Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient

Let $A\to C$ be a unital ring map and $(t_i{)}_{i\in J}$ a family of variables. By [L1] there is a unique ring homomorphism $\bar{\cdot }:A[t_i]\to C[t_i]$ restricting to $A\to C$ on constants and sending $t_i\mapsto t_i$ for every $i$, so that $\bar{f}={\sum }_a{\bar{c}}_a{t}^a$ for $f={\sum }_a{c}_a{t}^a$; and [L1] applied to the $A$-algebra maps $\bar{\cdot }$ and $C\hookrightarrow C[t_i]$ yields a unique $A$-algebra homomorphism $\phi :A[t_i]{\otimes }_{A}C\to C[t_i]$ with $\phi (x\otimes 1)=\bar{x}$ and $\phi (1\otimes c)=c$, namely $\phi (f\otimes c)=\bar{f}c$; by [L2] $A[t_i]{\otimes }_{A}C$ is a commutative $A$-algebra for this structure.

Let $M\subseteq A$ be multiplicative. By [L4] the map $\mathrm{\Phi }:(M^{-1}A){\otimes }_{A}C\to {\bar{M}}^{-1}C$, $(a\mathrm{/}s)\otimes c\mapsto ac\mathrm{/}s$, is an isomorphism of $M^{-1}A$-modules, and it is multiplicative and unital on elementary tensors: $((a\mathrm{/}s)\otimes c)((a^{\mathrm{\prime }}\mathrm{/}{s}^{\mathrm{\prime }})\otimes c^{\mathrm{\prime }})=(a{a}^{\mathrm{\prime }}\mathrm{/}(s{s}^{\mathrm{\prime }}))\otimes c{c}^{\mathrm{\prime }}$ by [L2], and the localisation arithmetic of [L4] gives $(ac\mathrm{/}s)(a^{\mathrm{\prime }}{c}^{\mathrm{\prime }}\mathrm{/}{s}^{\mathrm{\prime }})=a{a}^{\mathrm{\prime }}c{c}^{\mathrm{\prime }}\mathrm{/}(s{s}^{\mathrm{\prime }})$; hence $\mathrm{\Phi }$ is an isomorphism of commutative rings. It preserves degrees when $A$ is graded, $C$ is graded, $t\in A_1$ is homogeneous and $M=\{t^j:j\ge 0\}$, because $\mathrm{\Phi }((a\mathrm{/}{t}^j)\otimes c)=(a\otimes c)\mathrm{/}{t}^j$ shifts both sides by the same amount in the gradings of [L4].

By [L1] applied over $C$, with the commutative $C$-algebra structure $c\mapsto 1\otimes c$ on $A[t_i]{\otimes }_{A}C$ given by [L2], there is a unique ring homomorphism $\psi :C[t_i]\to A[t_i]{\otimes }_{A}C$ restricting to $c\mapsto 1\otimes c$ and sending $t_i\mapsto t_i\otimes 1$ for every $i$. It is a $C$-algebra homomorphism, and $\psi (\bar{f})=f\otimes 1$ for every $f\in A[t_i]$; the latter identity is independent of the choice of $f$ because a coefficient in the kernel of $A\to C$ tensors to zero.

Apply step 1.2 to the ring map $S\to S_K$, $s\mapsto s\otimes 1$, and the multiplicative set $M=\{x_i^j:j\ge 0\}$: since ${\bar{M}}^{-1}{S}_K=(S_K{)}_{x_i}$, this identifies $S_{x_i}{\otimes }_S{S}_K$ with $(S_K{)}_{x_i}$, and the unit and associativity isomorphisms of [L2] identify $S_{x_i}{\otimes }_S{S}_K=S_{x_i}{\otimes }_S(S{\otimes }_{k}K)$ with $S_{x_i}{\otimes }_{k}K$; the composite is degree-preserving for the gradings in which $x_i$ has degree $1$ on both sides, as in step 1.2, so it restricts to the chart ring $((S_K{)}_{x_i}{)}_0\cong (S_{x_i}{)}_0{\otimes }_{k}K=A_i{\otimes }_{k}K$ of $X_K$ on $D_{+}(x_i)$ (Projective scheme of a homogeneous quotient and its standard affine charts). Since ${\dim }_k{A}_i<\mathrm{\infty }$ by [L7], this chart ring has $K$-dimension ${\dim }_k{A}_i$ by [L2].

