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Roughly speaking, Hensel's Lemma states that a polynomial $f \in O[X]$ over a certain local ring $(O,\mathfrak{m})$ which factors over the residue field $O/\mathfrak{m}$ into coprime polynomials also factors over $O$ in a compatible way. However, there are different versions of the lemma with different requirements on $O$. For example the statement holds true if $O$ is complete with respect to the $\mathfrak{m}$-adic topology, and it also holds true if $O$ is the valuation ring of a nontrivial non-archimedean absolute value on some field $K$, which is complete with respect to this absolute value.

Is there a more general version of Hensel's Lemma which implies both of the above statements?

Dune
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1 Answers1

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Let $(R, \mathfrak{m})$ be a local ring, $k = R/\mathfrak{m}$ its residue field, $S = \operatorname{Spec}R$, and $s$ the closed point of $S$. The following conditions are equivalent.

$(i)$ Every finite $R$-algebra $A$ is a direct product of local rings.

$(ii)$ The condition $(i)$ holds for $A = R[t]/(f (t))$ for any monic polynomial $f (t) \in R[t]$.

$(iii)$ For any finite $R$-algebra $A$, the canonical homomorphism $A \rightarrow A/\mathfrak{m}A$ induces a one-to-one correspondence between the set of idempotent elements in $A$ and the set of idempotent elements in $A/\mathfrak{m}A.$

$(iv)$ The condition $(iii)$ holds for $A = R[t]/(f (t))$ for any monic polynomial $f (t) \in R[t].$

$(v)$ For any monic polynomial $f (t) \in R[t]$ and any factorization $\overline{f}(t) = \overline{g}(t) \overline{h}(t)$, where $\overline{f}(t)$ is the image of $f (t)$ in $k[t]$, and $\overline{g}(t)$ and $\overline{h}(t)$ are relatively prime monic polynomials in $k[t]$, there exist uniquely determined polynomials $g(t)$ and $h(t)$ in $R[t]$ such that $f (t) = g(t)h(t)$, $\overline{g}(t)$ and $\overline{h}(t)$ are images of $g(t)$ and $h(t)$ in $k[t]$, respectively, and the ideal generated by $g(t)$ and $h(t)$ is $R[t].$

$(vi)$ The condition $(v)$ holds for any factorization $\overline{f} (t) = (t− \overline{a}) \overline{h}(t)$ such that $t − \overline{a}$ and $\overline{h}(t)$ are relatively prime monic polynomials in $k[t].$

$(vii)$ For any etale morphism $g : X \rightarrow S$, any section of $g_s : X \otimes _R k \rightarrow \operatorname{Spec} k$ is induced by a section $g.$

If a local ring $(R, \mathfrak{m})$ satisfies any (and hence every) of these properties, we call the ring henselian.

Examples: (1) A complete local ring is henselian.

(2) The convergent power series ring $\mathbb{C}\{z_1, z_2, \dots , z_n\}$ is also henselian. (This ring is not a complete local ring.)

The above definition is taken from Etale Cohomology Theory, Lei Fu. You can also look at here.

user26857
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Krish
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  • Is there an obvious reason why complete local rings and valuation rings of complete fields satisfy any of those conditions (other than (v) or (vi))? – Dune Dec 30 '14 at 10:45
  • @Dune: I don't know. Sorry!! But the conditions (v) and (vi) have a very beautiful geometric picture (namely, Newton-Raphson method). You can look at these two links: (1) http://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem?rq=1 (2) http://math.stackexchange.com/questions/709533/what-is-the-difference-between-hensel-lifting-and-the-newton-raphson-method – Krish Dec 30 '14 at 11:20
  • @Krish: No problem. :) I just wonder if there is a unifying way to show that these two kinds of rings are henselian. But nevertheless, thank you for your answer! – Dune Dec 30 '14 at 19:46