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Article items · 18 — Levine–Weibel relations split on clopen pure-dimensional components
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 20–23.
1 · Prerequisites
- Normal dimension strata and the actual residue class model
- Levine–Weibel zero cycles
- External Levine–Weibel quotient cycle map with exact generators
- The Axiom of Choice
- Affine schemes and their coordinate rings
- Locally finite type and finite type morphisms
- Weil divisor normal noetherian scheme
- Affine open subschemes
- An affine scheme of finite type over a field has a finitely generated coordinate ring
- Every algebra of finite type over a Noetherian ring is a Noetherian ring
- Affine finite-type source immerses into relative projective space
- The spectrum of a finite product ring is the disjoint union of the factor spectra
- Openness of the regular locus over a perfect field
- Regular equals smooth over a perfect field
- Every point of a Zariski-open set has a distinguished-open neighbourhood inside it
- A principal localization identifies its spectrum with a distinguished open
- Closed points of an affine scheme
- A maximal ideal of an affine algebra has finite residue field over the base field
- A clopen decomposition of the spectrum comes from a nontrivial idempotent
- A sheaf on a topological space
- Subalgebra generated by a subset, algebras of finite type, and module-finite algebras
- Every affine scheme is quasi-compact
- Smooth morphisms via local standard smooth presentations
- The Stacks Project, Noetherian sources give quasi-compact morphisms · Section 28.5, Lemma 28.5.8 and its displayed proof
- The Stacks Project, factoring a quasi-compact immersion · Lemma 29.3.2, statement and displayed proof
- Excision and K theory purity at a smooth closed point
- Motivic purity and lower-weight vanishing at a smooth closed point
- Perfect residue classes at smooth closed points
- Open excision for a descent theory
- Smooth motivic comparison
- A smooth point has the exact finite-type excision pair
- The actual smooth restriction arrow gives the point-supported cycle fiber
- Point-supported cycles give all-weight chain purity and low-weight zero
- Bloch cycle complexes and higher Chow groups
- Motivic and filtered K theory with support
- Support fibers with explicit cohomological indexing
- Coordinate faces of the algebraic simplex
- Finite-type field schemes lie in the qcqs source domain
- Schemes and morphisms over a base
- Smooth morphism of schemes
- The residue field at a point of an affine scheme
- Perfect fields: every irreducible polynomial is separable
- Grothendieck universe closure convention
- Every algebraic extension of a perfect field is separable
- A specified étale-natural comparison with supported cycles
- The globally normalized first-layer Picard map is determinant
- The actual first EM stage is the rank-zero K space on all qcqs schemes
- Motivic weight one
- Motivic filtration and image groups
- External EM low-weight maps with fixed positive normalization
- External determinant and pointed one-type inputs
- Decreasing spectral filtrations and associated graded
- Stable spectra and exact triangles
- External stable enhancement and smash product of spectra
- The actual smooth open pair and its choice scope
- A coherent comparison of the particular open restriction
- Excision and the exact fixed-support cycle fiber
- The divisor presentation and actual Picard surjection
- Divisibility survives the quotient and the support image
- Article scheme conventions
- Integral schemes
- The smooth locus of a morphism
- Motivic complexes and cohomology
- Cycle complexes with fixed support
- Normalization of an affine integral curve
- Top cycles supported on an integral curve
- Relative dimension of a smooth morphism at a point
- Valuative dimension and degreewise bounded towers
- Spectral sequence of a tower
- Actual cofiber triangles and source grading of the motivic tower
- Cohomological indexing of the actual graded groups
- The product on graded motivic groups is the source cup product
- Identifying all spectral-sequence pages with the source multiplicative pages
- Stage bounds imply completeness and finite multiplicative convergence
- External actual graded algebra and multiplicative tower spectral sequence
- External actual motivic stagewise boundedness
- Field
- A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p
- Affine schemes are contravariantly equivalent to commutative rings
- Quasi-compact and quasi-separated schemes
- Derived exact couple
- A map of exact couples induces a map of spectral sequences
- Multiplicative filtrations and graded products
- Homotopy exactness of a spectral fiber sequence
- Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes · v1, 15 September 2023 (title date 18 September): Theorem 1.1(1), p.3; section 2.2, pp.11–12; Definitions 4.9/4.23, pp.21 and 27–28; Theorems 4.10(1)/4.24(1), pp.21 and 28; Theorems 4.12(1)/4.27(1), pp.22–23 and 29.
