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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
- Stable spectra and exact triangles
- Decreasing spectral filtrations and associated graded
- Homotopy exactness of a spectral fiber sequence
- External K-flat sheaf resolution and pullback input
- External localization of one marked scheme and tensor family
- External input on derived enhancements
- The Axiom of Choice
- Perfect complexes on schemes and their support
- K-flat roofs give the full derived localization
- Strong monoidal transport from a product-base cocartesian family
- The Stacks Project, Derived Categories of Schemes, Compact and perfect objects · Remark 36.17.2; Proposition 36.17.1; Theorem 36.15.3 (09IS), together with Proposition 13.37.6 (09SR)
- Triangle K0 of perfect complexes equals split K0 of finite projectives
- Affine bundle K zero is the projective split group
- Exact equivalences preserve triangle K zero
- Cofiber of a direct-summand inclusion
- R. W. Thomason and Thomas Trobaugh, Higher Algebraic K-Theory of Schemes and of Derived Categories · §5, hypotheses of Theorem 5.1 p. 337; Propositions 5.2.2–5.2.4 pp. 338–339; general-qcqs completion Lemma 5.6.2 and 5.6.3 pp. 348–350; Euler relations 1.5.6–1.5.7 p. 261. PDF pages 91–93, 102–104 and 15.
- The Stacks Project, Derived Categories · Lemma 13.6.6 and Definition 13.6.7 (05RI)
- Stage projections, kernels and image groups
- Complex shifts and spectral homotopy indices
- Exact couple
- Derived exact couple
- Cohomological spectral sequence
- Multiplicative filtrations and graded products
- Valuative dimension and degreewise bounded towers
- Parviz Sahandi, Semistar-Krull and Valuative Dimension of Integral Domains · arXiv:0809.1305v3, page 1, Section 1, introductory paragraph on classical valuative dimension
- Enhanced affine perfect equivalence
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
Support fibers commute with graded triangles
Lemma
One coherent K-flat family and its supplied perfect fibers
Theorem
External qcqs classical perfect generator
Lemma
Affine perfect triangle classes and bundle Euler classes
Lemma
Graded pieces of a direct-sum tail tower
Lemma
Images of filtration stages
Theorem
External extension and morphism input for perfect complexes
Proposition
Spectral sequence of a tower
Lemma
Products and projections on image filtrations
Lemma
Dimension decreases under quotients
Lemma
A valuation overring has a center
Theorem
Noetherian domains have equal Krull and valuative dimensions
Lemma
Essential smallness of perfect and supported categories
Lemma
Coherent derived tensor and pullback on perfect complexes
Lemma