A counterpart to Nagata idealization

Abstract

Idealization of a module K over a commutative ring S produces a ring having K as an ideal, all of whose elements are nilpotent. We develop a method that under suitable field-theoretic conditions produces from an S-module K and derivation D : S → K a subring R of S that behaves like the idealization of K but is such that when S is a domain, so is R. The ring S is contained in the normalization of R but is finite over R only when R = S. We determine conditions under which R is Noetherian, Cohen-Macaulay, Gorenstein, a complete intersection or a hypersurface. When R is local, then its m-adic completion is the idealization of the m-adic completions of S and K.

The Ideal Completion of a Noetherian Local Domain

“…These notions of analytic extension and strongly analytic extension are defined in [18] by Olberding. Another important tool needed to prove the main theorem of this section is the idealization or trivialization of a module defined by Nagata in [17].…”

“…Let S be a domain with quotient field F , and let R be a subring of S. Let K be a torsion-free S-module, and let FK denote the divisible hull F ⊗ S K of K. Then R is strongly twisted by K if there is a derivation D F −→ FK such that: R = S ∩ D −1 K , D F generates FK as an F -vector space and S ⊆ KerD + sS for all 0 = s ∈ S. Olberding introduced the notion of strongly twisted subring in [18] to construct rings that behave analytically like the idealization of a module. Notice that, if S is an R-algebra and D R −→ L is a derivation, with L an S-module, then we can define an R-algebra structure on S L by r · s = r D r s = rs r + sD r for each r ∈ R, s ∈ S, and ∈ L. With this R-algebra structure, we have the following result.…”

“…By [18,Lemma 3.4], there exists a proper subring A of S such that A ⊆ S is strongly analytic. Therefore, for each nonzero element a ∈ A, there exists an isomorphism A/aA −→ S/aS given by b + aA −→ b + aS.…”

“…By Lemma [18,Lemma 3.4], there exists an A-linear derivation D F −→ F such that D S = F , where A is the ring constructed in the proof of (1). The ring R = S ∩ D −1 V is strongly twisted by V and D is the derivation that twists R [18, …”

“…In Section 6, we use the notions of analytic isomorphism and twisted ring introduced by Olberding in [18], along with the idealization of a module defined by Nagata in [17], to show that uncountable Noetherian local domains are not always complete in the ideal topology, and that there exist Noetherian local domains whose completions in the ideal topology are not Noetherian (Theorem 6.2).…”