2026/07/31 by Clare D'Cruz, Mousumi Mandal, Shruti Priya
Mathematics · #math.AC #msc:13A30 #msc:13D40 #msc:13E05 #msc:13H10
29 Pages. Comments are welcome
arxiv created 2026/07/31 · arxiv updated 2026/08/03
Let (R, \mathfrak m) be a Noetherian local ring of dimension d ≥ 1 with depth R ≥ d-1, and let I be an \mathfrak m-primary ideal. In this paper, we study bounds on the second Hilbert coefficient of I, denoted by e2(I). Under the assumption that the associated graded ring G(I) has depth at least d-1, we first establish a lower bound for e2(I). We then extend several known results from the Cohen-Macaulay case to this general setting and obtain upper bounds for e2(I) in terms of the sectional genus denoted by gs(I) and the Hilbert coefficients of I and those of a minimal reduction Q of I. We further analyze the extremal case when e2(I) attains this bound and relate it to the depth of G(I). In addition, for Buchsbaum local rings, we establish a sharp upper bound for e2(\mathfrak m) using the technique of S2-fication. Finally, in the Cohen-Macaulay case, we give sufficient conditions to ensure good properties on the depth of G(I) and of G(In) under the assumption that e2(I)=0.