Mukai transforms).

We’ve collected our main technical results in the following generalization. In the statement, the tensors and functors of ∞\infty-categories of quasi-coherent sheaves are calculated in the symmetric monoidal ∞\infty-category 𝒫​rL\mathcal{P}r^{\rm L} of presentable ∞\infty-categories with morphisms left adjoints (as developed in [L4, 4] and [L5, 5], see Section 2 for a precise summary). The tensors and functors of ∞\infty-categories of perfect complexes are calculated in the symmetric monoidal ∞\infty-category s​t{st} of kk-linear idempotent complete stable small ∞\infty-categories (as developed in Section 4.1 below).

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  1. (1)

    Theorem 1.2. For X→Y←X′X\rightarrow Y\leftarrow X^{\prime} maps of perfect stacks, there is a canonical equivalence

    QC⁡(X×YX′)≃QC⁡(X)⊗QC⁡(Y)QC⁡(X′)\qc(X\times_{Y}X^{\prime})\simeq\qc(X)\otimes_{\qc(Y)}\qc(X^{\prime})

    between the ∞\infty-category of sheaves on the derived fiber product and the tensor product of the ∞\infty-categories of sheaves on the factors.

    There is also a canonical equivalence

    Perf⁡(X×X′)≃Perf⁡(X)⊗Perf⁡(X′)\operatorname{Perf}(X\times X^{\prime})\simeq\operatorname{Perf}(X)\otimes\operatorname{Perf}(X^{\prime})

    for ∞\infty-categories of perfect complexes.

  2. (2)

    For X→YX\to Y a perfect morphism to a derived stack YY with affine diagonal, and X′→YX^{\prime}\to Y arbitrary, there is a canonical equivalence

    QC⁡(X×YX′)≃FunQC⁡(Y)⁡(QC⁡(X),QC⁡(X′))\qc(X\times_{Y}X^{\prime})\simeq\Fun_{\qc(Y)}(\qc(X),\qc(X^{\prime}))

    between the ∞\infty-category of sheaves on the derived fiber product and the ∞\infty-category of colimit-preserving QC⁡(Y)\qc(Y)-linear functors.

    When XX is a smooth and proper perfect stack, there is also a canonical equivalence

    Perf⁡(X×X′)≃Fun⁡(Perf⁡(X),Perf⁡(X′))\operatorname{Perf}(X\times X^{\prime})\simeq\Fun(\operatorname{Perf}(X),\operatorname{Perf}(X^{\prime}))

    for ∞\infty-categories of perfect complexes.

Our arguments in Section 4 first establish the pair of assertions about tensors, then deduce sufficient duality to conclude the pair of assertions about functors.

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Remark 1.3. The hypothesis that our stacks are perfect appears to be essential, and we do not expect the above theorem to hold in significantly greater generality. Alternatively, in complete generality, one should rather replace the notion of tensor product. Jacob Lurie has described (in private communication) a completed tensor product for stable ∞\infty-categories equipped with tt-structures, and such that passing to quasi-coherent sheaves takes fiber products of geometric stacks to the completed tensor product of ∞\infty-categories.

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5