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Supporting prerequisites · 05 — External actual EM graded algebra, Picard class and complex PBF
Presents 26 supporting prerequisites, covering dependency levels 17–31.
1 · Prerequisites
- Motivic complexes and cohomology
- Motivic projective bundle formula
- The Axiom of Choice
- Grothendieck universe closure convention
- External EM multiplicative motivic complexes and actual Chern map
- External EM projective bundle equivalence with its specified components
- Quotient and line projectivizations agree after dualizing
- Nisnevich covers and distinguished squares
- Derived global sections on a scheme site
- External classification of torsors by first cohomology
- Units and Picard groups in Nisnevich degree zero and one
- Frames and their transition units
- External inputs for flat descent of line bundles
- Dold-Kan equivalence for simplicial modules with explicit inverse
- External stable enhancement and smash product of spectra
- Quasi-compact and quasi-separated schemes
- Smooth morphism of schemes
- Invertible sheaves
- Locally free sheaves of finite rank
- Pullback of a module along a morphism of ringed spaces
- Projective bundle in the quotient convention
- A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes (arXiv v1) · Lemma 5.1 and its proof, Definition 5.2, equation (5.2) and Theorem 5.3, printed pp. 33–34; component maps, line convention, Theorem 5.24(1)–(2) and its proof, printed pp. 43–44.
- Stefan Schwede and Brooke Shipley, Stable model categories are categories of modules · Appendix B.1, construction of W and HC immediately before (B.1.9), equation (B.1.9), Remark B.1.10 and Theorem B.1.11, printed pp. 38–40.
- A unit-induced integral field coefficient zigzag
- The normalized class detects the actual Picard-space map on smooth schemes
- Exact primary unit, Picard and smooth-purity inputs for field compatibility
- External actual EM graded algebra, Picard class and complex PBF
- Smooth motivic comparison
- External actual augmented EM and smooth slice tower comparison
- Shifted Nisnevich units and the line-bundle groupoid
- The heart of the canonical t structure is equivalent to the original abelian category
- The prime subfield as the intersection of all subfields
- Perfect fields: every irreducible polynomial is separable
- Exact linear hyperdescent and derived-localization input
- Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes (primary arXiv HTML) · Corollary 6.26 (weight-one clause), statement and proof; Construction 4.35/(4.36); Remark 4.37; Theorem 4.39(4) and Remark 4.45
- External symmetric P¹ model, orientation criterion and fundamental stabilization
- Source-specific non-A¹ motivic spectrum framework and hypotheses
- The native smooth site over an ordinary base and transport of coefficients
- Bloch cycle complexes and higher Chow groups
- Cycle complexes with fixed support
- Motivic and filtered K theory with support
- Support fibers with explicit cohomological indexing
- Stable spectra and exact triangles
- A coherent spectral realization of derived abelian complexes
- Finite-type field schemes lie in the qcqs source domain
- Schemes and morphisms over a base
- Locally finite type and finite type morphisms
- Separated morphism of schemes
- Étale morphism of schemes
- Open immersions of schemes
- Closed points of an affine scheme
- The residue field at a point of an affine scheme
- Fibre product of schemes
- Existence of all scheme fibre products
- Closed immersions of schemes
- Finite type is affine-local on source and target
- Quasi-compact and quasi-separated morphisms
- Open and closed immersions are separated
- Separated morphisms compose
- Normalized simplicial chains, prism homotopies and the abelian trivial-fibration criterion
- Model categories and Quillen adjunctions
- External stable mapping and derived linear target interfaces
- Markus Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv:1207.4078v3 · §5.1 Theorem 5.8, Lemmas 5.11–5.16, pp.37–41; §5.2.1 Lemmas 5.17–5.18/Corollaries 5.19–5.20, pp.42–43; §5.2.2 Lemma 5.21/Corollary 5.22 and Lemma 5.23/Corollary 5.24, pp.43–45; §5.3 Proposition 5.29 and §6 Proposition 6.1, (15), (18), Lemmas 6.2–6.6/Theorem 6.7, pp.50–56; retained absolute-field inputs: Lemma 7.5/Theorem 7.6 p.58, §7.5 Theorem 7.18 pp.60–64, Definition 8.1/Theorem 8.2 pp.64–65, Corollary 9.13 p.76, Theorem 9.16/Lemma 9.17 p.77, Proposition 11.1/Remark 11.2 p.79.
- Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture Notes on Motivic Cohomology (2006) · Proposition 13.10/Example 13.11 pp.101–102; Proposition 14.8/Corollary 14.9 p.112; §16 pp.125–126, Theorem 16.7 pp.126–127; Properties 17.4 p.136, Example 17.12 pp.138–139; Lemma 19.4 p.159, Example 19.5/Proposition 19.6 p.160, Theorem 19.8/Corollary 19.9 p.161, Proposition 19.12 p.162, proof of Theorem 19.1 and Examples 19.13–19.14 p.163 (cached PDF page = printed page +12).
- Mark Hovey, Spectra and symmetric spectra in general model categories, arXiv:math/0004051v3 · Definition 1.2 p.5; Theorems 3.4/3.8 pp.13–15; Definition 6.3 p.30; §7 setup p.30, Theorems 7.8/7.10 pp.32–33
- Denis-Charles Cisinski and Frédéric Déglise, Triangulated categories of mixed motives, arXiv:0912.2110v4 (2019) · Definition 1.4.6 p.29; Example 5.1.10 pp.170–171; §5.2.25 p.194; §§5.3.20–5.3.24 pp.207–209; Remark 5.3.27 p.210; §5.3.35 and (5.3.35.1) p.213 (PDF indices: printed page +39).
- Frédéric Déglise, Motivic homotopy theory and stable homotopy groups, Séminaire Bourbaki no.1241 (June 2025) · §3.1 p.34; Definition 3.1 p.36; §3.2.3 p.39, including footnote 52 (PDF indices 33–35 and 38).
- Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes, primary arXiv HTML at 2508.09915v1 · §3.1 symmetric monoidal stable setup; §3.3 opening effective-subcategory definitions, (3.36), properties (1)–(4), and Remark 3.39 (3.40)
- Satoshi Kondo and Seidai Yasuda, Product structures in motivic cohomology and higher Chow groups (author manuscript; journal version JPAA 215 (2011), 511–522) · Manuscript p.1 perfect-field convention; §3, (3.1), Theorem 3.1 p.8; §3.1, (3.3)/(3.5) p.9 and Proposition 3.3 p.10.
- Eric M. Friedlander and Andrei Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), 773–875 · §12, definition of the Suslin cycle complex and Proposition 12.1 with its proof, printed pp.829–830 (PDF pages 57–58); Corollary 12.2 p.830 for its stated quasiprojective scope.
- Elmanto and Morrow, Motivic cohomology of equicharacteristic schemes, arXiv 2309.08463v1 · Theorem 5.3, map (5.2) and preceding negative-weight convention, printed p. 34; Theorem 5.24(1), map (5.15) and proof, pp. 43–44
- Annala and Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · Adjunction 1.3.12 and Lemma 1.3.13, pp. 9–10; §§2.1–2.2.10, pp. 14–15, especially Notations 2.2.7–2.2.9
- Jacob Lurie, Higher Topos Theory, author-hosted PDF · Proposition 5.1.2.2 and Corollary 5.1.2.3, pp. 318–319 (and dual limits); Lemma 5.1.5.2, pp. 324–325; Definition 6.2.2.6 and Proposition 6.2.2.7, p. 579
- Compactness of the smooth Zariski stack coefficients
- Elmanto–Morrow motivic complex input
- Classical realization of the native smooth field site
- Jacob Lurie, Higher Topos Theory (2009) · Definition 6.2.2.6 and Proposition 6.2.2.7, printed p. 579; proof of Proposition 6.2.2.7, pp. 582–583 (PDF pages 597, 600–601)
- Jacob Lurie, Higher Algebra (18 September 2017) · Theorem 7.1.2.13 and proof, printed pp. 1212–1213; Corollary 4.2.3.3(1)–(2), p. 527
- Universal Chern cohomology and countable derived limits
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF14April2025 · Conventions 2.1–2.2 (pp. 14–15), Remark 2.4.6 and Corollary 2.4.8 (p. 20), §§3.1.3–3.1.7 (pp. 21–22), Lemma 3.2.3 (p. 23), §4.3.1 (p. 29), Definition 4.4.1 through Corollary 4.4.7 (pp. 30–32)
- External formal universal Chern maps and supported construction
- Homotopy exactness of a spectral fiber sequence
- 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
- An actual non-A¹ P¹-spectrum for the supplied EM complexes
- Actual EM orientation and projective bundle maps on that model
- The source Chern classes are the supplied projective-relation classes
- External non-A¹ projective-relation filtered and Whitney theorem
- Motivic Chern classes
- Field
- Exact sequences of sheaves
- Kernel sheaves are objectwise, while cokernels and images are sheafified
- Derived schemes and the cotangent complex of a morphism
- External input on derived enhancements
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · Section 2.1.1–2.1.5, printed page 14; Definition 4.4.1 and Lemma 4.4.3(ii)–(iii), printed pages 30–31
- Full projective Quillen–Waldhausen comparison and the local free-class map
- Full affine K-space comparison and its sheafification
- A local ring is a nonzero commutative ring with a unique maximal ideal
- Charles Weibel, The K-book, Chapter IV · IV.1.1 and IV.1.1.1–1.1.2, printed pp. 2–3; Definitions IV.1.4–1.4.1, Theorem IV.1.5 and Lemma IV.1.6, pp. 4–5; Construction IV.1.9(ii), p. 6; Theorem IV.4.9 and the explicit maps etaₙ(g)=(g,1), pp. 42–43; Corollary IV.4.11.1, p. 45; Theorem IV.7.1, p. 61, and Corollary IV.7.2, p. 62.
