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Article items · 19 — Extending the geometric support method to higher dimensions
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 23–27.
1 · Prerequisites
- Motivic filtration and image groups
- Subgroup generated by smooth closed points
- The Axiom of Choice
- Affine schemes and their coordinate rings
- Affine schemes are contravariantly equivalent to commutative rings
- Every affine scheme is quasi-compact
- Affine schemes are quasi-separated
- Fields of characteristic zero, finite fields, and algebraically closed fields are perfect
- Every algebraic extension of a perfect field is separable
- Integral closure in an extension ring and integrally closed domains
- Perfect complexes on schemes and their support
- Isomorphism classes of vector bundles
- Motivic Chern classes
- An algebraically closed field: every nonconstant polynomial has a root in the field
- Schemes and morphisms over a base
- Algebras over a commutative ring, central structure maps, and algebra homomorphisms
- Grothendieck universe closure convention
- Motivic complexes and cohomology
- Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes
- Clopen rank extension and genuine representative independence
- The source Chern classes are the supplied projective-relation classes
- Extending a Whitney class to K zero
- Formal Chern arithmetic through degree three
- Quasi-compact and quasi-separated schemes
- Locally free sheaves of finite rank
- Exact sequences of sheaves
- Tensor product of sheaves of modules
- Locally Noetherian and Noetherian schemes
- Milnor K symbols and the Nisnevich sheaf
- Nisnevich covers and distinguished squares
- Derived global sections on a scheme site
- Normal dimension strata and the actual residue class model
- Finite clopen restrictions split the actual motivic image filtration
- Levine–Weibel relations split on clopen pure-dimensional components
- The quotient cycle map has exactly the smooth-point image
- Weight zero is the locally constant rank function
- External Levine–Weibel quotient cycle map with exact generators
- Clopen affine pieces and the zero-dimensional residue-point basis
- Levine–Weibel zero cycles
- Finite-variable polynomial algebras over fields are integrally closed
- A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are
- A polynomial ring in n variables over a field has dimension n
- A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point
- The spectrum of a finite product ring is the disjoint union of the factor spectra
- The stalk of the affine structure sheaf at a prime is the localization at that prime
- Amalendu Krishna, Murthy’s conjecture on 0-cycles, arXiv:1511.04221v3 · §1.9 and §2.1, printed pp. 6–8, equations (2.1)–(2.2) and the following K-theory convention.
- Local free chains detect natural perfect operations on every qcqs base
- External actual universal-class detection on Vect and Grass
- External formal universal Chern maps and supported construction
- An actual non-A¹ P¹-spectrum for the supplied EM complexes
- Motivic projective bundle formula
- Negative coefficients of the actual EM projective-line model
- External Quillen acyclicity and the actual local free-class model
- Smooth morphism of schemes
- Binary detection by the ordinary rank-zero K-stack comparison
- Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes
- Chern classes of a vector bundle on a smooth scheme
- Operational Chern classes and the Whitney formula
- Intersection with an invertible sheaf and the first Chern class
- Locally finite type and finite type morphisms
- Field
- Positive supported coordinates at a smooth threefold point
- External supported slice projection with positive generic coefficient
- External actual augmented EM and smooth slice tower comparison
- Augmented tower comparisons preserve supported graded projections
- Third edge homomorphism
- Motivic purity and lower-weight vanishing at a smooth closed point
- Motivic graded pieces and spectral sequence
- Support fibers commute with graded triangles
- Motivic and filtered K theory with support
- Stable spectra and exact triangles
- Finite-type field schemes lie in the qcqs source domain
- Closed points of an affine scheme
- The residue field at a point of an affine scheme
- Perfect fields: every irreducible polynomial is separable
- Closed immersions of schemes
- Quasi-projective morphisms before Proj
- Affine finite-type source immerses into relative projective space
- The inverse of a bijective group homomorphism is a group homomorphism
- Perfect K theory localization and open excision
- A unit-induced integral field coefficient zigzag
- The actual EM comparison commutes with the Picard first-Chern map
- The field comparison preserves refined smooth Gysin and the point class
- Kahn and Levine, Motives of Azumaya algebras, published final, JIMJ 9 (2010), 481–599 · Notation p. 487; §2.1, formulas (2.1)–(2.2), p. 494; Lemmas 2.3.1–2.3.2, pp. 499–500; Definition 3.1.5 and Lemmas 3.1.6–3.1.8, pp. 503–504; Definition 3.1.9, p. 505; §5.9, pp. 531–532; Lemma 6.2.1, p. 535; Lemmas 6.4.1–6.4.2, p. 539.
