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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
- Small sites and the plus construction
- Restriction of finite-support divisors
- Quotienting a compatible surjection
- Coordinate faces of the algebraic simplex
- Length multiplicity for one proper Cartier face
- Cycle coefficients and face intersections are open local
- External input for simplicial Bloch cycle groups
- External geometric inputs for affine reduction
- Free abelian group on a set
- Uniqueness from squares and cubes
- The square-cube condition implies reducedness
- Uniqueness of an initial object
- Intermediate rings in finite algebras
- Seminormalization and the finite-type interface
- Transporting an exact sequence through isomorphisms
- Degree-zero representatives modulo a boundary class
- Completion, divisors and degree on smooth curves
- Picard group of a scheme
- Over an algebraically closed field, every maximal ideal is an evaluation ideal
- Chain complex in an abelian category
- Homology object of a chain complex
- Mazza, Voevodsky and Weibel, Lecture Notes on Motivic Cohomology, December 2005 author PDF · Standing convention printed p.5; Definition19.10, Theorem19.11, Proposition19.12 and proof, printed pp.168–169 (PDF pp.179–180)
- Schemes and morphisms over a base
- Marc Levine, Techniques of localization in the theory of algebraic cycles · Primary 53-page preprint: §1.6–1.8 pp. 2–4; §2.2 p. 6; Theorem 6.12 and finite affine-cover reduction pp. 37–38; §7 pp. 43–44
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
Covering sieves admit common refinements
Lemma
Localization of a presented divisor class group
Lemma
Admissible face restrictions and their cycle identities
Lemma
Square-cube roots in a normal domain
Definition
Seminormality and seminormalization
Lemma
Picard group of an affine curve through its boundary
Lemma
The plus construction is a presheaf
Lemma
Alternating face boundaries square to zero
Definition
Bloch cycle complexes and higher Chow groups
Lemma
The plus presheaf is separated
Lemma
The plus construction turns a separated presheaf into a sheaf
Lemma
Universal mapping property of the plus construction
Lemma
Degree-zero Bloch cycles are ordinary cycles
Lemma
Negative homology of the cycle complex
Lemma
Face boundaries preserve closed support
Lemma
Open restriction of admissible cycles
Theorem
Separated raw-cycle Zariski descent input
Definition
Scheme sites and sheafification
Lemma
Boundaries in degree zero
Lemma
Reindexing a comparison with higher Chow groups
Lemma
The kernel of restriction is the supported subcomplex
Theorem