# Ag.algebraic geometry – Reading geometric properties of an appropriate scheme from its refinement

Let $$k$$ to be a field, $$X rightarrow mathrm {Spec} , k$$ to be an appropriate morphism.

• Yes $$X$$ is geometrically reduced, so $$mathcal {O} _X (X)$$ is the direct product of finely separable finite extensions of $$k$$.
• Yes $$X$$ is geometrically connected, then $$mathcal {O} _X (X)$$ is an irreducible geometrically $$k$$-algebra.

Are the reverse statements true?