Algebraic Approximation of Cohen-Macaulay Algebras

TL;DR

This paper demonstrates that Cohen-Macaulay algebras can be algebraically approximated while preserving key properties like Cohen-Macaulayness and minimal Betti numbers, with applications to flat homomorphisms.

Contribution

It introduces a method for algebraic approximation of Cohen-Macaulay algebras that maintains their essential algebraic invariants.

Findings

Preservation of Cohen-Macaulayness during approximation

Retention of minimal Betti numbers in approximations

Application to flat homomorphisms from power series rings

Abstract

This paper shows that Cohen-Macaulay algebras can be algebraically approximated in such a way that their Cohen-Macaulayness and minimal Betti numbers are preserved. This is achieved by showing that finitely generated modules over power series rings can be algebraically approximated in a manner that preserves their diagrams of initial exponents and their minimal Betti numbers. These results are also applied to obtain an approximation result for flat homomorphisms from rings of power series to Cohen-Macaulay algebras.