Current priority: compute the actual Nori comparison defect and prove D_F^2=0 through the full linear-plus-quadratic saturation identity, retaining double-weight-two cross terms.
Route status · Active routeAlgebraic geometry · algebraic cycles · motives · arithmetic geometry
Bloch–Beilinson Conjectures
Collaboration betaDo rational Chow groups carry a canonical finite filtration whose layers reflect topology, motives, and regulators?
Known results and sources
Research problem
Exact mathematical statement
For every smooth projective variety over a characteristic-zero subfield , and every codimension , construct a descending filtration on the rational Chow group
The expected package places homologically trivial cycles in , makes pullback, pushforward, correspondences, and products compatible with the filtration, and relates the graded pieces to cohomological or motivic Ext data through cycle-class and regulator maps. These expectations comprise a family of connected conjectures; no unconditional construction with the full package is known in general. Chow–Künneth projector formulations additionally require algebraicity and compatible lifting inputs.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Bloch–Beilinson Conjectures stands
The v14 packet keeps the Bloch–Beilinson filtration package open while reporting a substantial conditional reduction. Under explicit Nori, coniveau, continuity, and motivic-comparison inputs, it defines a candidate filtration, identifies its terminal step with the kernel of the Nori cycle map, reduces injectivity to residue-field odd cohomology, and expresses those groups through an exterior defect module generated by a quartic piece. The preferred open route is to establish actual quartic saturation, the equivalent weight-minus-four Ext comparison, or Tate nonresonance over every relevant residue field. Algebraicity of cohomological Künneth projectors remains an independent requirement for the stated Murre consequences. Included exact-arithmetic model computations are source-reported and were not reproduced by ProofAtlas.
Retain weight-two injection and first-prolongation injection as efficient sufficient screens, but not as necessary targets; quartic saturation is more permissive.
Route status · Narrowed routeInjectivity is conditionally reduced to vanishing of explicit fieldwise odd Nori groups, with the local/global distinction and low-dimensional base cases kept explicit.
Evidence posture · Reported reductionThe current work reports that p=0 is trivial, p=1 has no higher incoming differential, and curves have only p=0,1; hence the candidate filtration is complete for curves within the formal setup.
Evidence posture · Source-reported route statement · dependencies incompleteDetermine V_F, K_2,F, K_3,F, the linear map ell_F, the full cross-term quotient B_2,F, and q_F in finite Nori diagrams, then test quartic saturation as an R_F-module identity.
Task status · Ready to work onWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Bloch–Beilinson Conjectures in numbers
1.1kretained lines of mathematical investigation1,096 in the current working snapshot
- Argument development
- 1,014 · 93%
- Explored or eliminated routes
- 7 · 1%
- Computational analysis
- 8 · 1%
- Open obligations
- 23 · 2%
- Definitions and setup
- 44 · 4%
13selected mapped statements6routes investigated5reported milestones6open questions5contribution-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.
Recommended next task
Compute the actual quartic defect
Determine V_F, K_2,F, K_3,F, the linear map ell_F, the full cross-term quotient B_2,F, and q_F in finite Nori diagrams, then test quartic saturation as an R_F-module identity.
Suggested move: Apply rank-one, finite-rank, Koszul, and low-weight Stallings screens before calculating the full R_F-module defect.
What would count as progress
- All actual finite-diagram modules and maps are identified.
- Cross terms in B_2,F are retained.
- Quartic saturation is proved or a surviving R_F-module class is exhibited.
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
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
Scroll horizontally to explore the route
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.
Current priority: compute the actual Nori comparison defect and prove D_F^2=0 through the full linear-plus-quadratic saturation identity, retaining double-weight-two cross terms.
Route status · Active routeProve weight-one Ext agreement together with Ext^3 surjectivity and Ext^4 injectivity for every pure weight-minus-four coefficient over every relevant residue field.
Route status · Active routeIf the quartic module survives, use the actual reductive representation to prove its exterior descendants contain no relevant Tate summands.
Route status · Active routeIn parallel with the no-phantom problem, establish algebraicity of the cohomological Künneth projectors; only then use nilpotent lifting and separately address self-duality.
Route status · Active routeExplored alternatives
Other routes
Deferred fallback: if a Tate-visible defect survives, construct the filtered coniveau complex and prove its terminal transgression is rationally trivial by a global reciprocity law.
Route status · Route held in reserveRetain weight-two injection and first-prolongation injection as efficient sufficient screens, but not as necessary targets; quartic saturation is more permissive.
