Galois Action on Scheme
$begingroup$
Let $X$ be a $K$-scheme and $L vert K$ be a Galois extension with Galois group $G= Gal(L,K)$.
Let consider the base change $X_L := X otimes_K L:= X times_{Spec(K)} Spec(L)$. Since $X_L$ is a $L$-scheme $G$ can act on it.
My problems are following: I see some ways $G$ acting on $X_L$ and on it's structure sheaf but I'm not sure if they all coinside/corelate to each other and why?
- Let $g in G$ then it induce an automorphism $g: Spec(L) to Spec(L)$ and one can define the action of $g$ on $X_L$ via commutative diagram
$$
require{AMScd}
begin{CD}
X_L @>{bar{g}} >> X_L \
@VVprV @VVprV \
Spec(L) @>{g}>> Spec(L);
end{CD}
$$
or sugestively $bar{g}: id_X times g$
- Let $mathcal{F}$ be a $mathcal{O}_{X}$-module.
I heard that $G$ can induce canonically an "$mathcal{O}_{X_L}$-linear-action" on tnduced sheaf $mathscr{F}otimes mathcal{O}_L$ acting on local sections $mathcal{F}(U)$ for open $U$.
How concretely this action is described? Comes it from the same action as in case 1.?
In the sense of local action $(mathcal{F}(U) otimes_K L) ^g = mathcal{F}(U) otimes_K L^g$ ? So only on second summand? Or are these two actions different?
Espesially I don't see how could $G$ act on local sections of an arbitrary $mathcal{O}_{X}$-module $mathcal{F}$.
Futhermore this concept allows to define the so called sub-$mathcal{O}_{X_L}$-module $mathcal{F}^G subset mathcal{F}$ of invariants. But with respect to which action of $G$?
So the main point of my question is if 1. and 2. "generate" the same action and how this action extends to $mathcal{O}_{X_L}$-modules.
Is this exactly THE canonical Galois action on a scheme which in the literature often mentioned but nowhere explicitely described?
algebraic-geometry group-actions schemes
asked Dec 27 '18 at 1:17
KarlPeterKarlPeter
4521315
$endgroup$
|
show 5 more comments
$begingroup$
Let $X$ be a $K$-scheme and $L vert K$ be a Galois extension with Galois group $G= Gal(L,K)$.
Let consider the base change $X_L := X otimes_K L:= X times_{Spec(K)} Spec(L)$. Since $X_L$ is a $L$-scheme $G$ can act on it.
My problems are following: I see some ways $G$ acting on $X_L$ and on it's structure sheaf but I'm not sure if they all coinside/corelate to each other and why?
- Let $g in G$ then it induce an automorphism $g: Spec(L) to Spec(L)$ and one can define the action of $g$ on $X_L$ via commutative diagram
$$
require{AMScd}
begin{CD}
X_L @>{bar{g}} >> X_L \
@VVprV @VVprV \
Spec(L) @>{g}>> Spec(L);
end{CD}
$$
or sugestively $bar{g}: id_X times g$
- Let $mathcal{F}$ be a $mathcal{O}_{X}$-module.
I heard that $G$ can induce canonically an "$mathcal{O}_{X_L}$-linear-action" on tnduced sheaf $mathscr{F}otimes mathcal{O}_L$ acting on local sections $mathcal{F}(U)$ for open $U$.
How concretely this action is described? Comes it from the same action as in case 1.?
In the sense of local action $(mathcal{F}(U) otimes_K L) ^g = mathcal{F}(U) otimes_K L^g$ ? So only on second summand? Or are these two actions different?
Espesially I don't see how could $G$ act on local sections of an arbitrary $mathcal{O}_{X}$-module $mathcal{F}$.
Futhermore this concept allows to define the so called sub-$mathcal{O}_{X_L}$-module $mathcal{F}^G subset mathcal{F}$ of invariants. But with respect to which action of $G$?
So the main point of my question is if 1. and 2. "generate" the same action and how this action extends to $mathcal{O}_{X_L}$-modules.
Is this exactly THE canonical Galois action on a scheme which in the literature often mentioned but nowhere explicitely described?
algebraic-geometry group-actions schemes
asked Dec 27 '18 at 1:17
KarlPeterKarlPeter
4521315
$endgroup$
$begingroup$
Well 1 and 2 act on different things. 1 acts on the scheme $X_L$ by morphisms of schemes, and 2 acts on $mathcal{O}_{X_L}$-modules by morphisms of the same. They certainly aren't the same action thus you may want to ask, instead of whether or not they are the same action, whether each gives rise to the other or something like that.
