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Supporting prerequisites · 04 — Exact arbitrary-field open-natural classical cycle interface
Presents 26 supporting prerequisites, covering dependency levels 10–17.
1 · Prerequisites
- External canonical field restriction identifies the BEM and EM cdh models
- The Axiom of Choice
- Bloch cycle complexes and higher Chow groups
- Perfect fields: every irreducible polynomial is separable
- A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p
- Grothendieck universe closure convention
- Schemes and morphisms over a base
- Smooth morphism of schemes
- Separated morphism of schemes
- Locally finite type and finite type morphisms
- Quasi-compact and quasi-separated schemes
- Open immersions of schemes
- Closed immersions of schemes
- Every affine scheme is quasi-compact
- Quasi-separatedness and the diagonal
- 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); §1.6 notation; §3.1 definitions of SH and the Tate object; Definitions7.1/7.8
- 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.
- Derived category of an abelian category
- External input on derived enhancements
- Imported derived sections on a small site
- Scheme sites and sheafification
- Étale morphism of schemes
- Smoothness survives base change and composition
- Separated morphisms compose
- Open immersions are etale
- Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, served v1 HTML · §1.6; comparison (1.9), Corollary 1.11; proof of Theorem 6.3, equation (6.4) and its surrounding fixed-site qualification; Definition 7.1, Theorem 7.5, Definition 7.8 and Remark 7.9
- Mazza–Voevodsky–Weibel, Lecture notes on motivic cohomology, Rutgers served PDF · Definition 2.14 (printed p.16), Definition 3.1 and Lemma 3.2 (p.21); p.126 and Theorem 16.7 (pp.126–127); Definition 17.1 and Properties 17.4(1) (pp.135–136); Lemma 19.4 and the small-site construction (pp.159–160), Theorem 19.8, Proposition 19.12, Definition 19.13 and Example 19.14 (pp.161–163). PDF page = printed page + 12.
- The Stacks Project, Cohomology of unbounded complexes · Section 21.19, opening paragraph, additive-functor clause (2), computation formula and object-sections example (2), before Lemma 21.19.1
- The Stacks Project, Lemma 13.31.7 · Lemma 13.31.7, statement and full proof, particularly the K-injective quasi-isomorphism sentence
- The Stacks Project, Morphisms of finite type · Section 29.15, Definition 29.15.1 and Lemmas 29.15.3, 29.15.5 and 29.15.6 (tags 01T3, 01T5, 01T6)
- The Stacks Project, Noetherian schemes · Section 28.5, opening ring convention, Definition 28.5.1 and Lemmas 28.5.2–28.5.4
- The Stacks Project, Cancellation law for étale morphisms · Lemma 29.37.19 (tag 02GW), statement and short proof; older chapter numbering 29.36.18
- Commutative ring
- The source qcqs cdh site
- Derived global sections on a scheme site
- From constructible-category hyperdescent to the actual cohomology complex
- External stable mapping and derived linear target interfaces
- The Stacks Project, big and small étale sites · Definitions 34.4.6 and 34.4.8; Lemmas 34.4.13, 34.4.17–18; Lemma 34.4.20, statement and forward-construction proof.
- The Stacks Project, Lemma 59.99.1 · Lemma 59.99.1(2)(a), with (1)(a) and the two proofs following it.
- The Stacks Project, Lemma 21.20.7 · Lemma 21.20.7(2) and the first proof paragraph constructing the comparison from a module K-injective resolution and an abelian K-injective replacement.
- The Stacks Project, Lemma 18.16.2: Abelian lower shriek adjunction · Lemma 18.16.2, sheaf-level adjunction formula and proof.
- The Stacks Project, Lemma 18.16.3: Exactness of abelian lower shriek · Lemma 18.16.3, hypotheses (a)–(c), conclusion and proof.
- The Stacks Project, Section 21.19: Ringed-site K-injective resolutions · Opening paragraph, additive-functor clause (2), computation formula and object-sections example (2), before Lemma 21.19.1.
