---
date: 2022-02-23 18:45
modification date: Thursday 17th March 2022 21:12:57
title: "Hensel's Lemma"
aliases: [Henselian]

---

Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [etale cohomology](Unsorted/l-adic%20cohomology.md)

---

# Hensel

A [DVR](DVR.md) $V$ is **henselian** if for every finite extension $L$ of its fraction field, the [integral closure](integral%20closure) of $V$ in $L$ is a discrete valuation ring. 

Example: any [complete](Unsorted/adic%20completion.md) DVR. Occur as the local rings of points in the Nisnevich topology.


If $V$ is a local Henselian ring, $V$ is **strict** iff its residue field $\kappa(V)$ is [separably closed](separably%20closed). Idea: intermediate between the localization $A\localize{p}$ and the completion $A\complete{p}$. Occur as the local rings of [geometric points](geometric%20points) in the etale topology.


Idea:
![](attachments/Pasted%20image%2020220404010156.png)

Cauchy sequences that converge (in the completion) to the root of a polynomial are required to converge, but not every Cauchy sequence needs to converge.

![](attachments/Pasted%20image%2020220317211257.png)

![](attachments/Pasted%20image%2020220124223205.png)
![](attachments/Pasted%20image%2020220124223720.png)
![](attachments/Pasted%20image%2020220120125947.png)
![](attachments/Pasted%20image%2020220120130016.png)

- Slogan: If a polynomial $p(x)$ has a [simple root](simple%20root)  $r$ modulo a prime $p$, then $r$ corresponds to a unique root of $p(x)$ modulo any $p^n$ gotten by iteratively "lifting" solutions.
- 
- Setup: let $K\in \Field$ be [Complete ring](Complete%20ring) wrt a normalized [discrete valuation](discrete%20valuation) where $\OO_K$ is the [ring of integers](ring%20of%20integers.md) of $K$ with a [uniformizer](uniformizer) $\pi$ and let $\kappa(k) \da \OO_K/ \gens{\pi}$ be the [residue field](residue%20field).

	- $K$ is **Henselian** if $p \in \OO_K[x]$ where its reduction $\bar p \in \kappa(k)[x]$ has a simple root $k_0$, there is a lift $\tilde k_0 \in \OO_K$ with $p(\tilde k_0) = 0$.

- A [local ring](local%20ring) $R$ with maximal ideal $\mfm$ is called **Henselian** if Hensel's lemma holds. 
	- This means that if $p\in  R[x]$ is monic, then any factorization of its image $\bar p \in (R/\mfm)[x]$ into a product of coprime monic polynomials can be lifted to a factorization of $p$ in $R[x]$.

- A field with [valuation](valuation) is said to be **Henselian** if its [valuation ring](valuation%20ring.md) is Henselian.
- A Henselian local ring is called **strictly Henselian** if its residue field is [separably closed](separably%20closed.md).
- The Henselization of A is an algebraic substitute for the [completion](completion) of A
- See slogan: Henselian implies large. Can product points.
- If $R$ is a DVR, then its hensilization is $\hat{R} \intersect \sepcl(\ff(R))$.

# Examples
![](attachments/Pasted%20image%2020220124223626.png)