The Stacks project

Lemma 57.89.5. Let $K$ be a field. Let $X$ be a scheme of finite type over $K$. Let $x \in X$. Set $a = \text{trdeg}_ K(\kappa (x))$ and $d = \dim _ x(X)$. Then there is a map

\[ K(t_1, \ldots , t_ a)^{sep} \longrightarrow \mathcal{O}_{X, x}^{sh} \]

such that

  1. the residue field of $\mathcal{O}_{X, x}^{sh}$ is a purely inseparable extension of $K(t_1, \ldots , t_ a)^{sep}$,

  2. $\mathcal{O}_{X, x}^{sh}$ is a filtered colimit of finite type $K(t_1, \ldots , t_ a)^{sep}$-algebras of dimension $\leq d - a$.

Proof. We may assume $X$ is affine. By Noether normalization, after possibly shrinking $X$ again, we can choose a finite morphism $\pi : X \to \mathbf{A}^ d_ K$, see Algebra, Lemma 10.114.5. Since $\kappa (x)$ is a finite extension of the residue field of $\pi (x)$, this residue field has transcendence degree $a$ over $K$ as well. Thus we can find a finite morphism $\pi ' : \mathbf{A}^ d_ K \to \mathbf{A}^ d_ K$ such that $\pi '(\pi (x))$ corresponds to the generic point of the linear subspace $\mathbf{A}^ a_ K \subset \mathbf{A}^ d_ K$ given by setting the last $d - a$ coordinates equal to zero. Hence the composition

\[ X \xrightarrow {\pi ' \circ \pi } \mathbf{A}^ d_ K \xrightarrow {p} \mathbf{A}^ a_ K \]

of $\pi ' \circ \pi $ and the projection $p$ onto the first $a$ coordinates maps $x$ to the generic point $\eta \in \mathbf{A}^ a_ K$. The induced map

\[ K(t_1, \ldots , t_ a)^{sep} = \mathcal{O}_{\mathbf{A}^ a_ k, \eta }^{sh} \longrightarrow \mathcal{O}_{X, x}^{sh} \]

on étale local rings satisfies (1) since it is clear that the residue field of $\mathcal{O}_{X, x}^{sh}$ is an algebraic extension of the separably closed field $K(t_1, \ldots , t_ a)^{sep}$. On the other hand, if $X = \mathop{\mathrm{Spec}}(B)$, then $\mathcal{O}_{X, x}^{sh} = \mathop{\mathrm{colim}}\nolimits B_ j$ is a filtered colimit of étale $B$-algebras $B_ j$. Observe that $B_ j$ is quasi-finite over $K[t_1, \ldots , t_ d]$ as $B$ is finite over $K[t_1, \ldots , t_ d]$. We may similarly write $K(t_1, \ldots , t_ a)^{sep} = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of étale $K[t_1, \ldots , t_ a]$-algebras. For every $i$ we can find an $j$ such that $A_ i \to K(t_1, \ldots , t_ a)^{sep} \to \mathcal{O}_{X, x}^{sh}$ factors through a map $\psi _{i, j} : A_ i \to B_ j$. Then $B_ j$ is quasi-finite over $A_ i[t_{a + 1}, \ldots , t_ d]$. Hence

\[ B_{i, j} = B_ j \otimes _{\psi _{i, j}, A_ i} K(t_1, \ldots , t_ a)^{sep} \]

has dimension $\leq d - a$ as it is quasi-finite over $K(t_1, \ldots , t_ a)^{sep}[t_{a + 1}, \ldots , t_ d]$. The proof of (2) is now finished as $\mathcal{O}_{X, x}^{sh}$ is a filtered colimit1 of the algebras $B_{i, j}$. Some details omitted. $\square$

[1] Let $R$ be a ring. Let $A = \mathop{\mathrm{colim}}\nolimits _{i \in I} A_ i$ be a filtered colimit of finitely presented $R$-algebras. Let $B = \mathop{\mathrm{colim}}\nolimits _{j \in J} B_ j$ be a filtered colimit of $R$-algebras. Let $A \to B$ be an $R$-algebra map. Assume that for all $i \in I$ there is a $j \in J$ and an $R$-algebra map $\psi _{i, j} : A_ i \to B_ j$. Say $(i', j', \psi _{i', j'}) \geq (i, j, \psi _{i, j})$ if $i' \geq i$, $j' \geq j$, and $\psi _{i, j}$ and $\psi _{i', j'}$ are compatible. Then the collection of triples forms a directed set and $B = \mathop{\mathrm{colim}}\nolimits B_ j \otimes _{\psi _{i, j} A_ i} A$.

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