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Article items · 14 — Motivic filtration and image groups
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 13–14.
1 · Prerequisites
- A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p
- Affine schemes are contravariantly equivalent to commutative rings
- The Axiom of Choice
- Article scheme conventions
- Nonconnective K theory of perfect complexes
- Decreasing spectral filtrations and associated graded
- Multiplicative filtrations and graded products
- Elmanto–Morrow filtration input
- 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
- Perfect complexes on schemes and their support
- Nonconnective K theory in degree zero
- Affine perfect K zero and bundle K zero
- Compatible local sheaves glue uniquely up to unique isomorphism
- Schemes
- Modules on a ringed space
- Exact sequences of sheaves
- Grothendieck universe closure convention
- Commutative ring
- Affine schemes and their coordinate rings
- Stable spectra and exact triangles
- Smooth morphism of schemes
- Closed immersions of schemes
- Exact sequences and short exact sequences of modules
- The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
- Polynomial rings are flat and smooth
- Every affine scheme is quasi-compact
- Affine schemes are quasi-separated
- Finite-variable polynomial algebras over fields are Noetherian by finite generators
- A coherent closed-point skyscraper
- A surjective ring map induces a closed immersion of affine spectra
- Localisation of modules is exact
- The stalk of an associated sheaf is the localisation
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk
- Motivic complexes and cohomology
- Cohomology of shifted Nisnevich units
- Exact range of a cofiber connectivity bound
- Roots of unity and étale Tate twists
- The actual big cdh slice over a field
- Quasi-compact and quasi-separated schemes
- Scheme sites and sheafification
- Morphisms of schemes
- Fibre product of schemes
- Existence of all scheme fibre products
- The Stacks Project, compatible big and small étale sites · Definitions 34.4.6 and 34.4.8; Lemma 34.4.7; Lemmas 34.4.17(2) and 34.4.18, including their proofs
- The Stacks Project, inverse image on small étale sites · Definition 59.36.1; Equation 59.36.1.1; Lemma 59.36.2(1)–(2) and the proof of those parts
- The Stacks Project, pullback of modules on ringed topoi · Definition 18.13.1(2); Lemmas 18.13.2–3 with their proofs
- The Stacks Project, tensor products and pullback · Lemma 18.26.2, full statement and proof
- External canonical top-weight motivic-to-syntomic comparison
- The actual composite and its objectwise cofiber
- An independent full equivalence into the truncated cdh localization
- The source qcqs cdh site
- Exact cdh factorization inputs for the source étale map
- Finite-type field schemes lie in the qcqs source domain
- Canonical comparison and restriction from the prime field
- Picard group of a scheme
- External EM multiplicative motivic complexes and actual Chern map
- Cohomology factors through the derived category
- The prime subfield as the intersection of all subfields
- Schemes and morphisms over a base
- Pullback of a module along a morphism of ringed spaces
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes (arXiv:2309.08463v1) · Introduction, printed p. 2, prime-field convention; Section 5.1, Lemma 5.1 and Definition 5.2, printed pp. 33–34 (PDF page indices 32–33), especially the induced natural map immediately after (5.1).
- Complex shifts and spectral homotopy indices
- External spectral pairings and the integer sphere-interchange sign
- Projective bundle in the quotient convention
- Projective bundle represents line quotients
- Dual and base change for finite locally free sheaves
- Invertible sheaves
- Kernel sheaves are objectwise, while cokernels and images are sheafified
- Quasi-compact and quasi-separated morphisms
- Separated morphism of schemes
- Relative projective space from standard charts
- A local ring is a nonzero commutative ring with a unique maximal ideal
- Assuming the Axiom of Choice, a nonzero commutative ring is local exactly when its nonunits form an ideal, exactly when one of x and 1-x is a unit for every x
- The Stacks Project, Projective bundles · Section 27.21, Lemma 27.21.4
- The Stacks Project, Separation axioms · Section 26.21, paragraph after Definition 26.21.3; Remark 26.21.18(2) and its proof, with P = quasi-compact and quasi-separated
- Finite constant cdh sections
- Finite refinement for integer cdh sections
- Completely decomposed proper covers and the cdh topology
- An algebraically closed field: every nonconstant polynomial has a root in the field
- Field
- Locally finite type and finite type morphisms
- Finite type is affine-local on source and target
- A commutative ring is Noetherian exactly when every ideal is finitely generated, exactly when its ideals satisfy the ascending chain condition, and exactly when every nonempty set of ideals has a maximal member
- Every algebra of finite type over a Noetherian ring is a Noetherian ring
- The spectrum of a Noetherian ring is a Noetherian topological space
