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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 homotopy groups of a sequential prespectrum
- Singular complexes preserve the sequential group convention
- Omega replacement preserves sequential stable groups
- Omega towers and the suspension index
- Stable rotation and exact coherent products
- External stable enhancement and smash product of spectra
- External input on derived enhancements
- External affine derived and perfect comparisons
- Local perfection implies quasi-coherent cohomology
- Bounded finite-projective complexes and affine chain models
- Perfect complexes over a ring and its graded version
- The Axiom of Choice
- External local perfection and summand closure
- Restriction, stalks and cohomological support
- Derived category of an abelian category
- Cohomology maps induced by cochain maps
- Homotopic maps act equally on cohomology
- Blumberg–Gepner–Tabuada, A universal characterization of higher algebraic K-theory · §2.2 (Catₚerf domain), §3.1 (tensor universal property); Definition 9.6 and Theorem 9.8, PDF pp. 58–59
- Blumberg–Gepner–Tabuada, Uniqueness of the multiplicative cyclotomic trace · Proposition 5.9 and Corollary 5.10, PDF p. 23; Theorem 1.13, PDF p. 5
- Derived finite coefficients and the Bockstein sequence
- Reindexing cochains as chains
- Cohomology of an integer shift
- Reindexing commutes with shifts
- Reading spectrum homotopy from an Omega tower
- Eilenberg–Mac Lane spectrum of a chain complex
- Schemes and morphisms over a base
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
Definition
Stable spectra and exact triangles
Lemma
Enhanced affine perfect equivalence
Lemma
Stable supported perfect subcategories and retracts
Lemma
Cohomology descends to the derived category
Lemma
Homotopy exactness of a spectral fiber sequence
Definition
Perfect complexes on schemes and their support
Definition
Decreasing spectral filtrations and associated graded
Lemma
Homotopy groups of integer spectral shifts
Theorem
External nonconnective lax symmetric monoidal K-theory
Lemma
Cofiber of a direct-summand inclusion
Lemma
Functorial finite coefficients for derived complexes
Lemma
Stage projections, kernels and image groups
Lemma
Complex shifts and spectral homotopy indices
Definition
Multiplicative filtrations and graded products
Definition