Flat field extension commutes with coherent cohomology

Given: A field $k$, a field extension $K\mathrm{/}k$, a proper morphism $X\to \mathrm{Spec}k$, a coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the base change $g:X_K\to X$ and ${\mathcal{F}}_K=g^{\ast }\mathcal{F}$; the Axiom of Choice is inherited from the cited suppliers.

Properness unpacked: a proper morphism is separated, of finite type and universally closed; a morphism of finite type is quasi-compact; a quasi-compact morphism pulls quasi-compact open subsets back to quasi-compact open subsets; $\mathrm{Spec}k$ is quasi-compact and affine opens form a basis of every scheme, so $X$ is quasi-compact and admits a finite affine open cover $U_0,\dots ,U_r$, the empty cover occurring exactly when $X=\mathrm{\varnothing }$. (Proper morphisms, Locally finite type and finite type morphisms, Quasi-compact and quasi-separated morphisms, Every affine scheme is quasi-compact, Schemes, Affine open subschemes)

Properness is stable under arbitrary base change, so the projection $X_K\to \mathrm{Spec}K$ is proper; a proper morphism is separated and quasi-compact, so $X_K$ is quasi-compact and separated over $K$, in particular as a scheme. (Properness survives arbitrary base change, Proper morphisms)

Affine intersections: for affine opens of a scheme separated over an affine base, the intersection is affine, so every nonempty finite intersection $U_I={\bigcap }_{i\in I}{U}_i$ of the members of a finite affine open cover of the separated $k$-scheme $X$ is affine, say $U_I=\mathrm{Spec}{B}_I$ with $B_I={\mathcal{O}}_X(U_I)$, and an empty intersection is the affine scheme $\mathrm{Spec}0$. (Affine-overlap criterion for separatedness, The underlying space of an affine spectrum)

Base change of the cover: for every finite $I$ the fibre product $U_{I,K}:=U_I{\times }_{\mathrm{Spec}k}\mathrm{Spec}K$ is the affine scheme $\mathrm{Spec}(B_I{\otimes }_{k}K)$ over $\mathrm{Spec}K$, the projection $U_{I,K}\to X_K$ is an open immersion with image $g^{-1}{U}_I$, and the opens $U_{0,K},\dots ,U_{r,K}$ form a finite affine open cover of $X_K$ whose finite intersections are the $U_{I,K}$, so the base-changed cover has the same index set and the same nerve as $(U_i)$. (Affine fibre products are spectra of tensor products, Restricting fibre products to open subschemes)

Sections of the pullback on the affine charts: ${\mathcal{F}}_K$ is quasi-coherent, and for an affine open $U=\mathrm{Spec}B\subseteq X$ with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ the affine form of the pullback lemma gives a canonical isomorphism $g^{\ast }\mathcal{F}{\mathrm{\mid }}_{U_K}\cong \widetilde{((B{\otimes }_{k}K){\otimes }_{B}M)}\cong \widetilde{(M{\otimes }_{k}K)}$; under it the canonical morphism $g^{-1}\mathcal{F}\to {\mathcal{F}}_K$ sends a section $m\in \mathcal{F}(U)=M$ to $m\otimes 1$. Since this morphism is a morphism of sheaves, the identifications commute with restriction maps. In particular $\mathcal{F}(U_I)=M_I$ with ${\mathcal{F}}_K(U_{I,K})\cong M_I{\otimes }_{k}K$ for every finite $I$, the associated sheaf being evaluated on the affine scheme $U_{I,K}$. (Scheme pullback preserves quasi-coherence, Affine quasi-coherent sheaves are modules, Module sheaf on an affine scheme, Pullback of a module along a morphism of ringed spaces, Associativity of tensor products for compatible bimodules, The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$)

Flatness of a field extension: every module over the field $k$ is flat (Modules over a field are projective, flat, and injective), so $K$ is flat over $k$ and $-{\otimes }_{k}K$ preserves exact sequences (Flat and faithfully flat modules and ring homomorphisms); tensoring is right exact (Tensoring is right exact) and commutes with arbitrary direct sums and with the unit, $k{\otimes }_{k}K\cong K$ (Tensor products commute with arbitrary direct sums, The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$).

