Mixing
The big étale and Zariski topoi are generated by small sites
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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
asked Jan 3 at 17:30
W. RetherW. Rether
728417
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add a comment |
$begingroup$
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
asked Jan 3 at 17:30
W. RetherW. Rether
728417
$endgroup$
$begingroup$
Could you quickly copy the SGA definition of the étale site here?
$endgroup$
– Ingo Blechschmidt
Jan 5 at 19:32
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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 $bigcup alpha_i(Y_i)=Y$.
$endgroup$
– W. Rether
Jan 5 at 21:10
add a comment |
$begingroup$
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
asked Jan 3 at 17:30
W. RetherW. Rether
728417
$endgroup$
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
algebraic-geometry schemes topos-theory affine-schemes
asked Jan 3 at 17:30
W. RetherW. Rether
728417
asked Jan 3 at 17:30
W. RetherW. Rether
728417
asked Jan 3 at 17:30
W. RetherW. Rether
728417
asked Jan 3 at 17:30
W. RetherW. Rether
728417
asked Jan 3 at 17:30
W. RetherW. Rether
728417
728417
$begingroup$
Could you quickly copy the SGA definition of the étale site here?
$endgroup$
– Ingo Blechschmidt
Jan 5 at 19:32
$begingroup$
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 $bigcup alpha_i(Y_i)=Y$.
$endgroup$
– W. Rether
Jan 5 at 21:10
add a comment |
$begingroup$
Could you quickly copy the SGA definition of the étale site here?
$endgroup$
– Ingo Blechschmidt
Jan 5 at 19:32
$begingroup$
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 $bigcup alpha_i(Y_i)=Y$.
$endgroup$
– W. Rether
Jan 5 at 21:10
$begingroup$
Could you quickly copy the SGA definition of the étale site here?
$endgroup$
– Ingo Blechschmidt
Jan 5 at 19:32
$begingroup$
Could you quickly copy the SGA definition of the étale site here?
$endgroup$
– Ingo Blechschmidt
Jan 5 at 19:32
$begingroup$
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 $bigcup alpha_i(Y_i)=Y$.
$endgroup$
– W. Rether
Jan 5 at 21:10
$begingroup$
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 $bigcup alpha_i(Y_i)=Y$.
$endgroup$
– W. Rether
Jan 5 at 21:10
add a comment |
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$begingroup$
Could you quickly copy the SGA definition of the étale site here?
$endgroup$
– Ingo Blechschmidt
Jan 5 at 19:32
$begingroup$
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 $bigcup alpha_i(Y_i)=Y$.
$endgroup$
– W. Rether
Jan 5 at 21:10