Route status · Narrowed routeRoute statements and reductions
Statements the next route can inspect and build on
When the graded Lie comparison is an isomorphism in weight one, all relative excess-one defects form a graded module D_f over the exterior algebra on the common weight-one dual, generated in degree zero, with an exact linear-plus-quadratic presentation that retains cross terms.
Source-reported route statementExterior multiplication surjects from every degree-q defect to each higher defect, so D_f^2=0 forces D_f^{2r−2}=0 for every r≥2. The condition D_f^2=0 is the explicit quartic saturation identity.
Source-reported route statementUnder weight-one agreement and the stated continuity, invariant-exactness, image-reduction, and motivic comparison inputs, the current work identifies A_N^{2r−1}(F,Q(r)) with the R_F-invariant Tate part of D_F^{2r−2}.
Source-reported route statement · dependencies incompleteAfter weight-minus-one Ext^1 agreement, quartic saturation is equivalent to Ext^3 surjectivity and Ext^4 injectivity for all pure coefficients of weight −4; this reduces the fieldwise tower to ordinary and first higher Chow-group comparison data.
Source-reported route statement · dependencies incompleteConditional on the separated filtration package and algebraicity of the cohomological Künneth projectors, the current work derives nilpotence, orthogonal Chow-projector lifting, Murre B–D, and equality of the induced projector filtration with F_N. Self-duality requires a further compatible lifting argument.
Source-reported route statement · dependencies incompleteMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Compute the actual quartic defect
Determine V_F, K_2,F, K_3,F, the linear map ell_F, the full cross-term quotient B_2,F, and q_F in finite Nori diagrams, then test quartic saturation as an R_F-module identity.
Suggested move: Apply rank-one, finite-rank, Koszul, and low-weight Stallings screens before calculating the full R_F-module defect.Ready to work on
Test Tate-visible nonresonance
If the quartic defect is nonzero, determine whether any exterior descendant contains the Tate summand Q(−r) and prove uniform nonoccurrence when possible.
Suggested move: Decompose D_F^2 as an R_F-module and use actual weights, Hodge types, polarizations, parity, and monodromy rather than dimension counts.Ready to work on
Prove the quartic derived comparison
For every pure weight-minus-four coefficient, prove Ext^3 surjectivity and Ext^4 injectivity from the reference category to the Nori category, after the weight-one comparison is established.
Suggested move: Translate the comparison to the relevant Künneth summands of ordinary and first higher Chow groups where possible.Ready to work on
Solve the independent projector problem
Establish algebraicity of the cohomological Künneth projectors, then apply nilpotent lifting; add a separate transpose-compatible argument if a self-dual decomposition is claimed.
Suggested move: Keep this workstream separate from cycle-map injectivity and state precisely which Murre conclusions its completion unlocks.Ready to work on
Instantiate the residue-field comparison framework
Specify coherent reference and Nori categories over every coniveau residue field, prove pure equivalence, continuity, exact invariants, weight splittings, field functoriality, and weight-one comparison with correct variance.
Suggested move: Build the coherent comparison system N6a and M1–M5 over one controlled finite diagram and audit every map direction before generalization.Ready to work on
Build the chain-level global fallback
Only if a Tate-visible local quartic defect survives, construct the filtered coniveau model, derive the corrected higher differential, globalize a defect generator, and prove terminal cycles are rationally trivial through quartic-symbol reciprocity.
Suggested move: Defer this route until an actual Tate-visible quartic survivor is established.Blocked by the current route
Sourced mathematical context
The known mathematical landscape
Current statusOpen problem
Open in general. The conjectures predict a canonical finite, functorial, multiplicative filtration on rational Chow groups whose first step is homological triviality and whose graded pieces are controlled by cohomology or motivic Ext data. Murre's Chow–Künneth-projector formulation and several important classes of varieties furnish substantial partial results, but no unconditional construction with the full expected package is known for all smooth projective varieties.
