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Article items · 20 — Reduced bundle class identities
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 27–39.
1 · Prerequisites
- Motivic filtration and image groups
- The actual first EM stage is the rank-zero K space on all qcqs schemes
- The actual determinant fiber is a natural retract of the rank-zero K-space
- First Chern class factors through determinant
- Whitney identities and virtual Chern arithmetic
- The Axiom of Choice
- Isomorphism classes of vector bundles
- Exterior powers and determinant of an algebraic bundle
- Determinants and exact sequences
- Perfect complexes on schemes and their support
- Nonconnective K theory of perfect complexes
- Nonconnective K theory in degree zero
- External input on derived enhancements
- Motivic Chern classes
- Field
- Schemes and morphisms over a base
- Actual perfect Cartan through degree three on all qcqs field-schemes
- The globally normalized first-layer Picard map is determinant
- Stage projections, kernels and image groups
- Article scheme conventions
- Motivic complexes and cohomology
- External formal universal Chern maps and supported construction
- Grothendieck universe closure convention
- Graded filtration projections
- Subalgebra generated by a subset, algebras of finite type, and module-finite algebras
- Finite-variable polynomial algebras over fields are Noetherian by finite generators
- Every quotient and every localisation of a Noetherian ring is Noetherian
- The spectrum of a Noetherian ring is a Noetherian topological space
- Noetherian open subsets are quasi-compact
- Quasi-compact and quasi-separated schemes
- Rank-three alternating exterior class
- Subgroup generated by smooth closed points
- Affine bundles and finite projective modules
- Affine perfect K zero and bundle K zero
- The actual exterior-dual class in affine perfect K zero
- Dimension-specific point images and a precise lower-component obstruction
- Unique divisibility of the smooth-point subgroup
- Murthy splitting criterion for the exterior-dual obstruction
- An algebraically closed field: every nonconstant polynomial has a root in the field
- Affine schemes and their coordinate rings
- Locally finite type and finite type morphisms
- Krull dimension of a nonzero ring
- The height of a prime ideal
- Smooth morphisms via local standard smooth presentations
- Fields of characteristic zero, finite fields, and algebraically closed fields are perfect
- An affine scheme of finite type over a field has a finitely generated coordinate ring
- A field has only the zero ideal and itself, hence is Noetherian
- Every algebra of finite type over a Noetherian ring is a Noetherian ring
- Every affine scheme is quasi-compact
- Affine schemes are quasi-separated
- Normal noetherian schemes reduce to integral components
- Maximal ideals of an affine domain have full height
- The spectrum of a finite product ring is the disjoint union of the factor spectra
- A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point
- Perfect residue classes at smooth closed points
- A perfect affine module sheaf admits a finite projective resolution
- Motivic weight one
- Units and Picard groups in Nisnevich degree zero and one
- Determinant descends on the actual perfect K zero
- Locally free sheaves of finite rank
- Picard group of a scheme
- Nisnevich covers and distinguished squares
- Derived global sections on a scheme site
- External EM multiplicative motivic complexes and actual Chern map
- Elmanto–Morrow weight-one comparison input
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes (arXiv v1) · Section 5.1, Lemma 5.1 and its proof; Definition 5.2, equation (5.1); Corollary 7.11 and its proof; Introduction prime-field convention
- The Stacks Project, First cohomology and invertible sheaves, Lemma 21.6.1 (Tag 040E) · Lemma 21.6.1, first paragraph of proof, through the map from invertible sheaves to torsors and its composition with Lemma 21.4.3
- Actual second-stage comparison on every qcqs field-scheme
- Total Chern series for actual and virtual bundles
- An actual non-A¹ P¹-spectrum for the supplied EM complexes
- Actual EM orientation and projective bundle maps on that model
- 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
- Motivic and filtered K theory with support
