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Cdh descent, cdarc descent, and Milnor excision

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

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

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

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

  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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Authors and Affiliations

  1. Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA, 02138, USA

    Elden Elmanto

  2. Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040, Regensburg, Germany

    Marc Hoyois

  3. Institut for Matematiske Fag, Københavns Universitet, Universitetsparken 5, 2100, København, Denmark

    Ryomei Iwasa

  4. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan

    Shane Kelly

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  1. Elden Elmanto
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  2. Marc Hoyois
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  3. Ryomei Iwasa
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  4. Shane Kelly
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Correspondence to Elden Elmanto.

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Communicated by Vasudevan Srinivas.

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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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  • DOI: https://doi.org/10.1007/s00208-020-02083-5

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