Mixing
The small étale topos of a scheme is equivalent to the category of finite $pi_1(X,x)$-sets for every scheme...
$begingroup$
Recall Milne, Etale cohomology, Theorem I.5.3:
Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/Xto Sets$ ($FEt/X$ being the category of $X$-schemes finite and étale over $X$) defines an equivalence between the category $FEt/X$ and the category of finite sets on which $pi_1(X,x)$ acts continuously (on the left)$.
Now, in Jardine, Local Homotopy, one finds the following:
Example 2.20. If $G = {G_i}$ is a profinite group such that all transition maps $G_i to G_j$ are surjective, then the category $G − Set_d$ of discrete $G-sets$ is a Grothendieck topos. A discrete $G$-set is a set X equipped with a pro-map $G → Aut(X)$. The finite discrete $G$-sets form a generating set for this topos, and the full subcategory on the finite discrete G-sets is the site prescribed by Giraud’s theorem.
If the profinite group $G$ is the absolute Galois group of a field $k$, then the category $G−Set_d$ of discrete $G$-sets is equivalent to the category $Shv(et|_k)$ of sheaves on the étale site for $k$. More generally, if S is a locally Noetherian connected scheme with geometric point x, and the profinite group $pi_1(S,x)$ is the Grothendieck fundamental group, then the category of discrete $pi_1(S,x)$-sets is equivalent to the category of sheaves on the finite e ́tale site $fet|_S$ for the scheme $S$. See [1], [53].
where [1] is SGA1 and [53] is Milne.
Now my problem is that in Milne I find only the above theorem. My guess is that the last part of Jardine's example follows from Milne's theorem and the fact that étale sheaves are easily built from representable sheaves. But the fact that they are colimits of these latter should not suffice, since otherwise the argument would work for every presheaf.
So my idea would be to take Milne's theorem as "finitary" version and then to glue things to form étale sheaves on one side (not necessarily representable anymore) and $pi_1(X,x)$-discrete-sets on the other side (not necessarily finite anymore).
Any clues in this direction? At the moment, I am stuck here, so any help would be appreciated. Thank you un advance.
algebraic-geometry schemes fundamental-groups topos-theory etale-cohomology
asked Jan 1 at 21:23
W. RetherW. Rether
728417
$endgroup$
add a comment |
$begingroup$
Recall Milne, Etale cohomology, Theorem I.5.3:
Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/Xto Sets$ ($FEt/X$ being the category of $X$-schemes finite and étale over $X$) defines an equivalence between the category $FEt/X$ and the category of finite sets on which $pi_1(X,x)$ acts continuously (on the left)$.
Now, in Jardine, Local Homotopy, one finds the following:
Example 2.20. If $G = {G_i}$ is a profinite group such that all transition maps $G_i to G_j$ are surjective, then the category $G − Set_d$ of discrete $G-sets$ is a Grothendieck topos. A discrete $G$-set is a set X equipped with a pro-map $G → Aut(X)$. The finite discrete $G$-sets form a generating set for this topos, and the full subcategory on the finite discrete G-sets is the site prescribed by Giraud’s theorem.
If the profinite group $G$ is the absolute Galois group of a field $k$, then the category $G−Set_d$ of discrete $G$-sets is equivalent to the category $Shv(et|_k)$ of sheaves on the étale site for $k$. More generally, if S is a locally Noetherian connected scheme with geometric point x, and the profinite group $pi_1(S,x)$ is the Grothendieck fundamental group, then the category of discrete $pi_1(S,x)$-sets is equivalent to the category of sheaves on the finite e ́tale site $fet|_S$ for the scheme $S$. See [1], [53].
where [1] is SGA1 and [53] is Milne.
Now my problem is that in Milne I find only the above theorem. My guess is that the last part of Jardine's example follows from Milne's theorem and the fact that étale sheaves are easily built from representable sheaves. But the fact that they are colimits of these latter should not suffice, since otherwise the argument would work for every presheaf.
So my idea would be to take Milne's theorem as "finitary" version and then to glue things to form étale sheaves on one side (not necessarily representable anymore) and $pi_1(X,x)$-discrete-sets on the other side (not necessarily finite anymore).
Any clues in this direction? At the moment, I am stuck here, so any help would be appreciated. Thank you un advance.
algebraic-geometry schemes fundamental-groups topos-theory etale-cohomology
asked Jan 1 at 21:23
W. RetherW. Rether
728417
$endgroup$
1
$begingroup$
I would say that on this particular site, $F$ is a sheaf iff it is a presheaf which satisfy the axiom $F(Asqcup B)=F(A)sqcup F(B)$. Moreover, if $F$ has finite stalks, it is representable. So you should be able to prove that a sheaf is a disjoint union (in the category of sheaves) of representables.
$endgroup$
– Roland
Jan 2 at 9:20
add a comment |
$begingroup$
Recall Milne, Etale cohomology, Theorem I.5.3:
Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/Xto Sets$ ($FEt/X$ being the category of $X$-schemes finite and étale over $X$) defines an equivalence between the category $FEt/X$ and the category of finite sets on which $pi_1(X,x)$ acts continuously (on the left)$.
Now, in Jardine, Local Homotopy, one finds the following:
Example 2.20. If $G = {G_i}$ is a profinite group such that all transition maps $G_i to G_j$ are surjective, then the category $G − Set_d$ of discrete $G-sets$ is a Grothendieck topos. A discrete $G$-set is a set X equipped with a pro-map $G → Aut(X)$. The finite discrete $G$-sets form a generating set for this topos, and the full subcategory on the finite discrete G-sets is the site prescribed by Giraud’s theorem.
