Extension of scalars of a scheme along a field extension

Principal opens under base change. Let $A$ be a $k$-algebra and $f\in A$. The $k$-algebra map $A_f\to (A{\otimes }_{k}K{)}_{f\otimes 1}$, $a\mathrm{/}{f}^n\mapsto (a\otimes 1)(f\otimes 1{)}^{-n}$, is well defined by [F6], and after tensoring with $K$ gives a $(A{\otimes }_{k}K{)}_{f\otimes 1}$-linear map $A_f{\otimes }_{k}K\to (A{\otimes }_{k}K{)}_{f\otimes 1}$, because $(A{\otimes }_{k}K{)}_{f\otimes 1}$ is an $A_f$-algebra by [F7]. Conversely $a\otimes \lambda \mapsto (a\mathrm{/}1)\otimes \lambda$ is a $k$-algebra map $A{\otimes }_{k}K\to A_f{\otimes }_{k}K$ carrying $f\otimes 1$ to a unit, so by [F6] it factors through a map $(A{\otimes }_{k}K{)}_{f\otimes 1}\to A_f{\otimes }_{k}K$. The two maps are inverse on the generators $a\otimes \lambda$ and $1\mathrm{/}(f\otimes 1)$; composing with $\mathrm{Spec}$ by [F2], the principal open $D(f\otimes 1)$ of $\mathrm{Spec}(A{\otimes }_{k}K)$ is canonically $\mathrm{Spec}(A_f{\otimes }_{k}K)$, compatibly with further principal localisations $f\mapsto f^n$ and with the restriction maps of [F3].

Fibre rings. Let $A$ be a $k$-algebra and $\mathfrak{p}\in \mathrm{Spec}A$ with residue field $\kappa (\mathfrak{p})$ as in [F9]; then $\kappa (\mathfrak{p})=A_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{A}_{\mathfrak{p}}=A\mathrm{/}\mathfrak{p}{\otimes }_A{A}_{\mathfrak{p}}$, and the composite of the canonical isomorphisms $(A{\otimes }_{k}K){\otimes }_A\kappa (\mathfrak{p})\cong \kappa (\mathfrak{p}){\otimes }_A(A{\otimes }_{k}K)\cong (\kappa (\mathfrak{p}){\otimes }_{A}A){\otimes }_{k}K\cong \kappa (\mathfrak{p}){\otimes }_{k}K$ of [F5] and [F8] sends $(a\otimes \lambda )\otimes c$ to $(ca)\otimes \lambda$. Hence there is a canonical $\kappa (\mathfrak{p})$-linear isomorphism $(A{\otimes }_{k}K){\otimes }_A\kappa (\mathfrak{p})\cong \kappa (\mathfrak{p}){\otimes }_{k}K$.

Common principal neighbourhoods. For affine opens $U=\mathrm{Spec}A$, $V=\mathrm{Spec}B$ and $x\in U\cap V$, first take $x\in D_U(f)\subseteq V$ using the principal-open basis. Then take $x\in D_V(g)\subseteq D_U(f)$. The restriction of $g$ to $D_U(f)$ is $a\mathrm{/}{f}^r\in A_f$ by [F3], and its nonvanishing locus there is both $D_V(g)$ and $D_U(fa)$: the equality follows by applying the residue-field maps of the open immersion to this section. Thus $D_V(g)=D_U(fa)$ is principal in both original affines. This supplies the common principal refinements needed below, with independent denominators in the two coordinate rings.

Gluing. Let $X$ be a scheme over $k$ and let $\{U_i=\mathrm{Spec}{A}_i\}$ be a family of affine opens covering $X$; each $A_i$ is a $k$-algebra because $X\to \mathrm{Spec}k$ restricts to $U_i$. For each $i$ the $k$-algebra map $A_i\to A_i{\otimes }_{k}K$ gives by [F2] a morphism $\mathrm{Spec}(A_i{\otimes }_{k}K)\to U_i\subseteq X$, and the affine pieces over distinct $i$ are to be identified over the principal opens. Whenever $W$ is an open subscheme of $X$ which is principal in $U_i$ and in $U_j$, say $W=\mathrm{Spec}(A_i{)}_f=\mathrm{Spec}(A_j{)}_g$, the rings $(A_i{)}_f$ and $(A_j{)}_g$ are both $\mathrm{\Gamma }(W,{\mathcal{O}}_X)$ by [F3] and hence canonically equal, and the identity of rings induces by step 1.1 an identification of the corresponding open subschemes $\mathrm{Spec}((A_i{)}_f{\otimes }_{k}K)$ and $\mathrm{Spec}((A_j{)}_g{\otimes }_{k}K)$. These identifications are induced by identities of section rings and are therefore compatible: the identity and cocycle conditions hold on triple overlaps because all the identifications are the canonical comparison of $\mathrm{\Gamma }(W,{\mathcal{O}}_X){\otimes }_{k}K$ with itself. Since step 1.3 covers every overlap by such common principal opens, the data satisfy the hypotheses of [F1], which glues the schemes $\mathrm{Spec}(A_i{\otimes }_{k}K)$ to a scheme $X_K$ with open affine cover $\{\mathrm{Spec}(A_i{\otimes }_{k}K)\}$, and the morphisms to $X$ glue to $X_K\to X$. The maps to $\mathrm{Spec}K$ induced by $\lambda \mapsto 1\otimes \lambda$ also agree on overlaps, so they glue to the $K$-scheme structure on $X_K$.

