41 Chow Homology and Chern Classes

Section 41.1: Introduction
Section 41.2: Periodic complexes and Herbrand quotients
- Definition 41.2.1
- Definition 41.2.2
  - Equation 41.2.2.1
- Lemma 41.2.3
- Lemma 41.2.4
- Lemma 41.2.5
Section 41.3: Calculation of some multiplicities
- Lemma 41.3.1
- Lemma 41.3.2
- Lemma 41.3.3
- Lemma 41.3.4
Section 41.4: Preparation for tame symbols
- Lemma 41.4.1
- Lemma 41.4.2
- Lemma 41.4.3
- Lemma 41.4.4
Section 41.5: Tame symbols
- Equation 41.5.0.1
- Lemma 41.5.1
- Lemma 41.5.2
- Lemma 41.5.3
- Lemma 41.5.4
Section 41.6: A key lemma
- Lemma 41.6.1
- Lemma 41.6.2
- Lemma 41.6.3: Key Lemma reference
- Remark 41.6.4: Milnor K-theory
Section 41.7: Setup
- Situation 41.7.1
- Example 41.7.2
- Example 41.7.3
- Example 41.7.4
- Lemma 41.7.5
- Definition 41.7.6
Section 41.8: Cycles
- Definition 41.8.1
Section 41.9: Cycle associated to a closed subscheme
- Lemma 41.9.1
- Definition 41.9.2
Section 41.10: Cycle associated to a coherent sheaf
- Lemma 41.10.1
- Definition 41.10.2
- Lemma 41.10.3
- Lemma 41.10.4
Section 41.11: Preparation for proper pushforward
- Lemma 41.11.1
- Lemma 41.11.2
Section 41.12: Proper pushforward
- Definition 41.12.1
- Lemma 41.12.2
- Lemma 41.12.3
- Lemma 41.12.4
Section 41.13: Preparation for flat pullback
- Lemma 41.13.1
- Lemma 41.13.2
Section 41.14: Flat pullback
- Definition 41.14.1
- Lemma 41.14.2
- Lemma 41.14.3
- Lemma 41.14.4
Section 41.15: Push and pull
- Lemma 41.15.1
- Lemma 41.15.2
Section 41.16: Preparation for principal divisors
- Lemma 41.16.1
- Lemma 41.16.2
Section 41.17: Principal divisors
- Definition 41.17.1
- Lemma 41.17.2
Section 41.18: Principal divisors and pushforward
- Lemma 41.18.1
  - Equation 41.18.1.1
- Lemma 41.18.2
- Lemma 41.18.3
Section 41.19: Rational equivalence
- Definition 41.19.1
- Lemma 41.19.2
- Lemma 41.19.3
- Example 41.19.4
- Remark 41.19.5
Section 41.20: Rational equivalence and push and pull
- Lemma 41.20.1
- Lemma 41.20.2
- Lemma 41.20.3
Section 41.21: Rational equivalence and the projective line
- Lemma 41.21.1
- Lemma 41.21.2
- Lemma 41.21.3
Section 41.22: Chow groups and K-groups
- Lemma 41.22.1
- Lemma 41.22.2
- Lemma 41.22.3
- Lemma 41.22.4
- Remark 41.22.5
- Lemma 41.22.6
Section 41.23: The divisor associated to an invertible sheaf
- Definition 41.23.1
- Lemma 41.23.2
- Lemma 41.23.3
Section 41.24: Intersecting with an invertible sheaf
- Definition 41.24.1
- Lemma 41.24.2
- Lemma 41.24.3
  - Equation 41.24.3.1
- Lemma 41.24.4
Section 41.25: Intersecting with an invertible sheaf and push and pull
- Lemma 41.25.1
- Lemma 41.25.2
- Lemma 41.25.3
- Lemma 41.25.4
Section 41.26: The key formula
- Lemma 41.26.1: Key formula
Section 41.27: Intersecting with an invertible sheaf and rational equivalence
- Lemma 41.27.1
- Lemma 41.27.2
- Lemma 41.27.3
Section 41.28: Gysin homomorphisms
- Definition 41.28.1
- Remark 41.28.2
- Remark 41.28.3
- Lemma 41.28.4
- Lemma 41.28.5
- Remark 41.28.6
- Remark 41.28.7
- Lemma 41.28.8
- Lemma 41.28.9
Section 41.29: Gysin homomorphisms and rational equivalence
- Lemma 41.29.1
- Lemma 41.29.2
- Lemma 41.29.3
- Lemma 41.29.4
- Lemma 41.29.5
Section 41.30: Relative effective Cartier divisors