Finiteness on the two sides: for a scheme $Y$ proper over a field $F$ and a coherent $\mathcal{G}$ on $Y$, every $H^q(Y,\mathcal{G})$ is a finite-dimensional $F$-vector space and only finitely many are nonzero, so the Euler characteristic $\chi (Y,\mathcal{G})={\sum }_{q\ge 0}(-1{)}^q{\dim }_F{H}^q(Y,\mathcal{G})$ is a well-defined integer, equal to any finite truncation ${\sum }_{q=0}^{n-1}(-1{)}^q{\dim }_F{H}^q(Y,\mathcal{G})$ for a finite affine open cover with $n$ members. (Finite-dimensional coherent cohomology over a field, Euler characteristic of a coherent sheaf)

Dimension of an extension of scalars: if $V$ is a finite-dimensional vector space over a field $F$ and $F^{\mathrm{\prime }}\mathrm{/}F$ is a field extension, then ${\dim }_{F^{\mathrm{\prime }}}(V{\otimes }_F{F}^{\mathrm{\prime }})={\dim }_{F}V$: a finite basis $v_1,\dots ,v_n$ identifies $V$ with $F^n$, hence $V{\otimes }_F{F}^{\mathrm{\prime }}\cong F^n{\otimes }_F{F}^{\mathrm{\prime }}\cong (F^{\mathrm{\prime }}{)}^n$ by right exactness and the unit and direct-sum compatibilities of the tensor product, and the empty basis covers $V=0$. (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Linear combination of a finite list, and the span $\mathrm{span}(S)$ as the smallest linear subspace containing $S$, Tensoring is right exact, Tensor products commute with arbitrary direct sums, The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$)

The base-change map and its Čech description: in the Cartesian square over $\mathrm{Spec}K\to \mathrm{Spec}k$ the base-change map $g^{\ast }{R}^q{f}_{\ast }\mathcal{F}\to R^q{f}_{\ast }^{\mathrm{\prime }}{\mathcal{F}}_K$ is, on an affine open $V^{\mathrm{\prime }}=\mathrm{Spec}B$ lying over an affine open $V=\mathrm{Spec}A$, the extension of scalars of the pullback map $H^q(f^{-1}V,\mathcal{F})\to H^q(f^{\mathrm{\prime }-1}{V}^{\mathrm{\prime }},{\mathcal{F}}_K)$; it is functorial in the square and compatible with compositions of base changes. Here both bases are affine, so $g^{\ast }{R}^q{f}_{\ast }\mathcal{F}(\mathrm{Spec}K)=K{\otimes }_k{H}^q(X,\mathcal{F})$ and $R^q{f}_{\ast }^{\mathrm{\prime }}{\mathcal{F}}_K(\mathrm{Spec}K)=H^q(X_K,{\mathcal{F}}_K)$, and the map on global sections is the map ${\kappa }^q$ of the statement. Under the Čech comparison isomorphisms of [F10], the pullback of cohomology classes is computed by the pullback of Čech cochains: the comparison maps are the canonical maps $H^p(u{)}^{-1}{H}^p(w)$ assembled from the Čech–Godement double complex, whose Godement resolution is built from the stalks of the coefficient sheaf, and the pullback map is induced by the section pullback $\mathrm{\Gamma }(X,-)\Rightarrow \mathrm{\Gamma }(X_K,g^{-1}(-))$ composed with the canonical $g^{-1}\mathcal{F}\to {\mathcal{F}}_K$; by [F5] the latter sends a section $m$ over $U_I$ to $m\otimes 1$ over $U_{I,K}$, which is the termwise map $\gamma$ below, so the two canonical maps agree under the comparison isomorphisms. (Cohomology and base-change map, Higher direct images localize over an affine base, Variance of sheaf cohomology, Canonical map from fixed-cover Čech to sheaf cohomology, Godement resolution of an abelian sheaf)

Čech comparison: for a quasi-compact separated scheme $Y$ with a finite affine open cover whose finite intersections are affine and a quasi-coherent $\mathcal{G}$, the ordered Čech cohomology computed from the complex $C^{\bullet }$ is canonically isomorphic to sheaf cohomology, ${\check{H}}^q(\mathcal{U},\mathcal{G})\cong H^q(Y,\mathcal{G})$ for every $q\ge 0$, the complex is bounded with $C^p=0$ for $p<0$ and $p>r$, and the comparison is natural in the coefficient sheaf. (Cech cohomology computes quasi-coherent cohomology on a separated scheme, Ordered Čech cochain complex of a cover, Sheaf cohomology as right derived global sections)

The Axiom of Choice and the Axiom of Dependent Choice are the choice principles named in the statement; AC is inherited here from the Čech comparison and the finiteness corollary, and DC is inherited from the same suppliers. (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

Every field is a Noetherian ring because its only ideals are the zero ideal and the whole field; every finitely generated algebra over a Noetherian ring is Noetherian. (A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring)