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Article items · 16 — Explicit eventual-stage thresholds in every fixed homotopy degree
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 16–17.
1 · Prerequisites
- The Axiom of Choice
- The precise source stage bound and its homotopy interpretation
- General equicharacteristic scope and the standing filtration restriction
- Motivic filtration and image groups
- Stable spectra and exact triangles
- Grothendieck universe closure convention
- The actual smooth open pair and its choice scope
- Exact natural regular motivic-to-cdh comparison
- Exact arbitrary-field open-natural classical cycle interface
- Degree zero and negative homology of the cycle complex
- The smooth locus of a morphism
- Motivic complexes and cohomology
- Bloch cycle complexes and higher Chow groups
- Article scheme conventions
- Elmanto–Morrow motivic complex input
- Bottom supported Bloch groups on a codimension two curve
- Augmented tower comparisons preserve supported graded projections
- Support fibers commute with graded triangles
- Motivic and filtered K theory with support
- Cycle complexes with fixed support
- Decreasing spectral filtrations and associated graded
- Complex shifts and spectral homotopy indices
- Open restriction of admissible cycles
- Field
- Schemes and morphisms over a base
- Smooth morphism of schemes
- Locally finite type and finite type morphisms
- Integral schemes
- Closed immersions of schemes
- Open immersions of schemes
- Chain dimension and the empty-space convention
- Locally Noetherian and Noetherian schemes
- Quasi-compact and quasi-separated schemes
- The long exact sequence in homology
- Stefan Schwede and Brooke Shipley, Classification of stable model categories, arXiv:math/0108143v1 · Appendix B.1: Definition B.1.1, printed p.35; Theorem B.1.3(i), p.36; construction and (B.1.9), p.39; Remark B.1.10 and Theorem B.1.11 with proof, pp.39–40.
- The Stacks Project, Algebraic schemes · Definition 33.20.1 and the immediately following Noetherianity paragraph.
- The Stacks Project, Noetherian schemes · Definition 28.5.1 and Lemma 28.5.4 with its displayed proof.
- Coherent dual extension supplies global Swan controlling sections
- Finite-dimensional families of generated bundle functionals
- External relative Jacobian criterion and openness of smooth maps
- External finite-type dimension, rational-point and etale-fiber inputs
- Scheme-theoretic fibre
- Relative dimension of a smooth morphism at a point
- Differentials of a smooth morphism
- Cotangent space at a rational point
- Smoothness survives base change and composition
- Relative differentials commute with scheme base change
- A polynomial ring in finitely many indeterminates over an integral domain is an integral domain
- The underlying space of an affine spectrum
- Field-valued points and local-ring points
- Fibre product of schemes
- Kernels of surjections of vector bundles
- External normality and perfect-field smooth-locus inputs
- Geometric vector bundle with the sections convention
- Reduced induced closed subscheme construction
- Affine schemes and their coordinate rings
- Locally free sheaves of finite rank
- Dual and base change for finite locally free sheaves
- An algebraically closed field: every nonconstant polynomial has a root in the field
- Perfect fields: every irreducible polynomial is separable
- A polynomial ring in n variables over a field has dimension n
- Irreducibility via nonempty open subsets, connectedness and open subspaces
- Dimension can be computed on an open cover
- The actual exterior-dual class in affine perfect K zero
- Equality in affine K zero and stable isomorphism
- Isomorphism classes of vector bundles
- Exterior powers and determinant of an algebraic bundle
- Affine bundles and finite projective modules
- Operational Chern classes and the Whitney formula
- Chern classes of a vector bundle on a smooth scheme
- Intersection with an invertible sheaf and the first Chern class
- Locally finite presentation morphisms
- The internal Hom sheaf of two module sheaves
- Modules on a ringed space
- A sheaf on a topological space
- External affine étale vanishing at invertible primes
- Cancellation at the required dimension-bounded evaluation objects
- Slice localization and the actual root-of-unity coefficient map
- Derived global sections on a scheme site
- Roots of unity and étale Tate twists
- The source qcqs cdh site
- Subalgebra generated by a subset, algebras of finite type, and module-finite algebras
- Krull dimension of a nonzero ring
- Canonical truncation of a complex
- Canonical truncation is a complex and has the claimed cohomology
- Every affine scheme is quasi-compact
- Affine schemes are quasi-separated
- A smooth point has the exact finite-type excision pair
- Exact natural regular motivic-to-cdh comparison
- Fixed-site classical cycle comparison on every smooth ambient
- Support fibers with explicit cohomological indexing
- External input on derived enhancements
- Smooth morphisms via local standard smooth presentations
- Separated morphism of schemes
- Closed points of an affine scheme
- The residue field at a point of an affine scheme
- Actual cofiber triangles and source grading of the motivic tower
- Spectral sequence of a tower
- Excision and the exact fixed-support cycle fiber
- Actual Chow localization gives surjectivity of the motivic arrow
- Smooth motivic comparison
- Homotopy exactness of a spectral fiber sequence
- The actual smooth open pair and its Nisnevich square
- 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
- Nonconnective K theory of perfect complexes
- The actual enhanced restriction sequence and its K fiber
- Supported perfect complexes are invariant under an open neighbourhood
