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1

$begingroup$

Let $S$ be a connected scheme, and let $fcolon Xto S$ be a finite étale cover. Then there exists a finite étale cover $Yto S$ such that $Xtimes_S Ycong coprod_iin I Y$.

Replacing $Y$ by one of its connected components, we may assume that $Y$ is connected. Then it is easy to see that the action of $operatornameAut_S(X)$ on $Xtimes_S Y$ is just an action on the index set $I$.

I am wondering, if the same is still true in case $Y$ is not necessarily connected. It seems to me that an automorphism of $X$ over $S$ might map two connected components of a copy of $Y$ in $Xtimes_S Y$ into two different copies.

algebraic-geometry commutative-algebra

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asked Mar 20 at 17:38

user363120user363120

1,166310

$endgroup$

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1

$begingroup$

Let $S$ be a connected scheme, and let $fcolon Xto S$ be a finite étale cover. Then there exists a finite étale cover $Yto S$ such that $Xtimes_S Ycong coprod_iin I Y$.

Replacing $Y$ by one of its connected components, we may assume that $Y$ is connected. Then it is easy to see that the action of $operatornameAut_S(X)$ on $Xtimes_S Y$ is just an action on the index set $I$.

I am wondering, if the same is still true in case $Y$ is not necessarily connected. It seems to me that an automorphism of $X$ over $S$ might map two connected components of a copy of $Y$ in $Xtimes_S Y$ into two different copies.

algebraic-geometry commutative-algebra

share|cite|improve this question

asked Mar 20 at 17:38

user363120user363120

1,166310

$endgroup$

add a comment | 

$begingroup$

Let $S$ be a connected scheme, and let $fcolon Xto S$ be a finite étale cover. Then there exists a finite étale cover $Yto S$ such that $Xtimes_S Ycong coprod_iin I Y$.

Replacing $Y$ by one of its connected components, we may assume that $Y$ is connected. Then it is easy to see that the action of $operatornameAut_S(X)$ on $Xtimes_S Y$ is just an action on the index set $I$.

I am wondering, if the same is still true in case $Y$ is not necessarily connected. It seems to me that an automorphism of $X$ over $S$ might map two connected components of a copy of $Y$ in $Xtimes_S Y$ into two different copies.

algebraic-geometry commutative-algebra

share|cite|improve this question

asked Mar 20 at 17:38

user363120user363120

1,166310

$endgroup$

share|cite|improve this question

asked Mar 20 at 17:38

user363120user363120

1,166310

share|cite|improve this question

asked Mar 20 at 17:38

user363120user363120

1,166310

asked Mar 20 at 17:38

user363120user363120

1,166310

asked Mar 20 at 17:38

user363120user363120

1,166310