Generalized Behrend version for Grothendieck-Lefschetz trace formulaWhat are the automorphism groups of (principally polarized) abelian varieties?Behaviour of euler characteristics in characteristic p for finite etale coversIs there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula Further Questions Regarding Cohomolgoy Theory Of SheavesBase change for quotient stackRationality of zeta function and Grothendieck-Lefschetz fixed point formula, cohomology can be computed as the de Rham cohomologyWhat is the need for torsion in the definition of lisse sheaves?Universal homeomorphism of stacks and etale sitesmoduli stack of double covers of $mathbbP^1$ with one marked pointHigher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?
Generalized Behrend version for Grothendieck-Lefschetz trace formula
What are the automorphism groups of (principally polarized) abelian varieties?Behaviour of euler characteristics in characteristic p for finite etale coversIs there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula Further Questions Regarding Cohomolgoy Theory Of SheavesBase change for quotient stackRationality of zeta function and Grothendieck-Lefschetz fixed point formula, cohomology can be computed as the de Rham cohomologyWhat is the need for torsion in the definition of lisse sheaves?Universal homeomorphism of stacks and etale sitesmoduli stack of double covers of $mathbbP^1$ with one marked pointHigher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?
$begingroup$
[MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $mathbb F_q$,
$$#X(mathbb F_q) =sum_i (−1)^i Tr(Fr_X, H^i_c(X, mathbb Q_l)).$$
Also known is the version for general constructible l-adic sheaves $mathcal F$:
$$sum_xin X(mathbb F_q) Tr(Fr_x,mathcal F_x)=sum_i (−1)^i Tr(Fr_X, H^i_c(X, mathcal F)).$$
Thirdly, K. Behrend proved an analog for the first formula in the context of algebraic stacks (replacing the scheme $X$ by a Noetherian algebraic stack $mathcal X$).
Now my question is: is there a version of the second formula for an
algebraic stack $mathcal X$ (with nice hypotheses if necessary)?
It would seem natural, since the second formula is a generalization of the first, and the first is true in the context of algebraic stacks by Behrend's work. However, the second formula does not follow directly from the first in the case of schemes (as far as I know: I would be glad if it were true!), so I am not in the position to easily extend the proof of the second formula in the more general context of stacks.
Thank you in advance.
ag.algebraic-geometry etale-cohomology algebraic-stacks constructible-sheaves l-adic-sheaves
asked 8 hours ago
W. RetherW. Rether
165 bronze badges
$endgroup$
add a comment |
$begingroup$
[MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $mathbb F_q$,
$$#X(mathbb F_q) =sum_i (−1)^i Tr(Fr_X, H^i_c(X, mathbb Q_l)).$$
Also known is the version for general constructible l-adic sheaves $mathcal F$:
$$sum_xin X(mathbb F_q) Tr(Fr_x,mathcal F_x)=sum_i (−1)^i Tr(Fr_X, H^i_c(X, mathcal F)).$$
Thirdly, K. Behrend proved an analog for the first formula in the context of algebraic stacks (replacing the scheme $X$ by a Noetherian algebraic stack $mathcal X$).
Now my question is: is there a version of the second formula for an
algebraic stack $mathcal X$ (with nice hypotheses if necessary)?
It would seem natural, since the second formula is a generalization of the first, and the first is true in the context of algebraic stacks by Behrend's work. However, the second formula does not follow directly from the first in the case of schemes (as far as I know: I would be glad if it were true!), so I am not in the position to easily extend the proof of the second formula in the more general context of stacks.
Thank you in advance.
ag.algebraic-geometry etale-cohomology algebraic-stacks constructible-sheaves l-adic-sheaves
asked 8 hours ago
W. RetherW. Rether
165 bronze badges
$endgroup$
1
$begingroup$
The case of general constructible coefficients is not in Behrend's original 1993 paper as you say. But it's in his 2003 book "Derived $ell$-adic categories for algebraic stacks". (Also, you don't need $X$ proper in your formula.)