- The exact restriction kernel and its denominators
- Full faithfulness of the perfect triangulated quotient
- Every open perfect complex is a retract of a restriction
- Schemes
- Affine open subschemes
- Principal distinguished subsets of the prime spectrum
- 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 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
- A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p
- A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension
- tⁿ-1 is separable over K exactly when the characteristic does not divide n, and then a splitting field carries n distinct n-th roots of unity
- Perfect complexes on schemes and their support
- Nonconnective K theory in degree zero
- Cofiber and triangle presentations of K zero
- Inherited bounded t structure on supported perfect complexes
- Additivity of the supported coherent Euler class
- Postnikov inverse and the specified nonconnective degree-zero group
- Finite ideal-power filtrations and coherent sheaves on the closed subscheme
- Ideal-power dévissage and compatibility with coherent pushforward
- Regular points of locally Noetherian schemes
- Coherent module sheaves
- Projective modules and the lifting property
- Rational equivalence and the Chow group of cycles
- regular noetherian ring
- Fields of characteristic zero, finite fields, and algebraically closed fields are perfect
- Classical and scheme smoothness over a perfect field
- The stalk of the affine structure sheaf at a prime is the localization at that prime
- The mapping cone of a chain map
- A chain map is a quasi-isomorphism exactly when its cone is acyclic
- Modules over a ring form an abelian category
- Integral affine schemes
- Smoothness over a field by geometric regularity
- regular local domain induction
- Maximal ideals of an affine domain have full height
- The closed points of the prime spectrum are exactly the maximal ideals
- The height of a prime ideal
- M. Pavaman Murthy, Zero cycles and projective modules, ICTP SMR.761/7 preliminary lecture-note scan · Section 1 opening convention, printed p. 3 / PDF p. 5; Theorem 2.14(i)–(ii), printed pp. 13–14 / PDF pp. 15–16; Definition 3.5 and Remark 3.6(i)–(ii), printed p. 17 / PDF p. 19
- Charles Weibel, The K-book, author-hosted Chapter II · Resolution Theorem 7.6 and its proof, printed pp.65–66; Definition 7.7, p.66; Corollary 7.7.2 and Lemma 7.7.1, p.67
- The Stacks Project, Regular rings and global dimension · Lemma 10.110.8, Tag 00OE, equivalence of (1) and (2)
- The Stacks Project, Rings of finite global dimension · Lemma 10.109.6, Tag 0CXE, together with the projective-dimension conditions (1)–(4) of Lemma 10.109.4 (Tag 0CXC) in Section 10.109, Tag 00O2
- The ordinary Chow Chern map on isomorphism classes
- Irreducible components of a topological space
- Integral components of a smooth affine pure-dimensional scheme
- Bundles and Chow Chern maps on a finite clopen partition
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds (v5) · Printed p. 2, Section 1, Theorem 1 and the immediately following paragraph, displayed rank-three Chern map.
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds (v1) · Printed p. 2, Section 1, opening paragraph before Theorem 1.
- Local application interface for a critical-rank cancellation theorem
- Suslin cancellation over affine 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
- The spectrum of a Noetherian ring is a Noetherian topological space
- A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point
- Manoj Kumar Keshari, Cancellation problem for projective modules over affine algebras · Printed p. 1, Introduction: opening conventions, definition of cancellative, and Theorem 1.1(i)
- Chern coefficients of the projective relation
- Projective bundle represents line quotients
- Every actual motivic stage on the empty scheme is zero
- Valuative dimension and degreewise bounded towers
- Open excision for a descent theory
- A coherent spectral realization of derived abelian complexes
- Covering-sieve descent for the actual two-open square
- External Swan finite controlling-family Bertini theorem
- Irreducible topological spaces and irreducible subsets in the subspace topology
- The reduction of a scheme
- Base change of objects, morphisms and properties
- Restricting fibre products to open subschemes
- Surjectivity survives arbitrary base change
- Regular equals smooth over a perfect field
- one dimensional regular local rings are dvrs
- Every DVR is a PID
- Every principal ideal domain is a unique factorisation domain
- Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme
- On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group
- Curves over a field
- Closed-point local rings and function fields of integral curves
- Proper closed subsets of a curve are finite
- Rational sections of line bundles are Cartier divisors
- Coordinate faces of the algebraic simplex
- The degree [K:F]=dim_F K of a finite field extension
- Separable algebraic elements and separable extensions
- Finite field extensions and etaleness
- Étale morphism of schemes
- Affine morphisms are separated
- Closed immersions into affine schemes are quotient spectra
- Prime ideals of a quotient ring are exactly the prime ideals containing the ideal
- Multiplicative filtrations and graded products
- External actual graded algebra and multiplicative tower spectral sequence
- External spectral pairings and the integer sphere-interchange sign
- Abelian groups are monoidal under the tensor product
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Section 2.2, printed pp. 11–12, especially the monoidal paragraph on p. 12; Definition 4.9 and Theorem 4.10(1), p. 21; Definition 4.23, pp. 27–28, and Theorem 4.24(1), p. 28.
