Condition of pushward commutes with tensor product
1
$begingroup$
Let $f$ be a morphism between schemes. Is there a sufficient and necessary condition on $f$ such that $f_*$ commutes with $otimes$? i.e.
$$f_*Fotimes f_*Gcong f_*(Fotimes G)$$
for all coherent sheaves $F,G$.
In particular, I want to know that is it true for $f$ proper.
algebraic-geometry schemes coherent-sheaves
asked Dec 17 '18 at 14:24
User XUser X
33411
$endgroup$
add a comment |
1
$begingroup$
Let $f$ be a morphism between schemes. Is there a sufficient and necessary condition on $f$ such that $f_*$ commutes with $otimes$? i.e.
$$f_*Fotimes f_*Gcong f_*(Fotimes G)$$
for all coherent sheaves $F,G$.
In particular, I want to know that is it true for $f$ proper.
algebraic-geometry schemes coherent-sheaves
asked Dec 17 '18 at 14:24
User XUser X
33411
$endgroup$
1
$begingroup$
This is rarely true and properness is insufficient. For a simple example, take $f:mathbb{P}^1_kto k$ and $F=O(-1), G=O(1)$.
$endgroup$
– Mohan
Dec 17 '18 at 14:46
$begingroup$
@Mohan Emmm... I see. Thanks.
$endgroup$
– User X
Dec 17 '18 at 15:23
$begingroup$
Maybe we can say that a necessary and sufficient condition on $f$ is being a monomorphism. But my argument is incomplete : if $f$ into on closed points, so there is two closed points $xneq y$ such that $f(x)=f(y)$. Then take the skyscraper sheaves at $x$ and $y$. We have $kappa(x)otimeskappa(y)=0$ since they are supported on different points. But $f_*kappa(x)otimes f_*kappa(y)=kappa(f(x))otimes kappa(f(x))=kappa(f(x))$. Conversely, if moreover $A$ is affine (like most monomorphisms ?), then the problem reduces to a problem on modules which is easy.
$endgroup$
– Roland
Dec 17 '18 at 21:23
add a comment |
1
1
1
$begingroup$
Let $f$ be a morphism between schemes. Is there a sufficient and necessary condition on $f$ such that $f_*$ commutes with $otimes$? i.e.
$$f_*Fotimes f_*Gcong f_*(Fotimes G)$$
for all coherent sheaves $F,G$.
In particular, I want to know that is it true for $f$ proper.
algebraic-geometry schemes coherent-sheaves
asked Dec 17 '18 at 14:24
User XUser X
33411
$endgroup$
Let $f$ be a morphism between schemes. Is there a sufficient and necessary condition on $f$ such that $f_*$ commutes with $otimes$? i.e.
$$f_*Fotimes f_*Gcong f_*(Fotimes G)$$
for all coherent sheaves $F,G$.
In particular, I want to know that is it true for $f$ proper.
algebraic-geometry schemes coherent-sheaves
algebraic-geometry schemes coherent-sheaves
asked Dec 17 '18 at 14:24
User XUser X
33411
asked Dec 17 '18 at 14:24
User XUser X
33411
asked Dec 17 '18 at 14:24
User XUser X
33411
asked Dec 17 '18 at 14:24
User XUser X
33411
asked Dec 17 '18 at 14:24
User XUser X
33411
33411
1
$begingroup$
This is rarely true and properness is insufficient. For a simple example, take $f:mathbb{P}^1_kto k$ and $F=O(-1), G=O(1)$.
$endgroup$
– Mohan
Dec 17 '18 at 14:46
$begingroup$
@Mohan Emmm... I see. Thanks.
$endgroup$
– User X
Dec 17 '18 at 15:23
$begingroup$
Maybe we can say that a necessary and sufficient condition on $f$ is being a monomorphism. But my argument is incomplete : if $f$ into on closed points, so there is two closed points $xneq y$ such that $f(x)=f(y)$. Then take the skyscraper sheaves at $x$ and $y$. We have $kappa(x)otimeskappa(y)=0$ since they are supported on different points. But $f_*kappa(x)otimes f_*kappa(y)=kappa(f(x))otimes kappa(f(x))=kappa(f(x))$. Conversely, if moreover $A$ is affine (like most monomorphisms ?), then the problem reduces to a problem on modules which is easy.