- Levine, The homotopy coniveau tower, arXiv:math/0510334v1 (2005) · §8.3 pp.46–47 through (8.3.1); Theorem 9.0.3 and Remark 9.0.4 p.47.
- Elmanto and Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Remark 3.6; Construction 4.36; Theorem 6.1(2), Remark 6.3, Corollary 6.4 and footnote 12.
- Total Chern series for actual and virtual bundles
- Detection on a simultaneous flag tower
- Determinants and exact sequences
- The actual EM first Chern class is additive and natural
- Exterior powers and determinant of an algebraic bundle
- Picard group of a scheme
- Finite clopen common rank strata
- Elmanto–Morrow motivic complex input
- Derived Nisnevich descent on a finite clopen partition
- Finite derived products and motivic coefficient operations
- Third-stage graded isomorphism
- Rational splitting of the motivic tower
- Image filtration for a split tower
- Finite valuative dimension for noetherian and finite type schemes
- Article scheme conventions
- Finite type algebras over fields have bounded dimension
- Every localization is flat, and localizing a flat module preserves flatness
- The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums
- A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p
- General equicharacteristic scope and the standing filtration restriction
- Chain dimension and the empty-space convention
- Irreducible components as schemes
- A Noetherian space is a finite union of irreducible closed subsets
- Dimension of a finite closed union
- Dimension can be computed on an open cover
- Closed immersions into affine schemes are quotient spectra
- Integral schemes
- Flat and faithfully flat modules and ring homomorphisms
- Module homomorphism and isomorphism, kernel, image and cokernel
- Exact sequences and short exact sequences of modules
- Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel
- The Stacks Project, dimension of nonempty opens in irreducible field schemes · Definition 33.20.1; Lemma 33.20.3(3), statement and proof of (3).
- The Stacks Project, Noetherian topology and locally closed subschemes · Lemmas 28.5.5 and 28.5.6, including the displayed proof of 28.5.5 and the hint following 28.5.6.
- Actual EM orientation and projective bundle maps on that model
- Free-class detection on qcqs perfect K-space models
- Two-variable affine extension before hypercompletion
- K-stack mapping spaces and extension from smooth affine schemes
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated 14 April 2025 · Conventions 2.1–2.2, pp. 14–15; Lemma 3.2.3, p. 23; Definition 4.4.1, Lemmas 4.4.3–4.4.4, Corollary 4.4.5, Theorem 4.4.6 and Corollary 4.4.7, pp. 30–32; Lemmas 3.3.4–3.3.5, p. 24; §5.1.1–5.1.4, pp. 33–34, especially the displayed mapping equivalences in the proof of Theorem 5.1.4
- Coherent dual extension supplies global Swan controlling sections
- Swan chartwise integrality glues on the actual quasi-affine open
- External maximal-minor height bound for the component-scope example
- External normality and perfect-field smooth-locus inputs
- Coherent module sheaves
- A principal localization identifies its spectrum with a distinguished open
- Every algebra of finite type over a Noetherian ring is a Noetherian ring
- Affine space is normal
- Standard smooth presentations and locally standard smooth maps
- Locally standard smooth iff flat with geometrically regular fibres
- Polynomial lifting on punctured affine three-space
- The polynomial ring R[xᵢ:i∈ I] as finitely supported coefficient families on monomials
- Coherent sheaves on a locally Noetherian scheme
- Kernels of surjections of vector bundles
- Relative dimension of a smooth morphism at a point
- Étale morphism of schemes
- Finite and finite type etale schemes over an algebraically closed field
- Affine fibre products are spectra of tensor products
- Rank-nullity: dim_F V=nullityT+rankT
- A sheaf on a topological space
- The spectrum of a product ring is a disjoint union
- Primes of a localization at a prime
- A polynomial ring in finitely many indeterminates over an integral domain is an integral domain
- Stable triviality and stable bundle equivalence
- Determinant descends on the actual perfect K zero
- The actual first EM stage is the rank-zero K space on all qcqs schemes
- The globally normalized first-layer Picard map is determinant
- Motivic weight one
- Nonconnective K theory of perfect complexes
