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The Iyengar-Ma-Walker Dutta multiplicity conjecture for complete local domains of positive characteristic

Over a dd-dimensional Noetherian local ring RR, a short complex is a finite free complex in degrees 0,…,d0,\dots,d with finite-length homology and nonzero zeroth homology. In characteristic p>0p>0 its Dutta multiplicity is χ∞(F)=lim⁡np−nd∑i(−1)iℓ(Hi(ΦnF))\chi_\infty(F)=\lim_n p^{-nd}\sum_i(-1)^i\ell(H_i(\Phi^nF)), where Φn\Phi^n raises differential entries to pnp^n-th powers. Iyengar, Ma and Walker (2022, Conjecture 8.1) conjectured χ∞(F)≥e(R)\chi_\infty(F)\ge e(R) for short complexes over complete local rings, which would imply Lech's conjecture (their Proposition 8.3). For every complete Noetherian local domain of characteristic p>0p>0 and every short complex FF, is χ∞(F)≥e(R)\chi_\infty(F)\ge e(R)?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Commutative algebra; Dutta multiplicity of short complexes
Posed by
Srikanth B. Iyengar, Linquan Ma and Mark E. Walker, Multiplicities and Betti numbers in local algebra via lim Ulrich points, Algebra & Number Theory 16 (2022), Conjecture 8.1
Year posed
2022
Years open
4y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Corollary of the uniform complex estimate: for a complete Noetherian local domain RR of characteristic p>0p>0 and every short complex FF over RR, the Dutta multiplicity exists and χ∞(F)≥e(R)\chi_\infty(F)\ge e(R). This is the complete-domain, positive-characteristic case of IMW Conjecture 8.1; the general conjecture for all complete local rings (and mixed characteristic) is not claimed. The paper notes it is a consequence of, not an input to, its proof of Lech's conjecture.

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: Corollary (Dutta domain case) of the Lech manuscript was read against IMW Conjecture 8.1 as the paper cites it; it is the complete-domain, positive-characteristic case only. The challenge is not in the formalization catalogue (lean/formalization.yaml); it is found through lean/docs/194.md, and its solution module OAI.RingTheory.Multiplicity.Main (importing DuttaDomain) exists at the pinned commit. ComparatorChallenges/DuttaDomain.lean was read here: for every complete Noetherian local domain DD of characteristic pp and every short complex FF (free, finite, bounded in degrees −d..0-d..0, finite-length homology, nonzero H0H^0) it asserts finite-length homology of all Frobenius iterates, convergence of the Dutta sequence, and e(D)≤χ∞(F)e(D)\le\chi_\infty(F). This states the corollary. Not rebuilt here.

Sources

Changelog1 change

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