- Elmanto–Morrow motivic complex input
- Stable spectra and exact triangles
- Scheme sites and sheafification
- Quasi-compact and quasi-separated morphisms
- Perfect fields: every irreducible polynomial is separable
- Smooth morphism of schemes
- Locally finite presentation morphisms
- Smoothness survives base change and composition
- Finite type is affine-local on source and target
- A smooth point has the exact finite-type excision pair
- Normal dimension strata and the actual residue class model
- Finite-type field schemes lie in the qcqs source domain
- Coefficientwise perfect Cartan on every qcqs field-scheme
- Asakura–Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, 2019 journal version · Section 1.1, pp. 250–251; section 1.4, p. 254; section 5, pp. 269–272, especially (5.3.1); section 6, pp. 272–273, Proposition 6.1(1)
- Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv:1207.4078v3 · Theorem 9.16 and Lemma 9.17 (PDF page 77); Proposition 11.1 and Remark 11.2 (PDF page 79)
- Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Introduction, pp. 2–3, especially the opening of Theorem 1.1
- Formal Chern arithmetic through degree three
- The actual perfect K-space operations satisfy total Cartan
- External actual supported Riemann–Roch in codimension three
- Motivic projective bundle formula
- Motivic purity and lower-weight vanishing at a smooth closed point
- Positive supported coordinates at a smooth threefold point
- Relative projective space from standard charts
- Twisting sheaf on Proj
- Invertible twists for degree-one generated rings
- Koszul Complex Of A Sequence With Coefficients
- Regular Sequence On A Module
- Koszul Complex Resolves A Regular Quotient
- Localisation of modules is exact
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk
- Polynomial rings are flat and smooth
- If R is Noetherian then R[x₁,…,xₙ] is Noetherian for every n∈ℕ
- Affine and projective n-space have dimension n
- The field comparison preserves refined smooth Gysin and the point class
- Locally Noetherian and Noetherian schemes
- A specified étale-natural comparison with supported cycles
- Étale morphism of schemes
- Flat morphism of schemes
- Flat and faithfully flat modules and ring homomorphisms
- Relative dimension of a smooth morphism at a point
- Smooth maps have étale local affine-space form
- A maximal ideal of an affine algebra has finite residue field over the base field
- Étale stability
- Finite and finite type etale schemes over an algebraically closed field
- A principal localization identifies its spectrum with a distinguished open
- Affine schemes and affine morphisms are separated
- Alberto Navarro, On the higher Riemann–Roch without denominators, v1 · Section 1, bundle Chern/Whitney and additive-orientation paragraph, p. 3; sections 2.1–2.2, p. 4; Theorem 2.3(1),(3),(4), p. 5; Definition 2.4, p. 6; Theorem 3.1, pp. 7–8
- The perfect image filtration has the stated second projection and kernel
- High filtration vanishing
- First isomorphism theorem for groups: G/ker f congimf
- Motivic splitting principle
- Exterior-square line factors on a flag space
- Zeroth projection is rank
- First projection is first Chern class
- Motivic graded pieces and spectral sequence
- Low-stage homotopy equals its image
- Integral closure in an extension ring and integrally closed domains
- Critical-rank representatives of rank-zero classes
- The actual supported third edge sends the positive point class to its residue object
- Third edge homomorphism
- Third-stage graded isomorphism
- Finite clopen restrictions split the actual motivic image filtration
- Derived Nisnevich descent on a finite clopen partition
- Closed immersions of schemes
- Support fibers commute with graded triangles
- Closed points of an affine scheme
- The residue field at a point of an affine scheme
- Chain dimension and the empty-space convention
- Weil divisor normal noetherian scheme
- The inverse of a bijective group homomorphism is a group homomorphism
- Every algebraic extension of a perfect field is separable
- The actual EM point calculation through an étale chart
- Alberto Navarro, On the higher Riemann–Roch without denominators, arXiv:1901.09337v1 · Example 1.1, p. 3; §§2.1–2.2, p. 4; Theorem 2.3 and Definition 2.4, pp. 5–6; Theorem 3.1, p. 7.
