Cdh descent, cdarc descent, and Milnor excision

Abstract

We give necessary and sufficient conditions for a cdh sheaf to satisfy Milnor excision, following ideas of Bhatt and Mathew. Along the way, we show that the cdh \(\infty \)-topos of a quasi-compact quasi-separated scheme of finite valuative dimension is hypercomplete, extending a theorem of Voevodsky to nonnoetherian schemes. As an application, we show that if E is a motivic spectrum over a field k which is n-torsion for some n invertible in k, then the cohomology theory on k-schemes defined by E satisfies Milnor excision.

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Notes

  1. 1.

    The condition (G1) (resp. (G2)) asks that the horizontal (resp. vertical) morphisms in the square of Theorem A be injective (resp. surjective).

  2. 2.

    By a cofiltered limit of schemes, we will always mean the limit of a cofiltered diagram of schemes with affine transition morphisms.

  3. 3.

    By a blowup we will always mean a blowup with finitely presented center, so that blowups are proper.

  4. 4.

    However, if X is integral, then V need not be locally integral (unless X is geometrically unibranch). This is the reason for considering quasi-integral schemes.

  5. 5.

    More precisely, Rydh defines universally subtrusive morphisms, which are not necessarily qcqs. A v cover is thus a qcqs universally subtrusive morphism.

  6. 6.

    This statement only requires \(\mathcal {C} \) to be generated under colimits by cotruncated objects.

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Acknowledgements

We would like to thank Akhil Mathew and Bhargav Bhatt for some useful discussions about the results of [4] and Benjamin Antieau for communicating Theorem 3.4.5.

This work was partially supported by the National Science Foundation under grant DMS-1440140, while the first two authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the “Derived Algebraic Geometry” program in spring 2019.

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Correspondence to Elden Elmanto.

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Elmanto, E., Hoyois, M., Iwasa, R. et al. Cdh descent, cdarc descent, and Milnor excision. Math. Ann. 379, 1011–1045 (2021). https://doi.org/10.1007/s00208-020-02083-5

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