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1. • CommentRowNumber1.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 21st 2013
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI decided to add some content to the motivic pages here on the nLab. I started with [[Nisnevich site]]. More to come soon…

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

   I started with Nisnevich site. More to come soon…

2. • CommentRowNumber2.
   • CommentAuthoradeelkh
   • CommentTimeJul 21st 2013
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   Author: adeelkh
   Format: MarkdownItexGreat!

   Great!

3. • CommentRowNumber3.
   • CommentAuthorhilbertthm90
   • CommentTimeJul 21st 2013
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   Author: hilbertthm90
   Format: MarkdownItexWhere 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 $S$?

   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 $S$?

4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeJul 21st 2013
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   Author: Zhen Lin
   Format: MarkdownItexThere'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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 21st 2013
   • (edited Jul 21st 2013)
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItex@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.

   @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.