Lech's conjecture in dimension three

@article{Ma2016LechsCI,
  title={Lech's conjecture in dimension three},
  author={Linquan Ma},
  journal={arXiv: Commutative Algebra},
  year={2016}
}
  • Linquan Ma
  • Published 1 September 2016
  • Mathematics
  • arXiv: Commutative Algebra
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
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