Do higher etale homotopy groups of spectrum of a field always vanish?
$begingroup$
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $mathrm{Spec},k$ vanish?
If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.
I am not sure what happens for general fields.
ag.algebraic-geometry etale-covers
asked 11 hours ago
rorirori
282
New contributor
rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $mathrm{Spec},k$ vanish?
If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.
I am not sure what happens for general fields.
ag.algebraic-geometry etale-covers
asked 11 hours ago
rorirori
282
New contributor
rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $mathrm{Spec},k$ vanish?
If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.
I am not sure what happens for general fields.
ag.algebraic-geometry etale-covers
asked 11 hours ago
rorirori
282
New contributor
rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $mathrm{Spec},k$ vanish?
If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.
I am not sure what happens for general fields.
ag.algebraic-geometry etale-covers
ag.algebraic-geometry etale-covers
asked 11 hours ago
rorirori
282
New contributor
rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 11 hours ago
rorirori
282
New contributor
rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 11 hours ago
rorirori
282
New contributor
rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 11 hours ago
rorirori
282
asked 11 hours ago
rorirori
282
282
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rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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1 Answer
1
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$begingroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
answered 11 hours ago
Denis NardinDenis Nardin
8,62723461
$endgroup$
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
10 hours ago
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
10 hours ago
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
8 hours ago
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
8 hours ago
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
8 hours ago
add a comment |
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1 Answer
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1 Answer
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active
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$begingroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
answered 11 hours ago
Denis NardinDenis Nardin
8,62723461
$endgroup$
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
10 hours ago
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
10 hours ago
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
8 hours ago
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
8 hours ago
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
8 hours ago
add a comment |
$begingroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
answered 11 hours ago
Denis NardinDenis Nardin
8,62723461
$endgroup$
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
10 hours ago
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
10 hours ago
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
8 hours ago
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
8 hours ago
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
8 hours ago
add a comment |
$begingroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
answered 11 hours ago
Denis NardinDenis Nardin
8,62723461
$endgroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
answered 11 hours ago
Denis NardinDenis Nardin
8,62723461
answered 11 hours ago
Denis NardinDenis Nardin
8,62723461
answered 11 hours ago
Denis NardinDenis Nardin
8,62723461
answered 11 hours ago
Denis NardinDenis Nardin
8,62723461
8,62723461
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
10 hours ago
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
10 hours ago
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
8 hours ago
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
8 hours ago
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
8 hours ago
add a comment |
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
10 hours ago
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
10 hours ago
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
8 hours ago
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
8 hours ago
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
8 hours ago
1
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
10 hours ago
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
10 hours ago
2
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
10 hours ago
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
10 hours ago
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
8 hours ago
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
8 hours ago
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
8 hours ago
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
8 hours ago
2
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
8 hours ago
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
8 hours ago
add a comment |
rori is a new contributor. Be nice, and check out our Code of Conduct.
rori is a new contributor. Be nice, and check out our Code of Conduct.
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