Apery and micro-invariants of a one dimensional Cohen-Macaulay local ring and invariants of its tangent cone

TL;DR

This paper compares various invariants of one-dimensional Cohen-Macaulay local rings, establishes conditions for their equivalence, and provides explicit formulas and computations, especially for semigroup rings.

Contribution

It explicitly relates micro-invariants, Apery invariants, and tangent cone invariants, showing they coincide precisely when the tangent cone is Cohen-Macaulay.

Findings

Invariants coincide iff the tangent cone is Cohen-Macaulay

Explicit formulas relating different invariants are derived

Computations are provided for semigroup rings

Abstract

Given a one-dimensional Cohen-Macaulay local ring we compare several sets of invariants (micro-invariants, Apery invariants and invariants of the tangent cone) and give explicit formulas relating them. We show that, in fact, they coincide if and only if the tangent cone of the ring is Cohen-Macaulay. Some explicit computations are also given, particularly in the case of semigroup rings.