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Article items · 08 — Elmanto–Morrow smooth comparison input
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 12–14.
1 · Prerequisites
- Motivic complexes and cohomology
- Motivic filtration and image groups
- Bloch cycle complexes and higher Chow groups
- The Axiom of Choice
- Motivic cohomology of equicharacteristic schemes · Theorem 1.1(9), PDF page 4 of 74; §3.1, equation (3.1) and slice-filtration discussion, PDF pages 12–13 of 74
- Article scheme conventions
- Support fibers with explicit cohomological indexing
- Complex shifts and spectral homotopy indices
- Support fibers commute with graded triangles
- Stable spectra and exact triangles
- Scheme sites and sheafification
- External actual motivic stagewise boundedness
- General equicharacteristic scope and the standing filtration restriction
- External EM low-weight maps with fixed positive normalization
- Affine perfect K zero and bundle K zero
- Bertini functional with finite smooth zero locus
- Finite global generation on an affine scheme
- Kernels of surjections of vector bundles
- External finite-type dimension, rational-point and etale-fiber inputs
- Isomorphism classes of vector bundles
- Exterior powers and determinant of an algebraic bundle
- Affine bundles and finite projective modules
- Dual conventions for the exterior Euler class
- Euler class of a regular bundle functional
- External perfect-field Tate cancellation
- External classical coefficient, derived Yoneda and effective-localization interface
- Purity twists and cohomological shifts with explicit premises
- The hypercomplete derived-localization model on the actual general site
- Restriction of hyperdescent and arbitrary abstract blowup squares
- Derived global sections on a scheme site
- External actual graded algebra and multiplicative tower spectral sequence
- Affine bundle classes are actual projective-module classes
- Spectral K-zero equality gives an actual projective stabilization
- Common projective complements and cancellation by free modules
- Constant integer coefficients and exact derived evaluation
- Equivalence of the weight-zero pullback projection
- Exact source weight-zero and constant-sheaf input
- The actual composite and its objectwise cofiber
- Prime-field restriction and the intrinsic étale coefficient complex
- Affine bundle classes are actual projective-module classes
- Spectral K-zero equality gives an actual projective stabilization
- Common projective complements and cancellation by free modules
- The ordinary Chow Chern map on isomorphism classes
- Operational Chern classes and the Whitney formula
- Chern classes of a vector bundle on a smooth scheme
- Finite clopen common rank strata
- Representable cdh sheaves and seminormalization
- Maps to finite disjoint unions of the base
- Completely decomposed proper covers and the cdh topology
- Seminormality and seminormalization
- Nisnevich covers and distinguished squares
- Picard group of a scheme
- Units in a sheaf of rings form a sheaf
- Frames and their transition units
- Local sections of a torsor give a cocycle
- Compatible local torsor sections determine an isomorphism
- Tensor products multiply frame transitions
- External inputs for flat descent of line bundles
- External classification of torsors by first cohomology
- Bigraded cohomology and the projective bundle contract
- Natural unital bigraded motivic ring with negative zero extension
- The actual EM first Chern class is additive and natural
- Quotient and line projectivizations agree after dualizing
- The exact projective bundle map in every integer bidegree
- Projective bundle closure and naturality of the actual decomposition
- The actual big cdh slice over a field
- Restriction to the slice commutes with big-site localization
- A chosen root trivializes the actual Tate coefficient on the slice
- The root map through derived sections, truncation and localization
- The restricted BEM complex and the supplied EM finite quotient
- External BEM big-site finite Beilinson–Lichtenbaum comparison
- External canonical field restriction identifies the BEM and EM cdh models
- Degree zero and negative homology of the cycle complex
- Cohomology of equivalent derived complexes
- Cycle-complex degree translation
- Filtered equivalences preserve image filtrations
- Rank-two Chern map and its fixed-invariant fibers
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · Theorem 1, p. 2; §6, pp. 22–26
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · p. 2, Introduction, paragraph beginning For context, immediately after Theorem 1; compare v1 p. 2 opening paragraph
- Elmanto–Morrow motivic complex input
- Cohomology descends to the derived category
- Prime-field evaluation and compatibility with the standing theory
- Normalizing the comparison fiber for the dimension criterion
- The global dimension bound for the specified comparison fiber
- Gluing upper bounds through an actual fiber sequence
- Combining the actual motivic and cdh outer-term bounds
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem1.13 p.8 and Theorem8.1 p.53
- Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p.4 property(3), referring to EM23 Theorem1.13
- Chern coefficients of the projective relation
- Finite-dimensional families of generated bundle functionals
- External relative Jacobian criterion and openness of smooth maps
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
TheoremDependency level 12
Elmanto–Morrow smooth comparison input
DefinitionDependency level 13
Motivic and filtered K theory with support
LemmaDependency level 13
The precise source stage bound and its homotopy interpretation
LemmaDependency level 13
Weight zero is the locally constant rank function
LemmaDependency level 13
Coherent dual extension supplies global Swan controlling sections
LemmaDependency level 13
The actual exterior-dual class in affine perfect K zero
LemmaDependency level 13
Perfect-field cancellation removes exactly d coefficient twists
LemmaDependency level 13
The actual cdh-localization unit is an equivalence
LemmaDependency level 13
Actual cofiber triangles and source grading of the motivic tower
LemmaDependency level 13
Local application interface for a critical-rank cancellation theorem
LemmaDependency level 13
Weight-zero sections and cohomology on the chosen general site
LemmaDependency level 13
Cofiber range gives natural truncation and cohomology comparison
LemmaDependency level 13
Local application interface for a critical-rank cancellation theorem
LemmaDependency level 13
Chern data survive addition of trivial bundles
LemmaDependency level 13
Finite constant cdh sections
LemmaDependency level 13
Units and Picard groups in Nisnevich degree zero and one
LemmaDependency level 13
Open excision for a descent theory
PropositionDependency level 13
Motivic projective bundle formula
PropositionDependency level 13
Singular Beilinson–Lichtenbaum input
PropositionDependency level 13
Smooth motivic comparison
TheoremDependency level 13
Asok–Fasel Chern-compatible comparison and lifting input
TheoremDependency level 13
Mohan Kumar–Murthy rank-three Chow classification
TheoremDependency level 13
Motivic dimension vanishing
DefinitionDependency level 14
Motivic Chern classes
DefinitionDependency level 14
Images of support groups in global groups
LemmaDependency level 14
Explicit eventual-stage thresholds in every fixed homotopy degree
LemmaDependency level 14
The actual smooth open pair and its choice scope
LemmaDependency level 14
Augmented tower comparisons preserve supported graded projections
LemmaDependency level 14