Bounds for the first Hilbert coefficients of \(\mathfrak{m}\)-primary ideals (Q498644)

Let \((A,\mathfrak{m})\) be a Noetherian local ring of dimension \(d > 0\). For an \(\mathfrak{m}\)-primary ideal \(I\) in \(A\), define \(H_I(n)=\ell(A/I^{n+1})\) for \(n \geq 0\), where \(\ell(N)\) denote the length of the \(A\)-module \(N\). It is well known that \(H_I(n)=\sum_{j=0}^d (-1)^j e_j(I){n+d-j \choose d-j}\), for \(n \gg 0\), for some integers \(e_j(I)\). In the paper under review, the authors give a characterization for what class of local rings \((A,\mathfrak{m})\), for which the first Hilbert coefficients (Chern coefficients) of \(\mathfrak{m}\)-primary ideals range among only finitely many values. The main theorem (Theorem 1.1) shows that \(\{e_1(I) ~|~ I\) is an \(\mathfrak{m}\)-primary ideal in \(A\}\) is finite if and only if \(d=1\) and \(A/H_{\mathfrak{m}}^0(A)\) is analytically unramified, where \(H_{\mathfrak{m}}^0(A)\) denote the \(0^{th}\) local cohomology of \(A\).