# Lech’s multiplicity conjecture

> Hilbert–Samuel multiplicity is claimed never to drop along a flat local map, settling Lech's 1960 conjecture.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-194/
Last updated: 2026-10-14

**Hilbert–Samuel multiplicity is claimed never to drop along a flat local map, settling Lech's 1960 conjecture.**

**Trust:** Formal proof listed · **Papers:** 1 · **First paper:** September 23, 2026 · **Field:** Algebra · **Entry:** 194

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

The multiplicity of a ring at a point measures how singular that point is. Lech conjectured that it cannot go down along a flat map, a map where one ring varies evenly over another. Earlier proofs covered small dimensions and special rings. This paper claims to prove it in every case.

## At a glance

- **First step:** 1960 (Christer Lech, Note on multiplicities of ideals)
- **Significance:** Major. It would settle a conjecture of commutative algebra open since 1960, in full generality.
- **Trust:** Formal proof listed
- **Field:** Algebra (1 paper)

## The explanation

Take the curve xy = 0: two lines crossing at the origin. Study it very close to the crossing. Write down the monomials in x and y, but drop any of degree N or more. Since xy = 0, only 1, x, …, x^(N−1) and y, …, y^(N−1) survive. That makes 2N − 1. For N = 100 it is 199. Divide by N and let N grow, and you get 2. A single smooth line keeps only N monomials, so its limit is 1.

That limit is the Hilbert–Samuel multiplicity, written e. It measures how singular a point is. A smooth point has e = 1, and the crossing has e = 2. In general, the count grows like a multiple of N^d, where d is the dimension. One multiplies the count by d! and divides by N^d[1]. In dimension zero, e is simply the length, the number of independent quantities[2]. The setting is a Noetherian local ring: the algebraic model of the neighbourhood of a single point, with a finiteness condition.

Now add a third variable z to the crossing lines. The new ring describes two planes crossing along a line. Monomials of degree below N now number N². For N = 100 that is 10,000. Multiply by 2! and divide by N², and e is again 2. Adding a variable is the simplest flat map. A flat map from one local ring R to another, S, makes S vary evenly over R, with no sudden jumps. Christer Lech asked whether multiplicity can ever go down along such a map[3]. His conjecture says e(R) ≤ e(S) always.

Lech proved it when R has dimension at most two, and in some special cases[4]. Linquan Ma proved dimension three in equal characteristic, where the ring and its point share the same arithmetic[5]. Ma also proved a weaker inequality, with a factor max{1, d!/2^d}[5]. For d = 10 that factor is about 3,544. Ma also proved it in every dimension when R is a standard graded ring over a perfect field[6]. One popular method needed special sequences of modules over every ring. Yhee showed some two-dimensional rings have none[7].

This paper claims to prove Lech's conjecture with no restriction on dimension, residue field or characteristic[8].

The obstacle is a mismatch. Flatness controls powers of R's maximal ideal pushed into S. But e(S) uses powers of S's own maximal ideal. That difference blocks a direct comparison of the leading terms[9]. Two problems remained: keeping the factor exactly 1 in every dimension, and estimating lengths in mixed characteristic[10]. The paper claims to solve both with one estimate for chains of matrices, called free complexes. Its constants depend only on the number of parameters and the sizes of the matrices[11]. In characteristic p, the Frobenius map raises matrix entries to high powers without changing sizes[12]. In mixed characteristic, the paper uses perfectoid normalized lengths instead[13]. In residue characteristic zero, it shows a counterexample would give one in characteristic p, which cannot exist[14].

The same estimate gives a side result. For a complete local domain of characteristic p, a quantity called the Dutta multiplicity is at least e[15].

How sure is this? A computer proof checker called Lean has checked a supporting statement. It proves that side result, the Dutta comparison for complete local domains of characteristic p. The full conjecture in every characteristic is outside that check. No specialist review of the paper has been published.

