nLab Henselian pair

Idea

Recall that a Henselian ring is a local ring $R$ with maximal ideal $\mathfrak{m}$ satisfying the conclusion of Hensel's lemma. A Henselian pair is a generalisation of this concept to the case where the ring doesn’t have to be a local ring, and the maximal ideal is replaced by a more general ideal $I$.

Definition

A pair consists of a ring $R$ and an ideal $I\subset R$.

Definition

A henselian pair is a pair $(R, I)$ satisfying

1. $I$ is contained in the Jacobson radical of $R$, and
2. for any monic polynomial $f \in R[T]$ and factorization $\overline{f} = g_0h_0$ with $g_0, h_0 \in R/I[T]$ monic generating the unit ideal in $R/I[T]$, there exists a factorization $f = gh$ in $R[T]$ with $g, h$ monic and $g_0 = \overline{g}$ and $h_0 = \overline{h}$.

There are several equivalent characterizations, see the Stacks Project. Another characterization is the following (see Gabber):

Proposition

A pair $(R,I)$ is a henselian pair if and only if

1. $I$ is contained in the Jacobson radical of $R$, and
2. Let $f(t) = (t-1)t^N + g(t)$ where $g(t) \in IR[t]$ has degree $\leq N$. Then $f(t)$ has a (necessarily unique) root in $I+1$.

This characterization shows that the property of $(R,I)$ being a henselian pair essentially depends only on the ideal $I$, regarded as a non-unital ring, and not on $R$. Thus the property of $I$ being henselian may be axiomatized in terms of an additional set of algbraic operations on $I$, sending $(g_0,\dots, g_{N})$ to $i$ where $1+i$ is the root of $(t-1)t^N + g_{N}t^{N} + \dots + g_0$, and the resulting algebraic category is equivalent to a full subcategory of the category of nonunital rings.

Properties

The category of pairs has as morphisms $(R,I) \to (S,J)$ those ring maps $\phi\colon R\to S$ such that $\phi(I) \subset J$. The category of Henselian pairs is the obvious full subcategory.

Proposition

The inclusion map $HenselianPairs \to Pairs$ has a left adjoint

For a proof, see (Tag 0A02) in the Stacks Project. This Henselization functor restricts to the usual one on the subcategory of local rings and local homomorphisms.

References

• Stacks Project Tag 09XD
• Eberhard Scherzler, On henselian pairs, Communications in Algebra Volume 3 (1975) pp 391-404, doi:10.1080/00927877508822052
• Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics Volume 169 1970 doi:10.1007/BFb0069571
• Gabber, Ofer, $K$-Theory of Henselian Local Rings and Henselian Pairs, Algebraic $K$-Theory, Commutative Algebra, and Algebraic Geometry (Santa Margherita Ligure, 1989), 126:59–70. Contemp. Math. Amer. Math. Soc., Providence, RI, 1992. mathscinet