Galois Action on Coherent Sheaves Exact Functor
Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined action of $G$ on structure sheaf $mathcal{O}_X$ such that $mathcal{O}_{X/G}= mathcal{O}_X^G$).
Denote by $p: X to X/G$ the induced well defined projection morphism.
Firsly I heard that in this case $p$ is called a "Galois morphism". Why?
Now consider the exact sequence
$$0rightarrow mathcal{G} 'rightarrow mathcal{G} rightarrow mathcal{G}''rightarrow 0$$
of coherent sheaves over $X$.
Let assume that after applying the pushforward /direct image functor $p_*$ the sequence stay exact.
My second question is if there exist and how defined if exist an induced (canonical?) action of $G$ on any "pushed forward" coheherent sheaf $p_*mathcal{F}$
My intention is the following: If the last question has a positive answer then I can apply to the exact sequence
$$0rightarrow p_*mathcal{G} 'rightarrow p_*mathcal{G} rightarrow p_*mathcal{G}''rightarrow 0$$
in this case the functor $^G$ on sheaves over $X/G$ via $p_*mathcal{F} to (p_*mathcal{F})^G$
Then I'm looking for criterions which garantee the exactness of this functor.
sheaf-theory galois-cohomology projective-varieties
asked Dec 26 '18 at 23:07
KarlPeter
5951315
|
show 4 more comments
Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined action of $G$ on structure sheaf $mathcal{O}_X$ such that $mathcal{O}_{X/G}= mathcal{O}_X^G$).
Denote by $p: X to X/G$ the induced well defined projection morphism.
Firsly I heard that in this case $p$ is called a "Galois morphism". Why?
Now consider the exact sequence
$$0rightarrow mathcal{G} 'rightarrow mathcal{G} rightarrow mathcal{G}''rightarrow 0$$
of coherent sheaves over $X$.
Let assume that after applying the pushforward /direct image functor $p_*$ the sequence stay exact.
My second question is if there exist and how defined if exist an induced (canonical?) action of $G$ on any "pushed forward" coheherent sheaf $p_*mathcal{F}$
My intention is the following: If the last question has a positive answer then I can apply to the exact sequence
$$0rightarrow p_*mathcal{G} 'rightarrow p_*mathcal{G} rightarrow p_*mathcal{G}''rightarrow 0$$
in this case the functor $^G$ on sheaves over $X/G$ via $p_*mathcal{F} to (p_*mathcal{F})^G$
Then I'm looking for criterions which garantee the exactness of this functor.
sheaf-theory galois-cohomology projective-varieties
asked Dec 26 '18 at 23:07
KarlPeter
5951315
You said several times "Obviously", but some of them are not obvious at all. Is $X$ defined over $K$ ? Otherwise, what is the action of $Gal(mathbb{C}/K)$ ? What is the action of $G$ on a coherent sheaf on $X/G$ ? Except from the subgroups {identity} and {identity, conjugation}, do you have other examples of finite subgroup of $Gal(mathbb{C}/K)$ ? (there are none from Artin-Schreier theorem). Finally, why do you think this has to do with Hilbert 90 ?
– Roland
Dec 27 '18 at 17:12
@Roland: Yes, sorry, that was a not really sophisticated attempt to formulate the problem without explicitelly using the "Galois morphism" terminology because I'm till now quite unfamilar with it. Especially I don't see which connection the common Galois theory has with cases when the quotient $X/G$ is well defined as scheme/variery. I hope that it is now become a bit clearer.
– KarlPeter
Dec 28 '18 at 1:12
Ok this question makes much more sense to me than your previous one. Note that the sequence after taking $p_*$ will be exact since under your hypotheses, $p$ is finite.
– Roland
Dec 28 '18 at 9:14
@Roland: I think that it boils down to the statement that for every affine $f$ morphism the direct image $f_∗$ exact. Btw do you know a good reference for the proof of it /a sketch of the proof? I know a proof only for the case that $f$ is a closed immersion
– KarlPeter
Dec 28 '18 at 23:32
1
Well, $R^ip_*F$ is the sheaf associated to $Umapsto H^i(p^{-1}(U),F)$. If $p$ is affine, then $H^i(p ^{-1}(U),F)$ is zero. So $R^ip_*F$ is the sheaf associated to a presheaf which vanishes on a basis. This implies that $R^ip_*F$ is zero.
