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Let A be a noetherian local ring with dimension d and I be an ideal of A. Let E = E n n≥0 be a good I -filtration of submodules of an A-module E. Let H be an ideal of A containing I and F H E = n≥0 E n /HE n . Assume that E is a Cohen-Macaulay module with E/IE finite and Ann E = 0, and let J be a minimal reduction of I .In this paper we give conditions on E n ∩ JE/JE n−1 and HE n ∩ JE/JHE n−1 so, that F H E , has depth of at least d − 1.

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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 onedimensional 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.

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