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    • CommentRowNumber1.
    • CommentAuthoradeelkh
    • CommentTimeOct 20th 2014

    Started the page cd-structure.

    • CommentRowNumber2.
    • CommentAuthorMarc Hoyois
    • CommentTimeOct 21st 2014

    A minor thing, but it’s not true in general that the (∞,1)-topos of sheaves for a cd-structure is hypercomplete. The (∞,1)-topos of a coherent topological space is always given by a cd-structure but it may not be hypercomplete (see counterexample 6.5.4.5 in HTT).

    • CommentRowNumber3.
    • CommentAuthoradeelkh
    • CommentTimeOct 21st 2014
    • (edited Oct 21st 2014)

    Thanks for that correction. I guess I was misled by the fact that, for noetherian schemes, the Nisnevich topology is generated by a cd-structure, and the associated topos is hypercomplete, while both of these facts are false in general (right?).

    Do you know any way to characterize the toposes that arise from cd-structures? I guess a necessary condition is generation under contractible colimits by the Yoneda embedding, right?

    • CommentRowNumber4.
    • CommentAuthorMarc Hoyois
    • CommentTimeOct 21st 2014
    • (edited Oct 21st 2014)

    I don’t know any sufficient conditions. Another fairly strong necessary condition is that representables are compact in the (∞,1)-topos of sheaves.

    The Nisnevich topology on non-noetherian schemes is always a confusing topic. Let me try to shed some light on it. Originally, Nisnevich defined the following pretopology: {f i:X iX}\{f_i: X_i\to X\} is a covering if each f if_i is étale and if every point xXx\in X has a preimage in some X iX_i with the same residue field. Let’s call this topology Nis badNis_{bad}. In DAG XI, Lurie has a slightly different definition of the Nisnevich topology on affine schemes, which can be extended to all schemes by throwing in the Zariski topology; let’s call this topology NisNis. So we have

    ZarNisNis badet. Zar \leq Nis \leq Nis_{bad} \leq et.

    Now let’s fix a scheme XX and consider, for each of these topologies τ\tau, the (∞,1)-topos X τX_\tau of τ\tau-sheaves on the category Et XEt_X of étale XX-schemes (that’s a large category but nevertheless you get a topos). Then:

    • X etX Nis badX NisX ZarX_{et}\subset X_{Nis_{bad}} \subset X_{Nis} \subset X_{Zar}
    • If XX is locally noetherian, X Nis bad=X NisX_{Nis_{bad}}=X_{Nis}. In general they are different.
    • If XX is qcqs, then X ZarX_{Zar}, X NisX_{Nis}, and X etX_{et} are coherent. In general, X Nis badX_{Nis_{bad}} isn’t.
    • If X=lim αX αX=\lim_\alpha X^\alpha where the X αX^\alpha’s are qcqs, then X τ=lim αX τ αX_\tau=\lim_\alpha X^\alpha_\tau for τ=Zar,Nis,et\tau=Zar,Nis,et. Not so for τ=Nis bad\tau=Nis_{bad}.
    • If XX is qcqs, then X ZarX_{Zar} and X NisX_{Nis} can be described as presheaves on finitely presented étale XX-schemes satisfying a Mayer-Vietoris condition. Not so for X Nis badX_{Nis_{bad}} or X etX_{et}.
    • (Corollary) If XX is qcqs, then X ZarX_{Zar} and X NisX_{Nis} are compactly generated (by finitely presented étale XX-schemes). In general, X Nis badX_{Nis_{bad}} and X etX_{et} aren’t.
    • If XX is noetherian and of finite Krull dimension, then X ZarX_{Zar} and X Nis=X Nis badX_{Nis}=X_{Nis_{bad}} are hypercomplete.

    For these reasons, NisNis is often more appropriate than Nis badNis_{bad}. But Nis badNis_{bad} isn’t all that bad. For example, for any XX, the (∞,1)-topos X Nis bad X_{Nis_{bad}}^\wedge has a conservative family of points given by the henselizations of the local rings of étale XX-schemes. For XX qcqs, X Nis X_{Nis}^{\wedge} also has enough points by Lurie’s generalization of Deligne’s theorem, but I don’t know a nice description of the points.

    • CommentRowNumber5.
    • CommentAuthorMarc Hoyois
    • CommentTimeOct 21st 2014

    I changed a bit the definition of the associated topology, because what was defined before was not really a pretopology.

    I also upgraded Prop. 1 to (∞,1)-presheaves instead of just presheaves, and added a reference.

    • CommentRowNumber6.
    • CommentAuthoradeelkh
    • CommentTimeOct 22nd 2014

    That’s very helpful, thanks!

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