$endgroup$
– Dan Petersen
2 hours ago
add a comment |
$begingroup$
[MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $mathbb F_q$,
$$#X(mathbb F_q) =sum_i (−1)^i Tr(Fr_X, H^i_c(X, mathbb Q_l)).$$
Also known is the version for general constructible l-adic sheaves $mathcal F$:
$$sum_xin X(mathbb F_q) Tr(Fr_x,mathcal F_x)=sum_i (−1)^i Tr(Fr_X, H^i_c(X, mathcal F)).$$
Thirdly, K. Behrend proved an analog for the first formula in the context of algebraic stacks (replacing the scheme $X$ by a Noetherian algebraic stack $mathcal X$).
Now my question is: is there a version of the second formula for an
algebraic stack $mathcal X$ (with nice hypotheses if necessary)?
It would seem natural, since the second formula is a generalization of the first, and the first is true in the context of algebraic stacks by Behrend's work. However, the second formula does not follow directly from the first in the case of schemes (as far as I know: I would be glad if it were true!), so I am not in the position to easily extend the proof of the second formula in the more general context of stacks.
Thank you in advance.
ag.algebraic-geometry etale-cohomology algebraic-stacks constructible-sheaves l-adic-sheaves
asked 8 hours ago
W. RetherW. Rether
165 bronze badges
$endgroup$
[MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $mathbb F_q$,
$$#X(mathbb F_q) =sum_i (−1)^i Tr(Fr_X, H^i_c(X, mathbb Q_l)).$$
Also known is the version for general constructible l-adic sheaves $mathcal F$:
$$sum_xin X(mathbb F_q) Tr(Fr_x,mathcal F_x)=sum_i (−1)^i Tr(Fr_X, H^i_c(X, mathcal F)).$$
Thirdly, K. Behrend proved an analog for the first formula in the context of algebraic stacks (replacing the scheme $X$ by a Noetherian algebraic stack $mathcal X$).
Now my question is: is there a version of the second formula for an
algebraic stack $mathcal X$ (with nice hypotheses if necessary)?
It would seem natural, since the second formula is a generalization of the first, and the first is true in the context of algebraic stacks by Behrend's work. However, the second formula does not follow directly from the first in the case of schemes (as far as I know: I would be glad if it were true!), so I am not in the position to easily extend the proof of the second formula in the more general context of stacks.
Thank you in advance.
ag.algebraic-geometry etale-cohomology algebraic-stacks constructible-sheaves l-adic-sheaves
ag.algebraic-geometry etale-cohomology algebraic-stacks constructible-sheaves l-adic-sheaves
asked 8 hours ago
W. RetherW. Rether
165 bronze badges
asked 8 hours ago
W. RetherW. Rether
165 bronze badges
asked 8 hours ago
W. RetherW. Rether
165 bronze badges
asked 8 hours ago
W. RetherW. Rether
165 bronze badges
asked 8 hours ago
W. RetherW. Rether
165 bronze badges
165 bronze badges
1
$begingroup$
The case of general constructible coefficients is not in Behrend's original 1993 paper as you say. But it's in his 2003 book "Derived $ell$-adic categories for algebraic stacks". (Also, you don't need $X$ proper in your formula.)
$endgroup$
– Dan Petersen
2 hours ago
add a comment |
1
$begingroup$
The case of general constructible coefficients is not in Behrend's original 1993 paper as you say. But it's in his 2003 book "Derived $ell$-adic categories for algebraic stacks". (Also, you don't need $X$ proper in your formula.)
$endgroup$
– Dan Petersen
2 hours ago
1
1
$begingroup$
The case of general constructible coefficients is not in Behrend's original 1993 paper as you say. But it's in his 2003 book "Derived $ell$-adic categories for algebraic stacks". (Also, you don't need $X$ proper in your formula.)
$endgroup$
– Dan Petersen
2 hours ago
$begingroup$
The case of general constructible coefficients is not in Behrend's original 1993 paper as you say. But it's in his 2003 book "Derived $ell$-adic categories for algebraic stacks". (Also, you don't need $X$ proper in your formula.)
$endgroup$
– Dan Petersen
2 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
This is Theorem 4.2 of Shenghao Sun's paper L-Series of Artin stacks over finite fields (https://arxiv.org/pdf/1008.3689.pdf).
Let $f:mathscr X_0tomathscr Y_0$ be a morphism of $mathbb F_q$-algebraic stacks, and let $K_0in W^-,stra_m(mathscr X_0,overlinemathbb Q_ell)$ be a convergent complex of sheaves.
