Proposition 1.6.6, Etale Cohomology theory, Lei Fu
$begingroup$
I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu.
$mathrm{Proposition} 1.6.6$
Let $g:S'rightarrow S$ be a quasi-compact faithfully flat morphism, and let $mathcal F$ be a quasi-coherent $mathcal O_S$-module. Then $g^* mathcal F$ is locally of finite type(resp. locally of finite presentation,resp. locally free of finite rank) if and only if $mathcal F$ is so.
The author reduces the proof to the case where both S' and S are affine, but the detail is left to the reader.
How can I confirm this reduction? S' and S seem to be schemes but not necessarily Noetherian.
algebraic-geometry arithmetic-geometry flatness etale-cohomology descent
edited Jan 24 at 18:03
hardmath
29.1k953101
asked Jan 24 at 16:31
HardyHardy
11
$endgroup$
|
show 1 more comment
$begingroup$
I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu.
$mathrm{Proposition} 1.6.6$
Let $g:S'rightarrow S$ be a quasi-compact faithfully flat morphism, and let $mathcal F$ be a quasi-coherent $mathcal O_S$-module. Then $g^* mathcal F$ is locally of finite type(resp. locally of finite presentation,resp. locally free of finite rank) if and only if $mathcal F$ is so.
The author reduces the proof to the case where both S' and S are affine, but the detail is left to the reader.
How can I confirm this reduction? S' and S seem to be schemes but not necessarily Noetherian.
algebraic-geometry arithmetic-geometry flatness etale-cohomology descent
edited Jan 24 at 18:03
hardmath
29.1k953101
asked Jan 24 at 16:31
HardyHardy
11
$endgroup$
1
$begingroup$
I would cover $S$ with open affines (which are in particular quasi-compact). Their preimages are open and quasi-compact, since $g$ is quasi-compact. Then we cover these preimages with open affines which can be done by finitely many (quasi-compactness). Doesn't this work?
$endgroup$
– Paul K
Jan 24 at 18:12
$begingroup$
@PaulK I was thinking about the same thing, but I have no idea about how to use faithfully flatness. It seems for me that there's no morphism like $mathrm{Spec}A'rightarrow mathrm{Spec} A$ where $A'$ is faithfully flat over $A$.
$endgroup$
– Hardy
Jan 24 at 18:50
$begingroup$
What exactly do you mean?
$endgroup$
– Paul K
Jan 24 at 18:55
$begingroup$
@PaulK We may suppose $S=mathrm{Spec}A$, $S'$ can be covered by finite affine schemes $mathrm{Spec} A'_i$ and g is faithfully flat. Then we need to reduce the case $S'=mathrm{Spec} A' rightarrow mathrm{Spec} A$ is faithfully flat, but $A'_i$ is not necessarily faithfully flat over A. So, where can I use the case where S' and S are affine?
$endgroup$
– Hardy
Jan 24 at 19:09
1
$begingroup$
Now I got it. How about looking at the disjoint union of $A_i'$ which is an affine scheme by finiteness? Maybe that works?
$endgroup$
– Paul K
Jan 24 at 20:30
|
show 1 more comment
$begingroup$
I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu.
$mathrm{Proposition} 1.6.6$
Let $g:S'rightarrow S$ be a quasi-compact faithfully flat morphism, and let $mathcal F$ be a quasi-coherent $mathcal O_S$-module. Then $g^* mathcal F$ is locally of finite type(resp. locally of finite presentation,resp. locally free of finite rank) if and only if $mathcal F$ is so.
The author reduces the proof to the case where both S' and S are affine, but the detail is left to the reader.
How can I confirm this reduction? S' and S seem to be schemes but not necessarily Noetherian.
algebraic-geometry arithmetic-geometry flatness etale-cohomology descent
edited Jan 24 at 18:03
hardmath
29.1k953101
asked Jan 24 at 16:31
HardyHardy
11
$endgroup$
I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu.
$mathrm{Proposition} 1.6.6$
Let $g:S'rightarrow S$ be a quasi-compact faithfully flat morphism, and let $mathcal F$ be a quasi-coherent $mathcal O_S$-module. Then $g^* mathcal F$ is locally of finite type(resp. locally of finite presentation,resp. locally free of finite rank) if and only if $mathcal F$ is so.
