Mixing
Perversity and minimal extension functor
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Let $X$ be a stratified complex algebraic variety with smooth strata $U$, its inclusion is $j$ and $L$ is a local system on $U$. All the functors are derived if necessary.
Question 1 : Is it true that $j_*L[d]$ is perverse ?
What I tried : I can prove it when $D := X backslash U$ is a normal crossing divisor but I'm not sure about the general case.
Question 2 : What is the simplest example where $^pj_{!*}$ is different from $^pj_!$ and $^pj_*$ ?
What I tried : If $X = (Bbb A^1,0)$ , then $j_!L[1] = j_{!*}L[1]$ is just the extension by zero of $L$ in degree $-1$ and there is a short exact sequence $ 0 to j_!L[1] to j_*L[1] to W to 0$ where $W$ is a skyscraper sheaf supported on the origin.
algebraic-geometry sheaf-theory sheaf-cohomology
asked yesterday
student
1008
add a comment |
up vote
2
down vote
favorite
Let $X$ be a stratified complex algebraic variety with smooth strata $U$, its inclusion is $j$ and $L$ is a local system on $U$. All the functors are derived if necessary.
Question 1 : Is it true that $j_*L[d]$ is perverse ?
What I tried : I can prove it when $D := X backslash U$ is a normal crossing divisor but I'm not sure about the general case.
Question 2 : What is the simplest example where $^pj_{!*}$ is different from $^pj_!$ and $^pj_*$ ?
What I tried : If $X = (Bbb A^1,0)$ , then $j_!L[1] = j_{!*}L[1]$ is just the extension by zero of $L$ in degree $-1$ and there is a short exact sequence $ 0 to j_!L[1] to j_*L[1] to W to 0$ where $W$ is a skyscraper sheaf supported on the origin.
algebraic-geometry sheaf-theory sheaf-cohomology
asked yesterday
student
1008
Q1. If $p$ is the top perversity function, then $Rj_*L[d]$ is a $p-$perverse sheaf. Dually, if $p$ is zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following $j : U rightarrow X$ be an open affine immersion, then $Rj_*$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
– random123
16 hours ago
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(unirioja.es/cu/luhernan/jornadasmchagt/files/…) has the calculation I mentioned on page 4 or 5.
– random123
16 hours ago
@random123 : thanks for the really helfpul comments, maybe you want to make an answer ?
– student
14 hours ago
I am not sure if I have anything more to add to it.
– random123
14 hours ago
@random123 : I meant you could copy what you said in the answer box so I can upvote and accept the answer. If you prefer it like this, that's perfect too and thanks again for the help.
– student
14 hours ago
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $X$ be a stratified complex algebraic variety with smooth strata $U$, its inclusion is $j$ and $L$ is a local system on $U$. All the functors are derived if necessary.
Question 1 : Is it true that $j_*L[d]$ is perverse ?
What I tried : I can prove it when $D := X backslash U$ is a normal crossing divisor but I'm not sure about the general case.
Question 2 : What is the simplest example where $^pj_{!*}$ is different from $^pj_!$ and $^pj_*$ ?
What I tried : If $X = (Bbb A^1,0)$ , then $j_!L[1] = j_{!*}L[1]$ is just the extension by zero of $L$ in degree $-1$ and there is a short exact sequence $ 0 to j_!L[1] to j_*L[1] to W to 0$ where $W$ is a skyscraper sheaf supported on the origin.
algebraic-geometry sheaf-theory sheaf-cohomology
asked yesterday
student
1008
Let $X$ be a stratified complex algebraic variety with smooth strata $U$, its inclusion is $j$ and $L$ is a local system on $U$. All the functors are derived if necessary.
Question 1 : Is it true that $j_*L[d]$ is perverse ?
What I tried : I can prove it when $D := X backslash U$ is a normal crossing divisor but I'm not sure about the general case.
Question 2 : What is the simplest example where $^pj_{!*}$ is different from $^pj_!$ and $^pj_*$ ?
What I tried : If $X = (Bbb A^1,0)$ , then $j_!L[1] = j_{!*}L[1]$ is just the extension by zero of $L$ in degree $-1$ and there is a short exact sequence $ 0 to j_!L[1] to j_*L[1] to W to 0$ where $W$ is a skyscraper sheaf supported on the origin.
algebraic-geometry sheaf-theory sheaf-cohomology
algebraic-geometry sheaf-theory sheaf-cohomology
asked yesterday
student
1008
asked yesterday
student
1008
asked yesterday
student
1008
asked yesterday
student
1008
asked yesterday
student
1008
1008
Q1. If $p$ is the top perversity function, then $Rj_*L[d]$ is a $p-$perverse sheaf. Dually, if $p$ is zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following $j : U rightarrow X$ be an open affine immersion, then $Rj_*$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
– random123
16 hours ago
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(unirioja.es/cu/luhernan/jornadasmchagt/files/…) has the calculation I mentioned on page 4 or 5.
– random123
16 hours ago
@random123 : thanks for the really helfpul comments, maybe you want to make an answer ?
– student
14 hours ago
I am not sure if I have anything more to add to it.
– random123
14 hours ago
@random123 : I meant you could copy what you said in the answer box so I can upvote and accept the answer. If you prefer it like this, that's perfect too and thanks again for the help.
