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Article items · 10 — The support long exact sequence kills degree seven
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 15–17.
1 · Prerequisites
- The Axiom of Choice
- 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
- 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
- The source qcqs cdh site
- Motivic weight zero on all qcqs schemes
- External EM low-weight maps with fixed positive normalization
- Motivic filtration and image groups
- Affine perfect K zero and bundle K zero
- Etale morphisms are universally open and quasi-finite at every point
- Proper morphisms are closed
- Dévissage of K zero with regular ambient support
- Noetherian devissage for coherent proper pushforward
- Coherent sheaves on a locally Noetherian scheme
- A coherent closed-point skyscraper
- Isomorphism classes of vector bundles
- Motivic Chern classes
- Motivic and filtered K theory with support
- Perfect K theory localization and open excision
- Perfect complexes on schemes and their support
- Perfect residue classes at smooth closed points
- The actual exterior-dual class in affine perfect K zero
- A split trivial line annihilates the exterior and top Chow classes
- M. Pavaman Murthy, Zero cycles and projective modules, ICTP SMR.761/7 preliminary lecture-note scan · Introduction, printed pp. 1–2 / PDF pp. 3–4; Definition 3.5, Remark 3.6(i), Theorem 3.7 and its short proof, printed p. 17 / PDF p. 19
- Charles Weibel, The K-book, author-hosted Chapter II PDF (cached version identified by SHA256) · Resolution Theorem 7.6 and its proof, chapter/PDF pp. 65–66; Definition 7.7, pp. 66–67; Corollary 7.7.2 and the preceding truncated-cone proof, p. 67
- The ordinary Chow Chern map on isomorphism classes
- Cylinder and zero section of a smooth affine surface
- Chern data survive addition of trivial bundles
- Surface uniqueness from the cylinder and cancellation
- Surface existence from rank-three existence on the cylinder
- Local application interface for a critical-rank cancellation theorem
- Suslin cancellation over affine algebras
- Mohan Kumar–Murthy rank-three Chow classification
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · p. 2, Introduction, paragraph beginning For context, immediately after Theorem 1; compare v1 p. 2 opening paragraph
- Images of support groups in global groups
- The actual smooth open pair and its choice scope
- A coherent comparison of the particular open restriction
- Cycle complexes with fixed support
- Open excision for a descent theory
- Elmanto–Morrow motivic complex input
- 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
- Bertini functional with finite smooth zero locus
- An actual non-A¹ P¹-spectrum for the supplied EM complexes
- External actual EM graded algebra, Picard class and complex PBF
- External symmetric P¹ model, orientation criterion and fundamental stabilization
- The full top-weight map with its exact factorization
- An independent full equivalence into the truncated cdh localization
- Slice localization and the actual root-of-unity coefficient map
- Étale derived cdh locality on the actual qcqs site
- A smooth point has the exact finite-type excision pair
- The actual smooth restriction arrow gives the point-supported cycle fiber
- Bloch cycle complexes and higher Chow groups
- External finite-type dimension, rational-point and etale-fiber inputs
- Perfect-field cancellation removes exactly d coefficient twists
- External classical coefficient, derived Yoneda and effective-localization interface
- Gysin purity for a smooth closed pair
- Support fibers with explicit cohomological indexing
- Spectral sequence of a tower
- Actual cofiber triangles and source grading of the motivic tower
- Cohomological indexing of the actual graded groups
- The product on graded motivic groups is the source cup product
- External actual graded algebra and multiplicative tower spectral sequence
- The actual smooth open pair and its Nisnevich square
- A coherent comparison of the particular open restriction
- Every actual motivic stage on the empty scheme is zero
- Valuative dimension and degreewise bounded towers
- The local valuative convention matches the source convention
- General equicharacteristic scope and the standing filtration restriction
- The precise source stage bound and its homotopy interpretation
- Explicit eventual-stage thresholds in every fixed homotopy degree
- Degreewise boundedness and finite collections of eventual vanishings
- Motivic complexes and cohomology
- Restriction of the base field for weight one
- Elmanto–Morrow weight-one comparison input
- Low-degree consequences of a weight-one comparison
- Asok–Fasel Chern-compatible comparison and lifting input
- Smooth affine components and gluing a trivial line
- Determinants detect vanishing of the rank-one exterior-dual class
- The actual first Chern map is the Picard isomorphism on an affine curve
- The zero-dimensional critical-rank boundary
- Murthy top Chow cycle comparison with its actual exterior-dual map
- Affine bundles and finite projective modules
- Unimodular elements and trivial summands
- Chern classes of a vector bundle on a smooth scheme
- Normalization of an affine integral curve
- Top cycles supported on an integral curve
- Divisibility of the Picard group of an affine curve
- External determinant and pointed one-type inputs
- Weight zero is the locally constant rank function
- The EM weight-one Picard normalization is positive identity
- Units split the K one determinant
- Units and Picard groups in Nisnevich degree zero and one
- Bottom supported Bloch groups on a codimension two curve
- An actual non-A¹ P¹-spectrum for the supplied EM complexes
- Actual EM orientation and projective bundle maps on that model
- External non-A¹ projective-relation filtered and Whitney theorem
- Chern coefficients of the projective relation
- External non-A¹ projective-relation filtered and Whitney theorem
- External affine étale vanishing at invertible primes
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 15
The support long exact sequence kills degree seven
LemmaDependency level 15
Unique lifting from vanishing groups yields the Chern classification
LemmaDependency level 15
The actual first EM stage is the rank-zero K space on all qcqs schemes
LemmaDependency level 15
Generic rank and point-supported classes on a curve
Open problemDependency level 15
Rank-two uniqueness on singular threefolds
PropositionDependency level 15
Excision and K theory purity at a smooth closed point
TheoremDependency level 15
Murthy splitting criterion for the exterior-dual obstruction
TheoremDependency level 15
Classical smooth critical-rank classifications
DefinitionDependency level 16
Smooth-point K theory class map
LemmaDependency level 16
Excision and the exact fixed-support cycle fiber
LemmaDependency level 16
The actual supported Euler class has the generic length coordinate
LemmaDependency level 16
Componentwise curve construction and the scope of global irreducibility
LemmaDependency level 16
Actual EM orientation and projective bundle maps on that model
LemmaDependency level 16
Cancellation at the required dimension-bounded evaluation objects
LemmaDependency level 16
Point-supported cycles give all-weight chain purity and low-weight zero
LemmaDependency level 16
The identified support fiber is the classical relative Hom before Gysin
LemmaDependency level 16
Identifying all spectral-sequence pages with the source multiplicative pages
LemmaDependency level 16
Supported degree-seven vanishing
PropositionDependency level 16
Eventual homotopy vanishing of the motivic tower
PropositionDependency level 16
Motivic weight one
TheoremDependency level 16
Smooth threefold rank-two classification
TheoremDependency level 16
Smooth critical-rank splitting criterion
LemmaDependency level 17
The divisor presentation and actual Picard surjection
LemmaDependency level 17
Actual first-Chern and determinant maps of the first EM layer
LemmaDependency level 17
Positive generic cycles determine the curve-supported homomorphism
LemmaDependency level 17
The source Chern classes are the supplied projective-relation classes
LemmaDependency level 17
The source Chern classes are the supplied projective-relation classes
LemmaDependency level 17