Algebraic K-theory: The Homotopy Approach of Quillen and an Approach from Commutative Algebra

Contents
Preface
About the Author
Acknowledgment
1. Simplicial Sets
	1.1. Simplicial Sets
	1.2. Geometric Realization
		1.2.1. The CW Structure on |K|
		1.2.2. Simplicial Spaces
	1.3. Bisimplicial Sets
	1.4. The Homotopy Groups of Simplicial Sets
		1.4.1. Higher Homotopy Groups
	1.5. Exercises
2. Classifying Spaces of Categories
	2.1. The Classifying Spaces of Categories
		2.1.1. Properties of the classifying spaces
		2.1.2. Directed and filtering limit
		2.1.3. A key lemma on quasifibrations
	2.2. Exact Sequences of Homotopy Groups
		2.2.1. The Theorem A
		2.2.2. The Theorem B
		2.2.3. Fiberd and cofiberd category version of Theorems A and B
	2.3. Exercises
3. Quillen K-Theory
	3.1. Quillen’s Q-Construction
		3.1.1. Admissible layers
	3.2. K0(E) and π1(BQE, 0) of Exact Categories E
	3.3. Higher K-Groups of Exact categories
	3.4. Exact Sequences and Filtrations
		3.4.1. Additivity theorem
		3.4.2. Some examples
	3.5. The Resolution Theorem
		3.5.1. Extension closed subcategories
	3.6. Dévissage and Localization in Abelian Categories
		3.6.1. Semisimple Objects in abelian categories
		3.6.2. Quillen’s localization theorem
	3.7. K-Theory Spaces and Reformulations
	3.8. Exercises
4. The Agreement with Classical K-Theory
	4.1. Symmetric Monoidal Categories
	4.2. S−1S-Construction
	4.3. The Projection Functors
	4.4. The X-Coordinate Functor
	4.5. Split Exact Categories
		4.5.1. S−1S and QE
	4.6. Cofinality
	4.7. Agreement of Modern and Classical K-Theory
		4.7.1. The Whitehead Group
		4.7.2. The Agreement of Kc1(A) and K1(A)
		4.7.3. The Agreement of Kc2(A) and K2(A)
	4.8. Hc-Spaces
	4.9. Exercises
5. K-Theory of Rings
	5.1. K-Theory of Graded Rings
	5.2. Homotopy Invariance
	5.3. Filtered Rings
	5.4. Exercises
6. G-Theory of Schemes
	6.1. Preliminary Results
		6.1.1. Closed subschemes and the localization sequence
	6.2. Pullback and Pushforward
		6.2.1. Pullback maps
		6.2.2. Pushforward maps
		6.2.3. A projection formula
	6.3. G-Theory of Affine and Projective Bundles
	6.4. Filtration by Support
		6.4.1. Gersten conjecture
		6.4.2. The Chow groups
	6.5. Čech Cohomology Tools
		6.5.1. Application of Čech cohomology
		6.5.2. A spectral sequence
	6.6. Exercises
7. K-Theory of Projective Bundles
	7.1. The Canonical Resolution of Regular Sheaves on PE
	7.2. The Projective Bundle Theorem
	7.3. Exercises
8. Work of Swan on Quadric Hypersurfaces
	8.1. Hypersurfaces in Projective Spaces
	8.2. Canonical Resolution for Projective Schemes
		8.2.1. Truncation on R−1(X)
	8.3. Quadratic Spaces
		8.3.1. Clifford algebra
	8.4. Canonical Resolution and Minimal Resolution
	8.5. The Clifford Sequence
		8.5.1. C(q)-action on the Exterior Algebra
		8.5.2. Comparison of resolutions
	8.6. K-Theory of Quadric Hypersurfaces
		8.6.1. Graded interpretation
	8.7. The Affine Case
		8.7.1. Special case of q = q1 − T2
	8.8. Algebraic and Topological K-Theory of Spheres
9. Epilogue: K-Theory
	9.1. Introduction to the Epilogues
	9.2. Waldhausen K-Theory
		9.2.1. Exactness and Functorial properties
		9.2.2. Agreement with Quillen K-theory
		9.2.3. The chain complex categories
		9.2.4. Expected results
	9.3. K-Theory of Complicial Exact Categories
	9.4. Negative K-Theory
		9.4.1. Cofinality and idempotent completion
		9.4.2. Negative K-theory spectrum of exact categories
		9.4.3. Negative K-theory of complicial exact categories
	9.5. K-Theory of Schemes
		9.5.1. Quasi-projective schemes
10. Epilogue: Hermitian K-Theory
	10.1. Hermitian K-Theory of Exact Categories
		10.1.1. Exact categories with weak equivalences and duality
	10.2. dg Categories with Weak Equivalences and Duality
	10.3. Hermitian K-Theory of dg Categories
		10.3.1. Shifted dualities in dg categories
		10.3.2. The GW spectrum of dg categories
	10.4. Nonconnective Hermitian GW-Theory
		10.4.1. The category Sp of symmetric spectra
		10.4.2. The Bispetra BiSp
		10.4.3. The Karoubi GW-spectra
		10.4.4. Karoubi GW theory for quasi-projective schemes
		10.4.5. Gersten complex for the Karoubi GW-groups
		10.4.6. Further generality for regular schemes
	10.5. Nori Homotopy Obstructions
11. Epilogue: Triangulated Categories
	11.1. Basic Definitions
	11.2. Triangulated Witt Groups
		11.2.1. Localization
		11.2.2. Derived categories of exact categories and agreement
	11.3. Derived Witt Groups of Schemes
	11.4. Revisit Chow–Witt Groups
Appendices
	Appendix A. Category Theory and Exact Categories
		A.1. Main Definitions
			A.1.1. Classical and standard examples
			A.1.2. Pullback, Pushforward, kernel, and cokernel
			A.1.3. Equivalence and adjoint functors
		A.2. Additive and Abelian Categories
			A.2.1. Abelian categories
		A.3. Frequently Used Lemmas
			A.3.1. Pullback and Pushforward Lemmas
			A.3.2. The Snake Lemma
		A.4. Exact Categories
		A.5. Localization and Quotient Categories
			A.5.1. Calculus of fractions
			A.5.2. Quotient of abelian categories
			A.5.3. Quotient of exact categories
		A.6. Exercises
	Appendix B. Homotopy Theory
		B.1. Elements of Topological Spaces
			B.1.1. Compactly generated topologies
		B.2. Homotopy
			B.2.1. Relative homotopy groups
			B.2.2. Excision: Dold–Thom Theorem
		B.3. Fibrations
		B.4. Construction of Fibrations
		B.5. The Quasi-Fibrations
		B.6. Exercises
	Appendix C. CW Complexes
		C.1. Elements of CW Complexes
		C.2. Product of CW Complexes
		C.3. Frequently Used Results
			C.3.1. A triangle of fibrations
		C.4. Exercises
References
Index