– Roland
Dec 28 '18 at 23:55
|
show 4 more comments
Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined action of $G$ on structure sheaf $mathcal{O}_X$ such that $mathcal{O}_{X/G}= mathcal{O}_X^G$).
Denote by $p: X to X/G$ the induced well defined projection morphism.
Firsly I heard that in this case $p$ is called a "Galois morphism". Why?
Now consider the exact sequence
$$0rightarrow mathcal{G} 'rightarrow mathcal{G} rightarrow mathcal{G}''rightarrow 0$$
of coherent sheaves over $X$.
Let assume that after applying the pushforward /direct image functor $p_*$ the sequence stay exact.
My second question is if there exist and how defined if exist an induced (canonical?) action of $G$ on any "pushed forward" coheherent sheaf $p_*mathcal{F}$
My intention is the following: If the last question has a positive answer then I can apply to the exact sequence
$$0rightarrow p_*mathcal{G} 'rightarrow p_*mathcal{G} rightarrow p_*mathcal{G}''rightarrow 0$$
in this case the functor $^G$ on sheaves over $X/G$ via $p_*mathcal{F} to (p_*mathcal{F})^G$
Then I'm looking for criterions which garantee the exactness of this functor.
sheaf-theory galois-cohomology projective-varieties
asked Dec 26 '18 at 23:07
KarlPeter
5951315
Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined action of $G$ on structure sheaf $mathcal{O}_X$ such that $mathcal{O}_{X/G}= mathcal{O}_X^G$).
Denote by $p: X to X/G$ the induced well defined projection morphism.
Firsly I heard that in this case $p$ is called a "Galois morphism". Why?
Now consider the exact sequence
$$0rightarrow mathcal{G} 'rightarrow mathcal{G} rightarrow mathcal{G}''rightarrow 0$$
of coherent sheaves over $X$.
Let assume that after applying the pushforward /direct image functor $p_*$ the sequence stay exact.
My second question is if there exist and how defined if exist an induced (canonical?) action of $G$ on any "pushed forward" coheherent sheaf $p_*mathcal{F}$
My intention is the following: If the last question has a positive answer then I can apply to the exact sequence
$$0rightarrow p_*mathcal{G} 'rightarrow p_*mathcal{G} rightarrow p_*mathcal{G}''rightarrow 0$$
in this case the functor $^G$ on sheaves over $X/G$ via $p_*mathcal{F} to (p_*mathcal{F})^G$
Then I'm looking for criterions which garantee the exactness of this functor.
sheaf-theory galois-cohomology projective-varieties
sheaf-theory galois-cohomology projective-varieties
asked Dec 26 '18 at 23:07
KarlPeter
5951315
asked Dec 26 '18 at 23:07
KarlPeter
5951315
edited Dec 28 '18 at 1:05
asked Dec 26 '18 at 23:07
KarlPeter
5951315
asked Dec 26 '18 at 23:07
KarlPeter
5951315
asked Dec 26 '18 at 23:07
KarlPeter
5951315
5951315
You said several times "Obviously", but some of them are not obvious at all. Is $X$ defined over $K$ ? Otherwise, what is the action of $Gal(mathbb{C}/K)$ ? What is the action of $G$ on a coherent sheaf on $X/G$ ? Except from the subgroups {identity} and {identity, conjugation}, do you have other examples of finite subgroup of $Gal(mathbb{C}/K)$ ? (there are none from Artin-Schreier theorem). Finally, why do you think this has to do with Hilbert 90 ?
– Roland
Dec 27 '18 at 17:12
@Roland: Yes, sorry, that was a not really sophisticated attempt to formulate the problem without explicitelly using the "Galois morphism" terminology because I'm till now quite unfamilar with it. Especially I don't see which connection the common Galois theory has with cases when the quotient $X/G$ is well defined as scheme/variery. I hope that it is now become a bit clearer.