- Chern coefficients of the projective relation
- Scheme sites and sheafification
- Nonconnective K theory of perfect complexes
- Simplicial sets, homotopies and trivial Kan fibrations
- The tensor product of a right and a left chain complex is totalized by direct sums with the Koszul differential
- The Kunneth theorem for free complexes over a PID
- Locally Noetherian and Noetherian schemes
- Chain dimension and the empty-space convention
- Every subgroup of (ℤ, +) is ⟨ n ⟩ = nℤ for exactly one natural number n
- The integers have no zero divisors; multiplicative cancellation
- The spectrum of a Noetherian ring is a Noetherian topological space
- Noetherian open subsets are quasi-compact
- Filtered affine K-space continuity with the actual free-class arrow
- J. F. Jardine, Lectures on Homotopy Theory, Lecture 09 · Section 26, Theorem 26.1 and its proof opening, printed pp. 23–24 (PDF pages 23–24); Moore-complex Alexander–Whitney map (7), printed p. 26 (PDF page 26)
- The Stacks Project, Sites and Sheaves, Points · Definitions 7.32.1–7.32.2 and stalk equation (7.32.1.1)
- The Stacks Project, Modules on Sites, Stalks of modules · Lemmas 18.36.1(1)–(3) and 18.36.2(1)–(3), with their displayed proofs and intervening abelian-group construction
- R. W. Thomason and Thomas Trobaugh, Higher Algebraic K-Theory of Schemes and of Derived Categories (1990) · Appendix E, Lemma E.5(a)–(c) and proof, printed pp. 427–428 (PDF pages 181–182).
- External connective, nonconnective and automorphism comparisons
- Affine perfect K zero and bundle K zero
- Bounded finite-projective complexes and affine chain models
- Enhanced affine perfect equivalence
- Filtered colimits of spaces and based loops
- Normalized free chains preserve products of homology equivalences
- Derived free sheaf chains and unbounded Eilenberg targets
- Schemes
- The stalk of a presheaf at a point
- Two representatives in a filtered colimit of sets are equal exactly when they become equal at one common later stage
- Local comparison of coherent coefficient presheaves with derived sheaf Eilenberg targets
- Positive supported coordinates at a smooth threefold point
- Finite type algebras over fields have bounded dimension
- A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point
- Dimension can be computed on an open cover
- Regularity and dimensions for finite-type cycle arguments
- The actual smooth restriction arrow gives the point-supported cycle fiber
- Affine schemes and affine morphisms are separated
- Alberto Navarro, On the higher Riemann–Roch without denominators (arXiv:1901.09337v1) · Theorem 2.3(1),(3),(4), printed p.5; Definition 2.4 and §2.6, printed p.6; Theorem 2.7, trivial-bundle opening, printed p.7
- Markus Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains (arXiv:1207.4078v3) · §6, Proposition 6.1 and cycle pairing (15), printed pp.51–52; Lemmas 6.2–6.3, pp.52–53; Proposition 6.4, Lemmas 6.5–6.6, Theorem 6.7, pp.53–56 (PDF pages 50–55)
- Fabien Morel and Vladimir Voevodsky, A¹-homotopy theory of schemes, Publ. Math. IHES 90 (1999) · §3.1, printed pp.94–96; §3.2 Theorem 2.23/Proposition 2.24, printed p.115; Lemmas 2.27–2.28 and final Čech argument, pp.117–118 (PDF pages 50–52,71,73–74)
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Theorem 1.1(9), printed p.4; §2.2, pp.11–12; §3.1, (3.1)–(3.2), pp.12–13; Lemma 5.1, Definition 5.2 and Theorem 5.3, pp.33–34
- Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture Notes on Motivic Cohomology (2006) · Theorem 4.1/Corollary 4.2, pp.25–27; Properties 17.4(2), p.136; Example 17.12, pp.138–139; Theorem 19.1, p.159; Proposition 19.6, Theorem 19.8/Corollary 19.9 and Proposition 19.12, pp.160–162; Example 19.14/Proposition 19.16, pp.163–164
- Reduced free-chain adjunction for local derived Eilenberg targets
- Left and right flat modules over an arbitrary ring
- Bounded above flat tensor complexes preserve quasi isomorphisms
