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user395911
user395911

Let $G$ be a group scheme locally finite type and smooth over a base scheme $S$, and assume $S$ is normal and integral.

Then does the set of (geometrical) connected components of a group scheme form a group? or even point of a group scheme over $S$?

If $S=Spec k $ where $k$ is a field, then this is true and we have a theory of $\pi_0(G)$ using etale algebras over a field. I wonder what will happen for the generic case.

Let $G$ be a group scheme locally finite type and smooth over a base scheme $S$, and assume $S$ is normal and integral.

Then does (geometrical) connected components of a group scheme form a group? or even a group scheme over $S$?

If $S=Spec k $ where $k$ is a field, then this is true and we have a theory of $\pi_0(G)$ using etale algebras over a field. I wonder what will happen for the generic case.

Let $G$ be a group scheme locally finite type and smooth over a base scheme $S$, and assume $S$ is normal and integral.

Then does the set of (geometrical) connected components of a group scheme form a group? or even point of a group scheme over $S$?

If $S=Spec k $ where $k$ is a field, then this is true and we have a theory of $\pi_0(G)$ using etale algebras over a field. I wonder what will happen for the generic case.

Source Link
user395911
user395911

Does connected components of a group scheme form a group scheme?

Let $G$ be a group scheme locally finite type and smooth over a base scheme $S$, and assume $S$ is normal and integral.

Then does (geometrical) connected components of a group scheme form a group? or even a group scheme over $S$?

If $S=Spec k $ where $k$ is a field, then this is true and we have a theory of $\pi_0(G)$ using etale algebras over a field. I wonder what will happen for the generic case.