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.