Lech's conjecture in dimension three

2016/09/01 by Ma, Linquan
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1609.00095

Abstract

Let (R,m)→ (S,n) be a flat local extension of local rings. Lech conjectured in 1960 that there should be a general inequality e(R)≤ 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)≤ (d!/2^d)⋅ e(S) where d=dim R≥ 4, in equal characteristic.

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