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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
- Scheme sites and sheafification
- Étale morphism of schemes
- Bloch cycle complexes and higher Chow groups
- Degree-zero Bloch cycles are ordinary cycles
- Boundaries in degree zero
- Proper norm and graph formulas for principal divisors
- Regularity and dimensions for finite-type cycle arguments
- The kernel of restriction is the supported subcomplex
- Higher-cycle localization including nonseparated schemes
- The Axiom of Choice
- Separated raw-cycle Zariski descent input
- Kernel models the homotopy fiber when the cokernel is acyclic
- Marc Levine, Techniques of localization in the theory of algebraic cycles · §2.2 p.6; Theorems1.9 and6.12; §6.11 pp.37–38; §7 pp.43–44, proving Theorem1.7
- Mazza, Voevodsky and Weibel, Lecture Notes on Motivic Cohomology (xprova author PDF) · Definition19.10, Theorem19.11, Proposition19.12 and proof, printedpp168–169/PDF179–180
- Negative homology of the cycle complex
- Reindexing a comparison with higher Chow groups
- Face boundaries preserve closed support
- Open restriction of admissible cycles
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
Nisnevich covers and distinguished squares
Definition
Roots of unity and étale Tate twists
Lemma
The degree-zero homology quotient
Lemma
Classical Chow relations and Bloch boundaries
Lemma
Restriction modulo its kernel
Lemma
Matching families of abelian presheaves
Lemma
Units in a sheaf of rings form a sheaf
Lemma
Local sections of a torsor give a cocycle
Lemma
Actual raw simplicial cycles satisfy descent without separatedness
Definition
Completely decomposed proper covers and the cdh topology
Lemma
Degree zero and negative homology of the cycle complex
Definition
Cycle complexes with fixed support
Definition
Milnor K symbols and the Nisnevich sheaf
Lemma
Pointwise kernels of abelian sheaves
Lemma
Frames and their transition units
Lemma
Compatible local torsor sections determine an isomorphism
Lemma