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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
- Equivalent characterizations of a DVR
- embedding dimension and regular local ring
- The Stacks Project, Schemes smooth over fields · Lemma 33.25.8, tag 0B8X
- Affine bundles and finite projective modules
- Grothendieck groups of coherent sheaves and of vector bundles on a scheme
- The Axiom of Choice
- Exterior powers and determinant of an algebraic bundle
- Kernels of surjections of vector bundles
- A normal noetherian ring is a finite product of normal domains
- normal noetherian ring
- A Noetherian space is a finite union of irreducible closed subsets
- The stalk of the affine structure sheaf at a prime is the localization at that prime
- A normal Noetherian domain is the intersection of its height-one localizations
- 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
- Generic local rings of reduced schemes
- Fields are regular local rings
- Picard group of a scheme
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
Lemma
Discrete valuation rings are regular local
Lemma
Open subsets containing generic points are componentwise dense
Lemma
Excluding points of codimension zero and one
Lemma
Pointwise regularity and smoothness over a perfect field
Lemma
Finite projective complements
Definition
Unimodular elements and trivial summands
Lemma
Determinants and exact sequences
Lemma
Normal noetherian schemes reduce to integral components
Lemma
Equality in affine K zero and stable isomorphism
Lemma
Functions extend across codimension two
Lemma
Smooth density and normal codimension over a perfect field
Lemma