– KarlPeter
Dec 28 '18 at 1:12
Ok this question makes much more sense to me than your previous one. Note that the sequence after taking $p_*$ will be exact since under your hypotheses, $p$ is finite.
– Roland
Dec 28 '18 at 9:14
@Roland: I think that it boils down to the statement that for every affine $f$ morphism the direct image $f_∗$ exact. Btw do you know a good reference for the proof of it /a sketch of the proof? I know a proof only for the case that $f$ is a closed immersion
– KarlPeter
Dec 28 '18 at 23:32
1
Well, $R^ip_*F$ is the sheaf associated to $Umapsto H^i(p^{-1}(U),F)$. If $p$ is affine, then $H^i(p ^{-1}(U),F)$ is zero. So $R^ip_*F$ is the sheaf associated to a presheaf which vanishes on a basis. This implies that $R^ip_*F$ is zero.
– Roland
Dec 28 '18 at 23:55
|
show 4 more comments
You said several times "Obviously", but some of them are not obvious at all. Is $X$ defined over $K$ ? Otherwise, what is the action of $Gal(mathbb{C}/K)$ ? What is the action of $G$ on a coherent sheaf on $X/G$ ? Except from the subgroups {identity} and {identity, conjugation}, do you have other examples of finite subgroup of $Gal(mathbb{C}/K)$ ? (there are none from Artin-Schreier theorem). Finally, why do you think this has to do with Hilbert 90 ?
– Roland
Dec 27 '18 at 17:12
@Roland: Yes, sorry, that was a not really sophisticated attempt to formulate the problem without explicitelly using the "Galois morphism" terminology because I'm till now quite unfamilar with it. Especially I don't see which connection the common Galois theory has with cases when the quotient $X/G$ is well defined as scheme/variery. I hope that it is now become a bit clearer.
– KarlPeter
Dec 28 '18 at 1:12
Ok this question makes much more sense to me than your previous one. Note that the sequence after taking $p_*$ will be exact since under your hypotheses, $p$ is finite.
– Roland
Dec 28 '18 at 9:14
@Roland: I think that it boils down to the statement that for every affine $f$ morphism the direct image $f_∗$ exact. Btw do you know a good reference for the proof of it /a sketch of the proof? I know a proof only for the case that $f$ is a closed immersion
– KarlPeter
Dec 28 '18 at 23:32
1
Well, $R^ip_*F$ is the sheaf associated to $Umapsto H^i(p^{-1}(U),F)$. If $p$ is affine, then $H^i(p ^{-1}(U),F)$ is zero. So $R^ip_*F$ is the sheaf associated to a presheaf which vanishes on a basis. This implies that $R^ip_*F$ is zero.
– Roland
Dec 28 '18 at 23:55
You said several times "Obviously", but some of them are not obvious at all. Is $X$ defined over $K$ ? Otherwise, what is the action of $Gal(mathbb{C}/K)$ ? What is the action of $G$ on a coherent sheaf on $X/G$ ? Except from the subgroups {identity} and {identity, conjugation}, do you have other examples of finite subgroup of $Gal(mathbb{C}/K)$ ? (there are none from Artin-Schreier theorem). Finally, why do you think this has to do with Hilbert 90 ?
– Roland
Dec 27 '18 at 17:12
You said several times "Obviously", but some of them are not obvious at all. Is $X$ defined over $K$ ? Otherwise, what is the action of $Gal(mathbb{C}/K)$ ? What is the action of $G$ on a coherent sheaf on $X/G$ ? Except from the subgroups {identity} and {identity, conjugation}, do you have other examples of finite subgroup of $Gal(mathbb{C}/K)$ ? (there are none from Artin-Schreier theorem). Finally, why do you think this has to do with Hilbert 90 ?
– Roland
Dec 27 '18 at 17:12
@Roland: Yes, sorry, that was a not really sophisticated attempt to formulate the problem without explicitelly using the "Galois morphism" terminology because I'm till now quite unfamilar with it. Especially I don't see which connection the common Galois theory has with cases when the quotient $X/G$ is well defined as scheme/variery. I hope that it is now become a bit clearer.
