Base change of a morphism of schemes

Given $f:X\to Y$ and $g:Y^{\prime }\to Y$, the base change of $f$ along $g$ is the projectionProperties such as being proper and flat are stable under arbitrary base change.

Flatness of a projective family does not by itself force global sections to commute with base change. For a discrete valuation ring with uniformizer $\pi$, glue $\mathrm{Spec}R[\pi t,t^2,t^3]$ to $\mathrm{Spec}R[t^{-1}]$ along $\mathrm{Spec}R[t,t^{-1}]$. In the projective model with coordinates $(1,\pi t,t^2,t^3)$, the global ring is $R$. The special-fibre first chart has an additional square-zero element $a$ annihilated by $t^2,t^3$; it vanishes on the overlap and therefore extends by zero to a global section. The special-fibre global ring is $k[a]/(a^2)$, and base change from $R$ reaches only its constants.

For a scheme over $\mathbb{R}$, its base change of a morphism of schemes to $\mathbb{C}$ has scheme-theoretic fibre over a point $p$ equal to $\mathrm{Spec}(\kappa (p){\otimes }_{\mathbb{R}}\mathbb{C})$. This is the spectrum of $\kappa (p)[s]/(s^2+1)$. It has two points if $-1$ is a square in the residue field, and otherwise one point with a quadratic field extension. In both cases it is reduced, because the quadratic polynomial has no repeated root in characteristic zero. The distinction applies to generic points as well as closed points.

If $X\to Y$ is a surjective morphism of schemes, the projection $X{\times }_Y{Y}^{\prime }\to Y^{\prime }$ is surjective for every $Y^{\prime }\to Y$. For a point $y\prime$, choose a point $x$ over its base image and apply the point-lifting property of a scheme fibre product. No flatness assumption is necessary.