Etale Morphism of Affine Schemes
$begingroup$
Let $k$ be an arbitrary field and we consider the map
$$f: Spec(k[T, T^{-1},Y]/(Y^d - T)) to Spec(k[T, T^{-1}])$$
induced by canonical inclusion $varphi: k[T, T^{-1}] to k[T, T^{-1},Y]/(Y^d - T))$. I want rigorously see why this map is etale preferably using arguments from commutative algebra.
By definition a morphism $f:X to Y$ is etale iff it is * flat *(1), locally of finite type(2) and has the separable field condition(3):
Here (1), (2) and (3) mean:
(1) For every $x in X$ the induced morphism $f_x ^{#}:mathcal{O}_{Y, f(x)} to mathcal{O}_X$ is flat
(2) There exist an open affine neighbourhood $U_x =Spec(R)$ of $x$ and an o. a. n. $V_{f(x)}= Spec(S)$ of $f(x)$ with $f(U_x) subset V_{f(x)}$ such that the induced ring map $S to R$ is of finite presentation
(3) Let $m_x subset mathcal{O}_{X,x}$ the unique maximal ideal of local ring $mathcal{O}_{X,x}$ and respectively $m_{f(x)} subset mathcal{O}_{X,x}$ the unique maximal ideal of $mathcal{O}_{Y,f(x)}$: Then the induced finite field extension $mathcal{O}_{Y,f(x)}/m_{f(x)} to mathcal{O}_{X,x}/m_x$ is separable
I have problem to check the conditions (1) and (3):
To consider the induced map on stalks we must take an prime ideal $p subset k[T, T^{-1},Y]/(Y^d - T))$ resp $f(x_p) = varphi^{-1}(p)$ and consider the induced localized map
$$varphi_p: k[T, T^{-1}]_{varphi^{-1}(p)} to k[T, T^{-1},Y]/(Y^d - T))_p$$
Here we localize $k[T, T^{-1}]$ with respect to the multiplicative set $k[T, T^{-1}] backslash varphi^{-1}(p)$ and $k[T, T^{-1},Y]/(Y^d - T))$ wrt $k[T, T^{-1},Y]/(Y^d - T)) backslash p$.
What do we know about flatness of this map? Futhermore, what do we know about separableness of the induced field extension $k[T, T^{-1}]_{varphi^{-1}(p)}/m_{f(x_p)} subset (k[T, T^{-1},Y]/(Y^d - T))_p) / m_p$ of residue fields?
Could anybody expain how to argue here using facts about localizations of polynomial rings? I don't really know how I can analyze the induced localized map on stalks to verify flatness / separateness of residue fields.
What goes wrong with the canonical map
$$g:Spec(k[T, T^{-1},Y]/(Y^d - T)) to Spec(k[T])$$?
algebraic-geometry
asked Nov 24 '18 at 21:50
KarlPeterKarlPeter
7071315
$endgroup$
add a comment |
$begingroup$
Let $k$ be an arbitrary field and we consider the map
$$f: Spec(k[T, T^{-1},Y]/(Y^d - T)) to Spec(k[T, T^{-1}])$$
induced by canonical inclusion $varphi: k[T, T^{-1}] to k[T, T^{-1},Y]/(Y^d - T))$. I want rigorously see why this map is etale preferably using arguments from commutative algebra.
By definition a morphism $f:X to Y$ is etale iff it is * flat *(1), locally of finite type(2) and has the separable field condition(3):
Here (1), (2) and (3) mean:
(1) For every $x in X$ the induced morphism $f_x ^{#}:mathcal{O}_{Y, f(x)} to mathcal{O}_X$ is flat
(2) There exist an open affine neighbourhood $U_x =Spec(R)$ of $x$ and an o. a. n. $V_{f(x)}= Spec(S)$ of $f(x)$ with $f(U_x) subset V_{f(x)}$ such that the induced ring map $S to R$ is of finite presentation
(3) Let $m_x subset mathcal{O}_{X,x}$ the unique maximal ideal of local ring $mathcal{O}_{X,x}$ and respectively $m_{f(x)} subset mathcal{O}_{X,x}$ the unique maximal ideal of $mathcal{O}_{Y,f(x)}$: Then the induced finite field extension $mathcal{O}_{Y,f(x)}/m_{f(x)} to mathcal{O}_{X,x}/m_x$ is separable
I have problem to check the conditions (1) and (3):
To consider the induced map on stalks we must take an prime ideal $p subset k[T, T^{-1},Y]/(Y^d - T))$ resp $f(x_p) = varphi^{-1}(p)$ and consider the induced localized map
$$varphi_p: k[T, T^{-1}]_{varphi^{-1}(p)} to k[T, T^{-1},Y]/(Y^d - T))_p$$
Here we localize $k[T, T^{-1}]$ with respect to the multiplicative set $k[T, T^{-1}] backslash varphi^{-1}(p)$ and $k[T, T^{-1},Y]/(Y^d - T))$ wrt $k[T, T^{-1},Y]/(Y^d - T)) backslash p$.
