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Supporting prerequisites · 01 — Localization sequence for Chow groups and homotopy invariance of affine space
Presents 28 supporting prerequisites, covering published library prerequisites and dependency levels 0–1.
1 · Prerequisites
- Algebraic cycles and the cycle group of a scheme of finite type over a field
- The Axiom of Choice
- Rational equivalence and the Chow group of cycles
- Flat pullback of cycles and of rational equivalence
- The order function of a one-dimensional Noetherian local domain
- Proper pushforward of cycles and the norm formula
- Tame symbol reciprocity in dimension two (the Key Lemma)
- Stacks Project, Generically finite morphisms · Tag 0BAJ, displayed Lemma 29.52.6; online text retrieved 2026-11-09
- Stacks Project, Birational morphisms · Tag 0552, displayed Lemma 29.51.7; online text retrieved 2026-11-09
- Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv1207.4078v3 · Lemma6.6 and Theorem6.7 pp55–56; Theorem9.16/Lemma9.17 p77; Proposition11.1/Remark11.2 p79
- Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, arXiv2309.08463v1 · Theorem 3.5/Remark 3.6, printed pp.14–16; Constructions 4.34–4.36/Remark 4.37, pp.31–32; Lemma 5.1/Definition 5.2, pp.33–34; Lemmas 5.10/5.12 and Construction 5.11, p.36; Theorem 6.1(2), Remark 6.3/Corollary 6.4 and footnote 12, pp.44–45
- Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, arXiv2508.09915v1 · Proposition 3.15/Definition 3.16; Remarks 3.21/3.23; Lemma 3.27 proof (3.28); §4.1.2; Definitions 4.15/4.17, Remarks 4.18/4.20–4.23, Theorem 4.25/Remark 4.28; Lemma 4.33 and Construction 4.35/(4.36); Theorem 4.39/Remark 4.45; Corollaries 1.11/6.26; Theorem 6.3 coefficient-comparison paragraphs and (6.4) qualification; Proposition 7.4, Theorem 7.5/Remarks 7.6–7.10; Theorem 7.12/(7.14); Corollary 9.7/(9.8)
- Navarro, On the higher Riemann–Roch without denominators · Navarro §§2.1–2.2 p4; §2.6 p6, final regular-pair paragraph; Theorem3.1 pp7–8; smooth homotopy purity as cited there
- Morel–Voevodsky, A¹-homotopy theory of schemes, Publications Mathématiques IHES90(1999) · §3.2 Theorem2.23, printedp115/PDFp72; canonical blowup/purity maps in its proof; Proposition2.24
- Tom Bachmann, The very effective covers of KO and KGL over Dedekind schemes, EMS online-first PDF, DOI 10.4171/JEMS/1458 · Online-first PDF p.2: Theorem 1.1(1), Remark 1.2 and Example 1.3; p.3: Lemma 2.1 with proof and proof of Theorem 1.1(1)
- Marc Hoyois, The localization theorem for framed motivic spaces, Compositio157(2021),1–11 · Theorem18 and §4 introduction PDFp7; Lemma20 and Theorem21 with full proof PDFpp8–9
- Charles Weibel, The K-book, author-hosted standalone Chapter IV · IV.1.1.1–1.1.2 pp.3; IV.1.4–1.6 pp.4–5; IV.4.9–4.11.1 pp.42–45; IV.7.1–7.2 pp.61–62; IV.8.3.2–8.6 pp.68–70; Exercises 8.5–8.7 pp.74–75
- Charles Weibel, The K-book, author-hosted standalone Chapter V · Theorem V.2.2 and its displayed proof pp.13–14; V.2.7.2 p.18
- Annala–Iwasa, Motivic spectra and universality of K-theory (author PDF, 14April2025) · §1.3 Definitions1.3.4/1.3.8, Remark1.3.10; §2.1, §2.2 Definitions2.2.3/2.2.5, notation2.2.7–8 and coefficient adjunction2.2.10; Theorem2.4.5/Cor2.4.8; §3.1 Lemma3.1.4/Remarks3.1.5–6/Lemma3.1.7; §3.2 Definition3.2.1/Lemma3.2.3
- Annala–Iwasa, Motivic spectra and universality of K-theory (author PDF, 14April2025) · §1.3 Definitions1.3.4/1.3.8, Remark1.3.10; §2.1, §2.2 Definitions2.2.3/2.2.5, notation2.2.7–8 and coefficient adjunction2.2.10; Theorem2.4.5/Cor2.4.8; §3.1 Lemma3.1.4/Remarks3.1.5–6/Lemma3.1.7; §3.2 Definition3.2.1/Lemma3.2.3
