Some results on one-dimensional local rings

Sedighi, Hossein (1982) Some results on one-dimensional local rings. University of Southampton, Doctoral Thesis.

Record type: Thesis (Doctoral)

Abstract

The work of this thesis was motivated in the first place by Northcott's theory of dilations for one-dimensional local rings. In 1977, D. Kirby and M.R. Adranghi have introduced for an arbitrary commutative R with identity the lattice L(R) of extension rings. In 1978, D. Kirby had shown that with S(M) denoting the first neighbourhood ring of R there is a one-one correspondence between the rings S ,. L(R) which lengtl S(m)is'. and the graded ideals of the form ring R of order 2R and have no irrelevant component. In 1982, D. Kirby has established some general results which indicate the size of the problem. In 1982, D.Kirby and H.A. Tavallace have classified 1-dimensional Cohen-Macaulay local rings for which L(R) is finite, when R has embedding dimension two. Now there are two aspects of the study of 1-dimensional Cohen-Macaulay local rings R,M with which we are concerned. The first is the problem of finding for R the associated lattice L(R) of finite integral extension rings contained in the total quotient T(R). The second is to find canonical local equations relating generators of the maximal ideal in R, where we concentrate on those rings with embedding dimu-sion 3. We consider the cases of multiplicity 3 and 4, because embedding dimension R : e(R) always. As a special case we show that if R is cxnplete and R/M is algebraically closed, of characteristic 0 , then for multiplicity three the only rings R with finite lattice are those with generators x,y,z of M such that.(a) xy s yz s zx s 0. (b) y2-xr = xz s yz e O for some r i 2.(c) y2-xz s yz-x3 s z2-x2y - O. (d) y2-xz s yz-x2y a z2-x2z - 0. (e) y2-xz a yz-x4 s z2-x2y - 0.the lattice L(R) is found in all these cases. In the lengthier study of rings of multiplicity four we concentrate on classifying those rings with a finite lattice. We first consider the case in which the form ring R is a complete intersection and describe those rings with finite lattice when R/M is infinite. We conclude with a similar but less detailed description when R is not a complete intersection.

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