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Article items · 11 — Stage bounds imply completeness and finite multiplicative convergence
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 17–20.
1 · Prerequisites
- The Axiom of Choice
- Motivic filtration and image groups
- Spectral sequence of a tower
- Stable spectra and exact triangles
- External actual motivic stagewise boundedness
- Identifying all spectral-sequence pages with the source multiplicative pages
- External normality and perfect-field smooth-locus inputs
- Perfect residue classes at smooth closed points
- Affine perfect K zero and bundle K zero
- Smooth-point K theory class map
- Nonconnective K theory of perfect complexes
- Perfect K theory localization and open excision
- The actual first EM stage is the rank-zero K space on all qcqs schemes
- External determinant and pointed one-type inputs
- External EM low-weight maps with fixed positive normalization
- Motivic weight one
- Units split the K one determinant
- Units and Picard groups in Nisnevich degree zero and one
- Motivic and filtered K theory with support
- Smooth motivic comparison
- Gysin purity for a smooth closed pair
- Open excision for a descent theory
- 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
- Perfect-field cancellation removes exactly d coefficient twists
- The identified support fiber is the classical relative Hom before Gysin
- External perfect-field Tate cancellation
- External classical coefficient, derived Yoneda and effective-localization interface
- Bertini functional with finite smooth zero locus
- Finite-dimensional families of generated bundle functionals
- Coherent dual extension supplies global Swan controlling sections
- A common-core zero curve exists on a nonempty parameter open
- A rank-two family avoids each prescribed codimension-two locus
- Swan chartwise integrality glues on the actual quasi-affine open
- Componentwise curve construction and the scope of global irreducibility
- External Swan finite controlling-family Bertini theorem
- External relative Jacobian criterion and openness of smooth maps
- External maximal-minor height bound for the component-scope example
- External finite-type dimension, rational-point and etale-fiber inputs
- 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
- Asakura–Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, J. Math. Sci. Univ. Tokyo26(2019) · §5 pp269–272: Steps1–3, Proposition5.3 and formula(5.3.1); §6 pp272–273, definition and Proposition6.1(1)
- Annala–Iwasa, Motivic spectra and universality of K-theory (14April2025 PDF) · Definition4.4.1 p30, for vector bundles on S-stacks; Lemma3.2.3 p23; Lemma4.4.3 pp30–31
- The divisor presentation and actual Picard surjection
- The source Chern classes are the supplied projective-relation classes
- External non-A¹ projective-relation filtered and Whitney theorem
- Motivic Chern classes
- Levine–Weibel zero cycles
- External Levine–Weibel quotient cycle map with exact generators
- Excision and K theory purity at a smooth closed point
- 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
- Cycle complexes with fixed support
- Dévissage of K zero with regular ambient support
- Actual raw simplicial cycles satisfy descent without separatedness
- The source qcqs cdh site
- Slice localization and the actual root-of-unity coefficient map
- An independent full equivalence into the truncated cdh localization
- The full top-weight map with its exact factorization
- Cancellation at the required dimension-bounded evaluation objects
- Affine Artin bound and its consequence after localization
- Motivic complexes and cohomology
- Valuative dimension and degreewise bounded towers
- 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
- External actual graded algebra and multiplicative tower spectral 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.
- Stage projections, kernels and image groups
- Extending a Whitney class to K zero
- Derived Nisnevich descent on a finite clopen partition
- Finite derived products and motivic coefficient operations
- Formal Chern arithmetic through degree three
- 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
- Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes
- Detection on a simultaneous flag tower
- 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
- Support fibers commute with graded triangles
- Motivic dimension vanishing
- Eventual homotopy vanishing of the motivic tower
- Complex shifts and spectral homotopy indices
- Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p. 9, Lemma 4.2 and proof
- Article scheme conventions
- Derived global sections on a scheme site
- Motivic weight zero on all qcqs schemes
- Images of support groups in global groups
- Products and projections on image filtrations
- 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
- Generic rank and point-supported classes on a curve
- Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · pp. 12–13, opening of Section 6; p. 9, Lemma 4.2
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 17
Stage bounds imply completeness and finite multiplicative convergence
LemmaDependency level 17
Normal dimension strata and the actual residue class model
LemmaDependency level 17
The globally normalized first-layer Picard map is determinant
LemmaDependency level 17
Motivic purity and lower-weight vanishing at a smooth closed point
PropositionDependency level 17
Bertini curve functional with a prescribed avoidance locus
TheoremDependency level 17
External formal universal Chern maps and supported construction
LemmaDependency level 18
Divisibility survives the quotient and the support image
LemmaDependency level 18
Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes
LemmaDependency level 18
Levine–Weibel relations split on clopen pure-dimensional components
LemmaDependency level 18
Positive supported coordinates at a smooth threefold point
LemmaDependency level 18
The actual determinant fiber is a natural retract of the rank-zero K-space
LemmaDependency level 18
Smooth rank, first Chern, determinant and supported positive leading cycle
PropositionDependency level 18
Affine étale vanishing and the actual localized evaluation
PropositionDependency level 18
Motivic graded pieces and spectral sequence
DefinitionDependency level 19
Graded filtration projections
LemmaDependency level 19
Clopen rank extension and genuine representative independence
LemmaDependency level 19
Clopen rank extension and genuine representative independence
LemmaDependency level 19
Finite clopen restrictions split the actual motivic image filtration
LemmaDependency level 19
Divisibility of curve-supported top cohomology
LemmaDependency level 19
Second stage with integral curve support
LemmaDependency level 19
High filtration vanishing
LemmaDependency level 19
Weight-one K-one projection is determinant
LemmaDependency level 19
Smooth point classes lie in the top filtration piece
PropositionDependency level 19
Filtration projections and multiplication
DefinitionDependency level 20
Subgroup generated by smooth closed points
DefinitionDependency level 20
Total Chern series for actual and virtual bundles
LemmaDependency level 20
Low-stage homotopy equals its image
LemmaDependency level 20
Generic localization detects the point subgroup
LemmaDependency level 20