Constancy of the Hilbert-Samuel function (Q6655555)

The article under review investigates conditions under which the Hilbert-Samuel function remains locally constant on a locally Noetherian scheme.\N\NMore precisely, the authors establish (Theorem 1.1) that for a locally Noetherian scheme \( X \) with excellent local rings at every point, the Hilbert-Samuel function is locally constant if and only if \( X \) is normally flat along its reduction \( X_{\text{red}} \), and \( X_{\text{red}} \) itself is everywhere regular.\N\NThis result extends previous findings applicable only to reduced schemes, providing a broader criterion for the constancy of the Hilbert-Samuel function.\N\NThe proof of Theorem 1.1 makes use of Hironaka's characteristic polyhedron [\textit{H. Hironaka}, J. Math. Kyoto Univ. 7, 19--43 (1967; Zbl 0153.22302)], as well as Giraud presentations [\textit{J. Giraud}, Ann. Sci. Éc. Norm. Supér. (4) 8, 201--234 (1975; Zbl 0306.14004)].\N\NThe study also highlights the significance of the Hilbert-Samuel function and the multiplicity function as fundamental invariants in Noetherian schemes, particularly in the context of desingularization. Regarding the multiplicity function, the reader is referred to \textit{O. E. Villamayor U.} [Adv. Math. 262, 313--369 (2014; Zbl 1295.14015)] for a similar result (but under weaker hypothesis) concerning this function instead of the Hilbert-Samuel function when \(X\) is an equidimensional scheme of finite type over a perfect field.