[6][7][10]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintPelaez and Reyes described the Bloch–Beilinson filtration as still conjectural and proved that an unconditional orthogonal filtration recovers incidence equivalence in the stated zero-cycle setting, with…[10] Peer reviewedFu and Vial established a section-property framework and proved it for broad specified classes with motives of abelian type, supplying multiplicative Chow–Künneth structures in further special families…[9] Peer reviewedShen and Vial constructed a motivic Fourier decomposition of the Chow ring for the Hilbert square of a K3 surface and for the variety of lines on a very general cubic fourfold, providing a concrete…[8] Peer reviewedJannsen proved the full Murre package for further special finite-dimensional motives while emphasizing that the strongest form was known only for restricted classes and not for arbitrary smooth projective…[7]
Mathematical neighborhood
Related results and reusable starting points
Current focusBloch–Beilinson conjectures
Murre's conjectures ask for algebraic Chow–Künneth projectors, their prescribed action on Chow groups, independence of the induced filtration, and identification of its first step. For all smooth projective varieties, the complete Murre package provides the projector formulation of the Bloch–Beilinson filtration program.
[3][5]The expected mixed-motive category and its Ext groups motivate the graded pieces of the filtration. Constructing a candidate filtration inside one motivic realization does not by itself prove the full expected comparison, independence, or separatedness statements.
[5][6]Murre's projector route requires algebraic lifts of cohomological Künneth components. Their existence is part of the Chow–Künneth problem and is entangled with Grothendieck's standard conjectures; it cannot be inferred from filtration separatedness alone.
[3][6]The general filtration would imply Bloch's prediction for zero-cycles on complex surfaces of geometric genus zero. Proven instances of that surface conjecture are evidence for only the corresponding special cases, not for the full filtration program.
[7][13]The strongest Murre package is established for curves, rational surfaces, Brauer–Severi varieties, and certain additional motives covered by Jannsen's hypotheses; weaker subsets are known for more surfaces and threefolds.
[7]Beauville-type splittings, Fourier decompositions, and distinguished-cycle section properties construct graded Chow structures for abelian or hyperkähler-type families. They model expected Bloch–Beilinson behavior but are not a universal replacement for it.
[8][9]The orthogonal filtration is unconditionally finite and satisfies several analogous properties; its identified second step in the stated settings is a concrete partial structure, not an identification with the complete conjectural filtration in all codimensions and fields.
[10]Formal and computational footholds
Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.
- formal library support · partial resource linkedMathlib algebraic-cycle infrastructure
Mathlib defines algebraic cycles on schemes as locally finite coefficient functions on points and provides a basic pushforward API. This is useful groundwork only; it is not a definition of rational Chow groups, a Bloch–Beilinson filtration, or a formal statement or proof of the conjectures.
[11]
Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA statement-aligned formalization needs rational equivalence and Chow groups for smooth projective schemes, intersection products, pullback and proper pushforward, correspondences, and cycle-class maps into an appropriate Weil cohomology.
- Formalization targetThe Murre formulation additionally requires Chow motives, algebraic correspondences with composition, cohomological Künneth projectors, orthogonal idempotent Chow lifts, and their actions on Chow groups.
- Formalization targetThe Ext-graded formulation requires a precise formal category or realization of mixed motives, Tate twists, weight structures, derived Ext groups, and comparison maps; none may be replaced silently by a finite toy category.
- Formalization targetA formal candidate filtration must separately establish functoriality, multiplicativity, finite length or separatedness, independence from choices, and the exact first-step identification. Defining a descending family of subgroups is not statement alignment.
- Formalization targetNo problem-level formal statement or proof was identified in the scoped current Mathlib documentation and public Formal Conjectures repository search.
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 updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
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.
Correction detailsCorrected the research recordCorrection note
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.
Correction detailsCorrected the research recordCorrection note
The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.
How the route was assembled
Argument structure
These stages follow the mathematical order of the supplied argument.
Open Bloch–Beilinson target and quartic comparison frontierThe source reports a conditional Nori-filtration construction and compresses the no-phantom problem to an actual quartic motivic-comparison or Tate-nonresonance theorem over every relevant residue field, while projector algebraicity remains separate.
