nLab strict local ring

Idea

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).

Definition

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:

Definition

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 $\Delta(p) \in R[x]$ .

Definition

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:

Proposition

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 $\prod_{i=1}^n (1 + t_i x) = 1 + \sum_{i=1}^n a_i x^i$

Proof

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:

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.

References

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.