Why geometric properties differ from ordinary ones

Scalar extension on affine charts is tensor extension by Affine charts after extension of the ground field. For $\mathbb{C}\mathrm{/}\mathbb{R}$, the presentation formula Presentations and localization under base extension gives $\mathbb{C}{\otimes }_{\mathbb{R}}\mathbb{C}\cong \mathbb{C}[z]\mathrm{/}(z^2+1)\cong \mathbb{C}\times \mathbb{C}\mathrm{.}$ The last map follows from Chinese remainder theorem for pairwise comaximal ideals, since $(z-i)$ and $(z+i)$ differ by the unit $2i$. A one-point integral real scheme therefore becomes two disjoint points, losing connectedness, irreducibility and integrality.

For the transcendental element $u$ over ${\mathbb{F}}_p$, put $L={\mathbb{F}}_p(u)$ and $k={\mathbb{F}}_p(u^p)$. The elements $1,u,\dots ,u^{p-1}$ form a $k$-basis of $L$: independence follows after clearing denominators and comparing polynomial exponents modulo $p$; their span is closed under multiplication using $u^p\in k$, and is a field because multiplication by a nonzero element is an injective linear map on this finite-dimensional span and hence surjective. Therefore $L{\otimes }_{k}L\cong L[z]\mathrm{/}(z^p-u^p)=L[ϵ]\mathrm{/}({ϵ}^p),ϵ=z-u\mathrm{.}$ The class $ϵ$ is nonzero and nilpotent, so reducedness can also fail. These computations concern ordinary properties; they motivate testing the geometric fibre.