Asymptotic Lech's inequality

Abstract

We explore the classical Lech's inequality relating the Hilbert–Samuel multiplicity and colength of an m-primary ideal in a Noetherian local ring (R,m). We prove optimal versions of Lech's inequality for sufficiently deep ideals in characteristic p>0, and we conjecture that they hold in all characteristics.Our main technical result shows that if (R,m) has characteristic p>0 and Rˆ is reduced, equidimensional, and has an isolated singularity, then for any sufficiently deep m-primary ideal I, the colength and Hilbert–Kunz multiplicity of I are sufficiently close to each other. More precisely, for all ε>0, there exists N≫0 such that for any I⊆R with ℓ(R/I)>N, we have (1−ε)ℓ(R/I)≤eHK(I)≤(1+ε)ℓ(R/I).