- The Stacks Project, Section 13.31: K-injective comparison and products · Definition 13.31.1; Lemma 13.31.2 and proof; Lemma 13.31.5 and its product-exactness argument; Lemma 13.31.7 proof, first paragraph.
- Local simplicial sheaf and additive chain model interfaces
- Canonical truncation of a complex
- Canonical truncation is a complex and has the claimed cohomology
- Roots of unity and étale Tate twists
- For every natural n, (ℤ/n,+) is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold
- Exact linear hyperdescent and derived-localization input
- The Stacks Project, the étale topology · Definitions 34.4.6 and 34.4.8; Lemmas 34.4.13, 34.4.16–18 and 34.4.20, including the forward construction in its proof
- The Stacks Project, Lemma 59.99.1 · Lemma 59.99.1(2)(a), and the complete proof including the module counterexample
- The Stacks Project, Lemma 21.20.1 · Complete statement and one-paragraph proof of Lemma 21.20.1
- External canonical top-weight motivic-to-syntomic comparison
- The actual composite and its objectwise cofiber
- Exact cdh factorization inputs for the source étale map
- The actual finite-quotient map to cdh-local motivic cohomology
- Finite-type field schemes lie in the qcqs source domain
- Exactness of ordinary derived cdh sheafification and evaluation
- Existence of all scheme fibre products
- Quasi-compactness is local on the target and survives base change
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Theorem 1.1 opening, printed p.3 (PDF index 2); Section 2.1, printed p.11 (index 10); Section 3.3, (3.4)-(3.5), printed p.14 (index 13), and Theorem 3.5(3), printed p.15 (index 14); Theorem 4.33(1) and its full proof, printed p.31 (index 30).
- Elden Elmanto, Zero cycles on singular varieties (joint work with Matthew Morrow), Oberwolfach Report 36/2025 · Theorem 3 opening and part (1), with footnote 1, printed p.1994 (PDF index 45); Theorem 4 and its complete proof, especially the first, invertible-prime paragraph, printed p.1995 (PDF index 46).
- The Stacks Project, Sites and Sheaves, section 7.25: Localization · Section 7.25 opening and Definition 7.25.1; proof of Lemma 7.25.4 at the first sheafification square, and footnote 1.
- Units and Picard groups in Nisnevich degree zero and one
- Full projective Quillen–Waldhausen comparison and the local free-class map
- External connective, nonconnective and automorphism comparisons
- Bounded finite-projective complexes and affine chain models
- Enhanced affine perfect equivalence
- Affine perfect K zero and bundle K zero
- Nonconnective K theory of perfect complexes
- Qcqs scheme sites and affine detection for space-valued sheafification
- Locally free sheaves of finite rank
- Exact sequences of sheaves
- Pullback of a module along a morphism of ringed spaces
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk
- Perfect complexes on schemes and their support
- Coherent derived tensor and pullback on perfect complexes
- Charles Weibel, The K-book, author-hosted Chapter V · Theorem V.2.2 and its displayed proof, printed pp. 13–14
- Charles Weibel, The K-book, author-hosted Chapter IV · IV.8.3.2 and IV.8.5, p. 68; Exercises IV.8.5–8.7, pp. 74–75
- Blumberg–Gepner–Tabuada, A universal characterization of higher algebraic K-theory, arXiv:1001.2282v4 · Definition 7.1 and S construction/core map, pp. 44–45; Lemma 7.11/Corollary 7.12, pp. 48–49; Proposition 9.32 and Remark 9.33, p. 64; Definition 9.6/Theorem 9.8, p. 58, used via the enhanced supplier
- Asakura–Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, 2019 · Definition 5.1(1),(3), printed pp. 268–269; Steps 1–2 and Proposition 5.3, p. 270 (earlier construction record: Steps 1–3, pp. 269–271)
- Jacob Lurie, Higher Topos Theory, author-hosted PDF · Proposition 6.2.2.7, printed p. 579, proof pp. 582–583; hypercompletion paragraph after Proposition 6.5.2.8, p. 669; Proposition 6.5.2.14 and proof, pp. 670–671; Remark 6.5.2.2, p. 666
- The actual big cdh slice over a field
- Dugger, Hollander and Isaksen, Hypercovers and simplicial presheaves · Section 3, generalized-cover paragraph (p. 7); Definitions 4.2 and 4.3 (p. 8), Definition 4.8 and Lemma 4.9 (p. 10); Definition 6.1 and Theorem 6.2 (p. 16); Proposition 6.6 and Corollary 7.1 with its following observation (pp. 18–19).
- Jacob Lurie, Higher Topos Theory · Proposition 5.1.1.1 and Section 5.1.2, opening paragraph (1) (pp. 316–317); Proposition 5.2.6.2 (p. 360); Corollary 5.1.2.3 (p. 319), also applied to opposite categories; Definition 5.5.0.1 (p. 456); Theorem 5.5.1.1(6) (p. 458); Corollaries 5.5.2.4 and 5.5.2.9 (pp. 464–466); Proposition 5.5.3.6 (p. 467); Proposition 5.5.4.15 (p. 482).
- External EM multiplicative motivic complexes and actual Chern map
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes (arXiv v1) · Theorem5.24(2), p.43, and its proof pp.43–44; preceding explicit component maps and line convention, p.43; Definition5.2 and negative-weight convention following(5.2), p.34
- Stable spectra and exact triangles
- Filtered categories and filtered colimits
- Filtered colimits commute with finite limits in Set
- Filtered colimits of spaces and based loops
- R. W. Thomason and Thomas Trobaugh, Higher Algebraic K-Theory of Schemes and of Derived Categories (1990) · Definition 3.1, printed pp. 312–313; Lemma 3.8 and Proposition 3.10 with proofs, p. 316; Proposition 3.20.1–3.20.2 and strict-perfect/affine proof, pp. 323–325; Appendix C.1, p. 418.
- Philip S. Hirschhorn, Notes on Homotopy Colimits and Homotopy Limits, draft of November 23, 2014 · Section 14.2, Theorem 14.10, Proposition 14.11 and Corollary 14.12, printed p. 72 (zero-based PDF page 71).
- Jacob Lurie, Higher Topos Theory, author-hosted PDF · Proposition 5.1.2.2 and Corollary 5.1.2.3 with proofs, printed pp. 318–319 (zero-based PDF pages 335–336); Definition 5.3.1.7, p. 382; Example 5.3.1.8, p. 383; Proposition 5.3.1.18 and proof, pp. 386–387 (PDF pages 403–404); Proposition 4.1.1.8, p. 226, forward implications of proof on p. 226 (PDF page 243).
- Jacob Lurie, Higher Algebra, September 18, 2017 · Definition 1.4.2.8, printed p. 142; Notation 1.4.2.20, printed p. 146 (zero-based PDF pages 141 and 145).
- A chosen root trivializes the actual Tate coefficient on the slice
- The Stacks Project, big and small étale sites · Definitions 34.4.6 and 34.4.8; Lemmas 34.4.7, 34.4.13 and 34.4.16–18; Lemma 34.4.20, statement and forward construction through the composition identity.
- The Stacks Project, Lemma 59.99.1 · Lemma 59.99.1(2)(a), with (1)(a) and both proofs.
- The Stacks Project, Cohomology of unbounded complexes · Section 21.19, opening paragraph, additive-functor clause (2), computation formula and object-sections example (2), before Lemma 21.19.1.
- The Stacks Project, Lemma 21.20.1 · Lemma 21.20.1, full statement and one-line proof.
- The Stacks Project, Inverse image on small étale sites · Definition 59.36.1 and Lemma 59.36.2(1), with the proof of (1)–(2).
- The Stacks Project, K-injective complexes · Definition 13.31.1, Lemma 13.31.2 and the proof of Lemma 13.31.7.
- Isomorphism classes of vector bundles
- Equality in affine K zero and stable isomorphism
- Free stabilization passes through global sections
- Noetherian commutative rings and modules
- Affine schemes and their coordinate rings
- Locally Noetherian and Noetherian schemes
- Affine schemes are quasi-separated
- Free stabilization passes through global sections
- Sections, restrictions, and global sections of a presheaf
- Source-specific non-A¹ motivic spectrum framework and hypotheses
- Algebraic stacks and their inertia stacks
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated 14 April 2025 · §2.2.1–2.2.6, printed pp. 14–15; §5.1.1–5.1.4, printed pp. 33–34, particularly the mapping-space equivalences in the proof of Theorem 5.1.4; Lemmas 3.3.4–3.3.5, pp. 24–25
- Elmanto–Hoyois–Khan–Sosnilo–Yakerson, Modules over algebraic cobordism, author-hosted PDF · Appendix A, Proposition A.0.4 and its proof, Lemma A.0.5 and its proof, Example A.0.6(1), printed pp. 33–35
- The Stacks Project, the algebraic stack associated to a smooth groupoid · Section 94.17, Lemmas 94.17.1–94.17.2 and Theorem 94.17.3, statements and displayed proofs
- Elmanto–Morrow filtration input
- Motivic filtration and image groups
- Valuative dimension and degreewise bounded towers
- Spectral sequence of a tower
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §2.2 pp.11–12; Theorem4.12(1) p.22 and Theorem4.27(1) p.29; Theorem1.1(1) p.3
- Elmanto–Morrow motivic complex input
- A coherent spectral realization of derived abelian complexes
- Truncation and closure of the native derived smooth Zariski site
- External actual augmented EM and smooth slice tower comparison
- Jacob Lurie, Spectral Algebraic Geometry, February 2018 author manuscript · Definition 1.1.2.8 and Remark 1.1.2.10, p. 72; Definition 11.2.3.1, p. 978; Proposition 11.2.3.6, p. 979; Definition 11.2.5.5, p. 987
- Jacob Lurie, Higher Algebra, 18 September 2017 · Corollary 4.2.4.7, p. 531; Proposition 7.1.3.18 and proof, p. 1219; Definition 7.2.2.10 and Remark 7.2.2.12, p. 1240
- Can Yaylali, Notes on derived algebraic geometry, arXiv:2208.01506v1 · Remark 2.29, p. 16; Definition 2.53, p. 23; Definition 3.33, p. 32; Remark and Definition 3.35, pp. 32–33, properties (1)–(4); Definition 3.43, p. 36; Remarks 3.53–3.55, p. 39
- Jacob Lurie, Higher Topos Theory, 2009 · Definition 6.2.2.6 and Proposition 6.2.2.7, p. 579; proof pp. 582–583
- Sabin Cautis and Harold Williams, Tamely presented morphisms and coherent pullback, Mathematische Zeitschrift 311, article 88 (2025) · Section 2, conventions for CAlg and the Noetherian base; Section 4.4, paragraph immediately before Proposition 4.18
- The general linear group scheme and its coordinate ring
- Projective bundle represents line quotients
- Glue relative spectra of affine-local algebras
- Flat and faithfully flat modules and ring homomorphisms
- Fppf coverings and the fppf site
- Descent data, prestacks and stacks in groupoids over the fppf site
- Stacks Project, effective fpqc descent of quasi-coherent sheaves · Proposition 35.5.2 (tag 023T), full statement and displayed proof
- Stacks Project, finite locally free modules descend fpqc locally · Lemma 35.7.6 (tag 05B2), full statement and one-paragraph proof
- Stacks Project, fpqc covering families and the fppf-to-fpqc comparison · Section 34.9, Definition 34.9.1 (022B), Lemma 34.9.7 (022C), and Lemma 34.9.9
- The ordinary Chow Chern map on isomorphism classes
