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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
- Derived category of an abelian category
- Annihilators, torsion elements and the torsion subset of a module
- Exact sequences and short exact sequences of modules
- The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums
- Stable homotopy groups of a sequential prespectrum
- J. P. May, A Concise Course in Algebraic Topology · Chapter 16, §§2–4, printed pp. 124–127, especially the second theorem of §2 and the realization–singular adjunction in §4
- Mark Hovey, Spectra and symmetric spectra in general model categories · §4, Definition 4.2, Lemma 4.3, Proposition 4.4, Corollary 4.8 and Theorem 4.9, PDF pp. 17–20
- Jacob Lurie, Higher Algebra · Definitions 1.4.2.8 and 1.4.3.1; Corollary 1.4.2.17; Notation 1.4.2.20; Proposition 1.4.2.24 and Remark 1.4.2.25; Proposition 1.4.3.6; Proposition 1.4.4.4; Proposition 1.1.3.4; Corollary 4.8.2.19
- The Axiom of Choice
- The Stacks Project, Injectives, Remark 19.13.3 · Remark 19.13.3
- Jacob Lurie, Higher Algebra · Propositions 1.3.5.13–15 and 1.3.5.21; Lemma 1.2.4.6
- Jacob Lurie, Higher Topos Theory · Corollary 4.4.5.15, Proposition 4.4.5.17 and Remark 4.4.5.18
- Perfect complexes over a ring and its graded version
- The Stacks Project, Derived Categories of Schemes · Lemma 36.3.5
- The Stacks Project, Derived Categories of Schemes · Lemma 36.10.7
- The Stacks Project, Cohomology of Sheaves, Perfect complexes · Definition 20.49.1 and Lemmas 20.49.2, 20.49.3, 20.49.7 and 20.49.9
- Stable model categories are categories of modules · Appendix B.1, pp. 38–40, especially construction preceding equation (B.1.9) and Theorem B.1.11
- The Stacks Project, Cohomology of Sheaves, Flat resolutions · §20.26, Definition 20.26.2; Lemmas 20.26.3–6, 20.26.8–13 and 20.26.17; PDF chapter cohomology, pp. 55–59
- The Stacks Project, Cohomology of Sheaves, Derived pullback · §20.27, Lemmas 20.27.1–3 (tags 06YJ, 0D5S, 079U); §20.26 Lemma 20.26.8 (06YC)
- Blumberg–Gepner–Tabuada, Uniqueness of the multiplicative cyclotomic trace · §3, Proposition 3.5, PDF p. 13
- Jacob Lurie, Higher Algebra · Proposition 1.3.4.5 (statement p. 109, proof pp. 114–115); Proposition 1.3.2.10 and §1.3.5
- Brad Drew, Verdier quotients of stable quasi-categories are localizations · Theorem 1.3, PDF pp. 6–7; Definition 1.1
- Vladimir Hinich, Dwyer–Kan localization revisited · §1.1.2–1.1.3; Definition 2.1.1, Proposition 2.1.4 and complete proof §2.1.5–2.2.6; Definition 3.2.1 and Proposition 3.2.2; PDF pp. 4–5, 16–22
- Jacob Lurie, Higher Topos Theory · Remark 2.4.2.2 and its dual; Proposition 3.2.5.21; Hinich §2.1.5–2.1.6
- The Stacks Project, Localization of triangulated categories · Lemma 13.5.8, full statement and proof
- The Stacks Project, Derived categories · Lemma 13.11.2; Definition 13.11.3, tag 05RU
- The Stacks Project, Categories, Localization · Remark 4.27.15; right calculus of fractions preceding Lemma 4.27.11 (04VH), and Lemmas 4.27.13–14 (04VI,04VJ)
- The Stacks Project, Flat resolutions · Lemma 20.26.12; Lemma 20.26.6, tag 079S
- Jacob Lurie, Higher Topos Theory · Proposition 1.2.3.1, statement and displayed proof
- Vladimir Hinich, Dwyer–Kan localization revisited · §2.1.5, equivalence (31); Lemma 2.1.6; Proposition 2.1.4 and proof §2.2.2–2.2.6; Definition 3.2.1 and Proposition 3.2.2
- Jacob Lurie, Higher Algebra · Definition 2.0.0.7; Remark 2.1.2.19–20; Definition 2.1.3.7 and its preceding discussion
- Jacob Lurie, Higher Topos Theory · Notation 1.2.7.2 and Proposition 1.2.7.3; see Hinich §2.1.5 for straightening
- Algebraic K-theory of stable ∞-categories via binary complexes · Section 2.2, paragraph immediately preceding Lemma 2.12
- A universal characterization of higher algebraic K-theory · Definition 9.6 and Remark 9.33
- Charles Weibel, The K-book, Chapter III · Definition III.1.1 and Example III.1.1.1, PDF pp. 1–2
- Charles Weibel, The K-book, Chapter IV · Definition IV.1.1; Theorem IV.4.9 and explicit maps ηₙ(g)=(g,1), pp. 42–43; Theorem IV.4.11/Corollary IV.4.11.1, pp. 44–45; Theorem IV.7.1/Corollary IV.7.2, pp. 61–64; Exercises IV.7.8–7.9, printed and PDF p. 65; IV.8.3.2, IV.8.5 and Exercises IV.8.5–8.7, pp. 68,74–75
- Charles Weibel, The K-book, Chapter V · Theorem V.2.2 (Gillet–Waldhausen) and proof, pp. 13–14; §V.2.7.2, p. 18
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.
Items with no displayed connections
LemmaReindexing cochains as chainsLemmaCofiber and triangle presentations of K zeroLemmaExact equivalences preserve triangle K zeroLemmaAdditive equivalences preserve split K zeroLemmaDeterminant on the stable general linear groupLemmaCompatible local units glue uniquelyLemmaA right inverse makes a homomorphism surjective
3 · Items
Lemma
Derived finite coefficients and the Bockstein sequence
Lemma
Detecting injectivity by rationalization
Lemma
Cofinal tails and level invariance of stable groups
Theorem
External singular realization comparison
Theorem
External sequential simplicial Omega replacement
Theorem
External stable enhancement and smash product of spectra
Theorem
External input on derived enhancements
Theorem
External affine derived and perfect comparisons
Theorem
External local perfection and summand closure
Lemma
Reindexing cochains as chains
Lemma
Cohomology of an integer shift
Theorem
Eilenberg–Mac Lane spectrum of a chain complex
Theorem
External K-flat sheaf resolution and pullback input
Theorem
External localization of one marked scheme and tensor family
Theorem
External full-subcategory localization criterion
Theorem
External straightening and symmetric monoidal criteria
Lemma
Cofiber and triangle presentations of K zero
Lemma
Exact equivalences preserve triangle K zero
Lemma
Additive equivalences preserve split K zero
Theorem
Nonconnective K theory in degree zero
Lemma
Determinant on the stable general linear group
Lemma
Compatible local units glue uniquely
Lemma
A right inverse makes a homomorphism surjective
Theorem
External matrix, exact and bounded-projective K-theory comparisons
Lemma