$endgroup$
– jgon
Dec 27 '18 at 1:37
$begingroup$
@jgon:Ah sorry, I guess I forgot to explain some detail. If the action from 1. acts on $X_L$ then obviously induce an action on $mathcal{O}_{X_L}$ since the structure sheaf belongs to the data on $X_L$. The question is if the action from 1. then extends from the action on $mathcal{O}_{X_L}$ to arbitrary $mathcal{O}_{X_L}$-module and if it coinside with 2. The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:53
$begingroup$
The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:54
$begingroup$
sorry, I guess I wasn't very clear either. My issue with that idea is that in 1 it doesn't preserve open sets (as in $U$ and $gU$ aren't always the same set) (at least I don't think it does) it certainly doesn't preserve points. Also it acts by ring morphisms not module morphisms, so I'm not sure how you'd get the action in 2 to have those properties, but if it is possible, it would still not be an action I'd call the same as in 1.
$endgroup$
– jgon
Dec 27 '18 at 1:59
$begingroup$
Also your comments now leave me confused as to whether $mathcal{F}$ should be a module on $X$ or $X_L$
$endgroup$
– jgon
Dec 27 '18 at 2:00
|
show 5 more comments
$begingroup$
Let $X$ be a $K$-scheme and $L vert K$ be a Galois extension with Galois group $G= Gal(L,K)$.
Let consider the base change $X_L := X otimes_K L:= X times_{Spec(K)} Spec(L)$. Since $X_L$ is a $L$-scheme $G$ can act on it.
My problems are following: I see some ways $G$ acting on $X_L$ and on it's structure sheaf but I'm not sure if they all coinside/corelate to each other and why?
- Let $g in G$ then it induce an automorphism $g: Spec(L) to Spec(L)$ and one can define the action of $g$ on $X_L$ via commutative diagram
$$
require{AMScd}
begin{CD}
X_L @>{bar{g}} >> X_L \
@VVprV @VVprV \
Spec(L) @>{g}>> Spec(L);
end{CD}
$$
or sugestively $bar{g}: id_X times g$
- Let $mathcal{F}$ be a $mathcal{O}_{X}$-module.
I heard that $G$ can induce canonically an "$mathcal{O}_{X_L}$-linear-action" on tnduced sheaf $mathscr{F}otimes mathcal{O}_L$ acting on local sections $mathcal{F}(U)$ for open $U$.
How concretely this action is described? Comes it from the same action as in case 1.?
In the sense of local action $(mathcal{F}(U) otimes_K L) ^g = mathcal{F}(U) otimes_K L^g$ ? So only on second summand? Or are these two actions different?
Espesially I don't see how could $G$ act on local sections of an arbitrary $mathcal{O}_{X}$-module $mathcal{F}$.
Futhermore this concept allows to define the so called sub-$mathcal{O}_{X_L}$-module $mathcal{F}^G subset mathcal{F}$ of invariants. But with respect to which action of $G$?
So the main point of my question is if 1. and 2. "generate" the same action and how this action extends to $mathcal{O}_{X_L}$-modules.
Is this exactly THE canonical Galois action on a scheme which in the literature often mentioned but nowhere explicitely described?
algebraic-geometry group-actions schemes
asked Dec 27 '18 at 1:17
KarlPeterKarlPeter
4521315
$endgroup$
Let $X$ be a $K$-scheme and $L vert K$ be a Galois extension with Galois group $G= Gal(L,K)$.
Let consider the base change $X_L := X otimes_K L:= X times_{Spec(K)} Spec(L)$. Since $X_L$ is a $L$-scheme $G$ can act on it.
My problems are following: I see some ways $G$ acting on $X_L$ and on it's structure sheaf but I'm not sure if they all coinside/corelate to each other and why?
- Let $g in G$ then it induce an automorphism $g: Spec(L) to Spec(L)$ and one can define the action of $g$ on $X_L$ via commutative diagram
$$
require{AMScd}
begin{CD}
X_L @>{bar{g}} >> X_L \
@VVprV @VVprV \
Spec(L) @>{g}>> Spec(L);
end{CD}
$$
or sugestively $bar{g}: id_X times g$
- Let $mathcal{F}$ be a $mathcal{O}_{X}$-module.
I heard that $G$ can induce canonically an "$mathcal{O}_{X_L}$-linear-action" on tnduced sheaf $mathscr{F}otimes mathcal{O}_L$ acting on local sections $mathcal{F}(U)$ for open $U$.
How concretely this action is described? Comes it from the same action as in case 1.?
