# Lech's conjecture in dimension three

@article{Ma2016LechsCI, title={Lech's conjecture in dimension three}, author={Linquan Ma}, journal={arXiv: Commutative Algebra}, year={2016} }

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… Expand

#### 13 Citations

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