**What it doesn't show.** This does not say when equality e(R) = e(S) holds, and Lean has not checked the full theorem.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Major if it holds. Lean checks only a supporting characteristic-p statement; the main theorem is unchecked and no specialist review is published.

## Words used

- **Noetherian local ring:** The algebraic model of the neighbourhood of a single point, with a finiteness condition.
- **Hilbert–Samuel multiplicity:** A number e that measures how singular the point is; smooth points have e = 1.
- **Flat local homomorphism:** A map between local rings that makes the target vary evenly over the source.
- **Length:** A count of independent quantities in a module, like the dimension of a vector space.
- **Characteristic:** The arithmetic of a ring: characteristic p means p copies of 1 add to 0.
- **Free complex:** A chain of matrices where each product of neighbours is zero.
- **Dutta multiplicity:** A characteristic-p measure of a free complex, built by raising its entries to powers of p.

## The original paper

- [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf) - September 23, 2026

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/194.md)

## Quoted lines from the paper (15)

[1] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> For a nonzero Noetherian local ring $(A,\mathfrak a)$ of dimension $d$, its Hilbert--Samuel multiplicity is \[ e(A)=\lim_{N\longrightarrow\infty} \frac{d!\,\length_A(A/\mathfrak a^N)}{N^d}. \]

[2] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> In dimension zero this is simply the length of $A$.

[3] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> Lech asked whether this invariant can decrease under a flat local homomorphism.

[4] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> Lech's original work established the inequality when the source has dimension at most two and in important cases with complete-intersection closed fiber.

[5] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> Ma proved the conjecture for equal-characteristic sources of dimension three and established the general bound \[ e(R)\leq \max\{1,d!/2^d\}\,e(S),\qquad d=\dim R, \] in equal characteristic.

[6] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> In arbitrary dimension, Ma proved the conjecture when the source is a standard graded algebra over a perfect field, localized at its homogeneous maximal ideal; the flat local target need not be graded.

[7] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> Nor can one require weakly lim Ulrich sequences over every domain: Yhee constructed complete two-dimensional local domains admitting no such sequence.

[8] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> There is no restriction on the dimensions, the residue fields, or the characteristics of the rings.

[9] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> Even when the closed fiber is zero-dimensional, so that both filtrations have finite colength, this difference prevents a direct comparison of their leading terms from giving the asserted lower bound for $e(S)$.

[10] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> Two difficulties remain in passing from such results to the unrestricted theorem: retaining coefficient $1$ in every dimension, and obtaining the required length estimates in mixed characteristic.

[11] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> Our argument uses a single estimate for free complexes in both settings. Its constants depend only on the number of parameters and the ranks, even when the auxiliary ring or its length function changes.

[12] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> In characteristic $p$, Frobenius raises the differential entries to arbitrarily high order without changing their ranks.

[13] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> For the present problem we use perfection in characteristic $p$ and the perfectoid normalized lengths of Cai--Lee--Ma--Schwede--Tucker in mixed characteristic.

[14] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> Artin approximation and specialization preserve these identities in positive characteristic. A strict counterexample would therefore contradict the characteristic-$p$ case.

[15] From [Lech's multiplicity conjecture](https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf):

> Corollary … proves $\chi_\infty(F)\geq e(R)$ for every short complex over a complete Noetherian local domain of characteristic $p>0$.

## Other sources

- [C. Lech, “Note on multiplicities of ideals”, Arkiv för Matematik 4 (1960), 63–86](https://doi.org/10.1007/BF02591323)
- [L. Ma, “Lech's conjecture in dimension three”, Advances in Mathematics 322 (2017), 940–970, arXiv:1609.00095](https://arxiv.org/abs/1609.00095v3)
- [openai/math, Lean scope note for family 194 (2026)](https://github.com/openai/math/blob/main/lean/docs/194.md): “This selected statement is the complete-domain Dutta comparison.”
- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”

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