I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about ** algebraic varieties** (page 479). Since I still don't have the permission to add images I quote the relevant excerpt:

...The notion of properness has been introduced in 9.5/4. It means that the structural morphism $p: A \to Spec(K)$ is of finite type, separated, and universally closed. For the property of smoothness see 8.5/1. It follows from 8.5/15 in conjunction with 2.4/19 that all stalks $\mathcal{O}_{A,x}$ of a smooth $K$-group scheme $A$ are integral domains. Since abelian varieties are required to be irreducible, they give rise to integral schemes. Also let us mention that

for $K$-group schemes of finite type smooth is equivalent to geometrically reduced, which means that all stalks of the structure sheaf of $A×_K \bar{K}$ are reduced. In addition, let us point out thatfor $K$-group schemes of finite type the property irreducible can be checked after base change with $\bar{K}/K$so that we may replace irreducible by geometrically irreducible...

We fix an abelian variety $A$ over field $K$. By definition an abelian variety over $K$ is a proper smooth $K$-group scheme that is irreducible.

Following two questions:

Why is for a $K$-group scheme of finite type

equivalent to*smooth*?*geometrically reduced*Why under same conditions as in 1. (so $K$-group scheme of finite type) the property

is equivlaent to*irreducible*?*geometrically irreducible*

Remark: Here I previously asked this question in MSE: https://math.stackexchange.com/questions/3136827/abelian-varieties