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    • CommentRowNumber1.
    • CommentAuthorMarc Hoyois
    • CommentTimeJul 21st 2013

    I decided to add some content to the motivic pages here on the nLab.

    I started with Nisnevich site. More to come soon…

    • CommentRowNumber2.
    • CommentAuthoradeelkh
    • CommentTimeJul 21st 2013

    Great!

    • CommentRowNumber3.
    • CommentAuthorhilbertthm90
    • CommentTimeJul 21st 2013

    Where it says the Nisnevich site over a Noetherian scheme “usually refers to …” is this really what people mean? I don’t know anything about this topic, but I do use the etale site and the usual terminology there is different, so I find this inconsistency a little scary.

    By the etale site on SS, people usually mean the category of all schemes over SS with etale coverings as the topology (and similarly for Zariski if I’m not mistaken). Is it the case that for Nisnevich people only take the smooth schemes over SS rather than all schemes over SS?

    • CommentRowNumber4.
    • CommentAuthorZhen Lin
    • CommentTimeJul 21st 2013

    There’s a petit étale site whose objects are (some of the) schemes that are étale over the base; what you describe is the gros étale site.

    • CommentRowNumber5.
    • CommentAuthorMarc Hoyois
    • CommentTimeJul 21st 2013
    • (edited Jul 21st 2013)

    @hilbertthm90

    It’s certainly very common to consider the Nisnevich topology only on smooth schemes. When non-smooth schemes are thrown in, the Nisnevich topology becomes too coarse for “motivic purposes” and one usually uses a finer topology such as the cdh topology. I guess the idea is that motives of smooth schemes generate all motives.

    ETA: but as far as I know, no one ever says “Nisnevich site” by itself without precisely defining the underlying category first.

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