Chow groups with coefficients in a local system
$begingroup$
$newcommand{CH}{mathrm{CH}}
newcommand{F}{mathscr{F}}
$
Let $X$ be a smooth projective variety over a field $k$. Let $F$ be a local system on $X$, i.e. a locally constant sheaf (for Zariski topology).
How can we define the Chow groups of $X$ with coefficients in $F$ ?
Such a definition should satisfy the following conditions:
1) If $F$ is the constant sheaf $Bbb Z_X$ on $X$ (resp. $Bbb Q_X$), then we should find that $CH^j(X, F) = CH^j(X)$ is the usual Chow group (resp. $CH^j(X) otimes_{Bbb Z} Bbb Q$ is the Chow group taking equivalence classes of divisors with rational coefficients).
2) If $H^{bullet}$ is a Weil cohomology theory, we should get (functorial?) cycle class maps
$$CH^j(X, F) to H^{2j}(X, F).$$
I did not find any reference about such constructions.
Thank you for your advice.
algebraic-geometry reference-request sheaf-theory sheaf-cohomology local-systems
asked Nov 30 '18 at 10:35
WatsonWatson
15.8k92970
$endgroup$
add a comment |
$begingroup$
$newcommand{CH}{mathrm{CH}}
newcommand{F}{mathscr{F}}
$
Let $X$ be a smooth projective variety over a field $k$. Let $F$ be a local system on $X$, i.e. a locally constant sheaf (for Zariski topology).
How can we define the Chow groups of $X$ with coefficients in $F$ ?
Such a definition should satisfy the following conditions:
1) If $F$ is the constant sheaf $Bbb Z_X$ on $X$ (resp. $Bbb Q_X$), then we should find that $CH^j(X, F) = CH^j(X)$ is the usual Chow group (resp. $CH^j(X) otimes_{Bbb Z} Bbb Q$ is the Chow group taking equivalence classes of divisors with rational coefficients).
2) If $H^{bullet}$ is a Weil cohomology theory, we should get (functorial?) cycle class maps
$$CH^j(X, F) to H^{2j}(X, F).$$
I did not find any reference about such constructions.
Thank you for your advice.
algebraic-geometry reference-request sheaf-theory sheaf-cohomology local-systems
asked Nov 30 '18 at 10:35
WatsonWatson
15.8k92970
$endgroup$
1
$begingroup$
You can take the motivic cohomology $H^{2j}(X,mathcal{F}(j))$. But are you sure that there are locally constant sheaves for the Zariski topology ?
$endgroup$
– Roland
Nov 30 '18 at 11:10
$begingroup$
@Roland : thank you for your comment. I don't know motivic cohomology… But you may be right, I think that actually $F$ is a locally constant sheaf on $X(Bbb C)$ ($X$ being an algebraic variety over $k = Bbb Q$, say), for the analytic topology. What would happen in that case?
$endgroup$
– Watson
Nov 30 '18 at 11:20
$begingroup$
Well, you need to give it a "motivic" structure. I mean, if you want to get a cycle class map to étale cohomology, your local system needs to have some algebraic structure on it. Do you have a more precise question in mind ?
$endgroup$
– Roland
Nov 30 '18 at 11:26
$begingroup$
Dear @Roland, thank you for your comment. I see what you mean. Actually, my question arises from here, where « there is a local system $F$ on $Σ(G)$ over $E$ » and there is the « Chow group of $Σ(G)$ with coefficients in $F$ ». If I have time, I will add the notations to make the context clear.
$endgroup$
– Watson
Nov 30 '18 at 12:20
1
$begingroup$
By the way, Rost introduces the Chow group with coefficients here : math.uni-bielefeld.de/documenta/vol-01/16.pdf. I don't know if this is the same kind of coefficients as in your article.
$endgroup$
– Roland
Nov 30 '18 at 20:46
add a comment |
$begingroup$
$newcommand{CH}{mathrm{CH}}
newcommand{F}{mathscr{F}}
$
Let $X$ be a smooth projective variety over a field $k$. Let $F$ be a local system on $X$, i.e. a locally constant sheaf (for Zariski topology).
How can we define the Chow groups of $X$ with coefficients in $F$ ?
Such a definition should satisfy the following conditions:
1) If $F$ is the constant sheaf $Bbb Z_X$ on $X$ (resp. $Bbb Q_X$), then we should find that $CH^j(X, F) = CH^j(X)$ is the usual Chow group (resp. $CH^j(X) otimes_{Bbb Z} Bbb Q$ is the Chow group taking equivalence classes of divisors with rational coefficients).
2) If $H^{bullet}$ is a Weil cohomology theory, we should get (functorial?) cycle class maps
$$CH^j(X, F) to H^{2j}(X, F).$$
I did not find any reference about such constructions.
