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I decided to add some content to the motivic pages here on the nLab.
I started with Nisnevich site. More to come soon…
Great!
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 $S$, people usually mean the category of all schemes over $S$ 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 $S$ rather than all schemes over $S$?
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.
@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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