Affine restriction. Let $U=\mathrm{Spec}A$ be any affine open of $X$. By step 1.3 cover each $U_i\cap U$ by common principal opens $W=D_{U_i}(f)=D_U(h)$. Their section rings $(A_i{)}_f$ and $A_h$ are canonically identified by [F3]. Step 1.1 identifies the corresponding base-changed opens with $\mathrm{Spec}(\mathrm{\Gamma }(W,{\mathcal{O}}_X){\otimes }_{k}K)$ on either side. These opens cover the inverse image of $U$ in $X_K$ and cover $\mathrm{Spec}(A{\otimes }_{k}K)$: a principal cover remains a cover under inverse image, since $D(h\otimes 1)$ is precisely the inverse image of $D(h)$. The identifications agree on common refinements by step 1.1, so glue to an isomorphism over both $U$ and $\mathrm{Spec}K$ by [F1]. Hence the restriction is the asserted affine base change.

Independence of the cover. Let $\{U_i\}$ and $\{V_j\}$ be two affine covers of $X$. By step 1.3 the family of open subschemes $W$ of $X$ that are principal in some $U_i$ and in some $V_j$ covers $X$. For such a $W=\mathrm{Spec}B$, with $B=\mathrm{\Gamma }(W,{\mathcal{O}}_X)$ by [F3], the construction of step 2.1 attaches to $W$ the affine scheme $\mathrm{Spec}(B{\otimes }_{k}K)$ in the glueing over $\{U_i\}$ and, by the same computation, in the glueing over $\{V_j\}$: in both cases $W$ arises as a principal open of an affine chart, and the attached piece is $\mathrm{Spec}$ of the localisation of the chart ring tensored with $K$, which is $B{\otimes }_{k}K$ by step 1.1. Both glued schemes are therefore obtained by glueing the same family $\{\mathrm{Spec}(B{\otimes }_{k}K)\}$ along the same canonical identifications over principal opens of $W$. By the uniqueness clause of [F1], applied to the two open affine covers of $X_K$ and $X_K^{\mathrm{\prime }}$, the canonical chart identifications glue to an isomorphism $X_K\to X_K^{\mathrm{\prime }}$ over $X$ and $\mathrm{Spec}K$. Any other such isomorphism must preserve each inverse image of $W$; on its ring $\mathrm{\Gamma }(W,{\mathcal{O}}_X){\otimes }_{k}K$, the induced map fixes both factors because it is over $W$ and over $K$. It is therefore the identity by [F7]. These opens cover, proving uniqueness with both compatibilities.

Fibres. Let $x\in X$ and let $U=\mathrm{Spec}A\subseteq X$ be an affine open containing $x$. By step 3.1 the preimage of $U$ in $X_K$ is $\mathrm{Spec}(A{\otimes }_{k}K)$ over $U$, and the scheme over $\kappa (x)$ attached to the point $x$ of that affine piece is $\mathrm{Spec}((A{\otimes }_{k}K){\otimes }_A\kappa (x))$, which by step 1.2 is canonically $\mathrm{Spec}(\kappa (x){\otimes }_{k}K)$ over $\mathrm{Spec}\kappa (x)$. If $V=\mathrm{Spec}B$ is a second affine open containing $x$, choose $W$ principal in $U$ and in $V$ with $x\in W$, say $W=\mathrm{Spec}(A_f)=\mathrm{Spec}(B_g)$, using step 1.3 and [F3]; then $(A{\otimes }_{k}K){\otimes }_A{A}_f\cong A_f{\otimes }_{k}K$ and $(B{\otimes }_{k}K){\otimes }_B{B}_g\cong B_g{\otimes }_{k}K$ are canonically the same ring, so tensoring the identification with $\kappa (x)$ over the common ring $A_f=B_g=\mathrm{\Gamma }(W,{\mathcal{O}}_X)$ identifies $(A{\otimes }_{k}K){\otimes }_A\kappa (x)$ with $(B{\otimes }_{k}K){\otimes }_B\kappa (x)$ canonically. Hence the fibre is independent of the affine neighbourhood and is $\mathrm{Spec}(\kappa (x){\otimes }_{k}K)$ as asserted.

Transitivity. Let $K\subseteq L$ be a tower, and write $(X_K{)}_L$ for the construction applied over the base field $K$ to the extension $L\mathrm{/}K$. The affine pieces $\mathrm{Spec}(A_i{\otimes }_{k}K)$ constructed in step 2.1 form an affine cover of $X_K$, so the construction of $(X_K{)}_L$ glues the schemes $\mathrm{Spec}((A_i{\otimes }_{k}K){\otimes }_{K}L)$. The canonical isomorphisms $(A_i{\otimes }_{k}K){\otimes }_{K}L\cong A_i{\otimes }_{k}L$ of [F7] and [F8] are compatible with the transition identifications of step 2.1, because those are induced by identities of section rings; hence $X_L$, which is glued from the pieces $\mathrm{Spec}(A_i{\otimes }_{k}L)$, and $(X_K{)}_L$ are glued from corresponding pieces with corresponding identifications. By step 3.2 (applied to the two covers of the same scheme, or directly by the uniqueness clause of [F1]) the displayed chart isomorphisms glue to an isomorphism $(X_K{)}_L\to X_L$ over $X$ and $\mathrm{Spec}L$. It is unique with both compatibilities, by the same two-factor argument as step 3.2, proving clause 4.