Lech's conjecture on Hilbert–Samuel multiplicities

Let $(R,\mathfrak{m}) \to (S,\mathfrak{n})$ be a flat local extension of local rings. Here $e(R)$ and $e(S)$ denote their Hilbert–Samuel multiplicities.

Lech's conjecture. Then

$$e(R) \leq e(S).$$

This conjecture was formulated by Lech around 1960 and remains open in most cases. The paper proves it under additional hypotheses, including when $S$ is the localization of a standard graded ring over a field at the homogeneous maximal ideal and $\mathfrak{m}S$ is the localization of a homogeneous ideal that is $\mathfrak{n}$-primary.