A differential condition on the normal cone
Let R be a ring, P a proper ideal of R. P is said to be maximally \({\mathcal D}\)-differential for \({\mathcal D}\subset Der(P)\) if d(P)\(\subset P\) for \(d\in {\mathcal D}\) and if Q is any ideal such that \(P\subsetneqq Q\subsetneqq R\) then there exists \(d\in {\mathcal D}\) such hat d(Q)\(\not\subset Q\). A graded ring \(A=\oplus A_ n\) satisfies condition (C) iff \(A_+\) is maximally D(A)-differential. If P is a maximal differential ideal of R then \(gr_ P(R)\) satisfies condition (C). The author considers for a noetherian local ring R containing a field of characteristic zero and having the property to be a G-ring the following conditions: (i) P is maximally differential in R; (ii) \(gr_ P(R)\) satisfies condition (C); ((iii) P is permissible in R. Then: \[ (i)\to (ii)\to (iii)\nrightarrow (ii)\nrightarrow (i). \] This result shows that a criterion given by Seibt \(((iii)\to (i))\) [\textit{P. Seibt}, J. Algebra 66, 484-491 (1980; Zbl 0469.13002)] is wrong. Giving a structure theorem for noetherian graded rings A such that \(A_ 0\) is a complete local ring containing a field of characteristic zero which satisfies condition (C) (Theorem 2.8) the author shows: If R/P is complete then condition (C) on \(gr_ P(R)\) is equivalent to saying that Spec\((gr_ P(R))\) is the product of an affine algebroid space and a suitable tangent cone.
scientific article; zbMATH DE number 3955081
scientific article