The Stacks project

59.33 Stalks of the structure sheaf

In this section we identify the stalk of the structure sheaf at a geometric point with the strict henselization of the local ring at the corresponding “usual” point.

slogan

Lemma 59.33.1. Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$ lying over $s \in S$. Let $\kappa = \kappa (s)$ and let $\kappa \subset \kappa ^{sep} \subset \kappa (\overline{s})$ denote the separable algebraic closure of $\kappa $ in $\kappa (\overline{s})$. Then there is a canonical identification

\[ (\mathcal{O}_{S, s})^{sh} \cong (\mathcal{O}_ S)_{\overline{s}} \]

where the left hand side is the strict henselization of the local ring $\mathcal{O}_{S, s}$ as described in Theorem 59.32.8 and right hand side is the stalk of the structure sheaf $\mathcal{O}_ S$ on $S_{\acute{e}tale}$ at the geometric point $\overline{s}$.

Proof. Let $\mathop{\mathrm{Spec}}(A) \subset S$ be an affine neighbourhood of $s$. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$. With these choices we have canonical isomorphisms $\mathcal{O}_{S, s} = A_{\mathfrak p}$ and $\kappa (s) = \kappa (\mathfrak p)$. Thus we have $\kappa (\mathfrak p) \subset \kappa ^{sep} \subset \kappa (\overline{s})$. Recall that

\[ (\mathcal{O}_ S)_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathcal{O}(U) \]

where the limit is over the étale neighbourhoods of $(S, \overline{s})$. A cofinal system is given by those étale neighbourhoods $(U, \overline{u})$ such that $U$ is affine and $U \to S$ factors through $\mathop{\mathrm{Spec}}(A)$. In other words, we see that

\[ (\mathcal{O}_ S)_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits _{(B, \mathfrak q, \phi )} B \]

where the colimit is over étale $A$-algebras $B$ endowed with a prime $\mathfrak q$ lying over $\mathfrak p$ and a $\kappa (\mathfrak p)$-algebra map $\phi : \kappa (\mathfrak q) \to \kappa (\overline{s})$. Note that since $\kappa (\mathfrak q)$ is finite separable over $\kappa (\mathfrak p)$ the image of $\phi $ is contained in $\kappa ^{sep}$. Via these translations the result of the lemma is equivalent to the result of Algebra, Lemma 10.155.11. $\square$

Definition 59.33.2. Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$ lying over the point $s \in S$.

  1. The étale local ring of $S$ at $\overline{s}$ is the stalk of the structure sheaf $\mathcal{O}_ S$ on $S_{\acute{e}tale}$ at $\overline{s}$. We sometimes call this the strict henselization of $\mathcal{O}_{S, s}$ relative to the geometric point $\overline{s}$. Notation used: $\mathcal{O}_{S, \overline{s}}^{sh}$.

  2. The henselization of $\mathcal{O}_{S, s}$ is the henselization of the local ring of $S$ at $s$. See Algebra, Definition 10.155.3, and Theorem 59.32.8. Notation: $\mathcal{O}_{S, s}^ h$.

  3. The strict henselization of $S$ at $\overline{s}$ is the scheme $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh})$.

  4. The henselization of $S$ at $s$ is the scheme $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h)$.

Let $f : T \to S$ be a morphism of schemes. Let $\overline{t}$ be a geometric point of $T$ with image $\overline{s}$ in $S$. Let $t \in T$ and $s \in S$ be their images. Then we obtain a canonical commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(\mathcal{O}^ h_{T, t}) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{T, \overline{t}}) \ar[r] \ar[d] & T \ar[d]^ f \\ \mathop{\mathrm{Spec}}(\mathcal{O}^ h_{S, s}) \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \ar[r] & S } \]

of henselizations and strict henselizations of $T$ and $S$. You can prove this by choosing affine neighbourhoods of $t$ and $s$ and using the functoriality of (strict) henselizations given by Algebra, Lemmas 10.155.8 and 10.155.12.

Lemma 59.33.3. Let $S$ be a scheme. Let $s \in S$. Then we have

\[ \mathcal{O}_{S, s}^ h = \mathop{\mathrm{colim}}\nolimits _{(U, u)} \mathcal{O}(U) \]

where the colimit is over the filtered category of étale neighbourhoods $(U, u)$ of $(S, s)$ such that $\kappa (s) = \kappa (u)$.

Proof. This lemma is a copy of More on Morphisms, Lemma 37.34.5. $\square$

Remark 59.33.4. Let $S$ be a scheme. Let $s \in S$. If $S$ is locally Noetherian then $\mathcal{O}_{S, s}^ h$ is also Noetherian and it has the same completion:

\[ \widehat{\mathcal{O}_{S, s}} \cong \widehat{\mathcal{O}_{S, s}^ h}. \]

In particular, $\mathcal{O}_{S, s} \subset \mathcal{O}_{S, s}^ h \subset \widehat{\mathcal{O}_{S, s}}$. The henselization of $\mathcal{O}_{S, s}$ is in general much smaller than its completion and inherits many of its properties. For example, if $\mathcal{O}_{S, s}$ is reduced, then so is $\mathcal{O}_{S, s}^ h$, but this is not true for the completion in general. Insert future references here.

Lemma 59.33.5. Let $S$ be a scheme. The small étale site $S_{\acute{e}tale}$ endowed with its structure sheaf $\mathcal{O}_ S$ is a locally ringed site, see Modules on Sites, Definition 18.40.4.

Proof. This follows because the stalks $(\mathcal{O}_ S)_{\overline{s}} = \mathcal{O}^{sh}_{S, \overline{s}}$ are local, and because $S_{\acute{e}tale}$ has enough points, see Lemma 59.33.1, Theorem 59.29.10, and Remarks 59.29.11. See Modules on Sites, Lemmas 18.40.2 and 18.40.3 for the fact that this implies the small étale site is locally ringed. $\square$


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