Alphabeta MathPersonal research
Article items · 03 — Coordinate multiples of dual generators span all first jets
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 1–4.
1 · Prerequisites
- Finite global generation on an affine scheme
- Kernels of surjections of vector bundles
- External finite-type dimension, rational-point and etale-fiber inputs
- The Axiom of Choice
- External normality and perfect-field smooth-locus inputs
- Bigraded cohomology and the projective bundle contract
- Exterior powers and determinant of an algebraic bundle
- Grothendieck groups of coherent sheaves and of vector bundles on a scheme
- Affine bundles and finite projective modules
- Projective bundle represents line quotients
- Relative projective space from standard charts
- The relative projective-space diagonal is closed
- Uniqueness from squares and cubes
- Square-cube roots in a normal domain
- The square-cube condition implies reducedness
- Uniqueness of an initial object
- Intermediate rings in finite algebras
- Seminormalization and the finite-type interface
- Localization of a presented divisor class group
- Transporting an exact sequence through isomorphisms
- Degree-zero representatives modulo a boundary class
- Completion, divisors and degree on smooth curves
- Picard group of a scheme
- Over an algebraically closed field, every maximal ideal is an evaluation ideal
- Projectivity and rank of a free-summand complement
- Stable triviality as the kernel of a right-invertible matrix
- A torsion kernel in a uniquely divisible group vanishes
- An injection identifies its domain with its image
- Unique divisibility is invariant under group isomorphism
- Normal noetherian schemes reduce to integral components
- A normal Noetherian domain is the intersection of its height-one localizations
- Exterior Algebra Basis Monomials
- The smooth locus is open
- Height-one localizations of normal Noetherian domains are DVRs
- Finite-variable polynomial algebras over fields are Noetherian by finite generators
- The stalk of the affine structure sheaf at a prime is the localization at that prime
- Generic local rings of reduced schemes
- Fields are regular local rings
- Discrete valuation rings are regular local
- Open subsets containing generic points are componentwise dense
- Excluding points of codimension zero and one
- Pointwise regularity and smoothness over a perfect field
- Unimodular elements and trivial summands
- Admissible face restrictions and their cycle identities
- Alternating face boundaries square to zero
- Homology object of a chain complex
- Stable homotopy groups of a sequential prespectrum
- Singular complexes preserve the sequential group convention
- Omega replacement preserves sequential stable groups
- Omega towers and the suspension index
- Stable rotation and exact coherent products
- External stable enhancement and smash product of spectra
- Finite derived quotients of rational complexes vanish
- Derived finite coefficients preserve mapping cones
- Derived category of an abelian category
- Cohomology maps induced by cochain maps
- Homotopic maps act equally on cohomology
- Positive roots suffice for divisibility
- Roots descend through a surjective homomorphism
- Roots in a finite product of groups
- Picard groups of finite disjoint unions
- Connected components of a smooth affine curve
- Degree-zero Picard group as points of an abelian variety
- Divisibility of rational points of an abelian variety
- Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF page 13, paragraph citing Fulton Example 1.6.6
- Stable triviality and stable bundle equivalence
- 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; bibliography MK84/MK85, p. 36
- Finite projective modules are finitely presented
- Dimension of the maximal spectrum
- Local ranks in the affine bundle correspondence
- Serre free-summand theorem for modules
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 1
Coordinate multiples of dual generators span all first jets
LemmaDependency level 1
Normal affine threefolds have controlled nonsmooth strata
LemmaDependency level 1
Chern coefficients of the projective relation
LemmaDependency level 1
Determinants and exact sequences
LemmaDependency level 1
Finite projective complements
LemmaDependency level 1
Flag towers with line-bundle quotients
LemmaDependency level 1
Extending a Whitney class to K zero
DefinitionDependency level 2
Seminormality and seminormalization
LemmaDependency level 2
Picard group of an affine curve through its boundary
LemmaDependency level 2
Iterating a conditional free-summand criterion
LemmaDependency level 2
Matrix completion detects the actual nontrivial bundle
LemmaDependency level 2
Divisibility of an image with torsion kernel
LemmaDependency level 2
Functions extend across codimension two
LemmaDependency level 2
Exterior-square line factors on a flag space
LemmaDependency level 2
Equality in affine K zero and stable isomorphism
LemmaDependency level 2
Smooth density and normal codimension over a perfect field
LemmaDependency level 2
Finite-dimensional families of generated bundle functionals
LemmaDependency level 2
Formal Chern arithmetic through degree three
LemmaDependency level 2
A rank-two bundle with a trivial quotient and determinant
DefinitionDependency level 3
Bloch cycle complexes and higher Chow groups
DefinitionDependency level 3
Stable spectra and exact triangles
LemmaDependency level 3
Rational cofiber implies finite-coefficient equivalence
LemmaDependency level 3
Cohomology descends to the derived category
LemmaDependency level 3
The scheme of nontransverse zero jets has small dimension
LemmaDependency level 3
Line bundles are detected off codimension two
PropositionDependency level 3
Divisibility of the Picard group of an affine curve
TheoremDependency level 3
Mohan Kumar existence of a nontrivial stably trivial bundle on a complex affine fourfold
TheoremDependency level 3
Serre splitting above the dimension
DefinitionDependency level 4