– KarlPeter
Dec 28 '18 at 1:12
@Roland: Yes, sorry, that was a not really sophisticated attempt to formulate the problem without explicitelly using the "Galois morphism" terminology because I'm till now quite unfamilar with it. Especially I don't see which connection the common Galois theory has with cases when the quotient $X/G$ is well defined as scheme/variery. I hope that it is now become a bit clearer.
– KarlPeter
Dec 28 '18 at 1:12
Ok this question makes much more sense to me than your previous one. Note that the sequence after taking $p_*$ will be exact since under your hypotheses, $p$ is finite.
– Roland
Dec 28 '18 at 9:14
Ok this question makes much more sense to me than your previous one. Note that the sequence after taking $p_*$ will be exact since under your hypotheses, $p$ is finite.
– Roland
Dec 28 '18 at 9:14
@Roland: I think that it boils down to the statement that for every affine $f$ morphism the direct image $f_∗$ exact. Btw do you know a good reference for the proof of it /a sketch of the proof? I know a proof only for the case that $f$ is a closed immersion
– KarlPeter
Dec 28 '18 at 23:32
@Roland: I think that it boils down to the statement that for every affine $f$ morphism the direct image $f_∗$ exact. Btw do you know a good reference for the proof of it /a sketch of the proof? I know a proof only for the case that $f$ is a closed immersion
– KarlPeter
Dec 28 '18 at 23:32
1
1
Well, $R^ip_*F$ is the sheaf associated to $Umapsto H^i(p^{-1}(U),F)$. If $p$ is affine, then $H^i(p ^{-1}(U),F)$ is zero. So $R^ip_*F$ is the sheaf associated to a presheaf which vanishes on a basis. This implies that $R^ip_*F$ is zero.
– Roland
Dec 28 '18 at 23:55
Well, $R^ip_*F$ is the sheaf associated to $Umapsto H^i(p^{-1}(U),F)$. If $p$ is affine, then $H^i(p ^{-1}(U),F)$ is zero. So $R^ip_*F$ is the sheaf associated to a presheaf which vanishes on a basis. This implies that $R^ip_*F$ is zero.
– Roland
Dec 28 '18 at 23:55
|
show 4 more comments
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You said several times "Obviously", but some of them are not obvious at all. Is $X$ defined over $K$ ? Otherwise, what is the action of $Gal(mathbb{C}/K)$ ? What is the action of $G$ on a coherent sheaf on $X/G$ ? Except from the subgroups {identity} and {identity, conjugation}, do you have other examples of finite subgroup of $Gal(mathbb{C}/K)$ ? (there are none from Artin-Schreier theorem). Finally, why do you think this has to do with Hilbert 90 ?
– Roland
Dec 27 '18 at 17:12
@Roland: Yes, sorry, that was a not really sophisticated attempt to formulate the problem without explicitelly using the "Galois morphism" terminology because I'm till now quite unfamilar with it. Especially I don't see which connection the common Galois theory has with cases when the quotient $X/G$ is well defined as scheme/variery. I hope that it is now become a bit clearer.
– KarlPeter
Dec 28 '18 at 1:12
Ok this question makes much more sense to me than your previous one. Note that the sequence after taking $p_*$ will be exact since under your hypotheses, $p$ is finite.
– Roland
Dec 28 '18 at 9:14
@Roland: I think that it boils down to the statement that for every affine $f$ morphism the direct image $f_∗$ exact. Btw do you know a good reference for the proof of it /a sketch of the proof? I know a proof only for the case that $f$ is a closed immersion
– KarlPeter
Dec 28 '18 at 23:32
1
Well, $R^ip_*F$ is the sheaf associated to $Umapsto H^i(p^{-1}(U),F)$. If $p$ is affine, then $H^i(p ^{-1}(U),F)$ is zero. So $R^ip_*F$ is the sheaf associated to a presheaf which vanishes on a basis. This implies that $R^ip_*F$ is zero.
– Roland
Dec 28 '18 at 23:55