On torsion sheaf of a coherent sheaf of $dim X$
$begingroup$
$underline {Background}$:Let,$E$ be a coherent sheaf on a Noetherian,integral scheme $X$ and $dim E$=$dim X$.
Then we have the unique torsion filtration of that coherent sheaf as
$0subset T_0(E)subset....subset T_{dim X-1}(E) subset T_{dim X} (E)=E$
where,$T_i(E)$ is the maximal subsheaf of of $E$ of dimension $leq i$
we also have torsion subsheaf of $E$ ,denoted by $T(E)$ which is defined as
for any affine $SpecA$ in $X$, define $ T(E)(SpecA)$:={$m in E(SpecA)|exists sin A$ with $s$ nonzero such that $s.m=0$}
we also have $(T(E))_x=T(E_x)$
$underline {Question}$:How do we show that $T(E)=T_{dim X-1}(E)$
I only know that $T(E)$ is a subsheaf of $E$ ,and I am supposed to show the following
$T(E)$ is a maximal subsheaf of dimension $leq dim(X)-1$.
I have no clue how to show (i) it has dimension $leq dim(X)-1$ and
(ii) It is maximal among all such subsheaves.
Any help from anyone is welcome.
algebraic-geometry sheaf-theory coherent-sheaves
edited Jan 2 at 7:06
Henno Brandsma
114k348124
asked Jan 2 at 6:50
HARRYHARRY
889
$endgroup$
add a comment |
$begingroup$
$underline {Background}$:Let,$E$ be a coherent sheaf on a Noetherian,integral scheme $X$ and $dim E$=$dim X$.
Then we have the unique torsion filtration of that coherent sheaf as
$0subset T_0(E)subset....subset T_{dim X-1}(E) subset T_{dim X} (E)=E$
where,$T_i(E)$ is the maximal subsheaf of of $E$ of dimension $leq i$
we also have torsion subsheaf of $E$ ,denoted by $T(E)$ which is defined as
for any affine $SpecA$ in $X$, define $ T(E)(SpecA)$:={$m in E(SpecA)|exists sin A$ with $s$ nonzero such that $s.m=0$}
we also have $(T(E))_x=T(E_x)$
$underline {Question}$:How do we show that $T(E)=T_{dim X-1}(E)$
I only know that $T(E)$ is a subsheaf of $E$ ,and I am supposed to show the following
$T(E)$ is a maximal subsheaf of dimension $leq dim(X)-1$.
I have no clue how to show (i) it has dimension $leq dim(X)-1$ and
(ii) It is maximal among all such subsheaves.
Any help from anyone is welcome.
algebraic-geometry sheaf-theory coherent-sheaves
edited Jan 2 at 7:06
Henno Brandsma
114k348124
asked Jan 2 at 6:50
HARRYHARRY
889
$endgroup$
$begingroup$
What do you mean by $dim E$?
$endgroup$
– Armando j18eos
Jan 2 at 11:12
$begingroup$
@Armandoj18eos $dim E=dim(Supp( E))$
$endgroup$
– HARRY
Jan 2 at 13:11
add a comment |
$begingroup$
$underline {Background}$:Let,$E$ be a coherent sheaf on a Noetherian,integral scheme $X$ and $dim E$=$dim X$.
Then we have the unique torsion filtration of that coherent sheaf as
$0subset T_0(E)subset....subset T_{dim X-1}(E) subset T_{dim X} (E)=E$
where,$T_i(E)$ is the maximal subsheaf of of $E$ of dimension $leq i$
we also have torsion subsheaf of $E$ ,denoted by $T(E)$ which is defined as
for any affine $SpecA$ in $X$, define $ T(E)(SpecA)$:={$m in E(SpecA)|exists sin A$ with $s$ nonzero such that $s.m=0$}
we also have $(T(E))_x=T(E_x)$
$underline {Question}$:How do we show that $T(E)=T_{dim X-1}(E)$
I only know that $T(E)$ is a subsheaf of $E$ ,and I am supposed to show the following
$T(E)$ is a maximal subsheaf of dimension $leq dim(X)-1$.
