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Math-453-A2 · revision r1 · on Math-453

OpenAI claim summary for Math-453

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The proposed complex example is normal and complete, with Krull dimension three. Every nonzero finite module over it has smaller depth, directly opposing universal existence of a small Cohen--Macaulay module. This is presented as a claimed negative answer.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Small Cohen-Macaulay Conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Small Cohen-Macaulay Conjecture

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. A normal complete local domain of dimension three over $\mathbb C$ is claimed to admit no nonzero finite module of depth three.

Mathematical statement of the claimed result

Result 1.
The source claims a complete Noetherian normal local domain $R\supset\mathbb C$, with residue field $\mathbb C$, satisfying \[ \dim R=3,\qquad \forall M\ne0\text{ finitely generated over }R:\ \operatorname{depth}_RM<3. \] Its surface obstruction takes a smooth integral projective complex surface $X$, an ample globally generated divisor $H$, and the completed vertex ring $R(X,H)=\widehat{T(X,H)_{T_{>0}}}$, $T(X,H)=\bigoplus_{j\ge0}H^0(X,\mathcal O_X(jH))$. Existence of a small Cohen--Macaulay module would imply \[ K_X^2-2c_2(X)\le6H^2. \] For the constructed polarization at parameter $n=7$, its uncompleted three-dimensional vertex localization $T_{T_{>0}}$ also admits only the zero module among finitely generated modules of maximal Cohen--Macaulay depth.

Outline of the argument

The construction completes a polarized surface cone. Extending a hypothetical maximal Cohen--Macaulay module to a regular model, balancing slopes by modifications, and comparing Chern characters yields a numerical surface inequality; an explicit branched surface violates that inequality.

Formal verification

Family 195 has no Lean scope page in this release. Its selected result is not presented as formalized. 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