Thank you for your advice.
algebraic-geometry reference-request sheaf-theory sheaf-cohomology local-systems
asked Nov 30 '18 at 10:35
WatsonWatson
15.8k92970
$endgroup$
$newcommand{CH}{mathrm{CH}}
newcommand{F}{mathscr{F}}
$
Let $X$ be a smooth projective variety over a field $k$. Let $F$ be a local system on $X$, i.e. a locally constant sheaf (for Zariski topology).
How can we define the Chow groups of $X$ with coefficients in $F$ ?
Such a definition should satisfy the following conditions:
1) If $F$ is the constant sheaf $Bbb Z_X$ on $X$ (resp. $Bbb Q_X$), then we should find that $CH^j(X, F) = CH^j(X)$ is the usual Chow group (resp. $CH^j(X) otimes_{Bbb Z} Bbb Q$ is the Chow group taking equivalence classes of divisors with rational coefficients).
2) If $H^{bullet}$ is a Weil cohomology theory, we should get (functorial?) cycle class maps
$$CH^j(X, F) to H^{2j}(X, F).$$
I did not find any reference about such constructions.
Thank you for your advice.
algebraic-geometry reference-request sheaf-theory sheaf-cohomology local-systems
algebraic-geometry reference-request sheaf-theory sheaf-cohomology local-systems
asked Nov 30 '18 at 10:35
WatsonWatson
15.8k92970
asked Nov 30 '18 at 10:35
WatsonWatson
15.8k92970
asked Nov 30 '18 at 10:35
WatsonWatson
15.8k92970
asked Nov 30 '18 at 10:35
WatsonWatson
15.8k92970
asked Nov 30 '18 at 10:35
WatsonWatson
15.8k92970
15.8k92970
1
$begingroup$
You can take the motivic cohomology $H^{2j}(X,mathcal{F}(j))$. But are you sure that there are locally constant sheaves for the Zariski topology ?
$endgroup$
– Roland
Nov 30 '18 at 11:10
$begingroup$
@Roland : thank you for your comment. I don't know motivic cohomology… But you may be right, I think that actually $F$ is a locally constant sheaf on $X(Bbb C)$ ($X$ being an algebraic variety over $k = Bbb Q$, say), for the analytic topology. What would happen in that case?
$endgroup$
– Watson
Nov 30 '18 at 11:20
$begingroup$
Well, you need to give it a "motivic" structure. I mean, if you want to get a cycle class map to étale cohomology, your local system needs to have some algebraic structure on it. Do you have a more precise question in mind ?
$endgroup$
– Roland
Nov 30 '18 at 11:26
$begingroup$
Dear @Roland, thank you for your comment. I see what you mean. Actually, my question arises from here, where « there is a local system $F$ on $Σ(G)$ over $E$ » and there is the « Chow group of $Σ(G)$ with coefficients in $F$ ». If I have time, I will add the notations to make the context clear.
$endgroup$
– Watson
Nov 30 '18 at 12:20
1
$begingroup$
By the way, Rost introduces the Chow group with coefficients here : math.uni-bielefeld.de/documenta/vol-01/16.pdf. I don't know if this is the same kind of coefficients as in your article.
$endgroup$
– Roland
Nov 30 '18 at 20:46
add a comment |
1
$begingroup$
You can take the motivic cohomology $H^{2j}(X,mathcal{F}(j))$. But are you sure that there are locally constant sheaves for the Zariski topology ?
$endgroup$
– Roland
Nov 30 '18 at 11:10
$begingroup$
@Roland : thank you for your comment. I don't know motivic cohomology… But you may be right, I think that actually $F$ is a locally constant sheaf on $X(Bbb C)$ ($X$ being an algebraic variety over $k = Bbb Q$, say), for the analytic topology. What would happen in that case?
$endgroup$
– Watson
Nov 30 '18 at 11:20
$begingroup$
Well, you need to give it a "motivic" structure. I mean, if you want to get a cycle class map to étale cohomology, your local system needs to have some algebraic structure on it. Do you have a more precise question in mind ?
$endgroup$
– Roland
Nov 30 '18 at 11:26
$begingroup$
Dear @Roland, thank you for your comment. I see what you mean. Actually, my question arises from here, where « there is a local system $F$ on $Σ(G)$ over $E$ » and there is the « Chow group of $Σ(G)$ with coefficients in $F$ ». If I have time, I will add the notations to make the context clear.
$endgroup$
– Watson
Nov 30 '18 at 12:20
1
$begingroup$
By the way, Rost introduces the Chow group with coefficients here : math.uni-bielefeld.de/documenta/vol-01/16.pdf. I don't know if this is the same kind of coefficients as in your article.
