ProofAtlas

Representation theory · homological algebra · finite-dimensional algebras

Finitistic Dimension Conjecture

Collaboration beta

Each module with finite projective dimension has some finite resolution length. The conjecture asks whether, for each finite-dimensional algebra, all those finite lengths share one finite upper bound.

∀Afinite-dimensional,findim(A)<∞ ?
Known results and sources
Editorial diagram of correctly oriented finite projective resolutions over a finite-dimensional algebra, rising toward the open question findim(A)<∞.
The conjecture asks whether every finite-dimensional algebra has one finite upper bound on the projective dimensions that happen to be finite.

Research problem

Exact mathematical statement

For every finite-dimensional algebra AA over a field kk, is

findim(A):=sup{pdAM:M∈mod-A, pdAM<∞}<∞?\operatorname{findim}(A):=\sup\{\operatorname{pd}_A M:M\in\operatorname{mod}\text{-}A,\ \operatorname{pd}_A M<\infty\}<\infty?

The source also records the finite-dimensional duality findim(A)=finid(A)\operatorname{findim}(A)=\operatorname{finid}(A). The conjectured finiteness remains open.

Problem infographic

Problem at a glance

Problem-first explainer defining finitistic dimension and showing the source’s hypothetical X, S0, and S+ branches as unresolved rather than as a proof.
The handoff repairs an exhaustive branch split for a hypothetical counterexample, but explicitly proves none of the three terminal branches completely.

Current mathematical picture

Where work on Finitistic Dimension Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureWhole self-extension descent

The source computes the minimal presentation and finds original projective terms survive while total size strictly increases. A separately proved factor extraction preserving actual arbitrarily large depth, or a strict algebra reduction, could still be useful.

Route status · Narrowed route
Main reductionCurrent reduction

Assuming a counterexample, pass to a split-basic setting and analyze projective-dimension-one terminal presentations. Either one fixed projective pair carries unbounded depth, leading to S0/S+, or every fixed pair has finite depth and terminal size diverges, leading to X.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeClose terminal-pair explosion X.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Finitistic Dimension Conjecture in numbers

2.8kretained lines of mathematical investigation2,753 in the current working snapshot
Argument development
2,325 · 84%
Explored or eliminated routes
67 · 2%
Computational analysis
38 · 1%
Open obligations
128 · 5%
Definitions and setup
195 · 7%
7selected mapped statements1routes investigated3open questions3contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byAlex Rosenberg and Daniel Zelinsky, as attributed and publicized by Hyman BassProposed1960Status checkedAug 14, 2026
Source recordedAug 14, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

11 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

11 selected steps

Scroll horizontally to explore the route

Working route overview for Finitistic Dimension ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Are all finite projective dimensions uniformly bounded? — Depends on missing premiseAre all finite projectivedimensions uniformlybounded?Current reduction — Depends on missing premiseCurrent reductionExhaustive stationary/explosion split — Depends on missing premiseExhaustivestationary/explosion splitClosing target — Depends on missing premiseClosing targetExact self-extension dimension formula — Depends on missing premiseExact self-extensiondimension formulaProjective dimension survives scalar extension — Depends on missing premiseProjective dimensionsurvives scalar extensionTerminal profiles are generalized inverses — Depends on missing premiseTerminal profiles aregeneralized inversesWhole self-extension descent — stoppedWhole self-extension descentClose terminal-pair explosion X. — OpenClose terminal-pairexplosion X.Close the stationary zero-complexity branch S0. — OpenClose the stationaryzero-complexity branch S0.Close the positive-complexity branch S+. — OpenClose thepositive-complexity branchS+.
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeWhole self-extension descent

The source computes the minimal presentation and finds original projective terms survive while total size strictly increases. A separately proved factor extraction preserving actual arbitrarily large depth, or a strict algebra reduction, could still be useful.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Close terminal-pair explosion X.Suggested move: Derive a bounded deep core, asymptotic contradiction, or strict algebra reduction from the divergence of minimal terminal-pair size.
Ready to work on
02
Close the stationary zero-complexity branch S0.Suggested move: Audit rigid and nonrigid cases, build a finite-dimensional strict reduction if available, and prove reflection of high finite projective dimensions.
Ready to work on
03
Close the positive-complexity branch S+.Suggested move: Establish tangent or jet richness at arbitrarily large depths, then prove a separate factor extraction that preserves actual depth and lowers the data.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The little finitistic dimension conjecture remains open for general finite-dimensional algebras. Peer-reviewed work proves the conjecture under additional hypotheses or for special classes, including recent radical-preservation criteria, but does not supply a general proof.

[1][2]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedGiatagantzidis established new radical-preservation criteria and a finite-finitistic-dimension special class; this is not a general resolution.[2]
  2. Peer reviewedRickard proved that injectives generating the unbounded derived category implies the finitistic dimension conjecture for a finite-dimensional algebra, while retaining the general problem as open.[1]
  3. Historical sourceRickard records that Bass publicized the question in 1960 and attributed it to Rosenberg and Zelinsky.[1]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusFinitistic Dimension Conjecture
Stronger or generalized formBig finitistic dimension conjecture

The big finitistic dimension ranges over all modules; its finiteness implies the little conjecture, but it is a distinct and stronger question.

[1]
Dependency or reductionInjectives generate the unbounded derived category

Injective generation of the unbounded derived category is a sufficient condition for the little finitistic dimension conjecture in Rickard’s theorem.

[1]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetFormal homological-algebra infrastructure for minimal projective resolutions and projective dimension over finite-dimensional algebras.
  • Formalization targetA formal statement of the supremum over finitely generated modules of finite projective dimension.
  • Formalization targetMachine-checked reductions covering the general, rather than special-class, conjecture.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma4 of 74
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementAre all finite projective dimensions uniformly bounded?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementProjective dimension survives scalar extensionintermediate
  • retained route statementExact self-extension dimension formulaintermediate
  • retained route statementExhaustive stationary/explosion splitintermediate
  • retained route statementTerminal profiles are generalized inversesintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureWhole self-extension descentreported failure
  • Research targetClose terminal-pair explosion X.open
  • Research targetClose the stationary zero-complexity branch S0.open
  • Research targetClose the positive-complexity branch S+.open
  • Research targetExact finitistic-dimension questionsuperseded
  • Research targetAll three final branches remain opensuperseded
  • Narrowed routeWhole self-extension descentThe source computes the minimal presentation and finds original projective terms survive while total size strictly increases. A separately proved factor extraction preserving actual arbitrarily large depth, or a strict algebra reduction, could still be useful.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeClose terminal-pair explosion X.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointClose terminal-pair explosion X.

Finitistic Dimension Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Each module with finite projective dimension has some finite resolution length. The conjecture asks whether, for each finite-dimensional algebra, all those finite lengths share one finite upper bound.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references2 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Unbounded derived categories and the finitistic dimension conjecturepeer reviewed result · Jeremy Rickard · Advances in Mathematics · 2019-10-01 · DOI 10.1016/j.aim.2019.106735 · accessed Aug 14, 2026
  2. 2
    Radical preservation and the finitistic dimensionpeer reviewed result · Odysseas Giatagantzidis · Bulletin of the London Mathematical Society · 2025-11-03 · DOI 10.1112/blms.70222 · accessed Aug 14, 2026

Important qualifications

  • The collection focuses on the little finitistic dimension conjecture for finite-dimensional algebras, not every big-finitistic-dimension or Artin-algebra variant.
  • Recent results listed are special-class or conditional theorems and do not resolve the general conjecture.
  • No external computation or formal proof artifact was independently reproduced.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image