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Supporting prerequisites · 02 — Exact objectwise cofiber bound for the canonical source composite
Presents 28 supporting prerequisites, covering dependency levels 1–11.
1 · Prerequisites
- Exact cdh factorization inputs for the source étale map
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem4.33(1) and its complete proof, printed p.31; introductory equivalent truncation statement Theorem1.1(3), p.3
- Integer multiplication on rational complexes
- Derived finite coefficients and the Bockstein sequence
- Finite projective complements
- Affine bundle classes are actual projective-module classes
- Affine bundles and finite projective modules
- The Axiom of Choice
- Affine bundle classes are actual projective-module classes
- Cohomology descends to the derived category
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Proposition8.10 pp.56–57, with its complete proof; Lemma8.9 p.56
- Perfect complexes on schemes and their support
- Exterior powers and determinant of an algebraic bundle
- Knudsen–Mumford, The projectivity of the moduli space of stable curves I: Preliminaries on det and Div, Math. Scand. 39 (1976), 19–55 · Chapter I, pp.20–21 and31; Definition2 p.37 and remark p.38; perfect-complex definition and Proposition4 p.39; Definition4 pp.40–42; Theorem2 p.42; warning and Proposition7 p.44
- Scheme sites and sheafification
- Valuative dimension and degreewise bounded towers
- Elden Elmanto, Marc Hoyois, Ryomei Iwasa and Shane Kelly, Cdh descent, cdarc descent, and Milnor excision · Math.Ann.379 (2021),1011–1045; §2.3 definitions and Lemma2.3.1 p.9; Proposition2.3.2(1),(8),(9) pp.9–10
- Completely decomposed proper covers and the cdh topology
- Dense-open isomorphism for proper birational maps
- Seminormality and seminormalization
- Elmanto–Morrow motivic complex input
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §1 introduction footnote1 p.2; Theorem1.1 p.3; comparison map(1.2) p.4 and Theorem1.3(1) p.5; §3.3 definition(3.5) and Theorem3.5(1) pp.14–15; paragraph after Remark8.8 p.56
- External projective bundle input for the actual motivic complex
- Flag towers with line-bundle quotients
- Motivic complexes and cohomology
- Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Theorem 1.1 opening, printed p. 3; Theorem 5.24(2), printed p. 43, preceding projectivization convention, and proof conclusion p. 44; negative-weight convention p. 34
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Proposition8.12 p.57 and its proof p.58; Theorem8.2 p.54, footnote14, and its proof p.55; Theorem8.7 p.55 for identification of the intended fiber
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem1.1(9) p.4; Construction4.34 p.31; §6 equation(6.1) and Theorem6.1(2) p.44; Corollary6.4 and footnote12 p.45
- Multiplicative filtrations and graded products
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes (arXiv v1) · §2.2, pp.11–12; Definitions4.9/4.23, pp.21/27–28; Theorems4.10(1)/4.24(1); Lemma5.1 and its proof and Definition5.2, pp.33–34; multiplication and (5.2), p.34; line convention immediately before Theorem5.24, p.43
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem1.1(9) p.4; Construction4.34 p.31; §6 equation(6.1) and Theorem6.1(2) p.44; Corollary6.4 and footnote12 p.45
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Definition 4.14, Lemma 4.16 and proof, pp.24–25; Remark 4.30(2), Remark 4.31, Corollary 4.32 and proof, p.30
- External canonical field restriction identifies the BEM and EM cdh models
- Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes · Theorem7.5 and Remark7.9; Corollary1.11 and preceding canonical comparison(1.9); Remark3.41 maps(3.42)–(3.43); Remark4.12; Definitions4.15/4.17 and Remark4.18; Theorem1.1(8); Remark4.21; proof of Theorem6.3, paragraphs preceding (6.4); §3.2 mapping-presheaf construction; formula(4.7)
- Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture notes on motivic cohomology · Theorem16.7 and Suslin–Friedlander construction p.126; Lecture19 pp.159–163: Lemma19.4, Theorem19.8, Proposition19.12, proof of Theorem19.1, Definition19.13 and Example19.14
- Marc Levine, The homotopy coniveau tower (arXiv:math/0510334v1, 16 October 2005) · §§1.2–1.4, pp.7–8; §3.3, p.18; Theorem4.1.1, pp.20–21; Theorem6.4.1, p.35; §6.5, pp.37–38; Example8.2.2(3), pp.45–46; Theorem9.0.3 and Remark9.0.4, p.47; §§10.4–10.5, pp.59–61. These are v1 page and theorem numbers.
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
Theorem
Exact objectwise cofiber bound for the canonical source composite
Lemma
Finite derived quotients of rational complexes vanish
Lemma
Common projective complements and cancellation by free modules
Lemma
Free stabilization passes through global sections
Lemma
Common projective complements and cancellation by free modules
Lemma
Free stabilization passes through global sections
Lemma
Cofiber bounds imply low-degree cohomology isomorphisms
Theorem
External dimension criterion for a cdh-acyclic Nisnevich sheaf
Lemma
Exact range of a cofiber connectivity bound
Theorem
Perfect determinant with true-triangle additivity: precise external input
Lemma
A sheaf restricts injectively to a covering singleton
Theorem
External valuative dimension comparison on locally Noetherian schemes
Lemma
Completely decomposed proper maps are abstract-blowup covers
Lemma
Finite subintegral maps are cdh covers
Theorem
External prime-field cdh comparison and cdh dimension bound
Lemma
Flag detection over an arbitrary field by prime-field restriction
Lemma
The diagonal of a finite subintegral map is a cdh cover
Theorem
External geometric properties of the actual motivic comparison fiber
Theorem
Exact natural regular motivic-to-cdh comparison
Theorem
External EM multiplicative motivic complexes and actual Chern map
Theorem
Exact natural regular motivic-to-cdh comparison
Theorem
External EM affine syntomic filtration and actual obstruction map
Lemma
Sections match across a finite subintegral cover
Lemma
Affine characteristic-p syntomic amplitude without regularity
Theorem