Uniform Lech’s inequality

Abstract

Let ( R , m ) (R,\mathfrak {m}) be a Noetherian local ring of dimension d ≥ 2 d\geq 2 . We prove that if e ( R ^ r e d ) > 1 e(\widehat {R}_{red})>1 , then the classical Lech’s inequality can be improved uniformly for all m \mathfrak {m} -primary ideals, that is, there exists ε > 0 \varepsilon >0 such that e ( I ) ≤ d ! ( e ( R ) − ε ) ℓ ( R / I ) e(I)\leq d!(e(R)-\varepsilon )\ell (R/I) for all m \mathfrak {m} -primary ideals I ⊆ R I\subseteq R . This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of I I .