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Supporting prerequisites · 03 — Spectral K-zero equality gives an actual projective stabilization
Presents 29 supporting prerequisites, covering dependency levels 12–26.
1 · Prerequisites
- Isomorphism classes of vector bundles
- Equality in affine K zero and stable isomorphism
- Free stabilization passes through global sections
- The Axiom of Choice
- Free stabilization passes through global sections
- Elmanto–Morrow filtration input
- Motivic filtration and image groups
- Valuative dimension and degreewise bounded towers
- 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
- Rank-two Chern map and its fixed-invariant fibers
- Derived global sections on a scheme site
- Units and Picard groups in Nisnevich degree zero and one
- Motivic complexes and cohomology
- Motivic projective bundle formula
- Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, arXiv v1 · §2.2 pp11–12; Definitions4.9/4.23; Lemma5.1/Definition5.2 pp33–34; actual map(5.2) and Theorem5.3; Theorem5.24 pp43–44
- Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, arXiv v1 · §2.2 pp11–12; Definitions4.9/4.23; Lemma5.1/Definition5.2 pp33–34; actual map(5.2) and Theorem5.3; Theorem5.24 pp43–44
- 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
- Smooth motivic comparison
- External actual augmented EM and smooth slice tower comparison
- External symmetric P¹ model, orientation criterion and fundamental stabilization
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF14April2025 · Lemma4.4.4, Corollary4.4.5 and Theorem4.4.6 with proof pp31–32; Corollary4.4.7 and proof p32; Definition4.4.1/Lemma4.4.3 pp30–31; Lemma3.2.3 p23
- Universal Chern cohomology and countable derived limits
- External formal universal Chern maps and supported construction
- Stable spectra and exact triangles
- 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
- External matrix, exact and bounded-projective K-theory comparisons
- External connective, nonconnective and automorphism comparisons
- Charles Weibel, The K-book, Chapter IV · Definition1.1.1/Functoriality1.1.2 p3; Definitions1.4/1.4.1, Theorem1.5, Lemma1.6 pp4–5; Constructions1.9(ii) p6; §4 group-completion and plus comparisons4.9/4.11.1 pp38–45; Theorem IV.7.1, printed p61 (PDF page61), for group-completion/Q; Corollary IV.7.2, printed p62 (PDF page62), for the resulting plus/Q comparison
- Chern coefficients of the projective relation
- Nonconnective K theory of perfect complexes
- Full projective Quillen–Waldhausen comparison and the local free-class map
- Affine perfect K zero and bundle K zero
- Bounded finite-projective complexes and affine chain models
- Enhanced affine perfect equivalence
- Positive supported coordinates at a smooth threefold point
- Scheme sites and sheafification
- Full affine K-space comparison and its sheafification
- Whitney identities and virtual Chern arithmetic
- Alberto Navarro, On the higher Riemann–Roch without denominators · Example1.1 p3; §§2.1–2.2 p4; Definition2.4 p6; Theorem3.1 and proof pp7–8
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
Spectral K-zero equality gives an actual projective stabilization
Lemma
Spectral K-zero equality gives an actual projective stabilization
Theorem
External actual motivic stagewise boundedness
Lemma
Fixed determinant isolates the actual Chern fiber
Lemma
Cohomology of shifted Nisnevich units
Lemma
Chern-compatible presentation transports the exact fibers
Theorem
External actual EM graded algebra, Picard class and complex PBF
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
Negative coefficients of the actual EM projective-line model
Lemma
Actual EM orientation and projective bundle maps on that model
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
Theorem
Integral oriented field coefficient comparison with actual Picard and support maps
Lemma
Coefficientwise perfect Cartan on every qcqs field-scheme
Theorem