One-dimensional bad Noetherian domains

Olberding, Bruce

doi:10.1090/s0002-9947-2014-05921-7

Cited by 15 publications

(28 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A note on generality. The existence theorem (1.1) is our main source of examples for (strongly) twisted subrings in dimension > 1 (the articles [16,17] deal directly with the more tractable one-dimensional case). Moreover, there is a straightforward characterization in (2.9) of when a strongly twisted subring is Noetherian.…”

Section: ] Ismentioning

confidence: 99%

“…(2.5) The mapping f : R → S ⋆ K : r → (r, D(r)) is an analytic isomorphism along C; see [14,Theorem 4.6] or [16,Proposition 3.5].…”

Section: Twisted Subringsmentioning

confidence: 99%

“…The fact that I 2 = iI for a nonzerodivisor i ∈ I implies that I is stable, meaning that I is a projective module over its ring of endomorphisms; see [14,16] for more on the connection between stable ideals and twisted subrings.…”

Section: In Particular the Fiber Cone Ofmentioning

confidence: 99%

See 2 more Smart Citations

Prescribed subintegral extensions of local Noetherian domains

Olberding

2014

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

We show how subintegral extensions of certain local Noetherian domains S can be constructed with specified invariants including reduction number, Hilbert function, multiplicity and local cohomology. The construction behaves analytically like Nagata idealization but rather than a ring extension of S, it produces a subring R of S such that R ⊆ S is subintegral.Let i > 0, and choose 0 = c ∈ J i (J + I) ∩ C. By (2.4), the mapping g : R/cR → S/cS ⋆ K/cK induced by f is an isomorphism of rings. Therefore, from (3) we haveThus since g is an isomorphism, there is an isomorphism of A-modules:The second isomorphism is also an isomorphism of R-modules, where the R-module structure is the usual one on the direct sum. Indeed, since S = A + cS, we have R = A + cS ∩ R. Moreover, c ∈ J, and J = JS ∩ R, so R = A + cS ∩ R = A + J, and from this it follows that this isomorphism of A-modules is also an isomorphism of R-modules. This verifies (1). Now a similar argument using (4) and (5) instead of (3) shows that when I ⊆ J there is an isomorphism of R-modules for all i ≥ 0,and this verifies (2). Now to prove statement (1) of the theorem, we make use of the isomorphisms (1) and (2). We consider the case first where we make no assumption about whether I ⊆ J. Since S = R + JS, it follows that the lengths of the S/JS-modules, J i IS/J i+1 IS and J i−1 (I + J)K/J i (I + J)K, agree with their lengths as R/J-modules. Therefore, we have by (1) that for all n ≥ 1, H J,I (n) = length J n I/J n+1 I = length J n IS/J n+1 IS + length J n−1 (J + I)K/J n (J + I)K = H JS,IS (n) + H JS,(I+J)K (n − 1).

show abstract

Section: ] Ismentioning

confidence: 99%

“…(2.5) The mapping f : R → S ⋆ K : r → (r, D(r)) is an analytic isomorphism along C; see [14,Theorem 4.6] or [16,Proposition 3.5].…”

Section: Twisted Subringsmentioning

confidence: 99%

See 1 more Smart Citation

Prescribed subintegral extensions of local Noetherian domains

Olberding

2014

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our focus in this article is on the Noetherian case, but we develop characterizations in Sections 3 and 4 for the general one-dimensional case as well, since it requires little extra effort. In [26], examples of non-Noetherian one-dimensional stable domains are given.…”

Section: Preliminaries On Stable Ringsmentioning

confidence: 99%

Generic formal fibers and analytically ramified stable rings

Olberding¹

2013

Nagoya Mathematical Journal

Self Cite

View full text Add to dashboard Cite

Abstract. Let A be a local Noetherian domain of Krull dimension d. Heinzer, Rotthaus and Sally have shown that if the generic formal fiber of A has dimension d − 1, then A is birationally dominated by a one-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of A. We explore further this correspondence between prime ideals in the generic formal fiber and one-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

show abstract

“…(For example, the construction of Ferrand and Raynaud requires a priori that the pullback R of the derivation is Noetherian, a condition that can be hard to verify and one which seems to be the main obstacle to producing more examples with their method.) For more applications of some of these ideas to the case of dimension 1, see [19].…”

Section: Introductionmentioning

confidence: 99%

A counterpart to Nagata idealization

Olberding

2012

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

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.

show abstract

One-dimensional bad Noetherian domains

Cited by 15 publications

References 28 publications

Prescribed subintegral extensions of local Noetherian domains

Prescribed subintegral extensions of local Noetherian domains

Generic formal fibers and analytically ramified stable rings

A counterpart to Nagata idealization

Contact Info

Product

Resources

About