- Compatible local sheaves glue uniquely up to unique isomorphism
- Rational equivalence and the Chow group of cycles
- Chern classes of a vector bundle on a smooth scheme
- Flat pullback of cycles and of rational equivalence
- Chain dimension and the empty-space convention
- Irreducible components of a topological space
- Irreducible topological spaces and irreducible subsets in the subspace topology
- Algebraic cycles and the cycle group of a scheme of finite type over a field
- Restriction of a sheaf to an open subspace
- Integral schemes
- A gluing datum for sheaves on an open cover
- A sheaf on a topological space
- The intersection product and Chow ring of a smooth scheme
- Rank-two Chern map and its fixed-invariant fibers
- Injection, surjection, bijection
- External symmetric P¹ model, orientation criterion and fundamental stabilization
- Annala–Iwasa, Motivic spectra and universality of K-theory (14 April 2025) · §3.3.2–3.3.4, printed p. 24, especially the localization-generators description in the proof of Lemma 3.3.4
- The actual smooth open pair and its choice scope
- Article scheme conventions
- The smooth locus of a morphism
- Motivic complexes and cohomology
- Jacob Lurie, Higher Topos Theory (Princeton University Press, 2009) · Corollaries 5.1.2.3–5.1.2.4, printed p. 319 (PDF index 336); Section 5.1.5 opening and Lemmas 5.1.5.2–5.1.5.3, printed pp. 324–326 (PDF indices 341–343)
- Field
- Affine fibre products are spectra of tensor products
- Jacob Lurie, Higher Topos Theory, author-hosted PDF · Theorem 4.1.3.1, p. 236; Definition 4.3.2.2, pp. 271–272, and Lemma 4.3.2.13, pp. 282–283; Definition 5.5.8.1, p. 506, Proposition 5.5.8.6, p. 507, Lemma 5.5.8.11 and Remark 5.5.8.12, p. 508; Proposition 6.2.2.7, statement p. 579 and proof pp. 582–583
- f : A → B is a bijection if and only if there is a function g : B → A with g ∘ f = Δ_A and f ∘ g = Δ_B; such a g is unique, equals the inverse relation f⁻¹, and is itself a bijection
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 arbitrary-field open-natural classical cycle interface
Theorem
Fixed-site classical cycle comparison on every smooth ambient
Lemma
A compatible presheaf-complex model of étale derived sections
Lemma
Reduced free-chain adjunction for local derived Eilenberg targets
Lemma
Coherent étale Tate sections and root trivialization on a slice
Lemma
Canonical comparison and restriction from the prime field
Lemma
Cohomology of shifted Nisnevich units
Lemma
Full affine K-space comparison and its sheafification
Lemma
Hypercovers and derived hyperdescent on the big cdh slice
Theorem
External EM projective bundle equivalence with its specified components
Lemma
Filtered affine K-space continuity with the actual free-class arrow
Lemma
Coherent derived étale sections for the constant and Tate coefficients
Lemma
Spectral K-zero equality gives an actual projective stabilization
Lemma
Spectral K-zero equality gives an actual projective stabilization
Theorem
K-stack mapping spaces and extension from smooth affine schemes
Theorem
External actual motivic stagewise boundedness
Lemma
Coherent extension of Chern-target operations from smooth affines
Lemma
Classical realization of the native smooth field site
Lemma
Adapted frames identify the universal quotient-projective stack
Lemma
Bundles and Chow Chern maps on a finite clopen partition
Lemma
Fixed determinant isolates the actual Chern fiber
Lemma
Binary detection by the ordinary rank-zero K-stack comparison
Lemma
Covering-sieve descent for the actual two-open square
Lemma
Two-variable affine extension before hypercompletion
Lemma