The author reduces the proof to the case where both S' and S are affine, but the detail is left to the reader.
How can I confirm this reduction? S' and S seem to be schemes but not necessarily Noetherian.
algebraic-geometry arithmetic-geometry flatness etale-cohomology descent
algebraic-geometry arithmetic-geometry flatness etale-cohomology descent
edited Jan 24 at 18:03
hardmath
29.1k953101
asked Jan 24 at 16:31
HardyHardy
11
edited Jan 24 at 18:03
hardmath
29.1k953101
asked Jan 24 at 16:31
HardyHardy
11
edited Jan 24 at 18:03
hardmath
29.1k953101
edited Jan 24 at 18:03
hardmath
29.1k953101
edited Jan 24 at 18:03
hardmath
29.1k953101
29.1k953101
asked Jan 24 at 16:31
HardyHardy
11
asked Jan 24 at 16:31
HardyHardy
11
asked Jan 24 at 16:31
HardyHardy
11
11
1
$begingroup$
I would cover $S$ with open affines (which are in particular quasi-compact). Their preimages are open and quasi-compact, since $g$ is quasi-compact. Then we cover these preimages with open affines which can be done by finitely many (quasi-compactness). Doesn't this work?
$endgroup$
– Paul K
Jan 24 at 18:12
$begingroup$
@PaulK I was thinking about the same thing, but I have no idea about how to use faithfully flatness. It seems for me that there's no morphism like $mathrm{Spec}A'rightarrow mathrm{Spec} A$ where $A'$ is faithfully flat over $A$.
$endgroup$
– Hardy
Jan 24 at 18:50
$begingroup$
What exactly do you mean?
$endgroup$
– Paul K
Jan 24 at 18:55
$begingroup$
@PaulK We may suppose $S=mathrm{Spec}A$, $S'$ can be covered by finite affine schemes $mathrm{Spec} A'_i$ and g is faithfully flat. Then we need to reduce the case $S'=mathrm{Spec} A' rightarrow mathrm{Spec} A$ is faithfully flat, but $A'_i$ is not necessarily faithfully flat over A. So, where can I use the case where S' and S are affine?
$endgroup$
– Hardy
Jan 24 at 19:09
1
$begingroup$
Now I got it. How about looking at the disjoint union of $A_i'$ which is an affine scheme by finiteness? Maybe that works?
$endgroup$
– Paul K
Jan 24 at 20:30
|
show 1 more comment
1
$begingroup$
I would cover $S$ with open affines (which are in particular quasi-compact). Their preimages are open and quasi-compact, since $g$ is quasi-compact. Then we cover these preimages with open affines which can be done by finitely many (quasi-compactness). Doesn't this work?
$endgroup$
– Paul K
Jan 24 at 18:12
$begingroup$
@PaulK I was thinking about the same thing, but I have no idea about how to use faithfully flatness. It seems for me that there's no morphism like $mathrm{Spec}A'rightarrow mathrm{Spec} A$ where $A'$ is faithfully flat over $A$.
$endgroup$
– Hardy
Jan 24 at 18:50
$begingroup$
What exactly do you mean?
$endgroup$
– Paul K
Jan 24 at 18:55
$begingroup$
@PaulK We may suppose $S=mathrm{Spec}A$, $S'$ can be covered by finite affine schemes $mathrm{Spec} A'_i$ and g is faithfully flat. Then we need to reduce the case $S'=mathrm{Spec} A' rightarrow mathrm{Spec} A$ is faithfully flat, but $A'_i$ is not necessarily faithfully flat over A. So, where can I use the case where S' and S are affine?
$endgroup$
– Hardy
Jan 24 at 19:09
1
$begingroup$
Now I got it. How about looking at the disjoint union of $A_i'$ which is an affine scheme by finiteness? Maybe that works?
$endgroup$
– Paul K
Jan 24 at 20:30
1
1
$begingroup$
I would cover $S$ with open affines (which are in particular quasi-compact). Their preimages are open and quasi-compact, since $g$ is quasi-compact. Then we cover these preimages with open affines which can be done by finitely many (quasi-compactness). Doesn't this work?
