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Article items · 06 — The actual finite-quotient map to cdh-local motivic cohomology
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 9–11.
1 · Prerequisites
- Motivic complexes and cohomology
- Elmanto–Morrow motivic complex input
- Functorial finite coefficients for derived complexes
- Exact cdh factorization inputs for the source étale map
- Milnor K symbols and the Nisnevich sheaf
- External prime-field cdh comparison and cdh dimension bound
- Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes · Theorem7.5 and its full proof (following Proposition7.4); Definition7.8; Remark7.9; Theorem1.1(8) classical smooth-field identification. Compare EMv1 §3.3 definition(3.5), pp14–15.
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §3.3, definition(3.5), pp14–15
- Nonconnective K theory of perfect complexes
- Perfect complexes on schemes and their support
- Cofiber and triangle presentations of K zero
- Affine perfect triangle classes and bundle Euler classes
- Nonconnective K theory in degree zero
- The Axiom of Choice
- Prime-field evaluation and compatibility with the standing theory
- External geometric properties of the actual motivic comparison fiber
- External dimension criterion for a cdh-acyclic Nisnevich sheaf
- Derived finite coefficients and the Bockstein sequence
- The source qcqs cdh site
- A finite-family basis for the generated cdh topology
- Scheme sites and sheafification
- Pullback and composition of the cdh generating families
- Nisnevich covers and distinguished squares
- Picard group of a scheme
- External EM multiplicative motivic complexes and actual Chern map
- Multiplicative filtrations and graded products
- Projective bundle in the quotient convention
- Projective bundle represents line quotients
- Prime-field restriction preserves actual comparison diagrams
- Exact source weight-zero and constant-sheaf input
- The exact cdh-local-to-objectwise étale map
- Exact objectwise cofiber bound for the canonical source composite
- Perfect determinant with true-triangle additivity: precise external input
- Exterior powers and determinant of an algebraic bundle
- Flag towers with line-bundle quotients
- Projective bundle pullback detects cohomology classes
- Bigraded cohomology and the projective bundle contract
- Finite clopen common rank strata
- Derived Nisnevich descent on a finite clopen partition
- Finite derived products and motivic coefficient operations
- Finite rank strata detect Elmanto-Morrow motivic cohomology
- Line-quotient classes across clopen rank strata
- External projective bundle input for the actual motivic complex
- Regular equals smooth over a perfect field
- regular local residue field koszul resolution
- regular local rings are domains and cohen macaulay
- Finite-variable polynomial algebras over fields are Noetherian by finite generators
- Rational cofiber implies finite-coefficient equivalence
- Elmanto–Morrow rational comparison cofiber
- Decreasing spectral filtrations and associated graded
- Valuative dimension and degreewise bounded towers
- Cofiber of a direct-summand inclusion
- Graded pieces of a direct-sum tail tower
- Boundary maps vanish for a split tower
- Degeneration of the spectral sequence of a split tower
- Image filtration for a split tower
- Motivic cohomology of equicharacteristic schemes · September 2023 PDF v1, Theorems 4.12(1)–(3) and 4.27(1)–(3), printed pp. 22 and 29; Section 2.2, pp. 11–12; proof of Lemma 7.1, p. 49
- Motivic cohomology of equicharacteristic schemes · Version 1 (15 September 2023; internal date 18 September 2023): Theorem 1.1(1) p.3; Definition 4.9 p.21; Definition 4.23 pp.27–28; initial assertions of Theorems 4.12 p.22 and 4.27 p.29; §2.2 pp.11–12
- External EM affine syntomic filtration and actual obstruction map
- Elden Elmanto, Zero cycles on singular varieties (joint work with Matthew Morrow), Algebraic K-theory, Oberwolfach Report 36/2025 · Theorem 4 and its proof, printed p.1995/PDF47; notation and comparison Theorem3, printed1994/PDF46; application printed1996/PDF48
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Theorem 4.28 and proof, printed p.29; Definition 4.29, Remarks 4.30–4.31, map (4.7), Corollary 4.32 and proof, printed p.30
- Imported derived sections on a small site
- Negative cohomology of a nonnegative evaluated complex
- Article scheme conventions
- Extending a nonnegative filtration to integer indices
- Images of filtration stages
- Nesting of image filtration groups
- Naturality of image filtrations
- Products of filtration images
- Locally free sheaves of finite rank
- Finite subintegral maps are cdh covers
- Sections match across a finite subintegral cover
- A sheaf restricts injectively to a covering singleton
- External EM projective bundle equivalence with its specified components
- The actual smooth open pair and its Nisnevich square
- Exact natural regular motivic-to-cdh comparison
- Fixed-site classical cycle comparison on every smooth ambient
- Degree zero and negative homology of the cycle complex
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 9
The actual finite-quotient map to cdh-local motivic cohomology
RemarkDependency level 9
Reported alternative argument in arbitrary dimension
TheoremDependency level 9
External canonical field restriction identifies the BEM and EM cdh models
LemmaDependency level 10
Affine perfect K zero and bundle K zero
LemmaDependency level 10
Normalizing the comparison fiber for the dimension criterion
LemmaDependency level 10
The restricted BEM complex and the supplied EM finite quotient
LemmaDependency level 10
An independent full equivalence into the truncated cdh localization
LemmaDependency level 10
Constant-sheaf sections have finite local representatives
LemmaDependency level 10
Finite coproducts and refinement inside a sieve
LemmaDependency level 10
The actual EM first Chern class is additive and natural
LemmaDependency level 10
Natural unital bigraded motivic ring with negative zero extension
LemmaDependency level 10
Quotient and line projectivizations agree after dualizing
LemmaDependency level 10
Equivalence of the weight-zero pullback projection
LemmaDependency level 10
The actual composite and its objectwise cofiber
LemmaDependency level 10
Determinant descends on the actual perfect K zero
LemmaDependency level 10
Detection on a simultaneous flag tower
LemmaDependency level 10
Perfect residue classes at smooth closed points
PropositionDependency level 10
Finite-coefficient motivic to cdh comparison
PropositionDependency level 10
Rational splitting of the motivic tower
TheoremDependency level 10
Elmanto–Morrow filtration input
TheoremDependency level 10
External canonical top-weight motivic-to-syntomic comparison
DefinitionDependency level 11
Derived global sections on a scheme site
DefinitionDependency level 11
Motivic filtration and image groups
DefinitionDependency level 11
Isomorphism classes of vector bundles
LemmaDependency level 11
The global dimension bound for the specified comparison fiber
LemmaDependency level 11
The full top-weight map with its exact factorization
LemmaDependency level 11
Cdh sheaves invert finite subintegral maps
LemmaDependency level 11
The exact projective bundle map in every integer bidegree
LemmaDependency level 11