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

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

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.

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