$endgroup$
– Paul K
Jan 24 at 18:12
$begingroup$
I would cover $S$ with open affines (which are in particular quasi-compact). Their preimages are open and quasi-compact, since $g$ is quasi-compact. Then we cover these preimages with open affines which can be done by finitely many (quasi-compactness). Doesn't this work?
$endgroup$
– Paul K
Jan 24 at 18:12
$begingroup$
@PaulK I was thinking about the same thing, but I have no idea about how to use faithfully flatness. It seems for me that there's no morphism like $mathrm{Spec}A'rightarrow mathrm{Spec} A$ where $A'$ is faithfully flat over $A$.
$endgroup$
– Hardy
Jan 24 at 18:50
$begingroup$
@PaulK I was thinking about the same thing, but I have no idea about how to use faithfully flatness. It seems for me that there's no morphism like $mathrm{Spec}A'rightarrow mathrm{Spec} A$ where $A'$ is faithfully flat over $A$.
$endgroup$
– Hardy
Jan 24 at 18:50
$begingroup$
What exactly do you mean?
$endgroup$
– Paul K
Jan 24 at 18:55
$begingroup$
What exactly do you mean?
$endgroup$
– Paul K
Jan 24 at 18:55
$begingroup$
@PaulK We may suppose $S=mathrm{Spec}A$, $S'$ can be covered by finite affine schemes $mathrm{Spec} A'_i$ and g is faithfully flat. Then we need to reduce the case $S'=mathrm{Spec} A' rightarrow mathrm{Spec} A$ is faithfully flat, but $A'_i$ is not necessarily faithfully flat over A. So, where can I use the case where S' and S are affine?
$endgroup$
– Hardy
Jan 24 at 19:09
$begingroup$
@PaulK We may suppose $S=mathrm{Spec}A$, $S'$ can be covered by finite affine schemes $mathrm{Spec} A'_i$ and g is faithfully flat. Then we need to reduce the case $S'=mathrm{Spec} A' rightarrow mathrm{Spec} A$ is faithfully flat, but $A'_i$ is not necessarily faithfully flat over A. So, where can I use the case where S' and S are affine?
$endgroup$
– Hardy
Jan 24 at 19:09
1
1
$begingroup$
Now I got it. How about looking at the disjoint union of $A_i'$ which is an affine scheme by finiteness? Maybe that works?
$endgroup$
– Paul K
Jan 24 at 20:30
$begingroup$
Now I got it. How about looking at the disjoint union of $A_i'$ which is an affine scheme by finiteness? Maybe that works?
$endgroup$
– Paul K
Jan 24 at 20:30
|
show 1 more comment
0
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I would cover $S$ with open affines (which are in particular quasi-compact). Their preimages are open and quasi-compact, since $g$ is quasi-compact. Then we cover these preimages with open affines which can be done by finitely many (quasi-compactness). Doesn't this work?
$endgroup$
– Paul K
Jan 24 at 18:12
$begingroup$
@PaulK I was thinking about the same thing, but I have no idea about how to use faithfully flatness. It seems for me that there's no morphism like $mathrm{Spec}A'rightarrow mathrm{Spec} A$ where $A'$ is faithfully flat over $A$.
$endgroup$
– Hardy
Jan 24 at 18:50
$begingroup$
What exactly do you mean?
$endgroup$
– Paul K
Jan 24 at 18:55
$begingroup$
@PaulK We may suppose $S=mathrm{Spec}A$, $S'$ can be covered by finite affine schemes $mathrm{Spec} A'_i$ and g is faithfully flat. Then we need to reduce the case $S'=mathrm{Spec} A' rightarrow mathrm{Spec} A$ is faithfully flat, but $A'_i$ is not necessarily faithfully flat over A. So, where can I use the case where S' and S are affine?
$endgroup$
– Hardy
Jan 24 at 19:09
1
$begingroup$
Now I got it. How about looking at the disjoint union of $A_i'$ which is an affine scheme by finiteness? Maybe that works?
$endgroup$
– Paul K
Jan 24 at 20:30