$endgroup$
– Roland
Dec 17 '18 at 21:23
add a comment |
1
$begingroup$
This is rarely true and properness is insufficient. For a simple example, take $f:mathbb{P}^1_kto k$ and $F=O(-1), G=O(1)$.
$endgroup$
– Mohan
Dec 17 '18 at 14:46
$begingroup$
@Mohan Emmm... I see. Thanks.
$endgroup$
– User X
Dec 17 '18 at 15:23
$begingroup$
Maybe we can say that a necessary and sufficient condition on $f$ is being a monomorphism. But my argument is incomplete : if $f$ into on closed points, so there is two closed points $xneq y$ such that $f(x)=f(y)$. Then take the skyscraper sheaves at $x$ and $y$. We have $kappa(x)otimeskappa(y)=0$ since they are supported on different points. But $f_*kappa(x)otimes f_*kappa(y)=kappa(f(x))otimes kappa(f(x))=kappa(f(x))$. Conversely, if moreover $A$ is affine (like most monomorphisms ?), then the problem reduces to a problem on modules which is easy.
$endgroup$
– Roland
Dec 17 '18 at 21:23
1
1
$begingroup$
This is rarely true and properness is insufficient. For a simple example, take $f:mathbb{P}^1_kto k$ and $F=O(-1), G=O(1)$.
$endgroup$
– Mohan
Dec 17 '18 at 14:46
$begingroup$
This is rarely true and properness is insufficient. For a simple example, take $f:mathbb{P}^1_kto k$ and $F=O(-1), G=O(1)$.
$endgroup$
– Mohan
Dec 17 '18 at 14:46
$begingroup$
@Mohan Emmm... I see. Thanks.
$endgroup$
– User X
Dec 17 '18 at 15:23
$begingroup$
@Mohan Emmm... I see. Thanks.
$endgroup$
– User X
Dec 17 '18 at 15:23
$begingroup$
Maybe we can say that a necessary and sufficient condition on $f$ is being a monomorphism. But my argument is incomplete : if $f$ into on closed points, so there is two closed points $xneq y$ such that $f(x)=f(y)$. Then take the skyscraper sheaves at $x$ and $y$. We have $kappa(x)otimeskappa(y)=0$ since they are supported on different points. But $f_*kappa(x)otimes f_*kappa(y)=kappa(f(x))otimes kappa(f(x))=kappa(f(x))$. Conversely, if moreover $A$ is affine (like most monomorphisms ?), then the problem reduces to a problem on modules which is easy.
$endgroup$
– Roland
Dec 17 '18 at 21:23
$begingroup$
Maybe we can say that a necessary and sufficient condition on $f$ is being a monomorphism. But my argument is incomplete : if $f$ into on closed points, so there is two closed points $xneq y$ such that $f(x)=f(y)$. Then take the skyscraper sheaves at $x$ and $y$. We have $kappa(x)otimeskappa(y)=0$ since they are supported on different points. But $f_*kappa(x)otimes f_*kappa(y)=kappa(f(x))otimes kappa(f(x))=kappa(f(x))$. Conversely, if moreover $A$ is affine (like most monomorphisms ?), then the problem reduces to a problem on modules which is easy.
$endgroup$
– Roland
Dec 17 '18 at 21:23
add a comment |
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1
$begingroup$
This is rarely true and properness is insufficient. For a simple example, take $f:mathbb{P}^1_kto k$ and $F=O(-1), G=O(1)$.
$endgroup$
– Mohan
Dec 17 '18 at 14:46
$begingroup$
@Mohan Emmm... I see. Thanks.
$endgroup$
– User X
Dec 17 '18 at 15:23
$begingroup$
Maybe we can say that a necessary and sufficient condition on $f$ is being a monomorphism. But my argument is incomplete : if $f$ into on closed points, so there is two closed points $xneq y$ such that $f(x)=f(y)$. Then take the skyscraper sheaves at $x$ and $y$. We have $kappa(x)otimeskappa(y)=0$ since they are supported on different points. But $f_*kappa(x)otimes f_*kappa(y)=kappa(f(x))otimes kappa(f(x))=kappa(f(x))$. Conversely, if moreover $A$ is affine (like most monomorphisms ?), then the problem reduces to a problem on modules which is easy.
$endgroup$
– Roland
Dec 17 '18 at 21:23