- Bastiaan Cnossen, Stable Homotopy Theory and Higher Algebra · Section 16.6.2, Definition 16.6.6 and Observation 16.6.7 with its explanation; Theorem 16.6.1 for the ambient spectra model; linked Warning 4.2.11. HTML revision 6 November 2026.
- Rank-two Chern map and its fixed-invariant fibers
- Fixed determinant isolates the actual Chern fiber
- Chern-compatible presentation transports the exact fibers
- Vanishing end groups kill the middle obstruction group
- Group and abelian group
- Injection, surjection, bijection
- Noetherian devissage for coherent proper pushforward
- Coherent sheaves on a locally Noetherian scheme
- A coherent closed-point skyscraper
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk
- Filtered categories and filtered colimits
- Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties
- The direct sum of an indexed family of modules
- A module homomorphism vanishing on N factors uniquely through M/N
- Abelian groups and ℤ-modules have the same objects and morphisms
- Two finite-dimensional vector spaces over F are linearly isomorphic if and only if they have the same dimension
- Module length is additive in short exact sequences
- Subspaces of a Noetherian space and its compact open subsets
- The prime spectrum and vanishing sets
- Noetherian topological spaces via ACC on opens or DCC on closed subsets
- Restriction of a sheaf to an open subspace
- Every quotient and every localisation of a Noetherian ring is Noetherian
- Noetherian commutative rings and modules
- The stalk of a presheaf at a point
- Exact sequences of sheaves
- Finite type and finitely presented module sheaves
- Grothendieck groups of coherent sheaves and of vector bundles on a scheme
- Composition series and length of a module
- A coherent comparison of the particular open restriction
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 16
Explicit eventual-stage thresholds in every fixed homotopy degree
LemmaDependency level 16
A coherent comparison of the particular open restriction
LemmaDependency level 16
The supported second stage has the actual augmentation inverse
LemmaDependency level 16
A common-core zero curve exists on a nonempty parameter open
LemmaDependency level 16
A rank-two family avoids each prescribed codimension-two locus
LemmaDependency level 16
Determinants detect vanishing of the rank-one exterior-dual class
LemmaDependency level 16
A split trivial line annihilates the exterior and top Chow classes
LemmaDependency level 16
Affine Artin bound and its consequence after localization
LemmaDependency level 16
The actual smooth restriction arrow gives the point-supported cycle fiber
LemmaDependency level 16
Cohomological indexing of the actual graded groups
LemmaDependency level 16
The support long exact sequence kills degree seven
PropositionDependency level 16
Motivic projective bundle formula
PropositionDependency level 16
Perfect K theory localization and open excision
PropositionDependency level 16
Singular Beilinson–Lichtenbaum input
PropositionDependency level 16
Dévissage of K zero with regular ambient support
TheoremDependency level 16
Murthy top Chow cycle comparison with its actual exterior-dual map
TheoremDependency level 16
Mohan Kumar–Murthy rank-three Chow classification
TheoremDependency level 16
Critical-rank cancellation used in the article
DefinitionDependency level 17
Motivic Chern classes
LemmaDependency level 17
Degreewise boundedness and finite collections of eventual vanishings
LemmaDependency level 17
Excision and the exact fixed-support cycle fiber
LemmaDependency level 17
Swan chartwise integrality glues on the actual quasi-affine open
LemmaDependency level 17
The actual first Chern map is the Picard isomorphism on an affine curve
LemmaDependency level 17
Point-supported cycles give all-weight chain purity and low-weight zero
LemmaDependency level 17
The product on graded motivic groups is the source cup product
LemmaDependency level 17
Unique lifting from vanishing groups yields the Chern classification
LemmaDependency level 17
Generic rank and point-supported classes on a curve
LemmaDependency level 17