- Jacob Lurie, Higher Algebra (September 18, 2017) · Theorem 7.1.2.13 and proof, printed pp. 1212–1213; Theorem 7.1.2.1, first proof paragraph, p. 1206; Remark 7.1.2.9, p. 1211; Definition 7.1.1.2, p. 1199.
- An actual non-A¹ P¹-spectrum for the supplied EM complexes
- Actual EM orientation and projective bundle maps on that model
- Full affine K-space comparison and its sheafification
- External symmetric P¹ model, orientation criterion and fundamental stabilization
- External actual EM graded algebra, Picard class and complex PBF
- Negative coefficients of the actual EM projective-line model
- Elmanto–Morrow motivic complex input
- K-stack mapping spaces and extension from smooth affine schemes
- Coherent extension of Chern-target operations from smooth affines
- Universal Chern cohomology and countable derived limits
- Source-specific non-A¹ motivic spectrum framework and hypotheses
- Qcqs scheme sites and affine detection for space-valued sheafification
- Nonconnective K theory of perfect complexes
- A coherent spectral realization of derived abelian complexes
- The general linear group scheme and its coordinate ring
- Standard smooth presentations and locally standard smooth maps
- Locally standard smooth iff flat with geometrically regular fibres
- Adapted frames identify the universal quotient-projective stack
- Asakura–Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, J. Math. Sci. Univ. Tokyo26(2019) · §1.1 p. 250; Definitions 2.1–2.5 pp. 256–259; §4 pp. 265–268; Definition 5.1, Steps 1–3, Propositions 5.2–5.3 and (5.3.1), pp. 268–271; (5.4.1) and §6, Proposition 6.1(1), pp. 272–273
- Annala–Iwasa, Motivic spectra and universality of K-theory (14April2025 PDF) · Definition 4.4.1, printed p. 30 (relation on an S-stack in weight equal to bundle rank); Lemma 3.2.3, printed p. 23; Lemma 4.4.3, printed pp. 30–31
- Stage projections, kernels and image groups
- An algebraically closed field: every nonconstant polynomial has a root in the field
- Integral closure in an extension ring and integrally closed domains
- Derived category of an abelian category
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · arXiv:2309.08463v1 (arXiv stamp 15 September 2023; title-page date 18 September 2023), printed pp. 21–22 and 27–28, Definitions 4.9/4.23 and Theorems 4.10(1)–(2)/4.24(1)–(2)
- The source Chern classes are the supplied projective-relation classes
- External non-A¹ projective-relation filtered and Whitney theorem
- Motivic Chern classes
- Locally free sheaves of finite rank
- Exact sequences of sheaves
- Increasing and decreasing filtration of an object
- External input on derived enhancements
- Annala–Iwasa, Motivic spectra and universality of K-theory (author PDF dated April 14, 2025) · §§2.1.1–2.1.5, printed p. 14; Definition 4.4.1 and Lemma 4.4.3(ii)–(iii), including its proof, printed pp. 30–31.
- Jack Hall and David Rydh, Perfect complexes on algebraic stacks (July 6, 2017) · §1 opening, pp. 6–7; §1.4, Proposition 1.3 and its proof, pp. 10–11; Remark 1.7, p. 12.