Research stage 1Mapped research milestoneInitial research sequence
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
11 standing statements2 proposed statements5 mathematical milestones6 open questions1 narrowed routes7 conditional results1 completed special cases
- theorem candidate
2 of 13 2 - definition
1 of 13 1 - lemma
6 of 13 6 - reduction
3 of 13 3 - equivalence
1 of 13 1
Selected mathematical clusters3 mathematical clusters
Conjectural filtration and scopeThe open filtration target, its conditional Nori construction, finite-length reduction, and low-dimensional base cases.4 displayed rows
- retained route statementBloch–Beilinson filtration package
- retained route statementCandidate Nori filtrationconditional
- retained route statementTerminal filtration step equals the cycle-map kernelconditional
- retained route statementCurves and codimension one close in the formal setupspecial case
Coniveau and local defect reductionThe residue-field obstruction, corrected chain fallback, relative defect module, propagation, and exact local phantom formula.11 displayed rows · 4 routes included
- retained route statementConiveau reduction to residue-field odd groupsconditional
- retained route statementCorrected intrinsic d2 representativeintermediate
- retained route statementOne-sided two-jet transferintermediate
- retained route statementRelative exterior-defect moduleintermediate
- retained route statementQuartic propagation theoremintermediate
- retained route statementExact local phantom formulaconditional
- ComputationThe current work supplies six exact-rational-arithmetic programs for finite graded Lie-algebra and residue models, plus stored outputs and a runner that reportedly compares outputs byte-for-byte.The source reports that all six audits pass and support the corrected residue, two-jet, prolongation, and quartic-defect model calculations. ProofAtlas did not execute the scripts or independently reproduce the outputs by ProofAtlas. · reported unreproduced
- Active routeQuartic saturationCurrent priority: compute the actual Nori comparison defect and prove D_F^2=0 through the full linear-plus-quadratic saturation identity, retaining double-weight-two cross terms.
- Active routeWeight-minus-four derived comparisonProve weight-one Ext agreement together with Ext^3 surjectivity and Ext^4 injectivity for every pure weight-minus-four coefficient over every relevant residue field.
- Active routeTate-visible nonresonanceIf the quartic module survives, use the actual reductive representation to prove its exterior descendants contain no relevant Tate summands.
- Route held in reserveGlobal quartic-symbol reciprocityDeferred fallback: if a Tate-visible defect survives, construct the filtered coniveau complex and prove its terminal transgression is rationally trivial by a global reciprocity law.
Open mathematical frontierThe conditional no-phantom endpoint, independent projector route, six concrete obligations, and narrowed approaches that should not be repeated unchanged.19 displayed rows · 6 routes included
- retained route statementWeight-minus-four comparison criterionconditional
- retained route statementConditional master no-phantom theoremconditional
- retained route statementMurre consequences after projector algebraicityconditional
- Research targetInstantiate the residue-field comparison frameworkopen
- Research targetCompute the actual quartic defectopen
- Research targetProve the quartic derived comparisonopen
- Research targetTest Tate-visible nonresonanceopen
- Research targetBuild the chain-level global fallbackblocked
- Research targetSolve the independent projector problemopen
- Useful failureProve all local odd Nori Ext groups vanish using abstract nilpotent Lie theory.reported failure
- Useful failureAssume coniveau is represented by a strict one-step Cousin bicomplex.reported failure
- Useful failureRequire the full weight-two kernel K2 to vanish.reported failure
- Useful failureInfer algebraic Chow–Künneth projectors from separatedness alone.reported failure
- Active routeQuartic saturationCurrent priority: compute the actual Nori comparison defect and prove D_F^2=0 through the full linear-plus-quadratic saturation identity, retaining double-weight-two cross terms.
- Active routeWeight-minus-four derived comparisonProve weight-one Ext agreement together with Ext^3 surjectivity and Ext^4 injectivity for every pure weight-minus-four coefficient over every relevant residue field.
- Active routeTate-visible nonresonanceIf the quartic module survives, use the actual reductive representation to prove its exterior descendants contain no relevant Tate summands.
- Route held in reserveGlobal quartic-symbol reciprocityDeferred fallback: if a Tate-visible defect survives, construct the filtered coniveau complex and prove its terminal transgression is rationally trivial by a global reciprocity law.
- Active routeIndependent projector algebraicityIn parallel with the no-phantom problem, establish algebraicity of the cohomological Künneth projectors; only then use nilpotent lifting and separately address self-duality.
- Narrowed routeTwo-jet and first-prolongation screensRetain weight-two injection and first-prolongation injection as efficient sufficient screens, but not as necessary targets; quartic saturation is more permissive.
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
1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- All actual finite-diagram modules and maps are identified.
- Cross terms in B_2,F are retained.
- Quartic saturation is proved or a surviving R_F-module class is exhibited.
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Prepared starting pointCompute the actual quartic defect
Bloch–Beilinson Conjectures · ready to start
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Research contextPrepared context for any AI agent
Do rational Chow groups carry a canonical finite filtration whose layers reflect topology, motives, and regulators?
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- Current routes and known obstacles
- What a useful result should report
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Sources and references13 cited works · next context review by Nov 6, 2026
The mathematical context was checked on Aug 6, 2026. Status can be refreshed sooner after a material result or claim.
