p********e
发帖数: 16048 | 1 For a scheme morphism f:X--->Y
if f is dominant,ie the image of f is dense in Y
for any morphism Z--->Y
I am asking
f':X\product_{Y}Z---->Z
is f' a dominant morphism? | S******w
发帖数: 195 | 2 No.
Set rings R=Q, S=Z, T=Z/nZ;
u: S->R: a->a, v: S->T: a->a mod n;
X=Spec R, Y=Spec S, Z=Spec T.
Then u induces dominant f: X->Y, v induces morphism g: Z->Y;
f' is induced by T->(R tensor T over S, which is 0). So f' is not dominant.
【在 p********e 的大作中提到】
: For a scheme morphism f:X--->Y
: if f is dominant,ie the image of f is dense in Y
: for any morphism Z--->Y
: I am asking
: f':X\product_{Y}Z---->Z
: is f' a dominant morphism?
| f******h
发帖数: 104 | 3 Great example! On the other hand, surjection is stable
under base change.
【在 S******w 的大作中提到】
: No.
: Set rings R=Q, S=Z, T=Z/nZ;
: u: S->R: a->a, v: S->T: a->a mod n;
: X=Spec R, Y=Spec S, Z=Spec T.
: Then u induces dominant f: X->Y, v induces morphism g: Z->Y;
: f' is induced by T->(R tensor T over S, which is 0). So f' is not dominant.
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