Lech's multiplicity conjecture

For a nonzero Noetherian local ring $(A,\mathfrak a)$ of dimension $d$ , the Hilbert-Samuel multiplicity is $e(A)=\lim_N d!\,\ell_A(A/\mathfrak a^N)/N^d$ . Lech (1960) proved the inequality below when the source has dimension at most two and in complete-intersection cases; Ma proved it in equal characteristic for dimension three, for standard graded sources over perfect fields, and a bound $e(R)\le\max\{1,d!/2^d\}e(S)$ . Lech asked whether multiplicity can decrease under flat maps. If $(R,\mathfrak m)\to(S,\mathfrak n)$ is a flat local homomorphism of Noetherian local rings, is $e(R)\le e(S)$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Commutative algebra; Hilbert-Samuel multiplicity
Posed by: Christer Lech, Note on multiplicities of ideals, Ark. Mat. 4 (1960); Inequalities related to certain couples of local rings, Acta Math. 112 (1964)
Year posed: 1960
Years open: 66y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 45 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

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.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

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.

Lech's multiplicity conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion