The University of Southampton
University of Southampton Institutional Repository

On Cohen-Macaulay local rings of dimension one and embedding dimension two

On Cohen-Macaulay local rings of dimension one and embedding dimension two
On Cohen-Macaulay local rings of dimension one and embedding dimension two

In this thesis we classify one-dimensional Cohen-Macaulay local rings , (Q ,M) of embedding dimension two in terms of the lattices L(Q). For any such local. ring, L(Q) is the set of all rings between Q c and its total quotient ring which are also fractionari ideals. In fact any S c L(Q) is a finite integral extension of Q and the integral closure Q is the union of rings in L(Q). We classify such local rings Q which have a finite associated lattice, this implies Q has multiplicity 2 or 3. The rings Q with a finite lattice are said to be of types Ak(k>l), Dk(k>4), E5,E6 and E7, as they can be associated with the simple Lie Algebras Ak(k>,1) , Dk(ks4) , E5, E6 and E71. When Q/M is algebraically closed of characteristic zero and Q iscomplete, we will find canonical relations between generators x,y of 14, 1 i.e.Y2 = x +1 in case Ak, xy2 = x -1 in case Dk ,Y3 = x4 in case E5, y3 = x 3 y in case E6 and y3 = x' in case E7. Geometrically, when we consider algebraic plane curves, which have the origin as a singular point and then construct their local rings at origin, the above relations are the local equations of the curves in a neighbourhood of the origin.

University of Southampton
Tavallaee, Hamid Agha
Tavallaee, Hamid Agha

Tavallaee, Hamid Agha (1982) On Cohen-Macaulay local rings of dimension one and embedding dimension two. University of Southampton, Doctoral Thesis.

Record type: Thesis (Doctoral)

Abstract

In this thesis we classify one-dimensional Cohen-Macaulay local rings , (Q ,M) of embedding dimension two in terms of the lattices L(Q). For any such local. ring, L(Q) is the set of all rings between Q c and its total quotient ring which are also fractionari ideals. In fact any S c L(Q) is a finite integral extension of Q and the integral closure Q is the union of rings in L(Q). We classify such local rings Q which have a finite associated lattice, this implies Q has multiplicity 2 or 3. The rings Q with a finite lattice are said to be of types Ak(k>l), Dk(k>4), E5,E6 and E7, as they can be associated with the simple Lie Algebras Ak(k>,1) , Dk(ks4) , E5, E6 and E71. When Q/M is algebraically closed of characteristic zero and Q iscomplete, we will find canonical relations between generators x,y of 14, 1 i.e.Y2 = x +1 in case Ak, xy2 = x -1 in case Dk ,Y3 = x4 in case E5, y3 = x 3 y in case E6 and y3 = x' in case E7. Geometrically, when we consider algebraic plane curves, which have the origin as a singular point and then construct their local rings at origin, the above relations are the local equations of the curves in a neighbourhood of the origin.

This record has no associated files available for download.

More information

Published date: 1982

Identifiers

Local EPrints ID: 460290
URI: http://eprints.soton.ac.uk/id/eprint/460290
PURE UUID: 1966c0e7-e8fe-4a7f-9647-9c1eafcdd220

Catalogue record

Date deposited: 04 Jul 2022 18:18
Last modified: 04 Jul 2022 18:18

Export record

Contributors

Author: Hamid Agha Tavallaee

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×