- Locally Noetherian and Noetherian schemes
- A Noetherian space is a finite union of irreducible closed subsets
- Group and abelian group
- The integers as equivalence classes of pairs of naturals
- (ℤ, +) is an abelian group, (ℤ, ·) is a commutative monoid that is not a group, and its group of units is {1, -1}
- Monoid homomorphism and group homomorphism
- Group isomorphisms, automorphisms and the set Aut(G)
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Serre splitting above the dimension
- Finite projective complements
- Subalgebra generated by a subset, algebras of finite type, and module-finite algebras
- An affine scheme of finite type over a field has a finitely generated coordinate ring
- Every algebra of finite type over a principal ideal domain is a Noetherian ring
- Irreducible topological spaces and irreducible subsets in the subspace topology
- Cycle complexes with fixed support
- The two supported cycle degrees on a curve
- Length orders on an integral curve
- Principal divisors are endpoint boundaries
- Normalization and the curve divisor quotient
- Integral schemes
- Chain dimension and the empty-space convention
- Function field of an integral finite-type scheme
- The stalk of the affine structure sheaf at a prime is the localization at that prime
- Composition series and length of a module
- Quotient module M/N with scalar multiplication on additive cosets
- Normalization of an affine integral curve
- 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
- Projective modules and the lifting property
- The mapping cylinder of a chain map
- The mapping cylinder factors a chain map
- A chain homotopy equivalence is a quasi-isomorphism
- Every point of a Zariski-open set has a distinguished-open neighbourhood inside it
- A principal localization identifies its spectrum with a distinguished open
- The hypercomplete derived-localization model on the actual general site
- Every qcqs cdh generator is an arc covering family
- From constructible-category hyperdescent to the actual cohomology complex
- Restriction of hyperdescent and arbitrary abstract blowup squares
- The actual cdh-localization unit is an equivalence
- Derived global sections on a scheme site
- Exact hypercomplete arc descent for finite constructible coefficients
- For every natural n, (ℤ/n,+) is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold
- The Stacks Project, Lemma 59.105.1 · Complete displayed statement and proof.
- Bhargav Bhatt and Akhil Mathew, The arc-topology · Definition 1.2 and Remark 1.4, p. 3; Notation 5.8 and preceding finite-ring paragraph, p. 41; Theorem 5.13 and complete proof, pp. 42–43.
- Utsav Choudhury and Martin Gallauer Alves de Souza, Homotopy theory of dg sheaves · Section 3 opening, p. 8; Definitions 5.1/5.3, pp. 17–18; Theorem 5.7 statement, p. 18, and proof, pp. 19–20; Lemma 5.10 and proof, p. 19; Fact 5.15 and preceding paragraph, p. 22.
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Section 3.3, p. 14, equation (3.4) and immediately following topology and square-criterion paragraph.
- Prime-field restriction and the intrinsic étale coefficient complex
- The exact cdh-local-to-objectwise étale map
- The actual finite-quotient map to cdh-local motivic cohomology
- Cofiber range gives natural truncation and cohomology comparison
- Constant integer coefficients and exact derived evaluation
- Prime-field restriction preserves actual comparison diagrams
- Equivalence of the weight-zero pullback projection
- Weight-zero sections and cohomology on the chosen general site
- Bockstein obstruction to prime torsion
- Primewise detection of torsion
- Top-weight degree-five vanishing away from the characteristic
- Characteristic-prime degree-five vanishing for singular affine threefolds
- Annihilators, torsion elements and the torsion subset of a module
- The characteristic of a ring: the least n ≥ 1 with n · 1_R = 0 when one exists, and 0 otherwise
- Prime and composite integers: p is prime when p > 1 and its only positive divisors are 1 and p
- The characteristic of a ring is the additive order of 1_R, with 0 recording infinite order; n · 1_R = 0 holds exactly when char(R) ∣ n; and in an integral domain every nonzero element has the same additive order as 1_R
- External EM affine syntomic filtration and actual obstruction map
- Elden Elmanto (joint work with Matthew Morrow), Zero cycles on singular varieties, Oberwolfach Report 36/2025 · Printed pp. 1994–1996 (PDF pages 46–48, one-based): Theorem 1, Theorem 3(2), Theorem 4 and its proof, and affine torsion application
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · v1, printed pp. 24–25: Definition 4.14 and Lemma 4.16 with proof; printed pp. 29–30: Theorem 4.28, Definition 4.29, Remarks 4.30–4.31, (4.7), Corollary 4.32 with proof
- Levine–Weibel zero cycles
- Reduced affine schemes
- Fields of characteristic zero, finite fields, and algebraically closed fields are perfect
- Affine finite-type source immerses into relative projective space
- Irreducible components as schemes
- Existence and basic properties of irreducible components
- Irreducibility via nonempty open subsets, connectedness and open subspaces