$endgroup$
– Roland
Nov 30 '18 at 20:46
1
1
$begingroup$
You can take the motivic cohomology $H^{2j}(X,mathcal{F}(j))$. But are you sure that there are locally constant sheaves for the Zariski topology ?
$endgroup$
– Roland
Nov 30 '18 at 11:10
$begingroup$
You can take the motivic cohomology $H^{2j}(X,mathcal{F}(j))$. But are you sure that there are locally constant sheaves for the Zariski topology ?
$endgroup$
– Roland
Nov 30 '18 at 11:10
$begingroup$
@Roland : thank you for your comment. I don't know motivic cohomology… But you may be right, I think that actually $F$ is a locally constant sheaf on $X(Bbb C)$ ($X$ being an algebraic variety over $k = Bbb Q$, say), for the analytic topology. What would happen in that case?
$endgroup$
– Watson
Nov 30 '18 at 11:20
$begingroup$
@Roland : thank you for your comment. I don't know motivic cohomology… But you may be right, I think that actually $F$ is a locally constant sheaf on $X(Bbb C)$ ($X$ being an algebraic variety over $k = Bbb Q$, say), for the analytic topology. What would happen in that case?
$endgroup$
– Watson
Nov 30 '18 at 11:20
$begingroup$
Well, you need to give it a "motivic" structure. I mean, if you want to get a cycle class map to étale cohomology, your local system needs to have some algebraic structure on it. Do you have a more precise question in mind ?
$endgroup$
– Roland
Nov 30 '18 at 11:26
$begingroup$
Well, you need to give it a "motivic" structure. I mean, if you want to get a cycle class map to étale cohomology, your local system needs to have some algebraic structure on it. Do you have a more precise question in mind ?
$endgroup$
– Roland
Nov 30 '18 at 11:26
$begingroup$
Dear @Roland, thank you for your comment. I see what you mean. Actually, my question arises from here, where « there is a local system $F$ on $Σ(G)$ over $E$ » and there is the « Chow group of $Σ(G)$ with coefficients in $F$ ». If I have time, I will add the notations to make the context clear.
$endgroup$
– Watson
Nov 30 '18 at 12:20
$begingroup$
Dear @Roland, thank you for your comment. I see what you mean. Actually, my question arises from here, where « there is a local system $F$ on $Σ(G)$ over $E$ » and there is the « Chow group of $Σ(G)$ with coefficients in $F$ ». If I have time, I will add the notations to make the context clear.
$endgroup$
– Watson
Nov 30 '18 at 12:20
1
1
$begingroup$
By the way, Rost introduces the Chow group with coefficients here : math.uni-bielefeld.de/documenta/vol-01/16.pdf. I don't know if this is the same kind of coefficients as in your article.
$endgroup$
– Roland
Nov 30 '18 at 20:46
$begingroup$
By the way, Rost introduces the Chow group with coefficients here : math.uni-bielefeld.de/documenta/vol-01/16.pdf. I don't know if this is the same kind of coefficients as in your article.
$endgroup$
– Roland
Nov 30 '18 at 20:46
add a comment |
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$begingroup$
You can take the motivic cohomology $H^{2j}(X,mathcal{F}(j))$. But are you sure that there are locally constant sheaves for the Zariski topology ?
$endgroup$
– Roland
Nov 30 '18 at 11:10
$begingroup$
@Roland : thank you for your comment. I don't know motivic cohomology… But you may be right, I think that actually $F$ is a locally constant sheaf on $X(Bbb C)$ ($X$ being an algebraic variety over $k = Bbb Q$, say), for the analytic topology. What would happen in that case?
$endgroup$
– Watson
Nov 30 '18 at 11:20
$begingroup$
Well, you need to give it a "motivic" structure. I mean, if you want to get a cycle class map to étale cohomology, your local system needs to have some algebraic structure on it. Do you have a more precise question in mind ?
$endgroup$
– Roland
Nov 30 '18 at 11:26
$begingroup$
Dear @Roland, thank you for your comment. I see what you mean. Actually, my question arises from here, where « there is a local system $F$ on $Σ(G)$ over $E$ » and there is the « Chow group of $Σ(G)$ with coefficients in $F$ ». If I have time, I will add the notations to make the context clear.
$endgroup$
– Watson
Nov 30 '18 at 12:20
1
$begingroup$
By the way, Rost introduces the Chow group with coefficients here : math.uni-bielefeld.de/documenta/vol-01/16.pdf. I don't know if this is the same kind of coefficients as in your article.
$endgroup$
– Roland
Nov 30 '18 at 20:46