For $Spec \mathbb{Z}$ an affine scheme, the objects which the étale topos $Sh(X_{et})$ classifies are **strict local rings**. The points of this topos are *strict Henselian rings?* (Hakim, III.2-4) and (Wraith).

We need to knead the definition of strict Henselian ring into being geometric.

Let $p = \sum_{i=0}^{n} a_i x^i \in R[x]$ be a monic polynomial of degree $n$ (so $a_n = 1$).

If $R$ is a field, a simple root of $p$ is an $r \in R$ with $p(r) = 0$ and $(p/(x-r))(r) \neq 0$. We generalise this as:

More generally if $R$ is a ring, a *simple root* of $p$ is an $r \in R$ with $p(r) = 0$ and $p'(r)$ invertible.

If $R$ is a field, $p$ is *unramifiable* if in any field extension $R \subseteq Q$ in which we can write $p = \prod_i (x-s_i)$, some $s_i$ is a simple root.

To make this definition work for more general rings, some work must be done. Let $R'$ be the ring $R[s_1, \cdots, s_n]/\langle a_k - \sigma_k(\vec s) \mid k \in \{0,\cdots,n\} \rangle$, where $\sigma_k$ is the elementary symmetric polynomial. Then the *hyperdiscriminant polynomial* is the polynomial $\Delta(p) = \prod_{i=1}^n (1 + p'(s_i) x) \in R'[x]$. As this polynomial is symmetric in the $s_i$, the theory of symmetric polynomials tells us that $\Delta(p) \in R[x]$.

More generally if $R$ is a ring, $p$ *unramifiable* if $\Delta(p)-1$ is nonconstant i.e. $1 \in \langle \Delta_1(p), \cdots, \Delta_n(p) \rangle$ where $\Delta_i(p)$ is the $x_i$ coefficient of $\Delta(p)$

This is roughly justified by the following:

If $R$ is a local ring and $t_1, \cdots, t_n \in R$, then at least one of $t_1, \cdots, t_n$ is invertible iff $1 \in \langle a_1, \cdots, a_n \rangle$ where $a_i$ are defined so that $\prod_{i=1}^n (1 + t_i x) = 1 + \sum_{i=1}^n a_i x^i$

Each $a_i$ is a finite sum of a nonempty product of some $t_j$, so if $1$ is in the ideal some $t_j$ is invertible.

Conversely, if $t_j$ is invertible, substituting $x = -1/t_j$ gives $1 = \sum_{i=1}^n a_i (-1)\cdot(-1/t_j)^i$

This lets us arrive at our definition:

A ring $R$ is *separably closed* iff whenever $p$ is a monic polynomial of positive degree, if $p$ is unramifiable, $p$ has a simple root.

A *strict local ring* is a separably closed local ring.

III.2-4 of

- Monique Hakim,
*Topos annelés et schémas relatifs*

and

- Gavin Wraith,
*Generic Galois theory of local rings*

Section 21 of

- Ingo Blechschmidt,
*Using the internal language of toposes in algebraic geometry*

includes discussion of what some similar topoi classify, including the étale topology over $Sh(S)$ partially via the internal language.

See also this MO discussion

See also at *classifying topos – For strict local rings*

Last revised on May 5, 2023 at 09:40:24. See the history of this page for a list of all contributions to it.