The Stacks project

66.11 Residue fields and henselian local rings

For a decent algebraic space we can define the residue field and the henselian local ring at a point. For example, the following lemma tells us the residue field of a point on a decent space is defined.

Lemma 66.11.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the map

\[ \{ \mathop{\mathrm{Spec}}(k) \to X \text{ monomorphism where }k\text{ is a field}\} \longrightarrow |X| \]

This map is always injective. If $X$ is decent then this map is a bijection.

Proof. We have seen in Properties of Spaces, Lemma 64.4.11 that the map is an injection in general. By Lemma 66.5.1 it is surjective when $X$ is decent (actually one can say this is part of the definition of being decent). $\square$

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If a point $x \in |X|$ can be represented by a monomorphism $\mathop{\mathrm{Spec}}(k) \to X$, then the field $k$ is unique up to unique isomorphism. For a decent algebraic space such a monomorphism exists for every point by Lemma 66.11.1 and hence the following definition makes sense.

Definition 66.11.2. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The residue field of $X$ at $x$ is the unique field $\kappa (x)$ which comes equipped with a monomorphism $\mathop{\mathrm{Spec}}(\kappa (x)) \to X$ representing $x$.

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of decent algebraic spaces over $S$. Let $x \in |X|$ be a point. Set $y = f(x) \in |Y|$. Then the composition $\mathop{\mathrm{Spec}}(\kappa (x)) \to Y$ is in the equivalence class defining $y$ and hence factors through $\mathop{\mathrm{Spec}}(\kappa (y)) \to Y$. In other words we get a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(\kappa (x)) \ar[r]_-x \ar[d] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(\kappa (y)) \ar[r]^-y & Y } \]

The left vertical morphism corresponds to a homomorphism $\kappa (y) \to \kappa (x)$ of fields. We will often simply call this the homomorphism induced by $f$.

Lemma 66.11.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of decent algebraic spaces over $S$. Let $x \in |X|$ be a point with image $y = f(x) \in |Y|$. The following are equivalent

  1. $f$ induces an isomorphism $\kappa (y) \to \kappa (x)$, and

  2. the induced morphism $\mathop{\mathrm{Spec}}(\kappa (x)) \to Y$ is a monomorphism.

Proof. Immediate from the discussion above. $\square$

The following lemma tells us that the henselian local ring of a point on a decent algebraic space is defined.

Lemma 66.11.4. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. For every point $x \in |X|$ there exists an étale morphism

\[ (U, u) \longrightarrow (X, x) \]

where $U$ is an affine scheme, $u$ is the only point of $U$ lying over $x$, and the induced homomorphism $\kappa (x) \to \kappa (u)$ is an isomorphism.

Proof. We may assume that $X$ is quasi-compact by replacing $X$ with a quasi-compact open containing $x$. Recall that $x$ can be represented by a quasi-compact (mono)morphism from the spectrum a field (by definition of decent spaces). Thus the lemma follows from Lemma 66.8.3. $\square$

Definition 66.11.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in X$ be a point. An elementary étale neighbourhood is an étale morphism $(U, u) \to (X, x)$ where $U$ is a scheme, $u \in U$ is a point mapping to $x$, and $\kappa (x) \to \kappa (u)$ is an isomorphism. A morphism of elementary étale neighbourhoods $(U, u) \to (U', u')$ is defined as a morphism $U \to U'$ over $X$ mapping $u$ to $u'$.

If $X$ is not decent then the category of elementary étale neighbourhoods may be empty.

Lemma 66.11.6. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x$ be a point of $X$. The category of elementary étale neighborhoods of $(X, x)$ is cofiltered (see Categories, Definition 4.20.1).

Proof. The category is nonempty by Lemma 66.11.4. Suppose that we have two elementary étale neighbourhoods $(U_ i, u_ i) \to (X, x)$. Then consider $U = U_1 \times _ X U_2$. Since $\mathop{\mathrm{Spec}}(\kappa (u_ i)) \to X$, $i = 1, 2$ are both monomorphisms in the class of $x$ (Lemma 66.11.3) , we see that

\[ u = \mathop{\mathrm{Spec}}(\kappa (u_1)) \times _ X \mathop{\mathrm{Spec}}(\kappa (u_2)) \]

is the spectrum of a field $\kappa (u)$ such that the induced maps $\kappa (u_ i) \to \kappa (u)$ are isomorphisms. Then $u \to U$ is a point of $U$ and we see that $(U, u) \to (X, x)$ is an elementary étale neighbourhood dominating $(U_ i, u_ i)$. If $a, b : (U_1, u_1) \to (U_2, u_2)$ are two morphisms between our elementary étale neighbourhoods, then we consider the scheme

\[ U = U_1 \times _{(a, b), (U_2 \times _ X U_2), \Delta } U_2 \]

