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Scheme sites and cycle theories
Define the Nisnevich, cdh and étale interfaces and distinguish derived sections from localization. Build the higher-cycle and curve-divisor chains, with precise Levine–Weibel, Milnor-K and Euler terminology for the article's comparisons and questions.
1 · Prerequisites
- Cycle complexes with fixed support
- The dimension formula for affine domains
- The Axiom of Choice
- Frames and their transition units
- Evaluation preserves kernels
- Scheme sites and sheafification
- The Stacks Project, Section 18.3: Abelian sheaves · Lemma 18.3.1(1)-(4)
- The Stacks Project, Section 21.19: Cohomology of unbounded complexes · Opening construction, additive-functor clause (2), and object-sections example (2), before Lemma 21.19.1
- The Stacks Project, Section 21.2: Cohomology of sheaves · Opening paragraph and equation 21.2.0.1
- The Stacks Project, Lemma 13.31.4: Bounded below injective complexes · Lemma 13.31.4, statement and one-line proof
- Length orders on an integral curve
- External norm-divisor identity for integral curves
- Projective morphisms are proper
- Residue fields and normalization point degrees
- Normalization of an affine integral curve
- Bloch cycle complexes and higher Chow groups
- Equivalent characterizations of a DVR
- On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group
- Finite morphisms are proper
- Completely decomposed proper covers and the cdh topology
- The Stacks Project, First cohomology and torsors · Lemma 21.4.3 and its proof
- Nisnevich covers and distinguished squares
- Picard group of a scheme
- Units in a sheaf of rings form a sheaf
- Local sections of a torsor give a cocycle
- Compatible local torsor sections determine an isomorphism
- External inputs for flat descent of line bundles
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
The two supported cycle degrees on a curve
Lemma
Tensor products multiply frame transitions
Lemma
Evaluation preserves chain homotopies
Lemma
Imported derived sections on a small site
Lemma
Endpoint boundaries are norm divisors
Lemma
Negative cohomology of a nonnegative evaluated complex
Lemma
Principal divisors are endpoint boundaries
Definition
Derived global sections on a scheme site
Lemma
Normalization and the curve divisor quotient
Definition
Derived cdh localization
Lemma
Top cycles supported on an integral curve
Lemma
External classification of torsors by first cohomology
Lemma