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Affine bundles and normal geometry
Establish the scheme conventions, module–bundle correspondence, stable classes, determinants and normal-extension tools used by the article. Connect affine curves and residue fields to the later support arguments.
1 · Prerequisites
- Schemes and morphisms over a base
- Quasi-compact and quasi-separated schemes
- Locally free sheaves of finite rank
- 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
- The Axiom of Choice
- Coherent module sheaves
- Finite type and finitely presented module sheaves
- Exterior Algebra Of A Finite Free Module
- Exterior Algebra Basis Monomials
- Sheafification is left adjoint to the inclusion of sheaves into presheaves
- Picard group of a scheme
- A finite-type domain over a field has finite normalization
- A normal curve over a perfect field is nonsingular
- Integral elements over a nonzero base ring form a subring
- Integral extensions are transitive
- Injective integral extensions preserve Krull dimension
- Finite is affine and local on its target
- Lying over for integral ring maps
- Under an integral extension, a prime is maximal if and only if its contraction is maximal
- Over an algebraically closed field, every maximal ideal is an evaluation ideal
- Chinese remainder theorem for pairwise comaximal ideals
- normal noetherian ring
- A radical ideal in a Noetherian ring is a finite intersection of minimal primes
- The stalk of the affine structure sheaf at a prime is the localization at that prime
- The nilradical is the intersection of all prime ideals
- embedding dimension and regular local ring
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
Definition
Article scheme conventions
Lemma
Affine bundles and finite projective modules
Lemma
Finite global generation on an affine scheme
Lemma
Kernels of surjections of vector bundles
Definition
Exterior powers and determinant of an algebraic bundle
Definition
Stable triviality and stable bundle equivalence
Lemma
Normalization of an affine integral curve
Lemma
Residue fields and normalization point degrees
Lemma
A normal noetherian ring is a finite product of normal domains
Lemma
Generic local rings of reduced schemes
Lemma