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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
- Decreasing spectral filtrations and associated graded
- Graded pieces of a direct-sum tail tower
- Images of filtration stages
- Multiplicative filtrations and graded products
- Dimension decreases under quotients
- A Noetherian polynomial ring has dimension one larger
- Every algebra of finite type over a Noetherian ring is a Noetherian ring
- The Axiom of Choice
- Valuative dimension and degreewise bounded towers
- A valuation overring has a center
- Schemes and morphisms over a base
- Coherent derived tensor and pullback on perfect complexes
- Perfect complexes on schemes and their support
- Restriction, stalks and cohomological support
- Stable supported perfect subcategories and retracts
- One coherent K-flat family and its supplied perfect fibers
- Boundary maps vanish for a split tower
- Spectral sequence of a tower
- Article scheme conventions
- Nisnevich covers and distinguished squares
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Introduction, pp. 2–3, especially Theorem 1.1 and its opening statement; Section 4 for the construction
- Cohomology descends to the derived category
- Functorial finite coefficients for derived complexes
- Noetherian domains have equal Krull and valuative dimensions
- Essential smallness of perfect and supported categories
- External nonconnective lax symmetric monoidal K-theory
- Stable spectra and exact triangles
- Cofiber and triangle presentations of K zero
- Affine perfect triangle classes and bundle Euler classes
- Nonconnective K theory in degree zero
- Cofiber of a direct-summand inclusion
- Motivic cohomology of equicharacteristic schemes · September 2023 PDF v1, Theorems 4.12(1)–(3) and 4.27(1)–(3), printed pp. 22 and 29; Section 2.2, pp. 11–12; proof of Lemma 7.1, p. 49
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
Image filtration for a split tower
Lemma
Nesting of image filtration groups
Lemma
Naturality of image filtrations
Lemma
Products of filtration images
Lemma
Extending a nonnegative filtration to integer indices
Lemma
Prime quotients of noetherian rings
Lemma
Finite type algebras over fields have bounded dimension
Lemma
Valuative dimension can be computed on an affine cover
Lemma
Support ideals, mixed products and pullback
Lemma
Degeneration of the spectral sequence of a split tower
Lemma
Elmanto–Morrow motivic complex input
Definition
Motivic complexes and cohomology
Theorem
Finite valuative dimension for noetherian and finite type schemes
Lemma
Applying nonconnective K-theory to the enhanced interface
Definition
Nonconnective K theory of perfect complexes
Lemma
Affine perfect K zero and bundle K zero
Proposition