Where is the local structure theory of étale morphisms needed?
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In Stacks 02GH it is mentioned étale morphisms are locally standard étale. This seems to be the analogue of the local form of local diffeomorphisms in the smooth category. In the latter, local structure plays a big role.
Where is the local structure of étale morphisms needed further on in the theory? Which important proofs crucially require actually knowing an explicit local form?
algebraic-geometry schemes
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up vote
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In Stacks 02GH it is mentioned étale morphisms are locally standard étale. This seems to be the analogue of the local form of local diffeomorphisms in the smooth category. In the latter, local structure plays a big role.
Where is the local structure of étale morphisms needed further on in the theory? Which important proofs crucially require actually knowing an explicit local form?
algebraic-geometry schemes
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
In Stacks 02GH it is mentioned étale morphisms are locally standard étale. This seems to be the analogue of the local form of local diffeomorphisms in the smooth category. In the latter, local structure plays a big role.
Where is the local structure of étale morphisms needed further on in the theory? Which important proofs crucially require actually knowing an explicit local form?
algebraic-geometry schemes
In Stacks 02GH it is mentioned étale morphisms are locally standard étale. This seems to be the analogue of the local form of local diffeomorphisms in the smooth category. In the latter, local structure plays a big role.
Where is the local structure of étale morphisms needed further on in the theory? Which important proofs crucially require actually knowing an explicit local form?
algebraic-geometry schemes
algebraic-geometry schemes
asked 15 hours ago
Arrow
5,04911445
5,04911445
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One part where it is essential is that the etale site is a "topological invariant". More precisely, if $X_0 subset X$ is a subscheme with the same underlying space as $X$ (for example, some nonreduced structure), then there is an equivalence of sites: $Et/X to Et/X_0$, given by $Y mapsto Y_0 = Y times_X X_0$.
The proof of full faithfullness doesn't require the local structure, rather it goes through some elementary facts about sections of the projection maps from the product and some more "Hartshorne-style" facts about maps. To show essential surjectivity though, you construct it locally. If you can show that for any point $p$ in the $X_0$-Scheme $Y_0$, there is a neighborhood $U_0$ so that $U_0 = U times_X X_0$, then a standard glueing argument for this will complete the proof. However to show this local fact, you need (very crucially!) the local structure of an etale map. This is contained nicely in Lemma 2.3.11 in Lei Fu, Etale Cohomology Theory.
Dear DKS, is this an important result on the path to proving the Weil conjectures? I would appreciate some more details about its significance!
– Arrow
7 hours ago
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
5 hours ago
Dear DKS, please forgive my ignorance again, but is this result in a sense necessary for the proof of the Weil conjectures, or, say, the comparison theorem for étale cohomology? Thanks again!
– Arrow
3 hours ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
One part where it is essential is that the etale site is a "topological invariant". More precisely, if $X_0 subset X$ is a subscheme with the same underlying space as $X$ (for example, some nonreduced structure), then there is an equivalence of sites: $Et/X to Et/X_0$, given by $Y mapsto Y_0 = Y times_X X_0$.
The proof of full faithfullness doesn't require the local structure, rather it goes through some elementary facts about sections of the projection maps from the product and some more "Hartshorne-style" facts about maps. To show essential surjectivity though, you construct it locally. If you can show that for any point $p$ in the $X_0$-Scheme $Y_0$, there is a neighborhood $U_0$ so that $U_0 = U times_X X_0$, then a standard glueing argument for this will complete the proof. However to show this local fact, you need (very crucially!) the local structure of an etale map. This is contained nicely in Lemma 2.3.11 in Lei Fu, Etale Cohomology Theory.
Dear DKS, is this an important result on the path to proving the Weil conjectures? I would appreciate some more details about its significance!
– Arrow
7 hours ago
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
5 hours ago
Dear DKS, please forgive my ignorance again, but is this result in a sense necessary for the proof of the Weil conjectures, or, say, the comparison theorem for étale cohomology? Thanks again!
– Arrow
3 hours ago
add a comment |
up vote
1
down vote
One part where it is essential is that the etale site is a "topological invariant". More precisely, if $X_0 subset X$ is a subscheme with the same underlying space as $X$ (for example, some nonreduced structure), then there is an equivalence of sites: $Et/X to Et/X_0$, given by $Y mapsto Y_0 = Y times_X X_0$.
The proof of full faithfullness doesn't require the local structure, rather it goes through some elementary facts about sections of the projection maps from the product and some more "Hartshorne-style" facts about maps. To show essential surjectivity though, you construct it locally. If you can show that for any point $p$ in the $X_0$-Scheme $Y_0$, there is a neighborhood $U_0$ so that $U_0 = U times_X X_0$, then a standard glueing argument for this will complete the proof. However to show this local fact, you need (very crucially!) the local structure of an etale map. This is contained nicely in Lemma 2.3.11 in Lei Fu, Etale Cohomology Theory.
Dear DKS, is this an important result on the path to proving the Weil conjectures? I would appreciate some more details about its significance!
– Arrow
7 hours ago
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
5 hours ago
Dear DKS, please forgive my ignorance again, but is this result in a sense necessary for the proof of the Weil conjectures, or, say, the comparison theorem for étale cohomology? Thanks again!
– Arrow
3 hours ago
add a comment |
up vote
1
down vote
up vote
1
down vote
One part where it is essential is that the etale site is a "topological invariant". More precisely, if $X_0 subset X$ is a subscheme with the same underlying space as $X$ (for example, some nonreduced structure), then there is an equivalence of sites: $Et/X to Et/X_0$, given by $Y mapsto Y_0 = Y times_X X_0$.
The proof of full faithfullness doesn't require the local structure, rather it goes through some elementary facts about sections of the projection maps from the product and some more "Hartshorne-style" facts about maps. To show essential surjectivity though, you construct it locally. If you can show that for any point $p$ in the $X_0$-Scheme $Y_0$, there is a neighborhood $U_0$ so that $U_0 = U times_X X_0$, then a standard glueing argument for this will complete the proof. However to show this local fact, you need (very crucially!) the local structure of an etale map. This is contained nicely in Lemma 2.3.11 in Lei Fu, Etale Cohomology Theory.
One part where it is essential is that the etale site is a "topological invariant". More precisely, if $X_0 subset X$ is a subscheme with the same underlying space as $X$ (for example, some nonreduced structure), then there is an equivalence of sites: $Et/X to Et/X_0$, given by $Y mapsto Y_0 = Y times_X X_0$.
The proof of full faithfullness doesn't require the local structure, rather it goes through some elementary facts about sections of the projection maps from the product and some more "Hartshorne-style" facts about maps. To show essential surjectivity though, you construct it locally. If you can show that for any point $p$ in the $X_0$-Scheme $Y_0$, there is a neighborhood $U_0$ so that $U_0 = U times_X X_0$, then a standard glueing argument for this will complete the proof. However to show this local fact, you need (very crucially!) the local structure of an etale map. This is contained nicely in Lemma 2.3.11 in Lei Fu, Etale Cohomology Theory.
answered 9 hours ago
DKS
600312
600312
Dear DKS, is this an important result on the path to proving the Weil conjectures? I would appreciate some more details about its significance!
– Arrow
7 hours ago
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
5 hours ago
Dear DKS, please forgive my ignorance again, but is this result in a sense necessary for the proof of the Weil conjectures, or, say, the comparison theorem for étale cohomology? Thanks again!
– Arrow
3 hours ago
add a comment |
Dear DKS, is this an important result on the path to proving the Weil conjectures? I would appreciate some more details about its significance!
– Arrow
7 hours ago
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
5 hours ago
Dear DKS, please forgive my ignorance again, but is this result in a sense necessary for the proof of the Weil conjectures, or, say, the comparison theorem for étale cohomology? Thanks again!
– Arrow
3 hours ago
Dear DKS, is this an important result on the path to proving the Weil conjectures? I would appreciate some more details about its significance!
– Arrow
7 hours ago
Dear DKS, is this an important result on the path to proving the Weil conjectures? I would appreciate some more details about its significance!
– Arrow
7 hours ago
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
5 hours ago
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
5 hours ago
Dear DKS, please forgive my ignorance again, but is this result in a sense necessary for the proof of the Weil conjectures, or, say, the comparison theorem for étale cohomology? Thanks again!
– Arrow
3 hours ago
Dear DKS, please forgive my ignorance again, but is this result in a sense necessary for the proof of the Weil conjectures, or, say, the comparison theorem for étale cohomology? Thanks again!
– Arrow
3 hours ago
add a comment |
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