In the sense of local action $(mathcal{F}(U) otimes_K L) ^g = mathcal{F}(U) otimes_K L^g$ ? So only on second summand? Or are these two actions different?
Espesially I don't see how could $G$ act on local sections of an arbitrary $mathcal{O}_{X}$-module $mathcal{F}$.
Futhermore this concept allows to define the so called sub-$mathcal{O}_{X_L}$-module $mathcal{F}^G subset mathcal{F}$ of invariants. But with respect to which action of $G$?
So the main point of my question is if 1. and 2. "generate" the same action and how this action extends to $mathcal{O}_{X_L}$-modules.
Is this exactly THE canonical Galois action on a scheme which in the literature often mentioned but nowhere explicitely described?
algebraic-geometry group-actions schemes
algebraic-geometry group-actions schemes
asked Dec 27 '18 at 1:17
KarlPeterKarlPeter
4521315
asked Dec 27 '18 at 1:17
KarlPeterKarlPeter
4521315
edited Dec 27 '18 at 2:14
KarlPeter
asked Dec 27 '18 at 1:17
KarlPeterKarlPeter
4521315
asked Dec 27 '18 at 1:17
KarlPeterKarlPeter
4521315
asked Dec 27 '18 at 1:17
KarlPeterKarlPeter
4521315
4521315
$begingroup$
Well 1 and 2 act on different things. 1 acts on the scheme $X_L$ by morphisms of schemes, and 2 acts on $mathcal{O}_{X_L}$-modules by morphisms of the same. They certainly aren't the same action thus you may want to ask, instead of whether or not they are the same action, whether each gives rise to the other or something like that.
$endgroup$
– jgon
Dec 27 '18 at 1:37
$begingroup$
@jgon:Ah sorry, I guess I forgot to explain some detail. If the action from 1. acts on $X_L$ then obviously induce an action on $mathcal{O}_{X_L}$ since the structure sheaf belongs to the data on $X_L$. The question is if the action from 1. then extends from the action on $mathcal{O}_{X_L}$ to arbitrary $mathcal{O}_{X_L}$-module and if it coinside with 2. The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:53
$begingroup$
The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:54
$begingroup$
sorry, I guess I wasn't very clear either. My issue with that idea is that in 1 it doesn't preserve open sets (as in $U$ and $gU$ aren't always the same set) (at least I don't think it does) it certainly doesn't preserve points. Also it acts by ring morphisms not module morphisms, so I'm not sure how you'd get the action in 2 to have those properties, but if it is possible, it would still not be an action I'd call the same as in 1.
$endgroup$
– jgon
Dec 27 '18 at 1:59
$begingroup$
Also your comments now leave me confused as to whether $mathcal{F}$ should be a module on $X$ or $X_L$
$endgroup$
– jgon
Dec 27 '18 at 2:00
|
show 5 more comments
$begingroup$
Well 1 and 2 act on different things. 1 acts on the scheme $X_L$ by morphisms of schemes, and 2 acts on $mathcal{O}_{X_L}$-modules by morphisms of the same. They certainly aren't the same action thus you may want to ask, instead of whether or not they are the same action, whether each gives rise to the other or something like that.
$endgroup$
– jgon
Dec 27 '18 at 1:37
$begingroup$
@jgon:Ah sorry, I guess I forgot to explain some detail. If the action from 1. acts on $X_L$ then obviously induce an action on $mathcal{O}_{X_L}$ since the structure sheaf belongs to the data on $X_L$. The question is if the action from 1. then extends from the action on $mathcal{O}_{X_L}$ to arbitrary $mathcal{O}_{X_L}$-module and if it coinside with 2. The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:53
$begingroup$
The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:54
$begingroup$
sorry, I guess I wasn't very clear either. My issue with that idea is that in 1 it doesn't preserve open sets (as in $U$ and $gU$ aren't always the same set) (at least I don't think it does) it certainly doesn't preserve points. Also it acts by ring morphisms not module morphisms, so I'm not sure how you'd get the action in 2 to have those properties, but if it is possible, it would still not be an action I'd call the same as in 1.
$endgroup$
– jgon
Dec 27 '18 at 1:59
$begingroup$
Also your comments now leave me confused as to whether $mathcal{F}$ should be a module on $X$ or $X_L$
$endgroup$
– jgon
Dec 27 '18 at 2:00
$begingroup$
Well 1 and 2 act on different things. 1 acts on the scheme $X_L$ by morphisms of schemes, and 2 acts on $mathcal{O}_{X_L}$-modules by morphisms of the same. They certainly aren't the same action thus you may want to ask, instead of whether or not they are the same action, whether each gives rise to the other or something like that.