Using Properties of Spaces, Lemma 64.16.6 we see that $U \to X$ is étale. Moreover, in exactly the same manner as before we see that $U$ has a point $u$ such that $(U, u) \to (X, x)$ is an elementary étale neighbourhood. Finally, $U \to U_1$ equalizes $a$ and $b$ and the proof is finished. $\square$

Definition 66.11.7. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The henselian local ring of $X$ at $x$, is

\[ \mathcal{O}_{X, x}^ h = \mathop{\mathrm{colim}}\nolimits \Gamma (U, \mathcal{O}_ U) \]

where the colimit is over the elementary étale neighbourhoods $(U, u) \to (X, x)$.

Here is the analogue of Properties of Spaces, Lemma 64.22.1.

Lemma 66.11.8. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. Let $(U, u) \to (X, x)$ be an elementary étale neighbourhood. Then

\[ \mathcal{O}_{X, x}^ h = \mathcal{O}_{U, u}^ h \]

In words: the henselian local ring of $X$ at $x$ is equal to the henselization $\mathcal{O}_{U, u}^ h$ of the local ring $\mathcal{O}_{U, u}$ of $U$ at $u$.

Proof. Since the category of elementary étale neighbourhood of $(X, x)$ is cofiltered (Lemma 66.11.6) we see that the category of elementary étale neighbourhoods of $(U, u)$ is initial in the category of elementary étale neighbourhood of $(X, x)$. Then the equality follows from More on Morphisms, Lemma 37.31.5 and Categories, Lemma 4.17.2 (initial is turned into cofinal because the colimit definining henselian local rings is over the opposite of the category of elementary étale neighbourhoods). $\square$

Lemma 66.11.9. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$ lying over $x \in |X|$. The étale local ring $\mathcal{O}_{X, \overline{x}}$ of $X$ at $\overline{x}$ (Properties of Spaces, Definition 64.22.2) is the strict henselization of the henselian local ring $\mathcal{O}_{X, x}^ h$ of $X$ at $x$.

Proof. Follows from Lemma 66.11.8, Properties of Spaces, Lemma 64.22.1 and the fact that $(R^ h)^{sh} = R^{sh}$ for a local ring $(R, \mathfrak m, \kappa )$ and a given separable algebraic closure $\kappa ^{sep}$ of $\kappa $. This equality follows from Algebra, Lemma 10.153.6. $\square$

Lemma 66.11.10. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The residue field of the henselian local ring of $X$ at $x$ (Definition 66.11.7) is the residue field of $X$ at $x$ (Definition 66.11.2).

Proof. Choose an elementary étale neighbourhood $(U, u) \to (X, x)$. Then $\kappa (u) = \kappa (x)$ and $\mathcal{O}_{X, x}^ h = \mathcal{O}_{U, u}^ h$ (Lemma 66.11.8). The residue field of $\mathcal{O}_{U, u}^ h$ is $\kappa (u)$ by Algebra, Lemma 10.154.1 (the output of this lemma is the construction/definition of the henselization of a local ring, see Algebra, Definition 10.154.3). $\square$

Remark 66.11.11. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of decent algebraic spaces over $S$. Let $x \in |X|$ with image $y \in |Y|$. Choose an elementary étale neighbourhood $(V, v) \to (Y, y)$ (possible by Lemma 66.11.4). Then $V \times _ Y X$ is an algebraic space étale over $X$ which has a unique point $x'$ mapping to $x$ in $X$ and to $v$ in $V$. (Details omitted; use that all points can be represented by monomorphisms from spectra of fields.) Choose an elementary étale neighbourhood $(U, u) \to (V \times _ Y X, x')$. Then we obtain the following commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}) \ar[r] \ar[d] & U \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}) \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}^ h) \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}_{V, v}) \ar[r] & V \ar[r] & Y } \]

This comes from the identifications $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}$, $\mathcal{O}_{X, x}^ h = \mathcal{O}_{U, u}^ h$, $\mathcal{O}_{Y, \overline{y}} = \mathcal{O}_{V, v}^{sh}$, $\mathcal{O}_{Y, y}^ h = \mathcal{O}_{V, v}^ h$ see in Lemma 66.11.8 and Properties of Spaces, Lemma 64.22.1 and the functoriality of the (strict) henselization discussed in Algebra, Sections 10.153 and 10.154.


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