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Article items · 01 — Annihilators, torsion elements and the torsion subset of a module
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites and dependency level 0.
1 · Prerequisites
- Unital left and right modules over a ring; unqualified module means left module
- Left, right and two-sided ideals
- Zero divisor, and integral domain: a commutative ring with 1 ≠ 0 and no zero divisors
- Choice function
- Intersection with an invertible sheaf and the first Chern class
- Relative dimension of a smooth morphism at a point
- Smooth morphism of schemes
- Naturality of the Chow ring and the projection formula
- Operational Chern classes and the Whitney formula
- The intersection product and Chow ring of a smooth scheme
- The projective bundle formula for Chow groups
- Module homomorphism and isomorphism, kernel, image and cokernel
- Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel
- Modules on a ringed space
- Quasi-coherent module on a scheme
- Invariant basis number and the rank of a free module
- Every nonzero commutative ring has invariant basis number for finite bases
- The stalk of the affine structure sheaf at a prime is the localization at that prime
- Schemes
- Affine open subschemes
- Morphisms of schemes
- Gluing affine schemes along compatible open isomorphisms
- Dimension equals transcendence degree
- Function fields and dominant pullbacks on general varieties
- Global and local dimension of classical varieties
- Left and right Noetherian rings
- Noetherian modules: every submodule is finitely generated
- Generated submodule, cyclic and finitely generated modules, module basis and free module
- Submodule of a module
- The ideal generated by a subset and principal ideals
- In a commutative ring, (S) consists of finite sums ∑ rᵢ sᵢ, and (a)=Ra
- The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
- Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree
- Degree inequalities for sums and products over a commutative ring
- Polynomial rings in finitely many commuting indeterminates by iteration
- Field
- Every field is a commutative ring with 1 ≠ 0; it is an integral domain, and it is a commutative division ring
- The well-ordering principle
- Projective bundle in the quotient convention
- Picard group of a scheme
- Exterior Algebra Of A Finite Free Module
- Exterior Algebra Basis Monomials
- Sheafification is left adjoint to the inclusion of sheaves into presheaves
- Rational equivalence and the Chow group of cycles
- Affine quasi-coherent sheaves are modules
- Equivalent characterizations of projective modules
- Sections of the associated sheaf on basic opens
- The stalk of an associated sheaf is the localisation
- The prime spectrum is compact in the library's non-Hausdorff sense
- Distinguished-subset covers detect radicals
- The underlying space of an affine spectrum
- Coherent module sheaves
- Finite type and finitely presented module sheaves
- Regular equals smooth over a perfect field
- A Noetherian space is a finite union of irreducible closed subsets
- regular local rings are domains and cohen macaulay
- A Noetherian polynomial ring has dimension one larger
- Smoothness survives base change and composition
- Relative Jacobian criterion with its presentation hypothesis
2 · Logical flowchart
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3 · Items
DefinitionLibrary prerequisite
Annihilators, torsion elements and the torsion subset of a module
DefinitionLibrary prerequisite
The Axiom of Choice
DefinitionLibrary prerequisite
Chern classes of a vector bundle on a smooth scheme
DefinitionLibrary prerequisite
Divisible modules over an integral domain
DefinitionLibrary prerequisite
Exact sequences and short exact sequences of modules
DefinitionLibrary prerequisite
Locally free sheaves of finite rank
DefinitionLibrary prerequisite
Quasi-compact and quasi-separated schemes
DefinitionLibrary prerequisite
Schemes and morphisms over a base
LemmaLibrary prerequisite
Nonempty opens preserve irreducible dimension
LemmaLibrary prerequisite
Finite-variable polynomial algebras over fields are Noetherian by finite generators
DefinitionDependency level 0
Article scheme conventions
DefinitionDependency level 0
Bigraded cohomology and the projective bundle contract
DefinitionDependency level 0
Euler class group presentation
DefinitionDependency level 0
Exterior powers and determinant of an algebraic bundle
DefinitionDependency level 0
Levine–Weibel zero cycles
DefinitionDependency level 0
Stable triviality and stable bundle equivalence
LemmaDependency level 0
Affine bundles and finite projective modules
LemmaDependency level 0
Finite global generation on an affine scheme
LemmaDependency level 0
Finite projective modules are finitely presented
LemmaDependency level 0
Dimension of the maximal spectrum
LemmaDependency level 0
Roots in a finite product of groups
LemmaDependency level 0
Picard groups of finite disjoint unions
LemmaDependency level 0
Positive roots suffice for divisibility
LemmaDependency level 0
Connected components of a smooth affine curve
LemmaDependency level 0
Roots descend through a surjective homomorphism
LemmaDependency level 0
Unique divisibility is invariant under group isomorphism
LemmaDependency level 0
Primewise detection of torsion
LemmaDependency level 0