- Bertini functional with finite smooth zero locus
- Bertini curve functional with a prescribed avoidance locus
- Smooth-locus restriction properties
- A rank-two bundle with a trivial quotient and determinant
- Finite derived products and motivic coefficient operations
- The actual Chern series of a trivial bundle is the unit
- A sheaf on a topological space
- Every Zariski-closed subset has a unique radical defining ideal
- Quasi-coherent ideals and closed subschemes
- Clopen affine pieces and the zero-dimensional residue-point basis
- Daniel Kasprowski and Christoph Winges, Algebraic K-theory of stable infinity-categories via binary complexes · Section 2.1 opening (printed p. 2), formula (2.6) (p. 3), and Section 2.2 opening before Lemma 2.12 (p. 4).
- Jacob Lurie, Higher Algebra, September 18, 2017 · Remark 1.3.2.17 (printed p. 91); Proposition 1.3.5.13 and its proof (printed p. 127).
- Serre splitting above the dimension
- Excision and K theory purity at a smooth closed point
- Smooth point classes lie in the top filtration piece
- Open immersions of schemes
- Homotopy exactness of a spectral fiber sequence
- Finite-type schemes over a field are qcqs
- A coherent spectral realization of derived abelian complexes
- Morphisms of schemes
- Jacob Lurie, Higher Topos Theory (Princeton University Press, 2009) · Corollaries 5.1.2.3–5.1.2.4, printed p. 319; Section 5.1.5 opening and Lemmas 5.1.5.2–5.1.5.3, printed pp. 324–326 (PDF indices 336, 341–343).
- First-class matching and explicit perfect Chern arithmetic
- Critical-rank cancellation used in the article
- Third edge isomorphism
- Third edge on a curve complement
- Smooth density and normal codimension over a perfect field
- Generic localization detects the point subgroup
- External finite-type dimension, rational-point and etale-fiber inputs
- Subspaces of a Noetherian space and its compact open subsets
- Affine open subschemes
- Closed immersions are affine quotients and survive base change
- The Stacks Project, Algebraic schemes · Section 33.20, Noetherian observation after Definition 33.20.1; Lemma 33.20.3(3), (6), (10), and their proof paragraphs.
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 27
Reduced bundle class identities
LemmaDependency level 27
Second Chern additivity on the second piece
LemmaDependency level 27
Rank and determinant on the second piece
PropositionDependency level 27
Murthy splitting input in rank three
TheoremDependency level 27
Line bundles and first motivic Chern class
LemmaDependency level 28
The first two perfect Chern classes vanish on the third image piece in full scope
LemmaDependency level 28
Actual supported Chern operation and its smooth comparison
LemmaDependency level 29
Degree-three coefficient algebra on the lower-Chern kernel
LemmaDependency level 29
An integral P³ test fixes the supported RR coefficient as positive two
LemmaDependency level 29
The low-dimensional quotient follows from the stage sequence
LemmaDependency level 30
Third Chern additivity on the third piece
PropositionDependency level 30
Second projection is minus second Chern class
LemmaDependency level 31
Chern classes of the alternating exterior class
LemmaDependency level 31
Second Chern surjectivity in dimensions two and three
LemmaDependency level 31
Rank-three representatives with two zero Chern classes
LemmaDependency level 32
Supported RR and the actual support-to-global application
LemmaDependency level 32
Curve-and-point support reduction
LemmaDependency level 32
Prescribing two surface Chern classes
LemmaDependency level 33
Removing isolated points from the support
PropositionDependency level 33
Factor two in the supported third Chern map
TheoremDependency level 33
Motivic Chern classification on normal affine surfaces
CorollaryDependency level 34
Independent determinant and second-Chern prescription
LemmaDependency level 34
Lifting the corrected class to the curve-supported third stage
TheoremDependency level 35
Smooth points generate the third motivic piece
CorollaryDependency level 36
Third Chern isomorphism
CorollaryDependency level 37
Third Chern vanishing and a trivial summand
LemmaDependency level 37
Prescribing three Chern classes successively
TheoremDependency level 38
Motivic Chern classification on normal affine threefolds
CorollaryDependency level 39