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Chern constructions and support geometry
Develop flag detection and virtual Chern arithmetic alongside indexed support fibers, excision, Koszul classes and curve dévissage. Place the article's support definitions here and prepare the Bertini and exterior-Euler interfaces needed by its final geometric arguments.
1 · Prerequisites
- Support fibers with explicit cohomological indexing
- Stable spectra and exact triangles
- Article scheme conventions
- Motivic filtration and image groups
- Motivic complexes and cohomology
- Complex shifts and spectral homotopy indices
- Support fibers commute with graded triangles
- Scheme sites and sheafification
- The Axiom of Choice
- Triangulated-category axiom TR3
- Long exact Hom sequences of a distinguished triangle
- Finite ideal-power filtrations and coherent sheaves on the closed subscheme
- External degree-zero abelian dévissage
- Postnikov inverse and the specified nonconnective degree-zero group
- External EM low-weight maps with fixed positive normalization
- Affine perfect K zero and bundle K zero
- Relative cofibers and quotient complexes
- Contravariant Hom reverses relative cofiber triangles
- Purity twists and cohomological shifts with explicit premises
- Relative objects for empty and split pairs
- Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture notes on motivic cohomology (xprova PDF) · Theorem 15.15, printed p.130, PDF page 141. Corollary 15.13 and Exercise 15.14, printed pp.129–130, PDF pages 140–141. Sm/k convention: Lecture 1, printed p.5, PDF page 16.
- Mazza–Voevodsky–Weibel, motive and Tate-twist conventions · Definition 14.1, printed p.113/PDF p.124; Definition 14.2 and following tensor conventions pp.113–114; Lemma 14.4 and first Properties 14.5, p.114/PDF p.125.
- Nonconnective K theory of perfect complexes
- Perfect complexes on schemes and their support
- Nonconnective K theory in degree zero
- Cofiber and triangle presentations of K zero
- Inherited bounded t structure on supported perfect complexes
- Additivity of the supported coherent Euler class
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
Open excision for a descent theory
Definition
Motivic and filtered K theory with support
Lemma
Compatible comparisons transport relative objects and support fibers
Lemma
Ideal-power dévissage and compatibility with coherent pushforward
Lemma
Weight zero is the locally constant rank function
Proposition
Gysin purity for a smooth closed pair
Definition
Images of support groups in global groups
Proposition
Dévissage of K zero with regular ambient support
Lemma