Let \((R,\mathfrak{m})\) denote a local Noetherian ring and \(M\) a finitely generated \(R\)-module with \(r = \dim_R M\). Let \(I \subset R\) be an ideal of definition with respect to \(M\). Then the length \(\ell(M/I^nM)\) becomes for large \(n\) a polynomial, the Hilbert polynomial, \(P_M(n) = \sum\limits_{i=0}^{r} (-1)^i e_i(I,M) \binom{n+r-i}{i}\). The first Hilbert coefficient \(e_1(I,M)\) is also called the Chern coeficient of \(I\) relative to \(M\) see W. V. Vasconcelos [Mich. Math. J. 57, 725–743 (2008; Zbl 1234.13005)]. By the work of several authors, the values of \(e_1(I,R)\) carries various information about the ring itself. In the more general situation of a finitely generated \(R\)-module \(M\) the authors investigate \(e_1(I,M)\), where they simplify and make arguments in the ring case more transparent and give several fresh new arguments in the general case.
To be more precise, they investigate the following sets:
(a) \(\Lambda(M) = \{e_1(Q,M) | Q \text{ a parameter ideal of } M\}\).
(b) \(\Xi (M) = \{\chi_1(Q;M) | Q \text{ a parameter ideal of } M\}\), where \(\chi_1(Q;M)\) denotes the first partial Euler characteristic of the Koszul homology.
(c) \[ \Lambda(M) = \{e_1(\mathfrak{q},M) | \mathfrak{q} \text{ a parameter ideal of } M \text{ with the same integral closure as }Q\}. \] Among others, the authors characterize quasi-unmixed \(R\)-modules \(M\) to be (1) Cohen-Macaulay by \(0 \in \Lambda(M)\), (2) Buchsbaum by \(\#(\Lambda(M)) =1\) and (c) generalized Cohen-Macaulay by \(\Lambda(M)\) bounded. They use also the technique of homological degree in order to investigate bounds for these sets of numbers. In a final section they discuss the Buchsbaum-Rim coefficients.