- Derived Nisnevich descent on a finite clopen partition
- Perfect K theory localization and open excision
- Perfect complexes on schemes and their support
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk
- Finite derived products and motivic coefficient operations
- Motivic dimension vanishing
- Eventual homotopy vanishing of the motivic tower
- Complex shifts and spectral homotopy indices
- Locally Noetherian and Noetherian schemes
- Chain dimension and the empty-space convention
- Noetherian topological spaces via ACC on opens or DCC on closed subsets
- The spectrum of a Noetherian ring is a Noetherian topological space
- Noetherian open subsets are quasi-compact
- Dimension can be computed on an open cover
- A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point
- Krull dimension of a nonzero ring
- Derived global sections on a scheme site
- Motivic weight zero on all qcqs schemes
- Units split the K one determinant
- Weight zero is the locally constant rank function
- Actual first-Chern and determinant maps of the first EM layer
- External actual augmented EM and smooth slice tower comparison
- External supported slice projection with positive generic coefficient
- Bottom supported Bloch groups on a codimension two curve
- The actual supported Euler class has the generic length coordinate
- Positive generic cycles determine the curve-supported homomorphism
- Augmented tower comparisons preserve supported graded projections
- The supported second stage has the actual augmentation inverse
- Dévissage of K zero with regular ambient support
- Actual raw simplicial cycles satisfy descent without separatedness
- Open-natural smooth cycle layers by separated-open descent
- Closed immersions of schemes
- Open immersions of schemes
- Repeated roots in extension fields and separable polynomials
- Factor theorem over a commutative ring
- Generic points of irreducible closed subsets
- Picard group of a scheme
- Composition series and length of a module
- Quasi-projective morphisms before Proj
- Smooth adjunction identifies the absolute and relative augmented slice towers
- Schemes
- Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, v1 · §3.1 (3.1)–(3.2), pp.12–13; Theorem 3.5(1),(2),(8), Remark 3.6, pp.14–16; Construction 4.36, p.32; Theorem 6.1(2), Remark 6.3, Corollary 6.4 and footnote 12, pp.44–45.
- Kahn–Levine, Motives of Azumaya algebras, published final author PDF · Lemmas 2.3.1–2.3.2, pp.499–500; Definition 3.1.5, p.503; Lemma 6.2.1, p.535; Lemmas 6.3.4/6.4.2, pp.538–539; Theorem 6.5.5 and proof, pp.542–543.
- Marc Levine, The homotopy coniveau tower, arXiv:math/0510334v1 · Theorem 4.1.1 and Corollary 4.1.5, pp.20–23; Theorem 6.4.2 and §6.5, pp.36–38; Example 8.2.2(3), pp.45–46; §8.3 and Theorem 9.0.3/Remark 9.0.4, pp.46–47.
- The Stacks Project, Lemma 29.41.7, tag 0B3H · Lemma 29.41.7, statement and one-line proof.
- The Stacks Project, Lemma 29.41.6, tag 01VZ · Lemma 29.41.6, statement and proof.
- Images of support groups in global groups
- Smooth-point K theory class map
- Support fibers commute with graded triangles
- Separable algebraic elements and separable extensions
- Torsion freeness of top threefold motivic cohomology
- Supported degree-seven vanishing
- Quotient by a divisible subgroup
- Products and projections on image filtrations
- Universal property of the tensor product for balanced maps into abelian groups
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · v1, Section 2.2, printed pp. 11–12; Definition 4.9 and Theorem 4.10(1)–(2), pp. 21–22; Definition 4.23 and Theorem 4.24(1)–(2), pp. 27–28.
