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Math-2315 · mathematics · Algebra

Strong Nakayama Conjecture

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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.
Does every Artin algebra $A$ have finite finitistic dimension? For every nonzero finitely generated $A$-module $M$, must there be an integer $n\geq0$ such that $\operatorname{Ext}_A^n(M,A)\neq0$?

Formulation references: [S1], [S2].

Short English statement

Does every Artin algebra $A$ have finite finitistic dimension? For every nonzero finitely generated $A$-module $M$, must there be an integer $n\geq0$ such that $\operatorname{Ext}_A^n(M,A)\neq0$?

Sources

Catalogue entry

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

Contributions 1

Math-2315-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2315

The selected Tachikawa corollaries give a nonzero simple module with $\operatorname{Ext}^i(S,\Gamma)=0$ for every $i\ge0$, including $\operatorname{Hom}(S,\Gamma)=0$, and modules of every finite projective dimension. These refute both the strong-Nakayama and finitistic-dimension clauses. The separate infinite-finitistic-dimension manuscript settles the latter clause alone.

Strong Nakayama Conjecture

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: A single finite-dimensional complex algebra is claimed to have finite modules whose finite projective dimensions are unbounded; A finite-dimensional symmetric algebra is claimed to carry a nonprojective finite module without positive self-extensions, refuting Tachikawa second-conjecture rigidity. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Manuscript 1.
A finite-dimensional unital complex algebra $A$ is claimed to admit, for every $m\ge1$, a left module $N_m$, finite-dimensional over $\mathbb C$, satisfying \[ 2m-2\le\operatorname{pd}_AN_m<\infty. \] Thus its little finitistic dimension, the supremum of finite projective dimensions of finite-dimensional modules, is infinite. A second complex algebra $\Lambda$ satisfies \[ \operatorname{findim}\Lambda=\operatorname{Findim}\Lambda=\infty, \qquad \operatorname{findim}(\Lambda^{\mathrm{op}}) =\operatorname{Findim}(\Lambda^{\mathrm{op}})=0, \] where each displayed dimension is for left modules and the capitalized version allows all modules. The corollary also distinguishes generation of the unbounded derived categories by injectives on the two sides.
Manuscript 2.
Take $k=\mathbb F_2(q,H_1,H_2)$, with three algebraically independent indeterminates. The source claims a finite-dimensional symmetric unital algebra $A$ and a nonprojective finite-dimensional left module $M$ satisfying \[ \operatorname{Ext}^i_A(M,M)=0\qquad(i>0). \] For every extension field $K/k$, put $A_K=K\otimes_kA$, $M_K=K\otimes_kM$, and $\Gamma_K=\operatorname{End}_{A_K}(A_K\oplus M_K)^{\mathrm{op}}$. The homological corollary claims that its minimal injective resolution has only projective terms but infinite injective dimension; also a nonzero simple $S$ satisfies \[ \operatorname{Ext}^i_{\Gamma_K}(S,\Gamma_K)=0\qquad(i\ge0). \] It supplies a projective-injective Wakamatsu tilting module that is not tilting. The finitistic corollary provides a left module $C_n$, finite-dimensional as a vector space, for which $\operatorname{pd}_{\Gamma_K}C_n=n$ for each $n\ge1$, hence both left finitistic dimensions are infinite.

Outline of the argument

One fixed finitely presented algebra encodes arbitrarily long terminating processes through a twisted derived tensor functor. The subsequent finite-dimensional realization converts these tensor histories into ordinary projective resolutions, retaining unbounded termination lengths without changing the final algebra.

The construction strengthens complete-resolution vanishing on both sides of degree zero before passing to a trivial extension. A transfer calculation then produces a symmetric algebra and preserves the nonprojective module together with its positive Ext vanishing.

Formal verification

For family 198, the Lean scope includes unbounded finite projective dimensions over one complex algebra and a separate extreme left/right finitistic and injective-generation asymmetry. The listed Comparator files are FinitisticAsymmetry.lean, LittleFinitistic.lean. Named declarations include OAI.LittleFinitistic.Main.exists_counterexample, OAI.LittleFinitistic.extreme_left_right_asymmetry. For family 199, the documented entries cover the characteristic-two rational-function-field Auslander--Reiten example, including field-extension persistence, and a separate symmetric Tachikawa counterexample. The listed Comparator files are AuslanderReiten.lean, Tachikawa.lean. Named declarations include OAI.ArExplicit.Statement.main, OAI.Tachikawa.main_theorem. 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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