Then
(i) (Finiteness) $f_!K_0$ is a convergent complex of sheaves on
$mathscr Y_0,$ and
(ii) (Trace formula) $c_v(mathscr X_0,K_0)=c_v(mathscr Y_0,f_!K_0)$
for every integer $vge1.$
Here $c_v$ is the sum of the trace of the $v$th power of Frobenius. Applying this in the case where $mathscr Y_0$ is a point gets you what you want.
answered 8 hours ago
Will SawinWill Sawin
71.1k7 gold badges143 silver badges294 bronze badges
$endgroup$
add a comment |
$begingroup$
To amplify on Will's answer, the published version of the paper he mentions is
https://projecteuclid.org/euclid.ant/1513729758
answered 8 hours ago
Denis Chaperon de LauzièresDenis Chaperon de Lauzières
511 bronze badge
New contributor
Denis Chaperon de Lauzières is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
hmm 3 upvotes for something that is really a comment. I await Lauziere's first question about arcane topological properties of morphism's of schemes.
$endgroup$
– aginensky
6 hours ago
add a comment |
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2 Answers
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2 Answers
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votes
$begingroup$
This is Theorem 4.2 of Shenghao Sun's paper L-Series of Artin stacks over finite fields (https://arxiv.org/pdf/1008.3689.pdf).
Let $f:mathscr X_0tomathscr Y_0$ be a morphism of $mathbb F_q$-algebraic stacks, and let $K_0in W^-,stra_m(mathscr X_0,overlinemathbb Q_ell)$ be a convergent complex of sheaves.
Then
(i) (Finiteness) $f_!K_0$ is a convergent complex of sheaves on
$mathscr Y_0,$ and
(ii) (Trace formula) $c_v(mathscr X_0,K_0)=c_v(mathscr Y_0,f_!K_0)$
for every integer $vge1.$
Here $c_v$ is the sum of the trace of the $v$th power of Frobenius. Applying this in the case where $mathscr Y_0$ is a point gets you what you want.
answered 8 hours ago
Will SawinWill Sawin
71.1k7 gold badges143 silver badges294 bronze badges
$endgroup$
add a comment |
$begingroup$
This is Theorem 4.2 of Shenghao Sun's paper L-Series of Artin stacks over finite fields (https://arxiv.org/pdf/1008.3689.pdf).
Let $f:mathscr X_0tomathscr Y_0$ be a morphism of $mathbb F_q$-algebraic stacks, and let $K_0in W^-,stra_m(mathscr X_0,overlinemathbb Q_ell)$ be a convergent complex of sheaves.
Then
(i) (Finiteness) $f_!K_0$ is a convergent complex of sheaves on
$mathscr Y_0,$ and
(ii) (Trace formula) $c_v(mathscr X_0,K_0)=c_v(mathscr Y_0,f_!K_0)$
for every integer $vge1.$
Here $c_v$ is the sum of the trace of the $v$th power of Frobenius. Applying this in the case where $mathscr Y_0$ is a point gets you what you want.
answered 8 hours ago
Will SawinWill Sawin
71.1k7 gold badges143 silver badges294 bronze badges
$endgroup$
add a comment |
$begingroup$
This is Theorem 4.2 of Shenghao Sun's paper L-Series of Artin stacks over finite fields (https://arxiv.org/pdf/1008.3689.pdf).
Let $f:mathscr X_0tomathscr Y_0$ be a morphism of $mathbb F_q$-algebraic stacks, and let $K_0in W^-,stra_m(mathscr X_0,overlinemathbb Q_ell)$ be a convergent complex of sheaves.
Then
(i) (Finiteness) $f_!K_0$ is a convergent complex of sheaves on
$mathscr Y_0,$ and
(ii) (Trace formula) $c_v(mathscr X_0,K_0)=c_v(mathscr Y_0,f_!K_0)$
for every integer $vge1.$
Here $c_v$ is the sum of the trace of the $v$th power of Frobenius. Applying this in the case where $mathscr Y_0$ is a point gets you what you want.
answered 8 hours ago
Will SawinWill Sawin
71.1k7 gold badges143 silver badges294 bronze badges
$endgroup$
This is Theorem 4.2 of Shenghao Sun's paper L-Series of Artin stacks over finite fields (https://arxiv.org/pdf/1008.3689.pdf).