For $c\in C$ and $x\in A[t_i]{\otimes }_{A}C$ one has $\phi ((1\otimes c)x)=\phi (1\otimes c)\phi (x)=c\phi (x)$ by [L2], so $\phi$ is $C$-linear; hence $\phi \psi$ is a $C$-algebra endomorphism of $C[t_i]$ with $\phi \psi (t_i)=\phi (t_i\otimes 1)=t_i$ for every $i$, and the identity is a second such endomorphism, so $\phi \psi =\mathrm{i}\mathrm{d}$ by the uniqueness in [L1].

If $D_{+}(x_i)=\mathrm{\varnothing }$, then $A_i=0$: by [L7] a nonzero $A_i$ is Artinian with a maximal ideal and hence has a point. Thus $A_i{\otimes }_{k}K=0$, and both the original and base-changed charts are empty by step 2.2. If $D_{+}(x_i)\mathrm{\ne }\mathrm{\varnothing }$, [L7] gives $A_i\cong {\prod }_{x\in D_{+}(x_i)}{\mathcal{O}}_{X,x}$ with at least one factor; tensoring this isomorphism with $K$ over $k$ and using that a finite product is a finite direct sum together with [L2] gives an isomorphism of $K$-algebras $A_i{\otimes }_{k}K\cong {\prod }_{x\in D_{+}(x_i)}\left({\mathcal{O}}_{X,x}{\otimes }_{k}K\right)$.

A finite-dimensional $K$-algebra is Artinian with all primes maximal by [L6], so the chart ring $A_i{\otimes }_{k}K$ of step 2.2 is zero or of Krull dimension $0$; hence $X_K$ satisfies the chartwise hypothesis of [L7] and all the conclusions of that lemma apply to $X_K$, in particular finiteness of its point set and the finite disjoint-union decomposition into the spectra of the local rings ${\mathcal{O}}_{X_K,y}$.

Every element of $A[t_i]{\otimes }_{A}C$ is a finite sum of elementary tensors by [L2], and $\psi \phi$ and $\mathrm{i}\mathrm{d}$ are ring homomorphisms agreeing on every $t_i\otimes 1$ and every $1\otimes c$, since $\psi \phi (t_i\otimes 1)=t_i\otimes 1$ and $\psi \phi (1\otimes c)=1\otimes c$; indeed $f\otimes c={\sum }_a(a_a\otimes 1)(t^a\otimes 1)(1\otimes c)$ with $a_a\otimes 1=1\otimes {\bar{a}}_a$, so both maps are additive and multiplicative on a set of elements in terms of which every element is written; therefore $\psi \phi =\mathrm{i}\mathrm{d}$ and $\phi$ is an isomorphism of $A$-algebras $A[t_i]{\otimes }_{A}C⟶C[t_i],f\otimes c⟼\bar{f}c\mathrm{.}$

By step 2.2 the chart of $X_K$ is $\mathrm{Spec}(A_i{\otimes }_{k}K)$. If $D_{+}(x_i)=\mathrm{\varnothing }$, this chart is empty by step 3.2 and contributes no points. Otherwise step 3.2 has a nonempty finite product, so [L5] identifies its points with those of ${\coprod }_{x\in D_{+}(x_i)}\mathrm{Spec}({\mathcal{O}}_{X,x}{\otimes }_{k}K)$ and its local rings with the localizations of the factors ${\mathcal{O}}_{X,x}{\otimes }_{k}K$. The chart correspondence of [L7] then identifies the local ring of $X_K$ at a point of the chart with the localization of the chart ring at the corresponding prime; hence the points $y\in X_K$ lying over a given $x\in X$ are exactly the maximal ideals $\mathfrak{m}$ of $B_x:={\mathcal{O}}_{X,x}{\otimes }_{k}K$, and ${\mathcal{O}}_{X_K,y}\cong (B_x{)}_{\mathfrak{m}}$.

