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Article items · 07 — Characteristic-prime degree-five vanishing for singular affine threefolds
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 11–12.
1 · Prerequisites
- Motivic complexes and cohomology
- External canonical top-weight motivic-to-syntomic comparison
- Affine characteristic-p syntomic amplitude without regularity
- External affine étale vanishing at invertible primes
- Article scheme conventions
- A chosen root trivializes the actual Tate coefficient on the slice
- Derived finite coefficients and the Bockstein sequence
- Primewise detection of torsion
- Serre splitting above the dimension
- Finite projective complements
- Affine perfect K zero and bundle K zero
- The Axiom of Choice
- Nonconnective K theory of perfect complexes
- Exterior powers and determinant of an algebraic bundle
- Enhanced affine perfect equivalence
- Bounded finite-projective complexes and affine chain models
- Coherent derived tensor and pullback on perfect complexes
- External matrix, exact and bounded-projective K-theory comparisons
- External connective, nonconnective and automorphism comparisons
- Determinant on the stable general linear group
- Stable determinant commutes with ring maps
- Compatible local units glue uniquely
- Gluing compatible local determinant homomorphisms
- A right inverse makes a homomorphism surjective
- Finite subfamilies of Nisnevich covers on the finite-type site
- Pullback and composition of the cdh generating families
- A finite-family basis for the generated cdh topology
- Finite coproducts and refinement inside a sieve
- Constant-sheaf sections have finite local representatives
- External MVW proper-cover identification and two-stage refinement
- Completely decomposed proper covers and the cdh topology
- Scheme sites and sheafification
- Levine–Weibel zero cycles
- Amalendu Krishna, Murthy’s conjecture on 0-cycles, arXiv:1511.04221v3 (18 March 2019) · Theorems 1.3 and 1.4, printed/PDF p. 2; §2.1, pp. 6–8, quotient (2.1), cycle map (2.2), and the following K-theory convention
- Isomorphism classes of vector bundles
- Chern classes of a vector bundle on a smooth scheme
- Derived global sections on a scheme site
- Complex shifts and spectral homotopy indices
- Perfect complexes on schemes and their support
- Nonconnective K theory in degree zero
- Motivic filtration and image groups
- Multiplicative filtrations and graded products
- Stable spectra and exact triangles
- Kasprowski–Winges, Algebraic K-theory of stable infinity-categories via binary complexes · Section 2.1, formula (2.6), p.3; Section 2.2, paragraph immediately before Lemma 2.12, p.4
- Prime-field evaluation and compatibility with the standing theory
- The global dimension bound for the specified comparison fiber
- Gluing upper bounds through an actual fiber sequence
- External prime-field cdh comparison and cdh dimension bound
- Elmanto–Morrow filtration input
- The local valuative convention matches the source convention
- External valuative dimension comparison on locally Noetherian schemes
- The actual big cdh slice over a field
- Restriction to the slice commutes with big-site localization
- Smooth affine components and gluing a trivial line
- Rational equivalence and the Chow group of cycles
- The source qcqs cdh site
- Imported derived sections on a small site
- Exact linear hyperdescent and derived-localization input
- Projective bundle in the quotient convention
- Quotient and line projectivizations agree after dualizing
- Natural unital bigraded motivic ring with negative zero extension
- The actual EM first Chern class is additive and natural
- The exact projective bundle map in every integer bidegree
- Bigraded cohomology and the projective bundle contract
- A coherent comparison of the particular open restriction
- Cycle complexes with fixed support
- Degree zero and negative homology of the cycle complex
- The actual smooth open pair and its Nisnevich square
- Stable triviality and stable bundle equivalence
- Roots of unity and étale Tate twists
- Seminormality and seminormalization
- Cdh sheaves invert finite subintegral maps
- Huber–Kelly representable cdh sheafification input
- Finite-dimensional families of generated bundle functionals
- Finite global generation on an affine scheme
- Kernels of surjections of vector bundles
- Normal affine threefolds have controlled nonsmooth strata
- Coordinate multiples of dual generators span all first jets
- The scheme of nontransverse zero jets has small dimension
- One nonempty parameter open gives smooth-locus avoidance and transversality
- The selected zero fibers are finite etale schemes
- External normality and perfect-field smooth-locus inputs
- External finite-type dimension, rational-point and etale-fiber inputs
- Bockstein obstruction to prime torsion
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Construction4.34 (4.8)–(4.9); Construction4.36; Theorem6.1(2), Remark6.3 and Corollary6.4; Theorem3.5(1),(8) and Remark3.6, pp.14–16; footnote12 p.45
- Elmanto–Morrow motivic complex input
- Spectral sequence of a tower
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Version 1 (15 September 2023): §2.2 pp.11–12; Definition4.9 and Theorem4.10(1)–(2) pp.21–22; Definition4.23 pp.27–28 and Theorem4.24 p.28; Theorem4.12 pp.22–23 and Theorem4.27 p.29; §3.1 p.13.
- Daniel Dugger, Multiplicative structures on homotopy spectral sequences I · §5 Proposition5.1 and Proposition5.3 p.11; §6 Theorem6.1 p.13, Theorem6.2 and Proposition6.4 p.14; proofs pp.16–17
- Valuative dimension and degreewise bounded towers
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §2.2 pp.11–12; Theorem4.12(1) p.22 and Theorem4.27(1) p.29; Theorem1.1(1) p.3
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 11
Characteristic-prime degree-five vanishing for singular affine threefolds
LemmaDependency level 11
Top-weight degree-five vanishing away from the characteristic
LemmaDependency level 11
Characteristic-zero coefficient vanishing and torsion freeness
LemmaDependency level 11
Critical-rank representatives of rank-zero classes
LemmaDependency level 11
Units split the K one determinant
PropositionDependency level 11
Finite refinement for integer cdh sections
TheoremDependency level 11
Krishna's affine zero-cycle divisibility and kernel input
DefinitionDependency level 12
Rank-two Chern map and its fixed-invariant fibers
DefinitionDependency level 12
Rank-three alternating exterior class
DefinitionDependency level 12
Support fibers with explicit cohomological indexing
LemmaDependency level 12
Every actual motivic stage on the empty scheme is zero
LemmaDependency level 12
Combining the actual motivic and cdh outer-term bounds
LemmaDependency level 12
General equicharacteristic scope and the standing filtration restriction
LemmaDependency level 12
The root map through derived sections, truncation and localization
LemmaDependency level 12
The zero-dimensional critical-rank boundary
LemmaDependency level 12
The hypercomplete derived-localization model on the actual general site
LemmaDependency level 12
Projective bundle closure and naturality of the actual decomposition
LemmaDependency level 12
Actual Chow localization gives surjectivity of the motivic arrow
LemmaDependency level 12
Stable triviality fixes the K-zero class and determinant
LemmaDependency level 12
Constant integer coefficients and exact derived evaluation
LemmaDependency level 12
Prime-field restriction and the intrinsic étale coefficient complex
LemmaDependency level 12
The ordinary Chow Chern map on isomorphism classes
PropositionDependency level 12
Representable cdh sheaves and seminormalization
PropositionDependency level 12
Bertini functional with finite smooth zero locus
PropositionDependency level 12
Torsion freeness of top threefold motivic cohomology
TheoremDependency level 12
External actual augmented EM and smooth slice tower comparison
TheoremDependency level 12
External actual graded algebra and multiplicative tower spectral sequence
TheoremDependency level 12