- The source qcqs cdh site
- Modules on a ringed space
- Pullback carries a free module to the corresponding free module
- Nonconnective K theory in degree zero
- Decreasing spectral filtrations and associated graded
- Daniel Kasprowski and Christoph Winges, Algebraic K-theory of stable infinity-categories via binary complexes · Section 2.1 opening paragraph (printed p. 2), formula (2.6) (p. 3), and Section 2.2 opening paragraph before Lemma 2.12 (p. 4).
- Torsion freeness of top threefold motivic cohomology
- Detecting injectivity by rationalization
- An affine scheme of finite type over a field has a finitely generated coordinate ring
- Noetherian commutative rings and modules
- Perfect residue classes at smooth closed points
- Smooth rank, first Chern, determinant and supported positive leading cycle
- Smooth motivic comparison
- Relative projective space from standard charts
- Projective bundle in the quotient convention
- Projective space is of finite type over its base
- Relative Jacobian criterion with its presentation hypothesis
- Locally finite presentation morphisms
- Finite-type schemes over fields are qcqs
- The underlying space of an affine spectrum
- Rational equivalence and the Chow group of cycles
- Asakura–Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, 2019 journal version · Definition 5.1 and Section 5 Steps 1–3, Proposition 5.3 and formula (5.3.1), printed pp. 269–271
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · Definition 4.4.1 and Lemma 4.4.3 with its displayed proof, printed pp. 30–31
- Torsion freeness after removing a smooth-locus integral curve
- Mohan Kumar existence of a nontrivial stably trivial bundle on a complex affine fourfold
- Stable triviality as the kernel of a right-invertible matrix
- Matrix completion detects the actual nontrivial bundle
- Stable triviality fixes the K-zero class and determinant
- Scheme-theoretic varieties
- Fields and ℤ are Noetherian, and so are their polynomial rings in finitely many variables
- Thomas Brazelton, Morgan Opie and Tariq Syed, On Algebraic Vector Bundles of Rank 2 over Smooth Affine Fourfolds · Example 6.6, printed/PDF pp. 33–34, opening existence assertion and p=2 geometric description; section-6 field restriction, p. 32
- The stable free-class comparison is locally integrally acyclic
- Local integral chains detect global natural maps into Chern targets
- Source-specific non-A¹ motivic spectrum framework and hypotheses
- Universal Chern cohomology and countable derived limits
- Cauchy multiplication makes R⟦ x⟧ a commutative ring containing R[x] as the finitely supported subring
- A formal power series is a unit exactly when its constant coefficient is a unit
- Bertini functional with finite smooth zero locus
- Finite-dimensional families of generated bundle functionals
- A common-core zero curve exists on a nonempty parameter open
- A rank-two family avoids each prescribed codimension-two locus
- External finite-type dimension, rational-point and etale-fiber inputs
- The internal Hom sheaf of two module sheaves
- Restriction of a sheaf to an open subspace
- Base change of objects, morphisms and properties
- Clopen reduction of the smooth-point comparison
- Divisibility of an image with torsion kernel
- Unique divisibility is invariant under group isomorphism
- Krishna's affine zero-cycle divisibility and kernel input
- Normal noetherian schemes reduce to integral components
- A field has only the zero ideal and itself, hence is Noetherian
- A clopen decomposition of the spectrum comes from a nontrivial idempotent
- Affine open subschemes
- Principal localisation R_f={1,f,f²,…}⁻¹R
- Subalgebra generated by a subset, algebras of finite type, and module-finite algebras
- Amalendu Krishna, Murthy’s conjecture on 0-cycles · v3, Theorems 1.3–1.4, Corollary 1.5; §1.9 finite-type convention; §2.1, equations (2.1)–(2.2) and following K-theory convention
- Euler class group presentation
- Affine perfect K zero and bundle K zero
- Affine bundle K zero is the projective split group
- Krishna, Murthy’s conjecture on 0-cycles, arXiv:1511.04221v3, 18 March 2019 · Section 1.1, height-d smooth maximal ideals and classical K₀; Section 2.1, opening component convention, quotient (2.1) and cycle map (2.2); Section 7 opening field convention; Definition 7.1; Corollary 7.6; Theorem 7.7 and its proof (PDF p. 34).
