Lech's conjecture in dimension three

Let (R,m)→(S,n) be a flat local extension of local rings. Lech conjectured around 1960 that there should be a general inequality e(R)≤e(S) on the Hilbert-Samuel multiplicities [24]. This conjecture is known when the base ring R has dimension less than or equal to two [24], 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)≤(d!/2d)⋅e(S) where d=dim⁡R≥4, in equal characteristic.