- Charles Weibel, The K-book, author-hosted Chapter IV · Printed pp. 3–6; Definitions 1.1.1, 1.4 and 1.4.1; Lemma 1.6; Construction 1.9(ii)
- Friedlander–Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology (2002) · Appendix B, printed pp. 860–861 (PDF pages 88–89), normalization paragraph, shuffle formula and normalized-quotient clause before Proposition B.2
- The Stacks Project, Stalks of modules · Section 18.36, Lemmas 18.36.1–18.36.2 and proofs, and intervening construction of abelian stalks
- The Stacks Project, Exactness at a conservative family of points · Lemma 18.14.4, statement and proof
- R. W. Thomason and Thomas Trobaugh, Higher Algebraic K-Theory of Schemes and of Derived Categories (1990) · Appendix E, E.1 and Lemma E.5(a)–(c) with proof, printed pp. 427–428; PDF pages 181–182
- Coefficientwise perfect Cartan on every qcqs field-scheme
- Extending a Whitney class to K zero
- Perfect residue classes at smooth closed points
- Every algebraic extension of a perfect field is separable
- Perfect K theory localization and open excision
- fields as regular local rings
- Joël Riou, Algebraic K-theory, A1-homotopy and Riemann–Roch theorems, arXiv:0907.2710v2 · Conventions §1.1, p.3; Theorem 1.1.6 and Definition 1.2.2, p.5; Lemma 1.2.6 and Remark 1.2.7, pp.6–7; Propositions 3.1.2, 3.1.3, 3.1.5 and Corollary 3.1.4 with their proofs, pp.11–13; Definition 6.2.1.1 and Theorem 6.2.1.2 with proof, p.31. Opening of §6.2.3, p.33 (HZ levels and projective-line loop equivalences).
- Alberto Navarro, On the higher Riemann–Roch without denominators, arXiv:1901.09337v1 · §1 conventions, p.2; oriented Chern properties and Example 1.1, p.3; §§2.1–2.2, p.4; §3.5, p.9.
- Markus Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv:1207.4078v3 · §6, equation (18) and Lemmas 6.5–6.6, pp.54–56; Theorem 6.7, p.56; Theorem 7.6, p.58; Theorem 9.16 and Lemma 9.17, p.77.
- Regular Sequence On A Module
- Projective bundle represents line quotients
- Dual and base change for finite locally free sheaves
- Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv:1207.4078v3 · Lemma 6.6/Theorem 6.7, pp.55–56; Theorem 9.16/Lemma 9.17, p.77; Proposition 11.1/Remark 11.2, p.79
- Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, arXiv:2508.09915v1 · Lemma 4.33 and Construction 4.35/(4.36); Theorem 4.39(1),(4); Theorem 7.5/Remarks 7.6,7.9–7.10; Theorem 7.12(1)–(2)/(7.14); Corollaries 1.11 and 9.7/(9.8)
- Actual perfect Cartan through degree three on all qcqs field-schemes
- Rank and determinant on the second piece
- Motivic filtration and image groups
- First-class matching and explicit perfect Chern arithmetic
- First Chern class factors through determinant
- An algebraically closed field: every nonconstant polynomial has a root in the field
- The actual determinant fiber is a natural retract of the rank-zero K-space
- Flag towers with line-bundle quotients
- Detection on a simultaneous flag tower
- Filtration projections and multiplication
- Whitney identities and virtual Chern arithmetic
- The actual perfect K-space operations satisfy total Cartan
- Reduced bundle class identities
- The actual first EM stage is the rank-zero K space on all qcqs schemes
- Perfect complexes on schemes and their support
- Nonconnective K theory in degree zero
- Hilbert basis theorem: if R is Noetherian then R[x] is Noetherian
- Dimension and purity of an affine polynomial cylinder
- The Stacks Project, Noetherian schemes · Definition 28.5.1; Lemmas 28.5.2–28.5.5, Section 28.5 (Tag 01OU)
- The Stacks Project, Grassmannians · Section 27.22, Lemma 27.22.1 and its proof; Definition 27.22.2
- Actual second-stage comparison on every qcqs field-scheme
- Graded filtration projections
- Stage projections, kernels and image groups
- Second Chern additivity on the second piece
- Integral closure in an extension ring and integrally closed domains
- Affine schemes and their coordinate rings
- A smooth point has the exact finite-type excision pair
- Subalgebra generated by a subset, algebras of finite type, and module-finite algebras
- Finite-variable polynomial algebras over fields are Noetherian by finite generators
- Every quotient and every localisation of a Noetherian ring is Noetherian
- External finite-type dimension, rational-point and etale-fiber inputs
- Alberto Navarro, On the higher Riemann–Roch without denominators, arXiv:1901.09337v1 · §1 pp.2–3, especially Example 1.1; §§2.1–2.2 p.4; Theorem 2.3 p.5, Definition 2.4 and §2.6 p.6, Theorem 2.7 p.7; Theorem 3.1 and Lemmas 3.2–3.3 pp.7–8; §3.5 p.9
- Masanori Asakura and Kanetomo Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, J. Math. Sci. Univ. Tokyo 26 (2019), 249–334 · §5, (5.3.1) and Definition 5.4(2), pp.270–271; opening of §6 and (6.0.1), p.272; §8 opening and Definition 8.3, pp.282–284; opening of §9 before Theorem 9.1, p.285
- The Stacks Project, Regular ideal sheaves and regular immersions · Definition 31.21.2 (063D); Lemmas 31.21.3–31.21.4 (063E,063H); Definition 31.22.1 (063J); Lemmas 31.22.2 and 31.22.5 (063K,063M)
- External actual graded algebra and multiplicative tower spectral sequence
- The local simplicial model retains actual sections on a finite-dimensional noetherian Nisnevich site
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated 14 April 2025 · Section 2.1 and Notations 2.2.7–2.2.9 / coefficient adjunction 2.2.10, pp. 14–15; Definition 4.4.1 through Corollary 4.4.7, pp. 30–32
- Asakura–Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, J. Math. Sci. Univ. Tokyo 26 (2019) · Definition 5.1, Section 5 Steps 1–3, Propositions 5.2–5.3, formula (5.3.1), and naturality square (5.4.1), pp. 269–272
- Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 (15 September 2023 header; 18 September 2023 internal date) · Theorem 1.1 opening and part (1), printed p. 3
- Jacob Lurie, Higher Topos Theory, author-hosted PDF · Lemma 5.1.5.2, printed pp. 324–325; Definition 6.2.2.6 and Proposition 6.2.2.7, p. 579, proof pp. 582–583
- An integral P³ test fixes the supported RR coefficient as positive two
- Actual supported Chern operation and its smooth comparison
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
External actual EM graded algebra, Picard class and complex PBF
Lemma
The actual EM comparison commutes with the Picard first-Chern map
Lemma
An actual non-A¹ P¹-spectrum for the supplied EM complexes
Lemma
A specified étale-natural comparison with supported cycles
Lemma
Negative coefficients of the actual EM projective-line model
Lemma
Actual EM orientation and projective bundle maps on that model
Lemma
Actual EM coefficient mapping and smooth-site comparison
Theorem
External actual universal-class detection on Vect and Grass
Lemma
Finite-rank detection of stable pointed motivic classes
Lemma
Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes
Theorem
External Quillen acyclicity and the actual local free-class model
Lemma
Projective relations transport the actual universal Chern operations
Lemma
The stable free-class comparison is locally integrally acyclic
Lemma
Local free chains detect natural perfect operations on every qcqs base
Lemma
The field comparison preserves refined smooth Gysin and the point class
Lemma
Local integral chains detect global natural maps into Chern targets
Lemma
Free-class detection on qcqs perfect K-space models
Lemma
Independent Navarro and EM Chern operations on the smooth separated site
Theorem
Integral oriented field coefficient comparison with actual Picard and support maps
Lemma
Apply the actual perfect Cartan identity to the second image
Lemma
The reduced universal bundle calculation identifies the second projection
Lemma
The perfect image filtration has the stated second projection and kernel
Theorem
External actual supported Riemann–Roch in codimension three
Lemma
The second-stage projection is minus the actual second Chern map on its homotopy fiber
Lemma