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Article items · 09 — A rank-two family avoids each prescribed codimension-two locus
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 14–15.
1 · Prerequisites
- Coherent dual extension supplies global Swan controlling sections
- Finite-dimensional families of generated bundle functionals
- Kernels of surjections of vector bundles
- External normality and perfect-field smooth-locus inputs
- External finite-type dimension, rational-point and etale-fiber inputs
- 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
- The Axiom of Choice
- Operational Chern classes and the Whitney formula
- Chern classes of a vector bundle on a smooth scheme
- Article scheme conventions
- Motivic and filtered K theory with support
- Open excision for a descent theory
- Elmanto–Morrow motivic complex input
- Actual cofiber triangles and source grading of the motivic tower
- Complex shifts and spectral homotopy indices
- Spectral sequence of a tower
- Motivic complexes and cohomology
- The actual smooth open pair and its Nisnevich square
- A coherent comparison of the particular open restriction
- Cycle complexes with fixed support
- The ordinary Chow Chern map on isomorphism classes
- Cylinder and zero section of a smooth affine surface
- Chern data survive addition of trivial bundles
- Serre splitting above the dimension
- Finite constant cdh sections
- Finite refinement for integer cdh sections
- Scheme sites and sheafification
- Maps to finite disjoint unions of the base
- 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
- The source qcqs cdh site
- 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
- The actual composite and its objectwise cofiber
- 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
- Nonconnective K theory of perfect complexes
- Stable spectra and exact triangles
- 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
- Relative cofibers and quotient complexes
- Contravariant Hom reverses relative cofiber triangles
- Purity twists and cohomological shifts with explicit premises
- Compatible comparisons transport relative objects and support fibers
- Relative objects for empty and split pairs
- Support fibers with explicit cohomological indexing
- Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture notes on motivic cohomology (xprova PDF) · Theorem 15.15, printed p.130, PDF page 141. Corollary 15.13 and Exercise 15.14, printed pp.129–130, PDF pages 140–141. Sm/k convention: Lecture 1, printed p.5, PDF page 16.
- Mazza–Voevodsky–Weibel, motive and Tate-twist conventions · Definition 14.1, printed p.113/PDF p.124; Definition 14.2 and following tensor conventions pp.113–114; Lemma 14.4 and first Properties 14.5, p.114/PDF p.125.
- 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
- Projective modules and the lifting property
- 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 in Section 10.109, Tag 00O2
- Local application interface for a critical-rank cancellation theorem
- Suslin cancellation over affine algebras
- Manoj Kumar Keshari, Cancellation problem for projective modules over affine algebras · p. 1, Introduction: definition of cancellative and Theorem 1.1(i); p. 16, reference [20]
- Flag towers with line-bundle quotients
- Detection on a simultaneous flag tower
- Motivic projective bundle formula
- Every actual motivic stage on the empty scheme is zero
- Valuative dimension and degreewise bounded towers
- Motivic filtration and image groups
- Explicit eventual-stage thresholds in every fixed homotopy degree
- Cohomology of shifted Nisnevich units
- Exact range of a cofiber connectivity bound
- 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
- Bottom supported Bloch groups on a codimension two curve
- Augmented tower comparisons preserve supported graded projections
- Support fibers commute with graded triangles
- A common-core zero curve exists on a nonempty parameter open
- External Swan finite controlling-family Bertini theorem
- 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
- Intersection with an invertible sheaf and the first Chern class
- Rational equivalence and the Chow group of cycles
- External actual EM graded algebra, Picard class and complex PBF
- External symmetric P¹ model, orientation criterion and fundamental stabilization
- Roots of unity and étale Tate twists
- Derived global sections on a scheme site
- Exact natural regular motivic-to-cdh comparison
- Fixed-site classical cycle comparison on every smooth ambient
- External actual graded algebra and multiplicative tower spectral sequence
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 14
A rank-two family avoids each prescribed codimension-two locus
LemmaDependency level 14
Determinants detect vanishing of the rank-one exterior-dual class
LemmaDependency level 14
A split trivial line annihilates the exterior and top Chow classes
LemmaDependency level 14
A smooth point has the exact finite-type excision pair
LemmaDependency level 14
Cohomological indexing of the actual graded groups
LemmaDependency level 14
Excision and the exact fixed-support cycle fiber
LemmaDependency level 14
Surface existence from rank-three existence on the cylinder
LemmaDependency level 14
Surface uniqueness from the cylinder and cancellation
LemmaDependency level 14
Integer cdh sections on seminormal schemes
PropositionDependency level 14
Étale derived cdh locality on the actual qcqs site
PropositionDependency level 14
Prime-to-characteristic étale comparison
PropositionDependency level 14
Motivic weight zero on all qcqs schemes
PropositionDependency level 14
Perfect K theory localization and open excision
PropositionDependency level 14
Gysin purity for a smooth closed pair
PropositionDependency level 14
Dévissage of K zero with regular ambient support
TheoremDependency level 14
Murthy top Chow cycle comparison with its actual exterior-dual map
TheoremDependency level 14
Critical-rank cancellation used in the article
TheoremDependency level 14
Motivic splitting principle
LemmaDependency level 15
Degreewise boundedness and finite collections of eventual vanishings
LemmaDependency level 15
Low-degree consequences of a weight-one comparison
LemmaDependency level 15
A coherent comparison of the particular open restriction
LemmaDependency level 15
The supported second stage has the actual augmentation inverse
LemmaDependency level 15
Swan chartwise integrality glues on the actual quasi-affine open
LemmaDependency level 15
The actual first Chern map is the Picard isomorphism on an affine curve
LemmaDependency level 15
An actual non-A¹ P¹-spectrum for the supplied EM complexes
LemmaDependency level 15
Slice localization and the actual root-of-unity coefficient map
LemmaDependency level 15
The actual smooth restriction arrow gives the point-supported cycle fiber
LemmaDependency level 15