What do we know about flatness of this map? Futhermore, what do we know about separableness of the induced field extension $k[T, T^{-1}]_{varphi^{-1}(p)}/m_{f(x_p)} subset (k[T, T^{-1},Y]/(Y^d - T))_p) / m_p$ of residue fields?
Could anybody expain how to argue here using facts about localizations of polynomial rings? I don't really know how I can analyze the induced localized map on stalks to verify flatness / separateness of residue fields.
What goes wrong with the canonical map
$$g:Spec(k[T, T^{-1},Y]/(Y^d - T)) to Spec(k[T])$$?
algebraic-geometry
asked Nov 24 '18 at 21:50
KarlPeterKarlPeter
7071315
$endgroup$
1
$begingroup$
For flatness note that $k[t,t^{-1}]$ is a PID and hence to check flatness is equivalent to checking that the module $k[t,t^{-1},y]/(y^d-t)$ is torsion free, which it is since $(y^d-t)$ is an irreducible polynomial. To check that is unramified, maybe it would be better to use that criteria that $Omega_{X/Y} = 0 Leftrightarrow Xrightarrow Y$ is unramified.
$endgroup$
– random123
Nov 25 '18 at 6:43
add a comment |
$begingroup$
Let $k$ be an arbitrary field and we consider the map
$$f: Spec(k[T, T^{-1},Y]/(Y^d - T)) to Spec(k[T, T^{-1}])$$
induced by canonical inclusion $varphi: k[T, T^{-1}] to k[T, T^{-1},Y]/(Y^d - T))$. I want rigorously see why this map is etale preferably using arguments from commutative algebra.
By definition a morphism $f:X to Y$ is etale iff it is * flat *(1), locally of finite type(2) and has the separable field condition(3):
Here (1), (2) and (3) mean:
(1) For every $x in X$ the induced morphism $f_x ^{#}:mathcal{O}_{Y, f(x)} to mathcal{O}_X$ is flat
(2) There exist an open affine neighbourhood $U_x =Spec(R)$ of $x$ and an o. a. n. $V_{f(x)}= Spec(S)$ of $f(x)$ with $f(U_x) subset V_{f(x)}$ such that the induced ring map $S to R$ is of finite presentation
(3) Let $m_x subset mathcal{O}_{X,x}$ the unique maximal ideal of local ring $mathcal{O}_{X,x}$ and respectively $m_{f(x)} subset mathcal{O}_{X,x}$ the unique maximal ideal of $mathcal{O}_{Y,f(x)}$: Then the induced finite field extension $mathcal{O}_{Y,f(x)}/m_{f(x)} to mathcal{O}_{X,x}/m_x$ is separable
I have problem to check the conditions (1) and (3):
To consider the induced map on stalks we must take an prime ideal $p subset k[T, T^{-1},Y]/(Y^d - T))$ resp $f(x_p) = varphi^{-1}(p)$ and consider the induced localized map
$$varphi_p: k[T, T^{-1}]_{varphi^{-1}(p)} to k[T, T^{-1},Y]/(Y^d - T))_p$$
Here we localize $k[T, T^{-1}]$ with respect to the multiplicative set $k[T, T^{-1}] backslash varphi^{-1}(p)$ and $k[T, T^{-1},Y]/(Y^d - T))$ wrt $k[T, T^{-1},Y]/(Y^d - T)) backslash p$.
What do we know about flatness of this map? Futhermore, what do we know about separableness of the induced field extension $k[T, T^{-1}]_{varphi^{-1}(p)}/m_{f(x_p)} subset (k[T, T^{-1},Y]/(Y^d - T))_p) / m_p$ of residue fields?
Could anybody expain how to argue here using facts about localizations of polynomial rings? I don't really know how I can analyze the induced localized map on stalks to verify flatness / separateness of residue fields.
What goes wrong with the canonical map
$$g:Spec(k[T, T^{-1},Y]/(Y^d - T)) to Spec(k[T])$$?
algebraic-geometry
asked Nov 24 '18 at 21:50
KarlPeterKarlPeter
7071315
$endgroup$
Let $k$ be an arbitrary field and we consider the map
$$f: Spec(k[T, T^{-1},Y]/(Y^d - T)) to Spec(k[T, T^{-1}])$$
induced by canonical inclusion $varphi: k[T, T^{-1}] to k[T, T^{-1},Y]/(Y^d - T))$. I want rigorously see why this map is etale preferably using arguments from commutative algebra.