- External determinant and pointed one-type inputs
- Filtration projections and multiplication
- Flag towers with line-bundle quotients
- Flag detection over an arbitrary field by prime-field restriction
- External EM low-weight maps with fixed positive normalization
- Homotopy exactness of a spectral fiber sequence
- External input on derived enhancements
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk
- A coherent spectral realization of derived abelian complexes
- Full affine K-space comparison and its sheafification
- Graded filtration projections
- Smooth-locus restriction properties
- Finite-variable polynomial algebras over fields are Noetherian by finite generators
- Every quotient and every localisation of a Noetherian ring is Noetherian
- The spectrum of a Noetherian ring is a Noetherian topological space
- Noetherian open subsets are quasi-compact
- Every Zariski-open subset is a union of distinguished opens
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
Open problemDependency level 23
Extending the geometric support method to higher dimensions
Open problemDependency level 23
Removing normality from the article results
PropositionDependency level 23
Whitney identities and virtual Chern arithmetic
RemarkDependency level 23
Reported alternative argument in arbitrary dimension
RemarkDependency level 23
Distinguishing smooth-point and motivic filtration notation
LemmaDependency level 24
Dimension-specific point images and a precise lower-component obstruction
LemmaDependency level 24
Actual perfect Cartan through degree three on all qcqs field-schemes
LemmaDependency level 24
The actual Chern series of a trivial bundle is the unit
LemmaDependency level 24
The actual supported third edge sends the positive point class to its residue object
LemmaDependency level 24
Agreement with the existing projective-relation and virtual bundle first classes
PropositionDependency level 24
Rational third-edge comparison
LemmaDependency level 25
Coefficientwise perfect Cartan on every qcqs field-scheme
LemmaDependency level 25
Componentwise curve construction and the scope of global irreducibility
LemmaDependency level 25
Stable triviality forces vanishing of the actual Chern coefficients
LemmaDependency level 25
Canonical first motivic class on every perfect K zero class
PropositionDependency level 25
Third edge isomorphism
RemarkDependency level 25
Levine–Weibel cycle image with pure-dimensional and component conventions
RemarkDependency level 25
Second Chern class need not be additive on the first piece
RemarkDependency level 25
Third edge on a curve complement
CounterexampleDependency level 26
A nontrivial rank-two bundle with trivial Chern classes on a smooth complex affine fourfold
LemmaDependency level 26
The actual perfect K-space operations satisfy total Cartan
PropositionDependency level 26
Bertini curve functional with a prescribed avoidance locus
PropositionDependency level 26
First Chern class factors through determinant
PropositionDependency level 26
Unique divisibility of the smooth-point subgroup
RemarkDependency level 26
Euler-group comparison and the article's expectation
LemmaDependency level 27
First-class matching and explicit perfect Chern arithmetic
LemmaDependency level 27
Actual second-stage comparison on every qcqs field-scheme
LemmaDependency level 27