Equivariant SheafSome questions on the basics of invertible sheavesRelative spectrum of a quasi-coherent algebra.For an algebraic group $G$ acting on a scheme $X$, $H^0(X,F)$ of a $G$-linearized $mathcalO_X$-module $F$ has the structure of a $G$-representationMorphism $XtomathsfR_T/S(X)$ to a Weil restrictionUnderstanding the pullback of a sheaf of differentialsDefinition of Geometric Quotient, some questionsWhat is the definition of an equivariant connection$H(G') otimes_H(G) V$ and $f^-1 mathcal G otimes_f^-1mathcal O_Y mathcal O_X$, the connectionOn morphisms of $mathcal O_X $ modulesHow can we understand a group-scheme action as a natural transformation
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Equivariant Sheaf
Some questions on the basics of invertible sheavesRelative spectrum of a quasi-coherent algebra.For an algebraic group $G$ acting on a scheme $X$, $H^0(X,F)$ of a $G$-linearized $mathcalO_X$-module $F$ has the structure of a $G$-representationMorphism $XtomathsfR_T/S(X)$ to a Weil restrictionUnderstanding the pullback of a sheaf of differentialsDefinition of Geometric Quotient, some questionsWhat is the definition of an equivariant connection$H(G') otimes_H(G) V$ and $f^-1 mathcal G otimes_f^-1mathcal O_Y mathcal O_X$, the connectionOn morphisms of $mathcal O_X $ modulesHow can we understand a group-scheme action as a natural transformation
$begingroup$
This question concerns the meaning of the data defining a so called equivariant sheaf $F$ on a scheme X. Let denote by $sigma: G times_S X to X$ an action of a group scheme $G$ on $X$ . Then a $O_X$-module $F$ is called equivariant if there exist in isomorphism $phi: sigma^* F simeq p_2^*F$ of $mathcalO_G times_S X$ and additionally the "cocycle" condition $p_23^* phi circ (1_G times sigma)^* phi = (m times 1_X)^* phi$ is satisfied where $p_23, 1_G times sigma, m times 1_X$ a maps between $G times G times X$ and $G times X$.
My simple question is what is a pullback of a sheaf morphsim as occuring in the cocycle condition concretely? Namely when $phi: F to G$ is a morphism on sheaves on $X$ and $f: Y to X$ is a morphism of schemes how is the pullback $f^*phi$ defined?
Is it established by following comutative diagram?
$$
requireAMScd
beginCD
F @>phi >> G \
@VVV @VVV \
f^*F @>f^*phi>>f^*G
endCD
$$
what are the vertical maps? locally $f^*F$ has the shape $O_Y otimes_f^-1O_Xf^-1F$. what bothers me is how can I obtain the arrow $F to O_Y otimes_f^-1O_Xf^-1F$? as a side note: is the approach by the diagram above correct at all?
algebraic-geometry sheaf-theory
asked Mar 21 at 15:11
KarlPeterKarlPeter
6311316
$endgroup$
add a comment |
$begingroup$
This question concerns the meaning of the data defining a so called equivariant sheaf $F$ on a scheme X. Let denote by $sigma: G times_S X to X$ an action of a group scheme $G$ on $X$ . Then a $O_X$-module $F$ is called equivariant if there exist in isomorphism $phi: sigma^* F simeq p_2^*F$ of $mathcalO_G times_S X$ and additionally the "cocycle" condition $p_23^* phi circ (1_G times sigma)^* phi = (m times 1_X)^* phi$ is satisfied where $p_23, 1_G times sigma, m times 1_X$ a maps between $G times G times X$ and $G times X$.
My simple question is what is a pullback of a sheaf morphsim as occuring in the cocycle condition concretely? Namely when $phi: F to G$ is a morphism on sheaves on $X$ and $f: Y to X$ is a morphism of schemes how is the pullback $f^*phi$ defined?
Is it established by following comutative diagram?
$$
requireAMScd
beginCD
F @>phi >> G \
@VVV @VVV \
f^*F @>f^*phi>>f^*G
endCD
$$
what are the vertical maps? locally $f^*F$ has the shape $O_Y otimes_f^-1O_Xf^-1F$. what bothers me is how can I obtain the arrow $F to O_Y otimes_f^-1O_Xf^-1F$? as a side note: is the approach by the diagram above correct at all?
algebraic-geometry sheaf-theory
asked Mar 21 at 15:11
KarlPeterKarlPeter
6311316
$endgroup$
add a comment |
$begingroup$
This question concerns the meaning of the data defining a so called equivariant sheaf $F$ on a scheme X. Let denote by $sigma: G times_S X to X$ an action of a group scheme $G$ on $X$ . Then a $O_X$-module $F$ is called equivariant if there exist in isomorphism $phi: sigma^* F simeq p_2^*F$ of $mathcalO_G times_S X$ and additionally the "cocycle" condition $p_23^* phi circ (1_G times sigma)^* phi = (m times 1_X)^* phi$ is satisfied where $p_23, 1_G times sigma, m times 1_X$ a maps between $G times G times X$ and $G times X$.