- Affine schemes are quasi-separated
- A field has only the zero ideal and itself, hence is Noetherian
- Global functions on Spec A recover A
- A locally ringed space
- A Noetherian local domain has dimension zero exactly when it is a field
- A field finitely generated as a k-algebra is a finite extension of k
- embedding dimension and regular local ring
- Affine perfect K zero and bundle K zero
- The spectrum of a field is a one-point affine scheme
- The Grothendieck ring of finite-dimensional vector spaces
- The stalk of a presheaf at a point
- Classical and scheme smoothness over a perfect field
- Extending a Whitney class to K zero
- Formal Chern arithmetic through degree three
- Motivic projective bundle formula
- Polynomial rings are flat and smooth
- Rational equivalence and the Chow group of cycles
- Elmanto and Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Theorem 1.1 opening, printed p. 3; prime-field convention, p. 2; Section 2.1, p. 11
- Bigraded cohomology and the projective bundle contract
- Formal power series over a commutative ring and the coefficient-extraction functional [xⁿ]
- Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes
- Finite clopen common rank strata
- Finite-clopen motivic restriction over every field
- The inverse of a bijective group homomorphism is a group homomorphism
- Integer cdh sections on seminormal schemes
- Constant cdh sections on the two article sites
- Finite-type schemes have finitely many clopen connected components
- Monoid homomorphism and group homomorphism
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem 1.1(1), printed p. 3; Definitions 4.9 and 4.23, printed pp. 21 and 27
- Isomorphism classes of vector bundles
- External nilpotence of oriented projective-relation Chern coefficients
- An actual non-A¹ P¹-spectrum for the supplied EM complexes
- Actual EM orientation and projective bundle maps on that model
- The source Chern classes are the supplied projective-relation classes
- Relative Jacobian criterion with its presentation hypothesis
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated 14 April 2025 · Definition 4.4.1 and Lemma 4.4.3(i), pp. 30–31; Lemma 3.1.8, p. 22
- Amalendu Krishna, Murthy’s conjecture on 0-cycles, arXiv:1511.04221v3 · §1.9 and §2 opening, printed p. 6; §2.1, pp. 6–7, quotient (2.1); p. 8, cycle map (2.2) and following K-theory convention; p. 1, projective-module K₀ convention.
- Generic rank and point-supported classes on a curve
- Subspaces of a Noetherian space and its compact open subsets
- Closed immersions are affine quotients and survive base change
- Every quotient and every localisation of a Noetherian ring is Noetherian
- The closed points of the prime spectrum are exactly the maximal ideals
- Every finite field extension is algebraic
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
LemmaDependency level 20
Levine–Weibel relations split on clopen pure-dimensional components
LemmaDependency level 20
Positive supported coordinates at a smooth threefold point
LemmaDependency level 20
The actual determinant fiber is a natural retract of the rank-zero K-space
LemmaDependency level 20
Divisibility of curve-supported top cohomology
PropositionDependency level 20
Motivic graded pieces and spectral sequence
TheoremDependency level 20
External formal universal Chern maps and supported construction
DefinitionDependency level 21
Graded filtration projections
LemmaDependency level 21
Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes
LemmaDependency level 21
Finite clopen restrictions split the actual motivic image filtration
LemmaDependency level 21
High filtration vanishing
LemmaDependency level 21
Weight-one K-one projection is determinant
LemmaDependency level 21
Smooth rank, first Chern, determinant and supported positive leading cycle
LemmaDependency level 21
Smooth point classes lie in the top filtration piece
PropositionDependency level 21
Torsion freeness after removing a smooth-locus integral curve
PropositionDependency level 21
Filtration projections and multiplication
DefinitionDependency level 22
Subgroup generated by smooth closed points
LemmaDependency level 22
Clopen affine pieces and the zero-dimensional residue-point basis
LemmaDependency level 22
Clopen rank extension and genuine representative independence
LemmaDependency level 22
Clopen rank extension and genuine representative independence
LemmaDependency level 22
Second stage with integral curve support
LemmaDependency level 22
Low-stage homotopy equals its image
LemmaDependency level 22
Third-stage graded isomorphism
LemmaDependency level 22
Zeroth projection is rank
DefinitionDependency level 23
Third edge homomorphism
DefinitionDependency level 23
Total Chern series for actual and virtual bundles
LemmaDependency level 23
Clopen reduction of the smooth-point comparison
LemmaDependency level 23
The quotient cycle map has exactly the smooth-point image
LemmaDependency level 23