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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
- The Stacks Project, Remark 19.13.3 (079Q) · Remark 19.13.3, final paragraph
- Jacob Lurie, Higher Algebra, September 18, 2017 · Definition 1.3.5.8; Propositions 1.3.5.9, 1.3.5.13–15 and 1.3.5.21
- Cofinal tails and level invariance of stable groups
- External sequential simplicial Omega replacement
- External stable enhancement and smash product of spectra
- External input on derived enhancements
- Perfect complexes over a ring and its graded version
- Affine bundles and finite projective modules
- External affine derived and perfect comparisons
- Reindexing cochains as chains
- Derived category of an abelian category
- External full-subcategory localization criterion
- External K-flat sheaf resolution and pullback input
- External localization of one marked scheme and tensor family
- External straightening and symmetric monoidal criteria
- Additive equivalences preserve split K zero
- The Stacks Project, Sheaves on Spaces · Definition 6.31.5(5)–(6); Lemma 6.31.8(2)–(4); Lemma 6.31.12
- The Stacks Project, Cohomology of Sheaves · §20.32 opening paragraphs; Lemma 20.32.8
- Determinant on the stable general linear group
- Compatible local units glue uniquely
- Cohomology of a cochain complex
- External singular realization comparison
- Kerodon, simplicial homotopy groups · Construction 3.2.2.4, Notation 3.2.2.5, Example 3.2.2.7; Theorem 3.2.2.10
- Kerodon, exactness of realization · Corollary 3.6.2.2, full statement and displayed proof
- Sequential prespectra, spectra, and adjoint structure maps
- Stable homotopy groups of a sequential prespectrum
- External local perfection and summand closure
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
Derived enhancements of module sheaves on ringed spaces
Lemma
Omega replacement preserves sequential stable groups
Lemma
Omega towers and the suspension index
Lemma
Stable rotation and exact coherent products
Lemma
Restriction, stalks and cohomological support
Lemma
Bounded finite-projective complexes and affine chain models
Lemma
Reindexing commutes with shifts
Lemma
K-flat roofs give the full derived localization
Lemma
Strong monoidal transport from a product-base cocartesian family
Lemma
Affine bundle K zero is the projective split group
Theorem
External exact extension by zero for module sheaves
Lemma
Stable determinant commutes with ring maps
Lemma
Gluing compatible local determinant homomorphisms
Lemma
Cohomology maps induced by cochain maps
Lemma
Singular sphere classes and ordered suspension
Lemma
Singular complexes preserve the sequential group convention
Lemma
Local perfection implies quasi-coherent cohomology
Lemma
Reading spectrum homotopy from an Omega tower
Lemma