Lech's conjecture in dimension three

TL;DR

This paper proves Lech's conjecture for dimension three in equal characteristic, establishing that the Hilbert-Samuel multiplicity of the base ring does not exceed that of the extension, and provides partial estimates in higher dimensions.

Contribution

It confirms Lech's conjecture in dimension three for equal characteristic rings and offers partial bounds in higher dimensions.

Findings

Lech's conjecture holds in dimension three for equal characteristic rings.

Established an inequality e(R) ≤ e(S) in the specified setting.

Provided partial estimates for higher dimensions, e.g., e(R) ≤ (d!/2^d)·e(S).

Abstract

Let $(R,m)\to (S,n)$ be a flat local extension of local rings. Lech conjectured in 1960 that there should be a general inequality $e(R)\leq e(S)$ on the Hilbert-Samuel multiplicities. This conjecture is known when the base ring $R$ has dimension less than or equal to two, and remains open in higher dimensions. In this paper, we prove Lech's conjecture in dimension three when $R$ has equal characteristic. In higher dimension, our method yields substantial partial estimate: $e(R)\leq (d!/2^d)\cdot e(S)$ where $d=\dim R\geq 4$, in equal characteristic.