On Semicontinuity of Multiplicities in Families

Ilya Smirnov

Documenta mathematica, Tome 25 (2020), pp. 381-399

Résumé

This paper investigates the behavior of Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity in families of ideals. We show that Hilbert-Samuel multiplicity is upper semicontinuous and that Hilbert-Kunz multiplicity is upper semicontinuous in families of finite type. As a consequence, F-rational signature, an invariant defined by Hochster and Yao as the infimum of relative Hilbert-Kunz multiplicities, is, in fact, a minimum. This gives a different proof for its main property: F-rational signature is positive if and only if the ring is F-rational. The tools developed in this paper can be also applied to families over Z and yield a solution to Claudia Miller's question on reduction mod p of Hilbert-Kunz function.

Détail

DOI : 10.4171/dm/751

Classification : 13A35, 13D40, 13H15, 14B05, 14B07, 14D06
Mots-clés : semicontinuity, multiplicity, Hilbert-Samuel polynomial, Hilbert-Kunz multiplicity, families