If the profinite group $G$ is the absolute Galois group of a field $k$, then the category $G−Set_d$ of discrete $G$-sets is equivalent to the category $Shv(et|_k)$ of sheaves on the étale site for $k$. More generally, if S is a locally Noetherian connected scheme with geometric point x, and the profinite group $pi_1(S,x)$ is the Grothendieck fundamental group, then the category of discrete $pi_1(S,x)$-sets is equivalent to the category of sheaves on the finite e ́tale site $fet|_S$ for the scheme $S$. See [1], [53].
where [1] is SGA1 and [53] is Milne.
Now my problem is that in Milne I find only the above theorem. My guess is that the last part of Jardine's example follows from Milne's theorem and the fact that étale sheaves are easily built from representable sheaves. But the fact that they are colimits of these latter should not suffice, since otherwise the argument would work for every presheaf.
So my idea would be to take Milne's theorem as "finitary" version and then to glue things to form étale sheaves on one side (not necessarily representable anymore) and $pi_1(X,x)$-discrete-sets on the other side (not necessarily finite anymore).
Any clues in this direction? At the moment, I am stuck here, so any help would be appreciated. Thank you un advance.
algebraic-geometry schemes fundamental-groups topos-theory etale-cohomology
asked Jan 1 at 21:23
W. RetherW. Rether
728417
$endgroup$
Recall Milne, Etale cohomology, Theorem I.5.3:
Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/Xto Sets$ ($FEt/X$ being the category of $X$-schemes finite and étale over $X$) defines an equivalence between the category $FEt/X$ and the category of finite sets on which $pi_1(X,x)$ acts continuously (on the left)$.
Now, in Jardine, Local Homotopy, one finds the following:
Example 2.20. If $G = {G_i}$ is a profinite group such that all transition maps $G_i to G_j$ are surjective, then the category $G − Set_d$ of discrete $G-sets$ is a Grothendieck topos. A discrete $G$-set is a set X equipped with a pro-map $G → Aut(X)$. The finite discrete $G$-sets form a generating set for this topos, and the full subcategory on the finite discrete G-sets is the site prescribed by Giraud’s theorem.
If the profinite group $G$ is the absolute Galois group of a field $k$, then the category $G−Set_d$ of discrete $G$-sets is equivalent to the category $Shv(et|_k)$ of sheaves on the étale site for $k$. More generally, if S is a locally Noetherian connected scheme with geometric point x, and the profinite group $pi_1(S,x)$ is the Grothendieck fundamental group, then the category of discrete $pi_1(S,x)$-sets is equivalent to the category of sheaves on the finite e ́tale site $fet|_S$ for the scheme $S$. See [1], [53].
where [1] is SGA1 and [53] is Milne.
Now my problem is that in Milne I find only the above theorem. My guess is that the last part of Jardine's example follows from Milne's theorem and the fact that étale sheaves are easily built from representable sheaves. But the fact that they are colimits of these latter should not suffice, since otherwise the argument would work for every presheaf.
So my idea would be to take Milne's theorem as "finitary" version and then to glue things to form étale sheaves on one side (not necessarily representable anymore) and $pi_1(X,x)$-discrete-sets on the other side (not necessarily finite anymore).
Any clues in this direction? At the moment, I am stuck here, so any help would be appreciated. Thank you un advance.
algebraic-geometry schemes fundamental-groups topos-theory etale-cohomology
algebraic-geometry schemes fundamental-groups topos-theory etale-cohomology
asked Jan 1 at 21:23
W. RetherW. Rether
728417
asked Jan 1 at 21:23
W. RetherW. Rether
728417
edited Jan 2 at 16:01
W. Rether
asked Jan 1 at 21:23
W. RetherW. Rether
728417
asked Jan 1 at 21:23
W. RetherW. Rether
728417
asked Jan 1 at 21:23
W. RetherW. Rether
728417
728417
1
$begingroup$
I would say that on this particular site, $F$ is a sheaf iff it is a presheaf which satisfy the axiom $F(Asqcup B)=F(A)sqcup F(B)$. Moreover, if $F$ has finite stalks, it is representable. So you should be able to prove that a sheaf is a disjoint union (in the category of sheaves) of representables.
$endgroup$
– Roland
Jan 2 at 9:20
add a comment |
1
$begingroup$
I would say that on this particular site, $F$ is a sheaf iff it is a presheaf which satisfy the axiom $F(Asqcup B)=F(A)sqcup F(B)$. Moreover, if $F$ has finite stalks, it is representable. So you should be able to prove that a sheaf is a disjoint union (in the category of sheaves) of representables.
$endgroup$
– Roland
Jan 2 at 9:20
1
1
$begingroup$
I would say that on this particular site, $F$ is a sheaf iff it is a presheaf which satisfy the axiom $F(Asqcup B)=F(A)sqcup F(B)$. Moreover, if $F$ has finite stalks, it is representable. So you should be able to prove that a sheaf is a disjoint union (in the category of sheaves) of representables.
$endgroup$
– Roland
Jan 2 at 9:20
$begingroup$
I would say that on this particular site, $F$ is a sheaf iff it is a presheaf which satisfy the axiom $F(Asqcup B)=F(A)sqcup F(B)$. Moreover, if $F$ has finite stalks, it is representable. So you should be able to prove that a sheaf is a disjoint union (in the category of sheaves) of representables.
$endgroup$
– Roland
Jan 2 at 9:20
add a comment |
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I would say that on this particular site, $F$ is a sheaf iff it is a presheaf which satisfy the axiom $F(Asqcup B)=F(A)sqcup F(B)$. Moreover, if $F$ has finite stalks, it is representable. So you should be able to prove that a sheaf is a disjoint union (in the category of sheaves) of representables.
$endgroup$
– Roland
Jan 2 at 9:20