My simple question is what is a pullback of a sheaf morphsim as occuring in the cocycle condition concretely? Namely when $phi: F to G$ is a morphism on sheaves on $X$ and $f: Y to X$ is a morphism of schemes how is the pullback $f^*phi$ defined?
Is it established by following comutative diagram?
$$
requireAMScd
beginCD
F @>phi >> G \
@VVV @VVV \
f^*F @>f^*phi>>f^*G
endCD
$$
what are the vertical maps? locally $f^*F$ has the shape $O_Y otimes_f^-1O_Xf^-1F$. what bothers me is how can I obtain the arrow $F to O_Y otimes_f^-1O_Xf^-1F$? as a side note: is the approach by the diagram above correct at all?
algebraic-geometry sheaf-theory
asked Mar 21 at 15:11
KarlPeterKarlPeter
6311316
$endgroup$
This question concerns the meaning of the data defining a so called equivariant sheaf $F$ on a scheme X. Let denote by $sigma: G times_S X to X$ an action of a group scheme $G$ on $X$ . Then a $O_X$-module $F$ is called equivariant if there exist in isomorphism $phi: sigma^* F simeq p_2^*F$ of $mathcalO_G times_S X$ and additionally the "cocycle" condition $p_23^* phi circ (1_G times sigma)^* phi = (m times 1_X)^* phi$ is satisfied where $p_23, 1_G times sigma, m times 1_X$ a maps between $G times G times X$ and $G times X$.
My simple question is what is a pullback of a sheaf morphsim as occuring in the cocycle condition concretely? Namely when $phi: F to G$ is a morphism on sheaves on $X$ and $f: Y to X$ is a morphism of schemes how is the pullback $f^*phi$ defined?
Is it established by following comutative diagram?
$$
requireAMScd
beginCD
F @>phi >> G \
@VVV @VVV \
f^*F @>f^*phi>>f^*G
endCD
$$
what are the vertical maps? locally $f^*F$ has the shape $O_Y otimes_f^-1O_Xf^-1F$. what bothers me is how can I obtain the arrow $F to O_Y otimes_f^-1O_Xf^-1F$? as a side note: is the approach by the diagram above correct at all?
algebraic-geometry sheaf-theory
algebraic-geometry sheaf-theory
asked Mar 21 at 15:11
KarlPeterKarlPeter
6311316
asked Mar 21 at 15:11
KarlPeterKarlPeter
6311316
asked Mar 21 at 15:11
KarlPeterKarlPeter
6311316
asked Mar 21 at 15:11
KarlPeterKarlPeter
6311316
asked Mar 21 at 15:11
KarlPeterKarlPeter
6311316
6311316
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The existence of the morphism $f^*phi$ follows from functoriality of $f^-1$. Once you have a morphism $f^-1phi$, you can tensor with $mathcalO_Y$ to get $f^*phi$.
Also note that $f^*mathcalF$ is exactly $O_Y otimes_f^-1O_Xf^-1F$ (not only locally).
Thus it doesn't really make sense to speak of a morphism of sheaves $mathcalFto f^*mathcalF$ as the former is a sheaf on $X$ while the latter is a sheaf on $ Y$.
But there is a natural morphism $mathcalFto f_*f^*mathcalF$ (which is the unit of the adjunction of $f^*$ and $f_*$).
answered Mar 21 at 21:10
NotoneNotone
8181413
$endgroup$
$begingroup$
yes your answer definitely answers my question but besides I have another question concerning the unit map $u: mathcalFto f_*f^*mathcalF$ (resp the corresponding counit map $c: mathcalG to f_*f^*mathcalG$) you have mentioned in the answer. Do you know a nice reference where is rigorously treated the question when $u$ or $c$ are isomorphisms? I know some well known sufficient criterions like for $c$ (when $f$ being an immersion) or for $u$ (when $f$ being projective with connected fibers). But these two exampled criterions seem to be a bit "scattered".
$endgroup$
– KarlPeter
Mar 21 at 22:12
$begingroup$
Is there a more deeper/ more intuitive approach to understand when $c$ or $u$ are isomorphisms? Is there maybe a geometric meaning in some simple cases? For example when we consider $F,G$ as quasi coherent over affine schemes? Then this reduces to statements about modules. Do we in these cases have a deeper understanding when $c,u$ are isomorphisms?
