nLab strict local ring

Idea

For SpecSpec \mathbb{Z} an affine scheme, the objects which the étale topos Sh(X et)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= i=0 na ix iR[x]p = \sum_{i=0}^{n} a_i x^i \in R[x] be a monic polynomial of degree nn (so a n=1a_n = 1).

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

Definition

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

If RR is a field, pp is unramifiable if in any field extension RQR \subseteq Q in which we can write p= i(xs i)p = \prod_i (x-s_i), some s is_i is a simple root.

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

Definition

More generally if RR is a ring, pp unramifiable if Δ(p)1\Delta(p)-1 is nonconstant i.e. 1Δ 1(p),,Δ n(p)1 \in \langle \Delta_1(p), \cdots, \Delta_n(p) \rangle where Δ i(p)\Delta_i(p) is the x ix_i coefficient of Δ(p)\Delta(p)

This is roughly justified by the following:

Proposition

If RR is a local ring and t 1,,t nRt_1, \cdots, t_n \in R, then at least one of t 1,,t nt_1, \cdots, t_n is invertible iff 1a 1,,a n1 \in \langle a_1, \cdots, a_n \rangle where a ia_i are defined so that i=1 n(1+t ix)=1+ i=1 na ix i\prod_{i=1}^n (1 + t_i x) = 1 + \sum_{i=1}^n a_i x^i

Proof

Each a ia_i is a finite sum of a nonempty product of some t jt_j, so if 11 is in the ideal some t jt_j is invertible.

Conversely, if t jt_j is invertible, substituting x=1/t jx = -1/t_j gives 1= i=1 na i(1)(1/t j) i1 = \sum_{i=1}^n a_i (-1)\cdot(-1/t_j)^i

This lets us arrive at our definition:

Definition

A ring RR is separably closed iff whenever pp is a monic polynomial of positive degree, if pp is unramifiable, pp has a simple root.

A strict local ring is a separably closed local ring.

References

III.2-4 of

and

Section 21 of

includes discussion of what some similar topoi classify, including the étale topology over Sh(S)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.