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Supporting prerequisites · 05 — External actual EM graded algebra, Picard class and complex PBF
Presents 26 supporting prerequisites, covering dependency levels 17–31.
1 · Prerequisites
- The actual EM Chern operation on the standard point
- Actual supported Chern operation and its smooth comparison
- Positive supported coordinates at a smooth threefold point
- A specified étale-natural comparison with supported cycles
- A smooth point has the exact finite-type excision pair
- Finite-type field schemes lie in the qcqs source domain
- Smooth maps have étale local affine-space form
- Relative dimension of a smooth morphism at a point
- A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension
- Perfect fields: every irreducible polynomial is separable
- Every algebraic extension of a perfect field is separable
- Every affine scheme is quasi-compact
- Every point of a Zariski-open set has a distinguished-open neighbourhood inside it
- A principal localization identifies its spectrum with a distinguished open
- Affine-local flatness
- Flat and faithfully flat modules and ring homomorphisms
- Coherent derived tensor and pullback on perfect complexes
- Affine fibre products are spectra of tensor products
- M⊗_RR/I≅ M/IM naturally
- The Axiom of Choice
- Nonconnective K theory of perfect complexes
- Smooth morphism of schemes
- Finite type is affine-local on source and target
- Locally finite type and finite type morphisms
- Quasi-compact and quasi-separated morphisms
- Affine schemes and affine morphisms are separated
- Open and closed immersions are separated
- Separated morphisms compose
- The Stacks Project, étale locality and flatness · Definition 29.37.1 and the following openness paragraph; Lemma 29.37.2 with restriction clause; Lemmas 29.37.3 and 29.37.12, with their short proofs
- The Stacks Project, fibres of étale morphisms · Lemma 29.37.7(1)–(2), complete statement and direct proof
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