– student
14 hours ago
add a comment |
Q1. If $p$ is the top perversity function, then $Rj_*L[d]$ is a $p-$perverse sheaf. Dually, if $p$ is zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following $j : U rightarrow X$ be an open affine immersion, then $Rj_*$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
– random123
16 hours ago
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(unirioja.es/cu/luhernan/jornadasmchagt/files/…) has the calculation I mentioned on page 4 or 5.
– random123
16 hours ago
@random123 : thanks for the really helfpul comments, maybe you want to make an answer ?
– student
14 hours ago
I am not sure if I have anything more to add to it.
– random123
14 hours ago
@random123 : I meant you could copy what you said in the answer box so I can upvote and accept the answer. If you prefer it like this, that's perfect too and thanks again for the help.
– student
14 hours ago
Q1. If $p$ is the top perversity function, then $Rj_*L[d]$ is a $p-$perverse sheaf. Dually, if $p$ is zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following $j : U rightarrow X$ be an open affine immersion, then $Rj_*$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
– random123
16 hours ago
Q1. If $p$ is the top perversity function, then $Rj_*L[d]$ is a $p-$perverse sheaf. Dually, if $p$ is zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following $j : U rightarrow X$ be an open affine immersion, then $Rj_*$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
– random123
16 hours ago
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(unirioja.es/cu/luhernan/jornadasmchagt/files/…) has the calculation I mentioned on page 4 or 5.
– random123
16 hours ago
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(unirioja.es/cu/luhernan/jornadasmchagt/files/…) has the calculation I mentioned on page 4 or 5.
– random123
16 hours ago
@random123 : thanks for the really helfpul comments, maybe you want to make an answer ?
– student
14 hours ago
@random123 : thanks for the really helfpul comments, maybe you want to make an answer ?
– student
14 hours ago
I am not sure if I have anything more to add to it.
– random123
14 hours ago
I am not sure if I have anything more to add to it.
– random123
14 hours ago
@random123 : I meant you could copy what you said in the answer box so I can upvote and accept the answer. If you prefer it like this, that's perfect too and thanks again for the help.
– student
14 hours ago
@random123 : I meant you could copy what you said in the answer box so I can upvote and accept the answer. If you prefer it like this, that's perfect too and thanks again for the help.
– student
14 hours ago
add a comment |
1 Answer
1
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oldest
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up vote
2
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accepted
Q1. If $p$ is the top perversity function, then $Rj_∗L[d]$ is a $p-$perverse sheaf. Dually, if $p$ sis zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following : Let $j:U→X$ be an open affine immersion then thhe functor $Rj_∗$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(https://www.unirioja.es/cu/luhernan/jornadasmchagt/files/3joana_cirici_beamer.pdf) has the calculation I mentioned on page 4 or 5.
answered 14 hours ago
random123
993719
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Q1. If $p$ is the top perversity function, then $Rj_∗L[d]$ is a $p-$perverse sheaf. Dually, if $p$ sis zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following : Let $j:U→X$ be an open affine immersion then thhe functor $Rj_∗$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(https://www.unirioja.es/cu/luhernan/jornadasmchagt/files/3joana_cirici_beamer.pdf) has the calculation I mentioned on page 4 or 5.
answered 14 hours ago
random123
993719
add a comment |
up vote
2
down vote
accepted
Q1. If $p$ is the top perversity function, then $Rj_∗L[d]$ is a $p-$perverse sheaf. Dually, if $p$ sis zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following : Let $j:U→X$ be an open affine immersion then thhe functor $Rj_∗$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(https://www.unirioja.es/cu/luhernan/jornadasmchagt/files/3joana_cirici_beamer.pdf) has the calculation I mentioned on page 4 or 5.
answered 14 hours ago
random123
993719
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Q1. If $p$ is the top perversity function, then $Rj_∗L[d]$ is a $p-$perverse sheaf. Dually, if $p$ sis zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following : Let $j:U→X$ be an open affine immersion then thhe functor $Rj_∗$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(https://www.unirioja.es/cu/luhernan/jornadasmchagt/files/3joana_cirici_beamer.pdf) has the calculation I mentioned on page 4 or 5.
answered 14 hours ago
random123
993719
Q1. If $p$ is the top perversity function, then $Rj_∗L[d]$ is a $p-$perverse sheaf. Dually, if $p$ sis zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following : Let $j:U→X$ be an open affine immersion then thhe functor $Rj_∗$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(https://www.unirioja.es/cu/luhernan/jornadasmchagt/files/3joana_cirici_beamer.pdf) has the calculation I mentioned on page 4 or 5.
answered 14 hours ago
random123
993719
answered 14 hours ago
random123
993719
answered 14 hours ago
random123
993719
answered 14 hours ago
random123
993719
993719
add a comment |
add a comment |
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Q1. If $p$ is the top perversity function, then $Rj_*L[d]$ is a $p-$perverse sheaf. Dually, if $p$ is zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following $j : U rightarrow X$ be an open affine immersion, then $Rj_*$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
– random123
16 hours ago
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(unirioja.es/cu/luhernan/jornadasmchagt/files/…) has the calculation I mentioned on page 4 or 5.
– random123
16 hours ago
@random123 : thanks for the really helfpul comments, maybe you want to make an answer ?
– student
14 hours ago
I am not sure if I have anything more to add to it.
– random123
14 hours ago
@random123 : I meant you could copy what you said in the answer box so I can upvote and accept the answer. If you prefer it like this, that's perfect too and thanks again for the help.
– student
14 hours ago