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

Given a one-dimensional equicharacteristic Cohen–Macaulay local ring A, Juan Elias introduced in 2001 the set of micro-invariants of A in terms of the first neighborhood ring. On the other hand, if A is a one-dimensional complete equicharacteristic and residually rational domain, Valentina Barucci and Ralf Fröberg defined in 2006 a new set of invariants in terms of the Apery set of the value semigroup of A. We give a new interpretation for these sets of invariants that allow to extend their definition to any one-dimensional Cohen–Macaulay ring. We compare these two sets of invariants with the one introduced by the authors for the tangent cone of a one-dimensional Cohen–Macaulay local ring and give explicit formulas relating them. We show that, in fact, they coincide if and only if the tangent cone G(A) is Cohen–Macaulay. Some explicit computations will also be given.