- Amalendu Krishna, Murthy’s conjecture on 0-cycles, arXiv:1511.04221v3 (18 March 2019) · §1.9 finite-type convention; §2 opening and §2.1, pp. 6–8, especially (2.1)–(2.2) and the image-independence paragraph immediately after (2.2); Corollary 7.6 and its proof, p. 34
- The Stacks Project, reduced immersions · Lemma 29.3.3 (tag 03DQ)
- Rahul Gupta and Amalendu Krishna, K-theory and 0-cycles on schemes, arXiv:1803.00837v4 · §1.10, p. 6; §2.1, p. 7; §3.2, p. 14, equation (3.6); proof of Theorem 4.2, p. 21
- The Stacks Project, schemes smooth over fields · Lemmas 33.25.3–33.25.4 and 33.25.8, §33.25 (tag 04QM)
- The Stacks Project, locally Noetherian normal schemes · Lemma 28.7.7, with Lemma 28.7.5 and its component argument
- 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 · v1, Theorem 1.13, printed p. 8; Theorem 8.1, printed p. 53; introduction footnote 1, printed p. 2
- Chern classes of a vector bundle on a smooth scheme
- Krull dimension of a nonzero ring
- Rational equivalence and the Chow group of cycles
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · Theorem 1, printed p. 2 (PDF page 2)
- Support fibers with explicit cohomological indexing
- Support fibers commute with graded triangles
- Nisnevich covers and distinguished squares
- Reduced induced closed subscheme construction
- Abelian sheafification and quotient sheaves on a site
- A coherent spectral realization of derived abelian complexes
- Open immersions of schemes
- Affine open subschemes
- Cochain complex in an abelian category
- A sheaf on a topological space
- 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
- The local valuative convention matches the source convention
- External valuative dimension comparison on locally Noetherian schemes
- Noetherian open subsets are quasi-compact
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · arXiv:2309.08463v1 (15 September 2023; internal date 18 September 2023), Introduction, footnote 1, printed/PDF page 2
- Hypercovers and derived hyperdescent on the big cdh slice
- Smooth affine components and gluing a trivial line
- The intersection product and Chow ring of a smooth scheme
- Algebraic cycles and the cycle group of a scheme of finite type over a field
- Exact linear hyperdescent and derived-localization input
- Canonical truncation of a complex
- Canonical truncation is a complex and has the claimed cohomology
- Slice categories, composition, and pullback along a morphism
- Quasi-separatedness and the diagonal
- Quasi-compactness is local on the target and survives base change
- Adjunction by unit, counit, and the triangle identities
- Coherent étale Tate sections and root trivialization on a slice
- External EM projective bundle equivalence with its specified components
- Cohomology of a cochain complex
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes (v1) · Introduction p.2; Definition 5.2 and convention following (5.2), p.34; component maps, Theorem 5.24(2) and proof, pp.43–44
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
DefinitionDependency level 13
Motivic filtration and image groups
DefinitionDependency level 13
Isomorphism classes of vector bundles
LemmaDependency level 13
The origin of the affine line has zero perfect K class
LemmaDependency level 13
Low-degree consequences of a weight-one comparison
LemmaDependency level 13
A chosen root trivializes the actual Tate coefficient on the slice
LemmaDependency level 13
The full top-weight map with its exact factorization
LemmaDependency level 13
The actual EM first Chern class is additive and natural
LemmaDependency level 13
Natural unital bigraded motivic ring with negative zero extension
LemmaDependency level 13
Quotient and line projectivizations agree after dualizing
LemmaDependency level 13
Integer cdh sections on seminormal schemes
LemmaDependency level 13
Critical-rank representatives of rank-zero classes
LemmaDependency level 13
Top cycles supported on an integral curve
LemmaDependency level 13
Units split the K one determinant
PropositionDependency level 13
Étale derived cdh locality on the actual qcqs site
PropositionDependency level 13
Prime-to-characteristic étale comparison
PropositionDependency level 13
Motivic weight zero on all qcqs schemes
PropositionDependency level 13
Torsion freeness of top threefold motivic cohomology
TheoremDependency level 13
Krishna's affine zero-cycle divisibility and kernel input
TheoremDependency level 13
Motivic dimension vanishing
DefinitionDependency level 14
Rank-two Chern map and its fixed-invariant fibers
DefinitionDependency level 14
Motivic and filtered K theory with support
DefinitionDependency level 14
Rank-three alternating exterior class
LemmaDependency level 14
Every actual motivic stage on the empty scheme is zero
LemmaDependency level 14
General equicharacteristic scope and the standing filtration restriction
LemmaDependency level 14
Restriction to the slice commutes with big-site localization
LemmaDependency level 14
The zero-dimensional critical-rank boundary
LemmaDependency level 14
Slice localization and the actual root-of-unity coefficient map
LemmaDependency level 14