Paper detail

The Chern coefficients of local rings

The Chern numbers of the title are the first coefficients (after the multiplicities) of the Hilbert functions of various filtrations of ideals of a local ring $(R, \mathfrak{m})$. For a Noetherian (good) filtration $\mathcal{A}$ of $\mathfrak{m}$-primary ideals, the positivity and bounds for $e_1(\mathcal{A})$ are well-studied if $R$ is Cohen-Macaulay, or more broadly, if $R$ is a Buchsbaum ring or mild generalizations thereof. For arbitrary geometric local domains, we introduce techniques based on the theory of maximal Cohen-Macaulay modules and of extended multiplicity functions to establish the meaning of the positivity of $e_1(\mathcal{A})$, and to derive lower and upper bounds for $e_1(\mathcal{A})$.