$endgroup$
– KarlPeter
Mar 21 at 22:12
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
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active
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votes
$begingroup$
The existence of the morphism $f^*phi$ follows from functoriality of $f^-1$. Once you have a morphism $f^-1phi$, you can tensor with $mathcalO_Y$ to get $f^*phi$.
Also note that $f^*mathcalF$ is exactly $O_Y otimes_f^-1O_Xf^-1F$ (not only locally).
Thus it doesn't really make sense to speak of a morphism of sheaves $mathcalFto f^*mathcalF$ as the former is a sheaf on $X$ while the latter is a sheaf on $ Y$.
But there is a natural morphism $mathcalFto f_*f^*mathcalF$ (which is the unit of the adjunction of $f^*$ and $f_*$).
answered Mar 21 at 21:10
NotoneNotone
8181413
$endgroup$
$begingroup$
yes your answer definitely answers my question but besides I have another question concerning the unit map $u: mathcalFto f_*f^*mathcalF$ (resp the corresponding counit map $c: mathcalG to f_*f^*mathcalG$) you have mentioned in the answer. Do you know a nice reference where is rigorously treated the question when $u$ or $c$ are isomorphisms? I know some well known sufficient criterions like for $c$ (when $f$ being an immersion) or for $u$ (when $f$ being projective with connected fibers). But these two exampled criterions seem to be a bit "scattered".
$endgroup$
– KarlPeter
Mar 21 at 22:12
$begingroup$
Is there a more deeper/ more intuitive approach to understand when $c$ or $u$ are isomorphisms? Is there maybe a geometric meaning in some simple cases? For example when we consider $F,G$ as quasi coherent over affine schemes? Then this reduces to statements about modules. Do we in these cases have a deeper understanding when $c,u$ are isomorphisms?
$endgroup$
– KarlPeter
Mar 21 at 22:12
add a comment |
$begingroup$
The existence of the morphism $f^*phi$ follows from functoriality of $f^-1$. Once you have a morphism $f^-1phi$, you can tensor with $mathcalO_Y$ to get $f^*phi$.
Also note that $f^*mathcalF$ is exactly $O_Y otimes_f^-1O_Xf^-1F$ (not only locally).
Thus it doesn't really make sense to speak of a morphism of sheaves $mathcalFto f^*mathcalF$ as the former is a sheaf on $X$ while the latter is a sheaf on $ Y$.
But there is a natural morphism $mathcalFto f_*f^*mathcalF$ (which is the unit of the adjunction of $f^*$ and $f_*$).
answered Mar 21 at 21:10
NotoneNotone
8181413
$endgroup$
$begingroup$
yes your answer definitely answers my question but besides I have another question concerning the unit map $u: mathcalFto f_*f^*mathcalF$ (resp the corresponding counit map $c: mathcalG to f_*f^*mathcalG$) you have mentioned in the answer. Do you know a nice reference where is rigorously treated the question when $u$ or $c$ are isomorphisms? I know some well known sufficient criterions like for $c$ (when $f$ being an immersion) or for $u$ (when $f$ being projective with connected fibers). But these two exampled criterions seem to be a bit "scattered".
$endgroup$
– KarlPeter
Mar 21 at 22:12
$begingroup$
Is there a more deeper/ more intuitive approach to understand when $c$ or $u$ are isomorphisms? Is there maybe a geometric meaning in some simple cases? For example when we consider $F,G$ as quasi coherent over affine schemes? Then this reduces to statements about modules. Do we in these cases have a deeper understanding when $c,u$ are isomorphisms?
$endgroup$
– KarlPeter
Mar 21 at 22:12
add a comment |
$begingroup$
The existence of the morphism $f^*phi$ follows from functoriality of $f^-1$. Once you have a morphism $f^-1phi$, you can tensor with $mathcalO_Y$ to get $f^*phi$.
Also note that $f^*mathcalF$ is exactly $O_Y otimes_f^-1O_Xf^-1F$ (not only locally).
Thus it doesn't really make sense to speak of a morphism of sheaves $mathcalFto f^*mathcalF$ as the former is a sheaf on $X$ while the latter is a sheaf on $ Y$.
But there is a natural morphism $mathcalFto f_*f^*mathcalF$ (which is the unit of the adjunction of $f^*$ and $f_*$).
answered Mar 21 at 21:10
NotoneNotone
8181413
$endgroup$
The existence of the morphism $f^*phi$ follows from functoriality of $f^-1$. Once you have a morphism $f^-1phi$, you can tensor with $mathcalO_Y$ to get $f^*phi$.
Also note that $f^*mathcalF$ is exactly $O_Y otimes_f^-1O_Xf^-1F$ (not only locally).
