(h\)-function, Hilbert-Kunz density function and Frobenius-Poincaré function

• Published in: arXiv.org 2023-10
• Main Authors: Cheng, Meng, Mukhopadhyay, Alapan
• Format: Article
• Language: English
• Subjects: Convexity; Density; Invariants; Questions
• ISSN: 2331-8422

Given ideals \(I,J\) of a noetherian local ring \((R, \mathfrak m)\) such that \(I+J\) is \(\mathfrak m\)-primary and a finitely generated module \(M\), we associate an invariant of \((M,R,I,J)\) called the \(h\)-function. Our results on \(h\)-function allow extensions of the theories of Frobenius-Poincaré functions and Hilbert-Kunz density functions from the known graded case to the local case, answering a question of Trivedi. When \(J\) is \(\mathfrak m\)-primary, we describe the support of the corresponding density function in terms of other invariants of \((R, I,J)\). We show that the support captures the \(F\)-threshold: \(c^J(I)\), under mild assumptions, extending results of Trivedi and Watanabe. The \(h\)-function treats Hilbert-Samuel, Hilbert-Kunz multiplicity and \(F\)-threshold on an equal footing. We develop the theory of \(h\)-functions in a more general setting which yields a density function for \(F\)-signature. A key to many results on \(h\)-function is a `convexity technique' that we introduce, which in particular proves differentiability of Hilbert-Kunz density function almost everywhere, thus contributing to another question of Trivedi.