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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
- Schemes and morphisms over a base
- Morphisms of schemes
- A Noetherian space is a finite union of irreducible closed subsets
- The Axiom of Choice
- Rational equivalence and the Chow group of cycles
- The Stacks Project, Curves, Remark 33.43.9 and preceding observation · Remark 33.43.9 and the observation immediately following it
- The Stacks Project, The Weil divisor class associated to an invertible module · Definition 31.28.4 and Lemma 31.28.7
- The Stacks Project, Curves and Degrees on curves · Lemmas 33.44.7 and 33.44.9; Curves, Lemma 33.43.8 at https://stacks.math.columbia.edu/tag/0A22
- The Stacks Project, Picard groups of curves · Section 59.69, opening paragraph before Lemma 59.69.1
- Composition series and length of a module
- Tor from a projective resolution of the right module
- Free abelian group on a set
- Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture Notes on Motivic Cohomology · Definition 17.1, printed p. 135, PDF p. 147; Definition 17A.1 and Example 17A.4, printed pp. 143–144, PDF pp. 155–156; convention on printed p. 3, PDF p. 15.
- The Stacks Project, Schemes smooth over fields · Lemmas 33.25.3 and 33.25.4.
- The Stacks Project, Normal schemes · Lemma 28.7.5.
- The Stacks Project, Absolute weak normalization and seminormalization · Lemma 29.48.7 (3)–(4) and its functoriality assertion
- The Stacks Project, Weak normalization · Lemmas 29.56.3 and 29.56.7, and the implication following Definition 29.56.9
- The Stacks Project, Normalization · Lemma 33.27.1(1)–(2)
- The Stacks Project, Absolute weak normalization and seminormalization · Lemma 29.48.9
- Module length is additive in short exact sequences
- The dimension formula for affine domains
- The Stacks Project, Principal divisors and pushforward · Lemma 42.18.1; Situation 42.7.1, tag 02QL, specialized to the spectrum of a field with dimension function zero
- The Stacks Project, Orders of vanishing · Lemma 10.121.8 and Definition 10.121.2, tag 02MD
- The Stacks Project, Fpqc descent of quasi-coherent sheaves · Proposition 35.5.2
- The Stacks Project, Descent of finiteness properties of modules · Lemma 35.7.6
- The Stacks Project, Meta theorem on quasi-coherent sheaves · Theorem 59.17.4, first conclusion
- The Stacks Project, The fppf topology · Lemma 34.7.2
- The Stacks Project, The V topology · Lemma 34.10.10, proof
- Stacks Project, Principal divisors and proper pushforward · Situation 42.7.1 (02QL); Lemma 42.18.1 (02RT) or Lemma 42.18.2 (02RQ), respectively; live online tag text retrieved and inspected 2026-11-09; no pinned commit asserted
- Stacks Project, Graph formula for a principal divisor · Situation 42.7.1 (02QL); Lemma 42.18.1 (02RT) or Lemma 42.18.2 (02RQ), respectively; live online tag text retrieved and inspected 2026-11-09; no pinned commit asserted
- Stacks Project, smooth schemes over fields · Lemma 33.25.3; live tag text inspected 2026-11-09
- Stacks Project, regular local rings · Lemma 10.106.2; live section text inspected 2026-11-09
- Stacks Project, finite-type dimension formulas · Lemmas 10.116.1-10.116.3; live section text inspected 2026-11-09
- Stacks Project, dimension formula for a proper dominant map · Lemma 29.53.4; live tag text inspected 2026-11-09
- Stacks Project, dimensions of Cartier divisor components · Lemma 42.16.1; live tag text inspected 2026-11-09
2 · Logical flowchart
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DefinitionEuler class group presentationDefinitionSmall sites and the plus constructionLemmaRestriction of finite-support divisorsLemmaQuotienting a compatible surjectionLemmaTransporting an exact sequence through isomorphismsLemmaDegree-zero representatives modulo a boundary classLemmaCoordinate faces of the algebraic simplexLemmaKernel models the homotopy fiber when the cokernel is acyclicLemmaUniqueness from squares and cubesLemmaThe square-cube condition implies reducednessLemmaUniqueness of an initial objectLemmaIntermediate rings in finite algebras
3 · Items
Lemma
Maps to finite disjoint unions of the base
Definition
Levine–Weibel zero cycles
Definition
Euler class group presentation
Definition
Small sites and the plus construction
Lemma
Restriction of finite-support divisors
Lemma
Quotienting a compatible surjection
Lemma
Transporting an exact sequence through isomorphisms
Lemma
Degree-zero representatives modulo a boundary class
Lemma
Completion, divisors and degree on smooth curves
Lemma
Coordinate faces of the algebraic simplex
Lemma
Length multiplicity for one proper Cartier face
Lemma
Cycle coefficients and face intersections are open local
Theorem
External input for simplicial Bloch cycle groups
Theorem
External geometric inputs for affine reduction
Lemma
Kernel models the homotopy fiber when the cokernel is acyclic
Lemma
Uniqueness from squares and cubes
Lemma
The square-cube condition implies reducedness
Lemma
Uniqueness of an initial object
Lemma
Intermediate rings in finite algebras
Theorem
Seminormalization and the finite-type interface
Lemma
Length orders on an integral curve
Theorem
External norm-divisor identity for integral curves
Lemma
External inputs for flat descent of line bundles
Theorem
Proper norm and graph formulas for principal divisors
Theorem