Global sections commute with extension of scalars over a field

Choose a finite affine open cover $X={\bigcup }_{j=1}^r{U}_j$, using quasi-compactness. Each $U_j\cap U_l$ is affine: it is the inverse image of the closed diagonal under $U_j{\times }_k{U}_l\to X{\times }_{k}X$, hence a closed subscheme of an affine scheme. The sheaf gluing axiom gives an exact sequence beginning with $0\to \mathrm{\Gamma }(X,{\mathcal{O}}_X)\to {\prod }_j\mathrm{\Gamma }(U_j,{\mathcal{O}}_X)\to {\prod }_{j,l}\mathrm{\Gamma }(U_j\cap U_l,{\mathcal{O}}_X)$, where the last arrow takes differences of restrictions.

Every $k$-module is a vector space and is flat: a short exact sequence of vector spaces splits by extending a basis, so tensoring with any vector space preserves its exactness. In the particular equalizer in step 1.1 this can also be checked using the finitely many linearly independent coefficients of each tensor, with only finite basis selections. Tensor that equalizer with $R$. Finite products commute with this tensor product. By [F1] the resulting rings are precisely the rings of the affine opens $(U_j{)}_R$ and their intersections. Their equalizer is the global-section ring of $X_R$ by the same sheaf gluing axiom. This identifies the natural map in the statement with an isomorphism. The coefficient argument uses no arbitrary basis choice.