Mixing
Where is the local structure theory of étale morphisms needed?
MO crosspost.
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
asked Nov 20 '18 at 14:05
Arrow
5,07621445
add a comment |
MO crosspost.
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
asked Nov 20 '18 at 14:05
Arrow
5,07621445
add a comment |
MO crosspost.
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
asked Nov 20 '18 at 14:05
Arrow
5,07621445
MO crosspost.
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 Nov 20 '18 at 14:05
Arrow
5,07621445
asked Nov 20 '18 at 14:05
Arrow
5,07621445
edited Dec 6 '18 at 15:02
asked Nov 20 '18 at 14:05
Arrow
5,07621445
asked Nov 20 '18 at 14:05
Arrow
5,07621445
asked Nov 20 '18 at 14:05
Arrow
5,07621445
5,07621445
add a comment |
add a comment |
1 Answer
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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.
answered Nov 20 '18 at 19:14
DKS
615412
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
Nov 20 '18 at 21:39
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
Nov 21 '18 at 0:01
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
Nov 21 '18 at 1:53
I'm not sure to be completely honest. I don't have my books on hand but the stacks project would probably give some statistics on this
– DKS
Nov 21 '18 at 17:02
1
Dear @DKS, interestingly, no one on MO has suggested your answer!
– Arrow
Dec 6 '18 at 15:03
add a comment |
Your Answer
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1 Answer
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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.
answered Nov 20 '18 at 19:14
DKS
615412
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
Nov 20 '18 at 21:39
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
Nov 21 '18 at 0:01
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
Nov 21 '18 at 1:53
I'm not sure to be completely honest. I don't have my books on hand but the stacks project would probably give some statistics on this
– DKS
Nov 21 '18 at 17:02
1
Dear @DKS, interestingly, no one on MO has suggested your answer!
– Arrow
Dec 6 '18 at 15:03
add a comment |
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 Nov 20 '18 at 19:14
DKS
615412
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
Nov 20 '18 at 21:39
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
Nov 21 '18 at 0:01
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
Nov 21 '18 at 1:53
I'm not sure to be completely honest. I don't have my books on hand but the stacks project would probably give some statistics on this
– DKS
Nov 21 '18 at 17:02
1
Dear @DKS, interestingly, no one on MO has suggested your answer!
– Arrow
Dec 6 '18 at 15:03
add a comment |
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 Nov 20 '18 at 19:14
DKS
615412
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 Nov 20 '18 at 19:14
DKS
615412
answered Nov 20 '18 at 19:14
DKS
615412
answered Nov 20 '18 at 19:14
DKS
615412
answered Nov 20 '18 at 19:14
DKS
615412
615412
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
Nov 20 '18 at 21:39
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
Nov 21 '18 at 0:01
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
Nov 21 '18 at 1:53
I'm not sure to be completely honest. I don't have my books on hand but the stacks project would probably give some statistics on this
– DKS
Nov 21 '18 at 17:02
1
Dear @DKS, interestingly, no one on MO has suggested your answer!
– Arrow
Dec 6 '18 at 15:03
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
Nov 20 '18 at 21:39
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
Nov 21 '18 at 0:01
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
Nov 21 '18 at 1:53
I'm not sure to be completely honest. I don't have my books on hand but the stacks project would probably give some statistics on this
– DKS
Nov 21 '18 at 17:02
1
Dear @DKS, interestingly, no one on MO has suggested your answer!
– Arrow
Dec 6 '18 at 15:03
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
Nov 20 '18 at 21:39
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
Nov 20 '18 at 21:39
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
Nov 21 '18 at 0:01
Not by itself, but it is an important foundational result in Étale cohomology. So it certainly is a part of that approach.
– DKS
Nov 21 '18 at 0:01
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
Nov 21 '18 at 1:53
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
Nov 21 '18 at 1:53
I'm not sure to be completely honest. I don't have my books on hand but the stacks project would probably give some statistics on this
– DKS
Nov 21 '18 at 17:02
I'm not sure to be completely honest. I don't have my books on hand but the stacks project would probably give some statistics on this
– DKS
Nov 21 '18 at 17:02
1
1
Dear @DKS, interestingly, no one on MO has suggested your answer!
– Arrow
Dec 6 '18 at 15:03
Dear @DKS, interestingly, no one on MO has suggested your answer!
– Arrow
Dec 6 '18 at 15:03
add a comment |
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