The big étale and Zariski topoi are generated by small sites












2














Grothendieck (SGA 4, Exposé VII, Corollaire 3.2) proves that if $X$ is a quasiseparated scheme then, although the étale site is not small, there exists a subsite of it which is small and has the same topos of sheaves. So the little étale topos is a well-defined category.



Now my questions are:



1) Can we please drop the quasiseparatedness hypothesis in some way? I'm afraid it is necessary to convert "locally of finite type+affine" into "of finite type" in the proof, but...



2) Does a similar argument work for the big étale site (and for the big Zariski site)? The point is that we lose that everything is locally finitely presented, so I think something more should be necessary, but I hope the conclusion is true as well.



Thank you in advance.










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  • Could you quickly copy the SGA definition of the étale site here?
    – Ingo Blechschmidt
    9 hours ago










  • Of course. To be precise, it is the little étale site. The category is the full subcategory of $Sch/X$ (schemes over $X$) spanned by étale schemes over $X$. The coverings are jointly surjective families of étale morphisms, i.e. an object $Yto X$ is covered by a family ${alpha_i:Y_ito Y mbox{ over } X}$ if $bigcup alpha_i(Y_i)=Y$.
    – W. Rether
    7 hours ago


















2














Grothendieck (SGA 4, Exposé VII, Corollaire 3.2) proves that if $X$ is a quasiseparated scheme then, although the étale site is not small, there exists a subsite of it which is small and has the same topos of sheaves. So the little étale topos is a well-defined category.



Now my questions are:



1) Can we please drop the quasiseparatedness hypothesis in some way? I'm afraid it is necessary to convert "locally of finite type+affine" into "of finite type" in the proof, but...



2) Does a similar argument work for the big étale site (and for the big Zariski site)? The point is that we lose that everything is locally finitely presented, so I think something more should be necessary, but I hope the conclusion is true as well.



Thank you in advance.










share|cite|improve this question






















  • Could you quickly copy the SGA definition of the étale site here?
    – Ingo Blechschmidt
    9 hours ago










  • Of course. To be precise, it is the little étale site. The category is the full subcategory of $Sch/X$ (schemes over $X$) spanned by étale schemes over $X$. The coverings are jointly surjective families of étale morphisms, i.e. an object $Yto X$ is covered by a family ${alpha_i:Y_ito Y mbox{ over } X}$ if $bigcup alpha_i(Y_i)=Y$.
    – W. Rether
    7 hours ago
















2












2








2







Grothendieck (SGA 4, Exposé VII, Corollaire 3.2) proves that if $X$ is a quasiseparated scheme then, although the étale site is not small, there exists a subsite of it which is small and has the same topos of sheaves. So the little étale topos is a well-defined category.



Now my questions are:



1) Can we please drop the quasiseparatedness hypothesis in some way? I'm afraid it is necessary to convert "locally of finite type+affine" into "of finite type" in the proof, but...



2) Does a similar argument work for the big étale site (and for the big Zariski site)? The point is that we lose that everything is locally finitely presented, so I think something more should be necessary, but I hope the conclusion is true as well.



Thank you in advance.










share|cite|improve this question













Grothendieck (SGA 4, Exposé VII, Corollaire 3.2) proves that if $X$ is a quasiseparated scheme then, although the étale site is not small, there exists a subsite of it which is small and has the same topos of sheaves. So the little étale topos is a well-defined category.



Now my questions are:



1) Can we please drop the quasiseparatedness hypothesis in some way? I'm afraid it is necessary to convert "locally of finite type+affine" into "of finite type" in the proof, but...



2) Does a similar argument work for the big étale site (and for the big Zariski site)? The point is that we lose that everything is locally finitely presented, so I think something more should be necessary, but I hope the conclusion is true as well.



Thank you in advance.







algebraic-geometry schemes topos-theory affine-schemes






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asked 2 days ago









W. Rether

723417




723417












  • Could you quickly copy the SGA definition of the étale site here?
    – Ingo Blechschmidt
    9 hours ago










  • Of course. To be precise, it is the little étale site. The category is the full subcategory of $Sch/X$ (schemes over $X$) spanned by étale schemes over $X$. The coverings are jointly surjective families of étale morphisms, i.e. an object $Yto X$ is covered by a family ${alpha_i:Y_ito Y mbox{ over } X}$ if $bigcup alpha_i(Y_i)=Y$.
    – W. Rether
    7 hours ago




















  • Could you quickly copy the SGA definition of the étale site here?
    – Ingo Blechschmidt
    9 hours ago










  • Of course. To be precise, it is the little étale site. The category is the full subcategory of $Sch/X$ (schemes over $X$) spanned by étale schemes over $X$. The coverings are jointly surjective families of étale morphisms, i.e. an object $Yto X$ is covered by a family ${alpha_i:Y_ito Y mbox{ over } X}$ if $bigcup alpha_i(Y_i)=Y$.
    – W. Rether
    7 hours ago


















Could you quickly copy the SGA definition of the étale site here?
– Ingo Blechschmidt
9 hours ago




Could you quickly copy the SGA definition of the étale site here?
– Ingo Blechschmidt
9 hours ago












Of course. To be precise, it is the little étale site. The category is the full subcategory of $Sch/X$ (schemes over $X$) spanned by étale schemes over $X$. The coverings are jointly surjective families of étale morphisms, i.e. an object $Yto X$ is covered by a family ${alpha_i:Y_ito Y mbox{ over } X}$ if $bigcup alpha_i(Y_i)=Y$.
– W. Rether
7 hours ago






Of course. To be precise, it is the little étale site. The category is the full subcategory of $Sch/X$ (schemes over $X$) spanned by étale schemes over $X$. The coverings are jointly surjective families of étale morphisms, i.e. an object $Yto X$ is covered by a family ${alpha_i:Y_ito Y mbox{ over } X}$ if $bigcup alpha_i(Y_i)=Y$.
– W. Rether
7 hours ago












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