I have no clue how to show (i) it has dimension $leq dim(X)-1$ and
(ii) It is maximal among all such subsheaves.
Any help from anyone is welcome.
algebraic-geometry sheaf-theory coherent-sheaves
edited Jan 2 at 7:06
Henno Brandsma
114k348124
asked Jan 2 at 6:50
HARRYHARRY
889
$endgroup$
$underline {Background}$:Let,$E$ be a coherent sheaf on a Noetherian,integral scheme $X$ and $dim E$=$dim X$.
Then we have the unique torsion filtration of that coherent sheaf as
$0subset T_0(E)subset....subset T_{dim X-1}(E) subset T_{dim X} (E)=E$
where,$T_i(E)$ is the maximal subsheaf of of $E$ of dimension $leq i$
we also have torsion subsheaf of $E$ ,denoted by $T(E)$ which is defined as
for any affine $SpecA$ in $X$, define $ T(E)(SpecA)$:={$m in E(SpecA)|exists sin A$ with $s$ nonzero such that $s.m=0$}
we also have $(T(E))_x=T(E_x)$
$underline {Question}$:How do we show that $T(E)=T_{dim X-1}(E)$
I only know that $T(E)$ is a subsheaf of $E$ ,and I am supposed to show the following
$T(E)$ is a maximal subsheaf of dimension $leq dim(X)-1$.
I have no clue how to show (i) it has dimension $leq dim(X)-1$ and
(ii) It is maximal among all such subsheaves.
Any help from anyone is welcome.
algebraic-geometry sheaf-theory coherent-sheaves
algebraic-geometry sheaf-theory coherent-sheaves
edited Jan 2 at 7:06
Henno Brandsma
114k348124
asked Jan 2 at 6:50
HARRYHARRY
889
edited Jan 2 at 7:06
Henno Brandsma
114k348124
asked Jan 2 at 6:50
HARRYHARRY
889
edited Jan 2 at 7:06
Henno Brandsma
114k348124
edited Jan 2 at 7:06
Henno Brandsma
114k348124
edited Jan 2 at 7:06
Henno Brandsma
114k348124
114k348124
asked Jan 2 at 6:50
HARRYHARRY
889
asked Jan 2 at 6:50
HARRYHARRY
889
asked Jan 2 at 6:50
HARRYHARRY
889
889
$begingroup$
What do you mean by $dim E$?
$endgroup$
– Armando j18eos
Jan 2 at 11:12
$begingroup$
@Armandoj18eos $dim E=dim(Supp( E))$
$endgroup$
– HARRY
Jan 2 at 13:11
add a comment |
$begingroup$
What do you mean by $dim E$?
$endgroup$
– Armando j18eos
Jan 2 at 11:12
$begingroup$
@Armandoj18eos $dim E=dim(Supp( E))$
$endgroup$
– HARRY
Jan 2 at 13:11
$begingroup$
What do you mean by $dim E$?
$endgroup$
– Armando j18eos
Jan 2 at 11:12
$begingroup$
What do you mean by $dim E$?
$endgroup$
– Armando j18eos
Jan 2 at 11:12
$begingroup$
@Armandoj18eos $dim E=dim(Supp( E))$
$endgroup$
– HARRY
Jan 2 at 13:11
$begingroup$
@Armandoj18eos $dim E=dim(Supp( E))$
$endgroup$
– HARRY
Jan 2 at 13:11
add a comment |
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$begingroup$
What do you mean by $dim E$?
$endgroup$
– Armando j18eos
Jan 2 at 11:12
$begingroup$
@Armandoj18eos $dim E=dim(Supp( E))$
$endgroup$
– HARRY
Jan 2 at 13:11