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Article items · 12 — Zeroth projection is rank
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 20–24.
1 · Prerequisites
- Graded filtration projections
- Motivic filtration and image groups
- Motivic weight zero on all qcqs schemes
- Integer cdh sections on seminormal schemes
- Constant cdh sections on the two article sites
- The Axiom of Choice
- The actual first EM stage is the rank-zero K space on all qcqs schemes
- Motivic Chern classes
- Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes
- Clopen rank extension and genuine representative independence
- The source Chern classes are the supplied projective-relation classes
- Extending a Whitney class to K zero
- Formal Chern arithmetic through degree three
- Motivic complexes and cohomology
- Motivic and filtered K theory with support
- Homotopy exactness of a spectral fiber sequence
- Torsion freeness of top threefold motivic cohomology
- Supported degree-seven vanishing
- Divisibility of curve-supported top cohomology
- Quotient by a divisible subgroup
- Third-stage graded isomorphism
- Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 13, immediately before Proposition 6.1
- The stable free-class comparison is locally integrally acyclic
- Local integral chains detect global natural maps into Chern targets
- External actual universal-class detection on Vect and Grass
- External formal universal Chern maps and supported construction
- An actual non-A¹ P¹-spectrum for the supplied EM complexes
- Actual EM orientation and projective bundle maps on that model
- Motivic projective bundle formula
- Normal dimension strata and the actual residue class model
- External Levine–Weibel quotient cycle map with exact generators
- Levine–Weibel zero cycles
- Subgroup generated by smooth closed points
- Local free chains detect natural perfect operations on every qcqs base
- Negative coefficients of the actual EM projective-line model
- Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes
- Chern classes of a vector bundle on a smooth scheme
- Operational Chern classes and the Whitney formula
- Total Chern series for actual and virtual bundles
- Detection on a simultaneous flag tower
- Determinants and exact sequences
- The actual EM first Chern class is additive and natural
- Isomorphism classes of vector bundles
- Euler class group presentation
- Affine perfect K zero and bundle K zero
- Fasel–Yakerson, Vector bundles on singular surfaces and threefolds, arXiv:2610.22679v1 · PDF pp. 2–3, Theorem C and following Euler-group expectation
- Krishna, Murthy’s conjecture on 0-cycles, arXiv:1511.04221v3, 18 March 2019 · Definition 7.1, pp. 31–32; Lemmas 7.3–7.4, pp. 32–33; Corollary 7.6 and Theorem 7.7, p. 34; Theorem 1.6, p. 3
- Finite clopen restrictions split the actual motivic image filtration
- Levine–Weibel relations split on clopen pure-dimensional components
- Weight zero is the locally constant rank function
- Stable triviality and stable bundle equivalence
- Positive supported coordinates at a smooth threefold point
- External supported slice projection with positive generic coefficient
- External actual augmented EM and smooth slice tower comparison
- Augmented tower comparisons preserve supported graded projections
- Motivic purity and lower-weight vanishing at a smooth closed point
- Motivic graded pieces and spectral sequence
- Support fibers commute with graded triangles
- Kahn and Levine, Motives of Azumaya algebras, published final, JIMJ 9 (2010), 481–599 · Notation p.487; Lemmas 2.3.1–2.3.2 pp.499–500; Definition 3.1.5 p.503; §5.9 pp.531–532 before Lemma 5.9.1; Lemma 6.2.1 p.535.
- Levine, The homotopy coniveau tower, arXiv:math/0510334v1 (2005) · §8.3 pp.46–47 through (8.3.1); Theorem 9.0.3 and Remark 9.0.4 p.47.
- Elmanto and Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Remark 3.6; Construction 4.36; Theorem 6.1(2), Remark 6.3, Corollary 6.4 and footnote 12.
