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Structure theorems for local Noether lattices

A local Noether lattice of dimension n is regular if and only if its maximal element is a join of n principal elements. A set of n principal elements whose join is the maximal element is called a regular system of parameters. An element of a regular system of parameters is called a regular parameter.

The main results of this thesis describe the structure of distributive regular local Noether lattices, and relate the structure of certain broad classes of local Noether lattices to the structure of distributive regular local Noether lattices.

A distributive regular local Noether lattice of dimension n is isomorphic to RL(subscript n), the sublattice of the lattice of ideals of F[x[?] (F a field) consisting of all joins of products of powers of the principal ideals [F[x[?]. A local Noether lattice L of dimension n is regular if and only if there exists a sublattice L' of L isomorphic to RL[subscript n] and prime, primary, and principal elements in L' are prime, primary, and principal, respectively, in L.

A lattice L with a unique proper maximal element which has a minimal representation as a join of n principal elements is a distributive local Noether lattice if and only if it is isomorphic to RL[subscript n] /[theta] where [theta] is an equivalence relation which is first defined on the principal elements of RL[subscript n] and is then extended to all of RL[subscript n] by preserving join, and where, in addition, [theta] preserves multiplication and preserves the cancellation of principal elements in nonzero products.

A final result which shows the strength of the condition that a local Noether lattice be regular is an abstract characterization of RL[subscript3]. If L is a regular local Noether lattice with precisely three minimal primes, and if each minimal prime of L is a regular parameter, then L is isomorphic to RL[subscript3].

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Structure theorems for local Noether lattices