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Perfect K theory and motivic towers
Connect affine bundle classes to nonconnective perfect K theory, then develop tower exactness, products, coefficients and rational detection. Give the article's bundle-class, motivic-filtration and motivic-cohomology definitions their authoring homes after these suppliers.
1 · Prerequisites
- The Axiom of Choice
- Nonconnective K theory of perfect complexes
- Multiplicative filtrations and graded products
- Motivic cohomology of equicharacteristic schemes · Version 1 (15 September 2023; internal date 18 September 2023): Theorem 1.1(1) p.3; Definition 4.9 p.21; Definition 4.23 pp.27–28; initial assertions of Theorems 4.12 p.22 and 4.27 p.29; §2.2 pp.11–12
- Perfect complexes on schemes and their support
- External input on derived enhancements
- External exact extension by zero for module sheaves
- Stable spectra and exact triangles
- Andrew J. Blumberg, David Gepner and Gonçalo Tabuada, A universal characterization of higher algebraic K-theory · Definition 5.12, Proposition 5.13 and Proposition 5.14 pp. 35–36; Proposition 5.15 p. 36; Theorem 8.7 p. 55; Definition 9.6 and Theorem 9.8 p. 58; localizing invariant convention p. 4.
- Bounded finite-projective complexes and affine chain models
- Enhanced affine perfect equivalence
- Blumberg–Gepner–Tabuada, A universal characterization of higher algebraic K-theory (arXiv:1001.2282v4, 5 February 2013) · §7.1, Definition 7.1 and the one-skeleton map, pp. 44–45; Lemma 7.11/Corollary 7.12, pp. 48–49; Definition 9.6/Theorem 9.8, p. 58; Definition 9.30, p. 63; Proposition 9.32/Remark 9.33, p. 64
- Blumberg–Gepner–Tabuada, A universal characterization of higher algebraic K-theory (arXiv:1001.2282v4, 5 February 2013): Waldhausen presentations · §4.2, Definition 4.7/Propositions 4.8–4.10, pp. 26–27; Theorem 4.22, p. 30; Lemma 7.11, p. 48, and the unnumbered presentation paragraph immediately after the proof of Corollary 7.12, p. 49
- Article scheme conventions
- Locally free sheaves of finite rank
- Nonconnective K theory in degree zero
- Affine perfect K zero and bundle K zero
- Exterior powers and determinant of an algebraic bundle
- Coherent derived tensor and pullback on perfect complexes
- External matrix, exact and bounded-projective K-theory comparisons
- Determinant on the stable general linear group
- Stable determinant commutes with ring maps
- Compatible local units glue uniquely
- Gluing compatible local determinant homomorphisms
- A right inverse makes a homomorphism surjective
- Decreasing spectral filtrations and associated graded
- Extending a nonnegative filtration to integer indices
- Images of filtration stages
- Nesting of image filtration groups
- Naturality of image filtrations
- Products of filtration images
- External extension and morphism input for perfect complexes
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
Elmanto–Morrow filtration input
Lemma
The exact restriction kernel and its denominators
Lemma
Supported perfect complexes are invariant under an open neighbourhood
Theorem
External enhanced quotient criterion and localizing nonconnective K
Theorem
External connective, nonconnective and automorphism comparisons
Definition
Isomorphism classes of vector bundles
Lemma
Units split the K one determinant
Definition
Motivic filtration and image groups
Lemma
Full faithfulness of the perfect triangulated quotient
Lemma