Lim Ulrich sequences and Lech's conjecture

TL;DR

This paper proves Lech's conjecture for standard graded rings over perfect fields by introducing lim Ulrich sequences, which are sequences of modules that asymptotically behave like Ulrich modules, and showing their existence implies the conjecture.

Contribution

The paper introduces lim Ulrich and weakly lim Ulrich sequences and proves their existence in certain cases, leading to a proof of Lech's conjecture in all dimensions under specified conditions.

Findings

Lech's conjecture is proven for standard graded rings over perfect fields.

Weakly lim Ulrich sequences are constructed for all standard graded domains over perfect fields of positive characteristic.

Existence of these sequences implies the validity of Lech's conjecture.

Abstract

The long standing Lech's conjecture in commutative algebra states that for a flat local extension $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local rings, we have an inequality on the Hilbert--Samuel multiplicities: $e(R)\leq e(S)$. In general the conjecture is wide open when $\dim R>3$, even in equal characteristic. In this paper, we prove Lech's conjecture in all dimensions, provided $(R,\mathfrak{m})$ is a standard graded ring over a perfect field localized at the homogeneous maximal ideal. We introduce the notions of lim Ulrich and weakly lim Ulrich sequences. Roughly speaking these are sequences of finitely generated modules that are not necessarily Cohen--Macaulay, but asymptotically behave like Ulrich modules. We prove that the existence of these sequences imply Lech's conjecture. Though the existence of Ulrich modules is known in very limited cases, we construct weakly lim Ulrich sequences for all standard graded domains over perfect fields of positive characteristic.