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Hensel
A DVR \(V\) is henselian if for every finite extension \(L\) of its fraction field, the integral closure of \(V\) in \(L\) is a discrete valuation ring.
Example: any complete 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. Idea: intermediate between the localization \(A \left[ { \scriptstyle { {p}^{-1}} } \right]\) and the completion \(A{ {}_{ \widehat{p} } }\). Occur as the local rings of geometric points in the etale topology.
Idea: 
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.
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Slogan: If a polynomial \(p(x)\) has a simple root \(r\) modulo a prime \(p\), then \(r\) corresponds to a unique root of \(p(x)\) modulo any \(p^n\) gotten by iteratively “lifting” solutions.
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Setup: let \(K\in \mathsf{Field}\) be Complete ring wrt a normalized discrete valuation where \({\mathcal{O}}_K\) is the ring of integers of \(K\) with a uniformizer \(\pi\) and let \(\kappa(k) \coloneqq{\mathcal{O}}_K/ \left\langle{\pi}\right\rangle\) be the residue field.
- \(K\) is Henselian if \(p \in {\mathcal{O}}_K[x]\) where its reduction \(\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu \in \kappa(k)[x]\) has a simple root \(k_0\), there is a lift \(\tilde k_0 \in {\mathcal{O}}_K\) with \(p(\tilde k_0) = 0\).
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A local ring \(R\) with maximal ideal \({\mathfrak{m}}\) is called Henselian if Hensel’s lemma holds.
- This means that if \(p\in R[x]\) is monic, then any factorization of its image \(\mkern 1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu \in (R/{\mathfrak{m}})[x]\) into a product of coprime monic polynomials can be lifted to a factorization of \(p\) in \(R[x]\).
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A field with valuation is said to be Henselian if its valuation ring is Henselian.
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A Henselian local ring is called strictly Henselian if its residue field is separably closed.
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The Henselization of A is an algebraic substitute for the completion of A
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See slogan: Henselian implies large. Can product points.
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If \(R\) is a DVR, then its hensilization is \(\widehat{R} \cap \operatorname{cl}^{\mathrm{sep}} (\operatorname{ff}(R))\).
Examples