- Integral oriented field coefficient comparison with actual Picard and support maps
- Smooth motivic comparison
- Perfect K theory localization and open excision
- Open excision for a descent theory
- Projective relations transport the actual universal Chern operations
- Finite projective complements
- Affine bundles and finite projective modules
- Elmanto–Morrow motivic complex input
- Determinant descends on the actual perfect K zero
- The globally normalized first-layer Picard map is determinant
- Motivic weight one
- Nonconnective K theory of perfect complexes
- Derived global sections on a scheme site
- The source qcqs cdh site
- Rational splitting of the motivic tower
- Image filtration for a split tower
- Finite valuative dimension for noetherian and finite type schemes
- Article scheme conventions
- Nonempty opens preserve irreducible dimension
- Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p. 14, proof of Proposition 6.2 and Remark 6.3; p. 13, Proposition 6.1
- Smooth rank, first Chern, determinant and supported positive leading cycle
- Mohan Kumar existence of a nontrivial stably trivial bundle on a complex affine fourfold
- Stable triviality as the kernel of a right-invertible matrix
- Matrix completion detects the actual nontrivial bundle
- Stable triviality fixes the K-zero class and determinant
- Thomas Brazelton, Morgan Opie and Tariq Syed, On Algebraic Vector Bundles of Rank 2 over Smooth Affine Fourfolds · Example 6.6, printed/PDF pp. 33–34, opening existence assertion and p=2 geometric description; bibliography MK84/MK85, p. 36
- Scheme sites and sheafification
- External actual supported Riemann–Roch in codimension three
- Detecting injectivity by rationalization
- Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · Proposition 6.2 and its proof, PDF p. 14
- Perfect residue classes at smooth closed points
- Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 14, Remark 6.3 and proof of Proposition 6.2; p. 13, Proposition 6.1
- Coefficientwise perfect Cartan on every qcqs field-scheme
- Free-class detection on qcqs perfect K-space models
- External determinant and pointed one-type inputs
- Filtration projections and multiplication
- Flag towers with line-bundle quotients
- Flag detection over an arbitrary field by prime-field restriction
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
Zeroth projection is rank
PropositionDependency level 20
Whitney identities and virtual Chern arithmetic
PropositionDependency level 20
Torsion freeness after removing a smooth-locus integral curve
DefinitionDependency level 21
Third edge homomorphism
LemmaDependency level 21
The actual perfect K-space operations satisfy total Cartan
LemmaDependency level 21
The quotient cycle map has exactly the smooth-point image
LemmaDependency level 21
Actual perfect Cartan through degree three on all qcqs field-schemes
LemmaDependency level 21
The actual Chern series of a trivial bundle is the unit
LemmaDependency level 21
Agreement with the existing projective-relation and virtual bundle first classes
Open problemDependency level 21
Extending the geometric support method to higher dimensions
Open problemDependency level 21
Removing normality from the article results
RemarkDependency level 21
Euler-group comparison and the article's expectation
RemarkDependency level 21
Distinguishing smooth-point and motivic filtration notation
LemmaDependency level 22
Dimension-specific point images and a precise lower-component obstruction
LemmaDependency level 22
Stable triviality forces vanishing of the actual Chern coefficients
LemmaDependency level 22
The actual supported third edge sends the positive point class to its residue object
LemmaDependency level 22
Actual supported Chern operation and its smooth comparison
LemmaDependency level 22
Canonical first motivic class on every perfect K zero class
PropositionDependency level 22
Rational third-edge comparison
RemarkDependency level 22
Second Chern class need not be additive on the first piece
CounterexampleDependency level 23
A nontrivial rank-two bundle with trivial Chern classes on a smooth complex affine fourfold
LemmaDependency level 23
An integral P³ test fixes the supported RR coefficient as positive two
PropositionDependency level 23
First Chern class factors through determinant
PropositionDependency level 23
Third edge isomorphism
RemarkDependency level 23
Levine–Weibel cycle image with pure-dimensional and component conventions
RemarkDependency level 23
Third edge on a curve complement
LemmaDependency level 24
First-class matching and explicit perfect Chern arithmetic
LemmaDependency level 24