nLab Henselian pair

Contents

Contents

Idea

Recall that a Henselian ring is a local ring RR with maximal ideal 𝔪\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 II.

Definition

A pair consists of a ring RR and an ideal IRI\subset R.

Definition

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

  1. II is contained in the Jacobson radical of RR, and
  2. for any monic polynomial fR[T]f \in R[T] and factorization f¯=g 0h 0\overline{f} = g_0h_0 with g 0,h 0R/I[T]g_0, h_0 \in R/I[T] monic generating the unit ideal in R/I[T]R/I[T], there exists a factorization f=ghf = gh in R[T]R[T] with g,hg, h monic and g 0=g¯g_0 = \overline{g} and h 0=h¯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)(R,I) is a henselian pair if and only if

  1. II is contained in the Jacobson radical of RR, and
  2. Let f(t)=(t1)t N+g(t)f(t) = (t-1)t^N + g(t) where g(t)IR[t]g(t) \in IR[t] has degree N\leq N. Then f(t)f(t) has a (necessarily unique) root in I+1I+1.

This characterization shows that the property of (R,I)(R,I) being a henselian pair essentially depends only on the ideal II, regarded as a non-unital ring, and not on RR. Thus the property of II being henselian may be axiomatized in terms of an additional set of algbraic operations on II, sending (g 0,,g N)(g_0,\dots, g_{N}) to ii where 1+i1+i is the root of (t1)t N+g Nt N++g 0(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)(S,J)(R,I) \to (S,J) those ring maps ϕ:RS\phi\colon R\to S such that ϕ(I)J\phi(I) \subset J. The category of Henselian pairs is the obvious full subcategory.

Proposition

The inclusion map HenselianPairsPairsHenselianPairs \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, KK-Theory of Henselian Local Rings and Henselian Pairs, Algebraic KK-Theory, Commutative Algebra, and Algebraic Geometry (Santa Margherita Ligure, 1989), 126:59–70. Contemp. Math. Amer. Math. Soc., Providence, RI, 1992. mathscinet

Last revised on March 7, 2018 at 03:39:09. See the history of this page for a list of all contributions to it.