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Lech's multiplicity conjecture

Theorem 1.1: for every flat local homomorphism $(R,\mathfrak m)\to(S,\mathfrak n)$ of nonzero Noetherian local rings, $e(R)\le e(S)$ , in all dimensions and characteristics. The key input is a uniform lower bound on the zeroth homology of finite free complexes with entries in high powers of a parameter ideal, valid for generalized length functions, applied with Frobenius perfection in characteristic $p$ and perfectoid lengths in mixed characteristic. The paper does not claim the stronger Iyengar-Ma-Walker conjecture for all complete local rings, only its complete-domain characteristic- $p$ case.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Lech's multiplicity conjecture · Lech's conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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VibeMathed record: Lech's multiplicity conjecture

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Lech's question as the paper states it from Lech (1960, 1964); it claims $e(R)\le e(S)$ for every flat local homomorphism of nonzero Noetherian local rings, with no restriction on dimension, residue fields or characteristic. The proof was not refereed. It uses the perfectoid normalized lengths of Cai-Lee-Ma-Schwede-Tucker in mixed characteristic and Eisenbud-Schreyer vector bundles. The family's Lean challenge (DuttaDomain, found via lean/docs/194.md, not in formalization.yaml) states only a supporting characteristic- $p$ comparison over complete local domains, not Lech's inequality, so this entry is left Unreviewed; that statement is entered separately.