Let $f:mathscr X_0tomathscr Y_0$ be a morphism of $mathbb F_q$-algebraic stacks, and let $K_0in W^-,stra_m(mathscr X_0,overlinemathbb Q_ell)$ be a convergent complex of sheaves.
Then
(i) (Finiteness) $f_!K_0$ is a convergent complex of sheaves on
$mathscr Y_0,$ and
(ii) (Trace formula) $c_v(mathscr X_0,K_0)=c_v(mathscr Y_0,f_!K_0)$
for every integer $vge1.$
Here $c_v$ is the sum of the trace of the $v$th power of Frobenius. Applying this in the case where $mathscr Y_0$ is a point gets you what you want.
answered 8 hours ago
Will SawinWill Sawin
71.1k7 gold badges143 silver badges294 bronze badges
answered 8 hours ago
Will SawinWill Sawin
71.1k7 gold badges143 silver badges294 bronze badges
answered 8 hours ago
Will SawinWill Sawin
71.1k7 gold badges143 silver badges294 bronze badges
answered 8 hours ago
Will SawinWill Sawin
71.1k7 gold badges143 silver badges294 bronze badges
71.1k7 gold badges143 silver badges294 bronze badges
add a comment |
add a comment |
$begingroup$
To amplify on Will's answer, the published version of the paper he mentions is
https://projecteuclid.org/euclid.ant/1513729758
answered 8 hours ago
Denis Chaperon de LauzièresDenis Chaperon de Lauzières
511 bronze badge
New contributor
Denis Chaperon de Lauzières is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
hmm 3 upvotes for something that is really a comment. I await Lauziere's first question about arcane topological properties of morphism's of schemes.
$endgroup$
– aginensky
6 hours ago
add a comment |
$begingroup$
To amplify on Will's answer, the published version of the paper he mentions is
https://projecteuclid.org/euclid.ant/1513729758
answered 8 hours ago
Denis Chaperon de LauzièresDenis Chaperon de Lauzières
511 bronze badge
New contributor
Denis Chaperon de Lauzières is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
hmm 3 upvotes for something that is really a comment. I await Lauziere's first question about arcane topological properties of morphism's of schemes.
$endgroup$
– aginensky
6 hours ago
add a comment |
$begingroup$
To amplify on Will's answer, the published version of the paper he mentions is
https://projecteuclid.org/euclid.ant/1513729758
answered 8 hours ago
Denis Chaperon de LauzièresDenis Chaperon de Lauzières
511 bronze badge
New contributor
Denis Chaperon de Lauzières is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
To amplify on Will's answer, the published version of the paper he mentions is
https://projecteuclid.org/euclid.ant/1513729758
answered 8 hours ago
Denis Chaperon de LauzièresDenis Chaperon de Lauzières
511 bronze badge
New contributor
Denis Chaperon de Lauzières is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 8 hours ago
Denis Chaperon de LauzièresDenis Chaperon de Lauzières
511 bronze badge
New contributor
Denis Chaperon de Lauzières is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 8 hours ago
Denis Chaperon de LauzièresDenis Chaperon de Lauzières
511 bronze badge
answered 8 hours ago
Denis Chaperon de LauzièresDenis Chaperon de Lauzières
511 bronze badge
511 bronze badge
New contributor
Denis Chaperon de Lauzières is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Denis Chaperon de Lauzières is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
hmm 3 upvotes for something that is really a comment. I await Lauziere's first question about arcane topological properties of morphism's of schemes.
$endgroup$
– aginensky
6 hours ago
add a comment |
$begingroup$
hmm 3 upvotes for something that is really a comment. I await Lauziere's first question about arcane topological properties of morphism's of schemes.
$endgroup$
– aginensky
6 hours ago
$begingroup$
hmm 3 upvotes for something that is really a comment. I await Lauziere's first question about arcane topological properties of morphism's of schemes.
$endgroup$
– aginensky
6 hours ago
$begingroup$
hmm 3 upvotes for something that is really a comment. I await Lauziere's first question about arcane topological properties of morphism's of schemes.
$endgroup$
– aginensky
6 hours ago
add a comment |
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$begingroup$
The case of general constructible coefficients is not in Behrend's original 1993 paper as you say. But it's in his 2003 book "Derived $ell$-adic categories for algebraic stacks". (Also, you don't need $X$ proper in your formula.)
$endgroup$
– Dan Petersen
2 hours ago