# Simply connectedness of algebraic group

$G$ is a semisimple algebraic group over $k$, if $G_{\bar k}$ is simply connected when we do base change to $\bar k$, can we descent the simply connectedness to $G$?

Here, simply connectedness means no nontrivial connected central isogeny onto $G$.

Can we say that simply connected algebraic group is geometrically connected? If then we can give an affirmative answer by considering the universal cover of $G$.

Welcome for any answer under further assumption that $\text{char }k=0$.

• What if we assume char=0?
– TJCM
Mar 25, 2010 at 2:54
• If we assume char 0 then the given definition of "simply connected" is correct, and the whole thing becomes an easy exercise in Galois descent. (The magic of the theory of connected semisimple groups is that it works in an essentially characteristic-free way if one defines things properly.) I recommend that if you are only interested in char. 0 then you treat this as an exercise in Galois descent and try to solve it for yourself accordingly via that technique (using a suitable "universal property" of simply connected cover over $\overline{k}$ so as to get the Galois descent datum on it). Mar 25, 2010 at 3:09