Number of points in the fibers of étale morphisms.
$begingroup$
Consider the following theorem$^{*}$:
Theorem of conservation of number: Suppose $f:X longrightarrow Y$ is an étale
covering, and $Y$ is connected. Then the number of points in a fibre $f^{-1}(y)$ is
independent of $y in Y$.
The way of proof is that locally on $Y$, the variety $X$
can be defined in $mathbb{A}^1 times Y$ by one equation, say $T^m + a_1T^{m-1} + ... + a_m = 0$,
where the $a_i in K[Y]$. Now, since $f$ is étale, all the roots of the equation
$T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of
them.
My question is why all the roots of the equation
$T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of
them $Longrightarrow$ the number of points in a fibre $f^{-1}(y)$ is
independent of $y in Y$? How to connect the simple $m$ roots of the equation with the number of points in a fiber?
$^*$Algebraic Geometry I - Algebraic Curves - Algebraic Manifolds and Schemes (I. R. Shafarevich). Section 5.5.
thanks
algebraic-geometry
asked Dec 18 '18 at 19:08
ManoelManoel
930518
$endgroup$
add a comment |
$begingroup$
Consider the following theorem$^{*}$:
Theorem of conservation of number: Suppose $f:X longrightarrow Y$ is an étale
covering, and $Y$ is connected. Then the number of points in a fibre $f^{-1}(y)$ is
independent of $y in Y$.
The way of proof is that locally on $Y$, the variety $X$
can be defined in $mathbb{A}^1 times Y$ by one equation, say $T^m + a_1T^{m-1} + ... + a_m = 0$,
where the $a_i in K[Y]$. Now, since $f$ is étale, all the roots of the equation
$T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of
them.
My question is why all the roots of the equation
$T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of
them $Longrightarrow$ the number of points in a fibre $f^{-1}(y)$ is
independent of $y in Y$? How to connect the simple $m$ roots of the equation with the number of points in a fiber?
$^*$Algebraic Geometry I - Algebraic Curves - Algebraic Manifolds and Schemes (I. R. Shafarevich). Section 5.5.
thanks
algebraic-geometry
asked Dec 18 '18 at 19:08
ManoelManoel
930518
$endgroup$
1
$begingroup$
A repeated root should result in ramification of the cover, so the map wouldn't be etale. Also the solutions of the equation are the points in the preimage, which is why the number of roots is the number of points in the fibre.
$endgroup$
– jgon
Dec 19 '18 at 15:36
$begingroup$
@jgon, Do you know where I can find proof of this fact, a reference?
$endgroup$
– Manoel
Dec 20 '18 at 1:59
add a comment |
$begingroup$
Consider the following theorem$^{*}$:
Theorem of conservation of number: Suppose $f:X longrightarrow Y$ is an étale
covering, and $Y$ is connected. Then the number of points in a fibre $f^{-1}(y)$ is
independent of $y in Y$.
The way of proof is that locally on $Y$, the variety $X$
can be defined in $mathbb{A}^1 times Y$ by one equation, say $T^m + a_1T^{m-1} + ... + a_m = 0$,
where the $a_i in K[Y]$. Now, since $f$ is étale, all the roots of the equation
$T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of
them.
My question is why all the roots of the equation
$T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of
them $Longrightarrow$ the number of points in a fibre $f^{-1}(y)$ is
independent of $y in Y$? How to connect the simple $m$ roots of the equation with the number of points in a fiber?
$^*$Algebraic Geometry I - Algebraic Curves - Algebraic Manifolds and Schemes (I. R. Shafarevich). Section 5.5.
thanks
algebraic-geometry
asked Dec 18 '18 at 19:08
ManoelManoel
930518
$endgroup$
Consider the following theorem$^{*}$:
Theorem of conservation of number: Suppose $f:X longrightarrow Y$ is an étale
covering, and $Y$ is connected. Then the number of points in a fibre $f^{-1}(y)$ is
independent of $y in Y$.
The way of proof is that locally on $Y$, the variety $X$
can be defined in $mathbb{A}^1 times Y$ by one equation, say $T^m + a_1T^{m-1} + ... + a_m = 0$,
where the $a_i in K[Y]$. Now, since $f$ is étale, all the roots of the equation
$T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of
them.
My question is why all the roots of the equation
$T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of
them $Longrightarrow$ the number of points in a fibre $f^{-1}(y)$ is
independent of $y in Y$? How to connect the simple $m$ roots of the equation with the number of points in a fiber?
$^*$Algebraic Geometry I - Algebraic Curves - Algebraic Manifolds and Schemes (I. R. Shafarevich). Section 5.5.
thanks
algebraic-geometry
algebraic-geometry
asked Dec 18 '18 at 19:08
ManoelManoel
930518
asked Dec 18 '18 at 19:08
ManoelManoel
930518
asked Dec 18 '18 at 19:08
ManoelManoel
930518
asked Dec 18 '18 at 19:08
ManoelManoel
930518
asked Dec 18 '18 at 19:08
ManoelManoel
930518
930518
1
$begingroup$
A repeated root should result in ramification of the cover, so the map wouldn't be etale. Also the solutions of the equation are the points in the preimage, which is why the number of roots is the number of points in the fibre.
$endgroup$
– jgon
Dec 19 '18 at 15:36
$begingroup$
@jgon, Do you know where I can find proof of this fact, a reference?
$endgroup$
– Manoel
Dec 20 '18 at 1:59
add a comment |
1
$begingroup$
A repeated root should result in ramification of the cover, so the map wouldn't be etale. Also the solutions of the equation are the points in the preimage, which is why the number of roots is the number of points in the fibre.
$endgroup$
– jgon
Dec 19 '18 at 15:36
$begingroup$
@jgon, Do you know where I can find proof of this fact, a reference?
$endgroup$
– Manoel
Dec 20 '18 at 1:59
1
1
$begingroup$
A repeated root should result in ramification of the cover, so the map wouldn't be etale. Also the solutions of the equation are the points in the preimage, which is why the number of roots is the number of points in the fibre.
$endgroup$
– jgon
Dec 19 '18 at 15:36
$begingroup$
A repeated root should result in ramification of the cover, so the map wouldn't be etale. Also the solutions of the equation are the points in the preimage, which is why the number of roots is the number of points in the fibre.
$endgroup$
– jgon
Dec 19 '18 at 15:36
$begingroup$
@jgon, Do you know where I can find proof of this fact, a reference?
$endgroup$
– Manoel
Dec 20 '18 at 1:59
$begingroup$
@jgon, Do you know where I can find proof of this fact, a reference?
$endgroup$
– Manoel
Dec 20 '18 at 1:59
add a comment |
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$begingroup$
A repeated root should result in ramification of the cover, so the map wouldn't be etale. Also the solutions of the equation are the points in the preimage, which is why the number of roots is the number of points in the fibre.
$endgroup$
– jgon
Dec 19 '18 at 15:36
$begingroup$
@jgon, Do you know where I can find proof of this fact, a reference?
$endgroup$
– Manoel
Dec 20 '18 at 1:59