Let $J\subseteq A[t_i]$ be an ideal. The sequence $J\to A[t_i]\to A[t_i]\mathrm{/}J\to 0$ is exact, so by [L3] the sequence $J{\otimes }_{A}C\to A[t_i]{\otimes }_{A}C\to (A[t_i]\mathrm{/}J){\otimes }_{A}C\to 0$ is exact and $(A[t_i]\mathrm{/}J){\otimes }_{A}C$ is the cokernel of the first map; under the isomorphism of step 4.1 that map has image the finite sums ${\sum }_k{\bar{j}}_k{c}_k$, which is $JC[t_i]$, the submodule of $C[t_i]$ generated by the products $jm$ with $j\in J$ and $m\in C[t_i]$ by [L3]; hence $(A[t_i]\mathrm{/}J){\otimes }_{A}C\cong C[t_i]\mathrm{/}JC[t_i]$.

Since $X_K$ is finite by step 3.3, its total length is the finite sum ${\mathrm{len}}_K(X_K)={\sum }_y{\mathrm{\ell }}_{{\mathcal{O}}_{X_K,y}}({\mathcal{O}}_{X_K,y})[\kappa (y):K]$ by [L7]; grouping the points by the point $x\in X$ over which they lie, using step 4.2, and applying the affine consistency in [L7] to the finite-dimensional $K$-algebra $B_x$, whose spectrum has exactly the points $y$ over $x$ with local rings $(B_x{)}_{\mathfrak{m}}$, gives ${\mathrm{len}}_K(X_K)={\sum }_x{\dim }_K{B}_x$.

Applying step 5.1 with $A=k$, $C=K$, the variables $x_0,\dots ,x_n$ and the ideal $I$ gives a $k$-algebra isomorphism $S_K=S{\otimes }_{k}K\cong K[x_0,\dots ,x_n]\mathrm{/}IK[x_0,\dots ,x_n]$ carrying $S_m{\otimes }_{k}K$ onto the degree-$m$ part. The extended ideal is generated by the images of all homogeneous elements of $I$ and so is homogeneous (homogeneous polynomial and homogeneous ideal); no finite homogeneous generating set is needed here. Hence the quotient is a standard graded $K$-algebra generated in degree one by the images of the variables, by the description of polynomial rings in [L1], and $X_K=\mathrm{Proj}(S_K)$ is defined in the sense of Projective scheme of a homogeneous quotient and its standard affine charts. By [L2] the degree-$m$ part of $S_K$ is $S_m{\otimes }_{k}K$, whence ${\dim }_K(S_K{)}_m={\dim }_k{S}_m$ for every $m$.

For every $x\in X$ one has ${\dim }_K{B}_x={\dim }_K({\mathcal{O}}_{X,x}{\otimes }_{k}K)={\dim }_k{\mathcal{O}}_{X,x}$ by [L2], and the affine consistency in [L7] applied to the finite-dimensional local $k$-algebra ${\mathcal{O}}_{X,x}$, whose only maximal ideal has residue field $\kappa (x)$, gives ${\dim }_k{\mathcal{O}}_{X,x}={\mathrm{\ell }}_{{\mathcal{O}}_{X,x}}({\mathcal{O}}_{X,x})[\kappa (x):k]$; hence ${\mathrm{len}}_K(X_K)={\sum }_x{\mathrm{\ell }}_{{\mathcal{O}}_{X,x}}({\mathcal{O}}_{X,x})[\kappa (x):k]={\mathrm{len}}_k(X)$ by [L7].

Claim 1 is steps 4.1 and 6.1, claim 2 is steps 2.2 and 3.3, and claim 3 is steps 5.2 and 6.2; the Axiom of Choice enters only through the Artinian-structure, prime-existence and prime-lifting suppliers cited in [L6] and [L7], and no finiteness, separability or algebraicness of the extension $K\mathrm{/}k$ was used.