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Math-2428 · mathematics · Algebra

Lech Multiplicity Conjecture for Flat Local Extensions

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Lech's conjecture: if $(R,\mathfrak m)\to(S,\mathfrak n)$ is a flat local extension of Noetherian local rings, then $e(R)\leq e(S)$.

Formulation references: [S1], [S2].

Short English statement

Lech's conjecture: if $(R,\mathfrak m)\to(S,\mathfrak n)$ is a flat local extension of Noetherian local rings, then $e(R)\leq e(S)$.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2428-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2428

Theorem 1.1 proves exactly the all-flat-local Hilbert–Samuel multiplicity comparison, without imposing a characteristic, ramification, dimension, or domain restriction. Within these assumptions, this is a claimed affirmative answer to the stated question.

Lech Multiplicity Conjecture for Flat Local Extensions

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For a flat local map of nonzero Noetherian rings, the proposed inequality is $e(R)\leq e(S)$ without a dimension or characteristic restriction.

Mathematical statement of the claimed result

Result 1.
For any flat local homomorphism $(R,\mathfrak m)\to(S,\mathfrak n)$ between nonzero Noetherian local rings, the claimed Hilbert--Samuel multiplicities satisfy \[ e_{\mathfrak m}(R)\le e_{\mathfrak n}(S). \] The theorem imposes no extra condition on residue fields, dimensions, or characteristic.

Outline of the argument

The argument replaces unavailable Ulrich modules by short free complexes and a uniform estimate for their Euler lengths. Cohen factorization connects those estimates to multiplicity, while normalized-length algebras provide the parameter calculations in positive and mixed characteristic.

Formal verification

For family 194, the Lean entry concerns positive-characteristic complete-domain Dutta multiplicity and its Frobenius limit; arbitrary-characteristic flat-local monotonicity is beyond that entry. The listed Comparator files are DuttaDomain.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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