Thus it doesn't really make sense to speak of a morphism of sheaves $mathcalFto f^*mathcalF$ as the former is a sheaf on $X$ while the latter is a sheaf on $ Y$.
But there is a natural morphism $mathcalFto f_*f^*mathcalF$ (which is the unit of the adjunction of $f^*$ and $f_*$).
answered Mar 21 at 21:10
NotoneNotone
8181413
answered Mar 21 at 21:10
NotoneNotone
8181413
answered Mar 21 at 21:10
NotoneNotone
8181413
answered Mar 21 at 21:10
NotoneNotone
8181413
8181413
$begingroup$
yes your answer definitely answers my question but besides I have another question concerning the unit map $u: mathcalFto f_*f^*mathcalF$ (resp the corresponding counit map $c: mathcalG to f_*f^*mathcalG$) you have mentioned in the answer. Do you know a nice reference where is rigorously treated the question when $u$ or $c$ are isomorphisms? I know some well known sufficient criterions like for $c$ (when $f$ being an immersion) or for $u$ (when $f$ being projective with connected fibers). But these two exampled criterions seem to be a bit "scattered".
$endgroup$
– KarlPeter
Mar 21 at 22:12
$begingroup$
Is there a more deeper/ more intuitive approach to understand when $c$ or $u$ are isomorphisms? Is there maybe a geometric meaning in some simple cases? For example when we consider $F,G$ as quasi coherent over affine schemes? Then this reduces to statements about modules. Do we in these cases have a deeper understanding when $c,u$ are isomorphisms?
$endgroup$
– KarlPeter
Mar 21 at 22:12
add a comment |
$begingroup$
yes your answer definitely answers my question but besides I have another question concerning the unit map $u: mathcalFto f_*f^*mathcalF$ (resp the corresponding counit map $c: mathcalG to f_*f^*mathcalG$) you have mentioned in the answer. Do you know a nice reference where is rigorously treated the question when $u$ or $c$ are isomorphisms? I know some well known sufficient criterions like for $c$ (when $f$ being an immersion) or for $u$ (when $f$ being projective with connected fibers). But these two exampled criterions seem to be a bit "scattered".
$endgroup$
– KarlPeter
Mar 21 at 22:12
$begingroup$
Is there a more deeper/ more intuitive approach to understand when $c$ or $u$ are isomorphisms? Is there maybe a geometric meaning in some simple cases? For example when we consider $F,G$ as quasi coherent over affine schemes? Then this reduces to statements about modules. Do we in these cases have a deeper understanding when $c,u$ are isomorphisms?
$endgroup$
– KarlPeter
Mar 21 at 22:12
$begingroup$
yes your answer definitely answers my question but besides I have another question concerning the unit map $u: mathcalFto f_*f^*mathcalF$ (resp the corresponding counit map $c: mathcalG to f_*f^*mathcalG$) you have mentioned in the answer. Do you know a nice reference where is rigorously treated the question when $u$ or $c$ are isomorphisms? I know some well known sufficient criterions like for $c$ (when $f$ being an immersion) or for $u$ (when $f$ being projective with connected fibers). But these two exampled criterions seem to be a bit "scattered".
$endgroup$
– KarlPeter
Mar 21 at 22:12
$begingroup$
yes your answer definitely answers my question but besides I have another question concerning the unit map $u: mathcalFto f_*f^*mathcalF$ (resp the corresponding counit map $c: mathcalG to f_*f^*mathcalG$) you have mentioned in the answer. Do you know a nice reference where is rigorously treated the question when $u$ or $c$ are isomorphisms? I know some well known sufficient criterions like for $c$ (when $f$ being an immersion) or for $u$ (when $f$ being projective with connected fibers). But these two exampled criterions seem to be a bit "scattered".
$endgroup$
– KarlPeter
Mar 21 at 22:12
$begingroup$
Is there a more deeper/ more intuitive approach to understand when $c$ or $u$ are isomorphisms? Is there maybe a geometric meaning in some simple cases? For example when we consider $F,G$ as quasi coherent over affine schemes? Then this reduces to statements about modules. Do we in these cases have a deeper understanding when $c,u$ are isomorphisms?
$endgroup$
– KarlPeter
Mar 21 at 22:12
$begingroup$
Is there a more deeper/ more intuitive approach to understand when $c$ or $u$ are isomorphisms? Is there maybe a geometric meaning in some simple cases? For example when we consider $F,G$ as quasi coherent over affine schemes? Then this reduces to statements about modules. Do we in these cases have a deeper understanding when $c,u$ are isomorphisms?
$endgroup$
– KarlPeter
Mar 21 at 22:12
add a comment |
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