- 1Lectures on Algebraic Cyclesoriginal source · Spencer Bloch · Duke University; second edition Cambridge University Press · Duke lectures 1979; original 1980 · DOI 10.1017/CBO9780511760693 · accessed Aug 6, 2026
- 2Height pairing between algebraic cyclesoriginal source · Alexander A. Beilinson · Springer · 1987 · DOI 10.1007/BFb0078364 · accessed Aug 6, 2026
- 3On a conjectural filtration on the Chow groups of an algebraic variety: Part I. The general conjectures and some examplespeer reviewed result · Jacob P. Murre · Indagationes Mathematicae · 1993-06-21 · DOI 10.1016/0019-3577(93)90038-Z · accessed Aug 6, 2026
- 4On a conjectural filtration on the Chow groups of an algebraic variety: Part II. Verification of the conjectures for threefolds which are the product of a surface and a curvepeer reviewed result · Jacob P. Murre · Indagationes Mathematicae · 1993 · DOI 10.1016/0019-3577(93)90039-2 · accessed Aug 6, 2026
- 5Motivic sheaves and filtrations on Chow groupspeer reviewed result · Uwe Jannsen · American Mathematical Society · 1994 · accessed Aug 6, 2026
- 6Lectures on the Theory of Pure Motivessurvey or monograph · Jacob P. Murre, Jan Nagel, Chris A. M. Peters · American Mathematical Society · 2013 · accessed Aug 6, 2026
- 7On Finite-dimensional Motives and Murre's Conjecturepeer reviewed result · Uwe Jannsen · Cambridge University Press · 2007 · DOI 10.1017/CBO9781107325968.008 · accessed Aug 6, 2026
- 8The Fourier Transform for Certain HyperKähler Fourfoldspeer reviewed result · Mingmin Shen, Charles Vial · Memoirs of the American Mathematical Society · 2016 · accessed Aug 6, 2026
- 9Distinguished cycles on varieties with motive of abelian type and the Section Propertypeer reviewed result · Lie Fu, Charles Vial · Journal of Algebraic Geometry · 2020 · ARXIV 1709.05644 · DOI 10.1090/jag/729 · accessed Aug 6, 2026
- 10Incidence equivalence and the Bloch-Beilinson filtrationpreprint · Pablo Pelaez, Araceli Reyes · arXiv · 2025-01-31 · ARXIV 2501.19147 · accessed Aug 6, 2026
- 11Mathlib.AlgebraicGeometry.AlgebraicCycle.Basicformalization · Mathlib community · accessed Aug 6, 2026
- 12List of conjecturesencyclopedia · Wikimedia Foundation · accessed Aug 6, 2026
- 13Chow groupencyclopedia · Wikimedia Foundation · accessed Aug 6, 2026
Important qualifications
- The name Bloch–Beilinson covers several closely related formulations. This record focuses on the functorial finite filtration on rational Chow groups and its Murre Chow–Künneth-projector formulation; it does not identify every Beilinson, Beilinson–Bloch, Bloch–Kato, height-pairing, or special-value conjecture with that statement.
- The historical year 1979 records Bloch's Duke lectures as reported by the 2013 Murre–Nagel–Peters monograph. Beilinson developed an independent formulation in work published later; the record does not claim a single-day priority chronology.
- The general conjectures remain open, while many special varieties and individual parts of Murre's package are known. The milestone list is representative rather than a comprehensive catalogue of such cases.
- Pelaez and Reyes (2025) is an arXiv preprint and remains in the current research map at preprint posture. Its unconditional orthogonal filtration has some expected Bloch–Beilinson properties but is not a solution of the general conjectures.
- The scoped recognition search found a Wikipedia list/reference but no current Clay, Hilbert, Smale, Erdős, Epoch/FrontierMath, named-prize, or comparable selective-list designation. Absence from this bounded record is not a global nonexistence claim.
- The scoped formalization search checked current Mathlib algebraic-cycle documentation and the public Google DeepMind Formal Conjectures repository. It found partial algebraic-cycle infrastructure but no problem-level Bloch–Beilinson statement or proof; this does not establish nonexistence in every proof assistant or private project.
- No canonical external computation, dataset, or certificate for the general conjectures was identified. The unreviewed source material's finite Lie-algebra scripts were not executed and are source-asserted packet material, not external administrative evidence.
- No submitted mathematical claim, contributor probability estimate, or packet computation was used as external authority. This record has no proof, novelty, review, acceptance, or publication authority.
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