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Lech’s multiplicity conjecture
Proves for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic.
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Lech's multiplicity conjecture
Sep 23, 2026·paper.pdf
We prove Lech's multiplicity conjecture: Hilbert–Samuel multiplicity cannot decrease under a flat local homomorphism of nonzero Noetherian local rings. The result holds in arbitrary dimension, with no restrictions on the residue fields or characteristics.