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By Thomas Keilen

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We recall that e0 (M) depends only on q and M, but it does not depend on the good q-filtration M. In contrast e1 (M) does depend on the filtration M. It is called by Vasconcelos tracking number for its tag position among the different filtrations having the same multiplicity. The coefficient e1 (M) is also called the Chern number (see [111]). In establishing the properties of e1 (M), we will need an ad hoc treatment of the one-dimensional case. 2, which describes the behaviour of the Hilbert coefficients modulo superficial elements.

2) A is Cohen–Macaulay and depth grm (A) ≥ r − 1. In the Cohen–Macaulay case we describe now another set of numerical characters of the filtered module M, which are important in the study of the Hilbert coefficients. Let M = {M j } j≥0 be a good q-filtration of M and J an ideal generated by an M-superficial sequence for q; then, for every j ≥ 0, we let w j (M) := λ (M j+1 + JM/JM) = λ (M j+1 /M j+1 ∩ JM). 9) The length of the abelian group JM ∩M j+1 /JM j will be denoted by vv j (M) since these groups are the homogeneous components of the Valabrega–Valla module VV (M) := (JM ∩ M j+1 /JM j ) j≥0 of M with respect to J, as defined in [109, Chap.

Then ei (M) = ei (Msat ) 0 ≤ i ≤ r − 1, er (M) = er (Msat ) + (−1)d λ (W ). 2 Bounds for e0 (M) and e1 (M) 24 We remark that, if dim(M) ≥ 1, the module M/W always has positive depth. This is the reason why, sometimes, we move our attention from the module M to the module M/W. This will be the strategy of the proof of the next proposition which gives, in the one-dimensional case, the promised upper bound for e1 . 4. Let M = {M j } j≥0 be a good q-filtration of a module M of dimension one. If a is an M-superficial element for q and N the (a)-adic filtration on M, then e1 (M) − e1 (N) ≤ ∑ λ (M j+1 /aM j ).

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