Prescribed subintegral extensions of local Noetherian domains

Abstract

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).

One-dimensional stable rings

“…Remarks 2.7. (1) Quadratic extensions were considered by Handelman [14] and Rush [36], and more recently in [31,32,33].…”

Trivial Ring Extension of Suitable-Like Conditions and some properties

“…Goto, 2013;Glaz, 1989;Huckaba, 1988;Gulliksen, 1974;Kourki, 2009;Levin, 1985;?, ? ;Olberding, 2014;Palmér, 1973;Popescu, 1985;Reiten, 1972;Roos, 1981;Salce, 2009).…”

Matlis’ semi-regularity in trivial ring extensions of integral domains

“…The ring R is also called the (Nagata) idealization of E over A and is denoted by A(+)E. This construction was first introduced, in 1962, by Nagata [33] in order to facilitate interaction between rings and their modules and also to provide various families of examples of commutative rings containing zero-divisors. The literature abounds of papers on trivial extensions dealing with the transfer of ring-theoretic notions in various settings of these constructions (see, for instance, [1,3,13,16,20,21,22,28,29,36,37,38,39,40,41,44]). For more details on commutative trivial extensions (or idealizations), we refer the reader to Glaz's and Huckaba's respective books [18,24], and also D. D. Anderson & Winders relatively recent and comprehensive survey paper [2].…”