- Utsav Choudhury and Martin Gallauer Alves de Souza, Homotopy theory of dg sheaves · §3 p.8 chain conventions; §5.1 Definitions5.1/5.3 pp.17–18; Theorem5.7 pp.18–19; Lemma5.10 p.19; Fact5.15 p.22
- Bhargav Bhatt and Akhil Mathew, The arc-topology · Definition1.2/Remark1.4 p.3; Definition1.5 p.4; §5.1 pp.37–38; Notation5.8 p.41; Proposition5.11 pp.41–42; Theorem5.13 pp.42–43
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §3.3 p.14, paragraph defining qcqs cdh topology and following square criterion
- Utsav Choudhury and Martin Gallauer Alves de Souza, Homotopy theory of dg sheaves · §3 p.8 chain conventions; §5.1 Definitions5.1/5.3 pp.17–18; Theorem5.7 pp.18–19; Lemma5.10 p.19; Fact5.15 p.22
- The Stacks Project, Extensions of valuation rings · Lemma15.125.5 and its full statement/proof
- The Stacks Project, Valuative criterion for properness · Lemma29.43.1, statement and proof
- Bhargav Bhatt and Akhil Mathew, The arc-topology · Definition1.2 p.3; Proposition2.1 and proof p.9
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem1.1(5) p.3; §3.3 p.14; Example4.11 p.22; Example4.26 p.28
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §3.3 pp.14–15, definition(3.5) and Theorem3.5(3); Theorem4.10(4) p.21 and proof p.22; Theorem4.24(3) p.28 and proof; Example4.26 p.28; Theorem4.33(1) proof p.31
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · §2.1.1–2.1.4 p. 14; Lemma 3.2.3 p. 23; Definition 4.4.1 and Lemma 4.4.3 pp. 30–31; Corollaries 4.4.5 and 4.4.7 and Theorem 4.4.6 pp. 31–32
- Stacks Project, derived limits and the Milnor exact sequence · Lemma 15.88.5 (tag 0919), statement and proof; Definition 13.34.1 (tag 08TB); Lemma 15.88.1(2)–(3) (tag 07KW)
- Stacks Project, derived limits: product-difference triangle · Definition 13.34.1
- Stacks Project, Rlim of abelian groups · Lemma 15.88.1(2)–(3), statement and proof
- Derived finite coefficients and the Bockstein sequence
- Affine bundles and finite projective modules
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
Localization sequence for Chow groups and homotopy invariance of affine space
Lemma
Integer multiplication on rational vector spaces
Lemma
Reindexing a chain complex cohomologically
Lemma
Cohomology after an even cochain shift
Lemma
Vanishing end groups kill the middle obstruction group
Lemma
An injection identifies its domain with its image
Lemma
Unique divisibility excludes torsion
Theorem
Dense-open isomorphism for proper birational maps
Theorem
Exact primary unit, Picard and smooth-purity inputs for field compatibility
Theorem
Full projective Quillen–Waldhausen comparison and the local free-class map
Theorem
External symmetric P¹ model, orientation criterion and fundamental stabilization
Theorem
External symmetric P¹ model, orientation criterion and fundamental stabilization
Theorem
Exact linear hyperdescent and derived-localization input
Theorem
Exact hypercomplete arc descent for finite constructible coefficients
Theorem
Exact linear hyperdescent and derived-localization input
Theorem
Exact valuation inputs for the topology comparison
Theorem
Exact source weight-zero and constant-sheaf input
Theorem
Exact cdh factorization inputs for the source étale map
Theorem
Universal Chern cohomology and countable derived limits
Lemma
A unit-induced integral field coefficient zigzag
Lemma
The normalized class detects the actual Picard-space map on smooth schemes
Lemma
Derived finite coefficients preserve mapping cones
Lemma
Integer multiplication on rational complexes
Lemma
Affine bundle classes are actual projective-module classes
Lemma