Cfrgtkky

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?

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.