By definition a morphism $f:X to Y$ is etale iff it is * flat *(1), locally of finite type(2) and has the separable field condition(3):
Here (1), (2) and (3) mean:
(1) For every $x in X$ the induced morphism $f_x ^{#}:mathcal{O}_{Y, f(x)} to mathcal{O}_X$ is flat
(2) There exist an open affine neighbourhood $U_x =Spec(R)$ of $x$ and an o. a. n. $V_{f(x)}= Spec(S)$ of $f(x)$ with $f(U_x) subset V_{f(x)}$ such that the induced ring map $S to R$ is of finite presentation
(3) Let $m_x subset mathcal{O}_{X,x}$ the unique maximal ideal of local ring $mathcal{O}_{X,x}$ and respectively $m_{f(x)} subset mathcal{O}_{X,x}$ the unique maximal ideal of $mathcal{O}_{Y,f(x)}$: Then the induced finite field extension $mathcal{O}_{Y,f(x)}/m_{f(x)} to mathcal{O}_{X,x}/m_x$ is separable
I have problem to check the conditions (1) and (3):
To consider the induced map on stalks we must take an prime ideal $p subset k[T, T^{-1},Y]/(Y^d - T))$ resp $f(x_p) = varphi^{-1}(p)$ and consider the induced localized map
$$varphi_p: k[T, T^{-1}]_{varphi^{-1}(p)} to k[T, T^{-1},Y]/(Y^d - T))_p$$
Here we localize $k[T, T^{-1}]$ with respect to the multiplicative set $k[T, T^{-1}] backslash varphi^{-1}(p)$ and $k[T, T^{-1},Y]/(Y^d - T))$ wrt $k[T, T^{-1},Y]/(Y^d - T)) backslash p$.
What do we know about flatness of this map? Futhermore, what do we know about separableness of the induced field extension $k[T, T^{-1}]_{varphi^{-1}(p)}/m_{f(x_p)} subset (k[T, T^{-1},Y]/(Y^d - T))_p) / m_p$ of residue fields?
Could anybody expain how to argue here using facts about localizations of polynomial rings? I don't really know how I can analyze the induced localized map on stalks to verify flatness / separateness of residue fields.
What goes wrong with the canonical map
$$g:Spec(k[T, T^{-1},Y]/(Y^d - T)) to Spec(k[T])$$?
algebraic-geometry
algebraic-geometry
asked Nov 24 '18 at 21:50
KarlPeterKarlPeter
7071315
asked Nov 24 '18 at 21:50
KarlPeterKarlPeter
7071315
edited Nov 24 '18 at 21:57
KarlPeter
asked Nov 24 '18 at 21:50
KarlPeterKarlPeter
7071315
asked Nov 24 '18 at 21:50
KarlPeterKarlPeter
7071315
asked Nov 24 '18 at 21:50
KarlPeterKarlPeter
7071315
7071315
1
$begingroup$
For flatness note that $k[t,t^{-1}]$ is a PID and hence to check flatness is equivalent to checking that the module $k[t,t^{-1},y]/(y^d-t)$ is torsion free, which it is since $(y^d-t)$ is an irreducible polynomial. To check that is unramified, maybe it would be better to use that criteria that $Omega_{X/Y} = 0 Leftrightarrow Xrightarrow Y$ is unramified.
$endgroup$
– random123
Nov 25 '18 at 6:43
add a comment |
1
$begingroup$
For flatness note that $k[t,t^{-1}]$ is a PID and hence to check flatness is equivalent to checking that the module $k[t,t^{-1},y]/(y^d-t)$ is torsion free, which it is since $(y^d-t)$ is an irreducible polynomial. To check that is unramified, maybe it would be better to use that criteria that $Omega_{X/Y} = 0 Leftrightarrow Xrightarrow Y$ is unramified.
$endgroup$
– random123
Nov 25 '18 at 6:43
1
1
$begingroup$
For flatness note that $k[t,t^{-1}]$ is a PID and hence to check flatness is equivalent to checking that the module $k[t,t^{-1},y]/(y^d-t)$ is torsion free, which it is since $(y^d-t)$ is an irreducible polynomial. To check that is unramified, maybe it would be better to use that criteria that $Omega_{X/Y} = 0 Leftrightarrow Xrightarrow Y$ is unramified.
$endgroup$
– random123
Nov 25 '18 at 6:43
$begingroup$
For flatness note that $k[t,t^{-1}]$ is a PID and hence to check flatness is equivalent to checking that the module $k[t,t^{-1},y]/(y^d-t)$ is torsion free, which it is since $(y^d-t)$ is an irreducible polynomial. To check that is unramified, maybe it would be better to use that criteria that $Omega_{X/Y} = 0 Leftrightarrow Xrightarrow Y$ is unramified.
$endgroup$
– random123
Nov 25 '18 at 6:43
add a comment |
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$begingroup$
For flatness note that $k[t,t^{-1}]$ is a PID and hence to check flatness is equivalent to checking that the module $k[t,t^{-1},y]/(y^d-t)$ is torsion free, which it is since $(y^d-t)$ is an irreducible polynomial. To check that is unramified, maybe it would be better to use that criteria that $Omega_{X/Y} = 0 Leftrightarrow Xrightarrow Y$ is unramified.
$endgroup$
– random123
Nov 25 '18 at 6:43