Affineness and properness descend under finite purely inseparable scalar extension

The projection $p:X_K\to X$ is finite faithfully flat and a universal homeomorphism. On an affine chart its ring map is finite free; after extending any residue field the spectrum of the purely inseparable tensor extension has exactly one point, since each element of $K$ has some $p$-power in $k$. It is therefore radicial and onto, and a finite onto map is closed, including after every base change. If $R$ is a local $k$-algebra, $R{\otimes }_{k}K$ is local: its finite integral extension has a unique prime over the maximal ideal, and every maximal ideal lies over that ideal. Thus the stalk at the unique point above $x\in X$ is ${\mathcal{O}}_{X,x}{\otimes }_{k}K$.

Assume $X_K$ affine and set $A=\mathrm{\Gamma }(X,{\mathcal{O}}_X)$. By [F1] and [F2], the canonical map $c:X\to \mathrm{Spec}A$ becomes the canonical affine isomorphism $X_K\cong \mathrm{Spec}(A{\otimes }_{k}K)$. By step 1.1 the two scalar-extension projections are homeomorphisms, so $c$ is a homeomorphism. On each stalk the map induced by $c$ becomes an isomorphism after tensoring with $K$, by the stalk description in step 1.1. Tensoring is exact and faithfully flat, so its kernel and cokernel vanish by [F2]. Thus $c$ is an isomorphism of locally ringed spaces and of schemes; $X$ is affine.

Assume instead $X_K$ proper. For an arbitrary $k$-scheme $T$ and closed subset $Z\subset X{\times }_{k}T$, its inverse image in $X_K{\times }_K{T}_K$ is closed. Its image in $T_K$ is closed by properness of $X_K$, and its image under the finite closed surjection $T_K\to T$ is exactly the image of $Z$ in $T$. Thus $X\to \mathrm{Spec}k$ is universally closed. Finite type and separatedness were given, so [F3] proves properness. AC is inherited from the scheme/global-section suppliers; no Galois action is assumed for the inseparable extension.