$endgroup$
– jgon
Dec 27 '18 at 1:37
$begingroup$
Well 1 and 2 act on different things. 1 acts on the scheme $X_L$ by morphisms of schemes, and 2 acts on $mathcal{O}_{X_L}$-modules by morphisms of the same. They certainly aren't the same action thus you may want to ask, instead of whether or not they are the same action, whether each gives rise to the other or something like that.
$endgroup$
– jgon
Dec 27 '18 at 1:37
$begingroup$
@jgon:Ah sorry, I guess I forgot to explain some detail. If the action from 1. acts on $X_L$ then obviously induce an action on $mathcal{O}_{X_L}$ since the structure sheaf belongs to the data on $X_L$. The question is if the action from 1. then extends from the action on $mathcal{O}_{X_L}$ to arbitrary $mathcal{O}_{X_L}$-module and if it coinside with 2. The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:53
$begingroup$
@jgon:Ah sorry, I guess I forgot to explain some detail. If the action from 1. acts on $X_L$ then obviously induce an action on $mathcal{O}_{X_L}$ since the structure sheaf belongs to the data on $X_L$. The question is if the action from 1. then extends from the action on $mathcal{O}_{X_L}$ to arbitrary $mathcal{O}_{X_L}$-module and if it coinside with 2. The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:53
$begingroup$
The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:54
$begingroup$
The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:54
$begingroup$
sorry, I guess I wasn't very clear either. My issue with that idea is that in 1 it doesn't preserve open sets (as in $U$ and $gU$ aren't always the same set) (at least I don't think it does) it certainly doesn't preserve points. Also it acts by ring morphisms not module morphisms, so I'm not sure how you'd get the action in 2 to have those properties, but if it is possible, it would still not be an action I'd call the same as in 1.
$endgroup$
– jgon
Dec 27 '18 at 1:59
$begingroup$
sorry, I guess I wasn't very clear either. My issue with that idea is that in 1 it doesn't preserve open sets (as in $U$ and $gU$ aren't always the same set) (at least I don't think it does) it certainly doesn't preserve points. Also it acts by ring morphisms not module morphisms, so I'm not sure how you'd get the action in 2 to have those properties, but if it is possible, it would still not be an action I'd call the same as in 1.
$endgroup$
– jgon
Dec 27 '18 at 1:59
$begingroup$
Also your comments now leave me confused as to whether $mathcal{F}$ should be a module on $X$ or $X_L$
$endgroup$
– jgon
Dec 27 '18 at 2:00
$begingroup$
Also your comments now leave me confused as to whether $mathcal{F}$ should be a module on $X$ or $X_L$
$endgroup$
– jgon
Dec 27 '18 at 2:00
|
show 5 more comments
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$begingroup$
Well 1 and 2 act on different things. 1 acts on the scheme $X_L$ by morphisms of schemes, and 2 acts on $mathcal{O}_{X_L}$-modules by morphisms of the same. They certainly aren't the same action thus you may want to ask, instead of whether or not they are the same action, whether each gives rise to the other or something like that.
$endgroup$
– jgon
Dec 27 '18 at 1:37
$begingroup$
@jgon:Ah sorry, I guess I forgot to explain some detail. If the action from 1. acts on $X_L$ then obviously induce an action on $mathcal{O}_{X_L}$ since the structure sheaf belongs to the data on $X_L$. The question is if the action from 1. then extends from the action on $mathcal{O}_{X_L}$ to arbitrary $mathcal{O}_{X_L}$-module and if it coinside with 2. The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:53
$begingroup$
The thing is that a $mathcal{O}_X$-module $mathcal{F}$ induce locally a $mathcal{O}_{X_L}$-module structure via $mathcal{F}(U) otimes_K L$ and here 1. can induce naively action.
$endgroup$
– KarlPeter
Dec 27 '18 at 1:54
$begingroup$
sorry, I guess I wasn't very clear either. My issue with that idea is that in 1 it doesn't preserve open sets (as in $U$ and $gU$ aren't always the same set) (at least I don't think it does) it certainly doesn't preserve points. Also it acts by ring morphisms not module morphisms, so I'm not sure how you'd get the action in 2 to have those properties, but if it is possible, it would still not be an action I'd call the same as in 1.
$endgroup$
– jgon
Dec 27 '18 at 1:59
$begingroup$
Also your comments now leave me confused as to whether $mathcal{F}$ should be a module on $X$ or $X_L$
$endgroup$
– jgon
Dec 27 '18 at 2:00