- Lemma 41.30.1
Section 41.31: Affine bundles
- Lemma 41.31.1
- Lemma 41.31.2
- Remark 41.31.3
- Lemma 41.31.4
- Lemma 41.31.5
- Lemma 41.31.6
- Lemma 41.31.7
Section 41.32: Bivariant intersection theory
- Definition 41.32.1 reference
- Lemma 41.32.2
- Lemma 41.32.3
- Lemma 41.32.4
- Remark 41.32.5
- Remark 41.32.6
- Example 41.32.7
- Remark 41.32.8
Section 41.33: Chow cohomology and the first chern class
- Definition 41.33.1
- Remark 41.33.2
- Lemma 41.33.3
- Definition 41.33.4
- Lemma 41.33.5
- Lemma 41.33.6
- Remark 41.33.7
- Remark 41.33.8
Section 41.34: Lemmas on bivariant classes
- Lemma 41.34.1 reference
- Lemma 41.34.2 reference
- Lemma 41.34.3
- Lemma 41.34.4
- Remark 41.34.5
Section 41.35: Projective space bundle formula
- Lemma 41.35.1
- Lemma 41.35.2: Projective space bundle formula
- Lemma 41.35.3
Section 41.36: The Chern classes of a vector bundle
- Definition 41.36.1
  - Equation 41.36.1.1
- Lemma 41.36.2
- Remark 41.36.3
  - Equation 41.36.3.1
Section 41.37: Intersecting with chern classes
- Definition 41.37.1
- Lemma 41.37.2
- Lemma 41.37.3
- Lemma 41.37.4
- Lemma 41.37.5
- Lemma 41.37.6
- Lemma 41.37.7
- Definition 41.37.8
- Lemma 41.37.9
- Remark 41.37.10
- Remark 41.37.11
Section 41.38: Polynomial relations among chern classes
- Lemma 41.38.1
  - Equation 41.38.1.1
Section 41.39: Additivity of chern classes
- Lemma 41.39.1
- Lemma 41.39.2
- Lemma 41.39.3
- Lemma 41.39.4
Section 41.40: Degrees of zero cycles
- Definition 41.40.1
- Lemma 41.40.2
- Lemma 41.40.3
- Lemma 41.40.4
Section 41.41: Cycles of given codimension
- Lemma 41.41.1
- Remark 41.41.2
Section 41.42: The splitting principle
- Lemma 41.42.1
- Lemma 41.42.2
- Lemma 41.42.3
- Remark 41.42.4
- Example 41.42.5
Section 41.43: Chern classes and sections
- Lemma 41.43.1
- Lemma 41.43.2
Section 41.44: The Chern character and tensor products
- Remark 41.44.1
- Lemma 41.44.2
- Lemma 41.44.3
- Lemma 41.44.4
Section 41.45: Chern classes and the derived category
- Lemma 41.45.1
- Definition 41.45.2
- Lemma 41.45.3
- Lemma 41.45.4
- Remark 41.45.5
- Lemma 41.45.6
- Lemma 41.45.7
- Lemma 41.45.8
Section 41.46: A baby case of localized chern classes
- Lemma 41.46.1
- Lemma 41.46.2
- Lemma 41.46.3
- Lemma 41.46.4
- Lemma 41.46.5
- Lemma 41.46.6
- Lemma 41.46.7
- Lemma 41.46.8
- Lemma 41.46.9
- Lemma 41.46.10
Section 41.47: Gysin at infinity
- Lemma 41.47.1
- Lemma 41.47.2
- Lemma 41.47.3
- Lemma 41.47.4
Section 41.48: Preparation for localized chern classes
- Lemma 41.48.1
- Lemma 41.48.2
- Lemma 41.48.3
- Lemma 41.48.4
- Lemma 41.48.5
- Lemma 41.48.6
- Lemma 41.48.7
Section 41.49: Localized chern classes
- Equation 41.49.0.1
- Lemma 41.49.1
- Lemma 41.49.2
- Lemma 41.49.3
- Definition 41.49.4
- Lemma 41.49.5
- Lemma 41.49.6
- Lemma 41.49.7
Section 41.50: Two technical lemmas
- Lemma 41.50.1
- Lemma 41.50.2
Section 41.51: Properties of localized chern classes
- Lemma 41.51.1
- Lemma 41.51.2
- Remark 41.51.3
- Lemma 41.51.4
- Lemma 41.51.5
- Lemma 41.51.6
Section 41.52: Blowing up at infinity
Section 41.53: Higher codimension gysin homomorphisms
- Lemma 41.53.1
- Lemma 41.53.2
- Lemma 41.53.3
- Lemma 41.53.4
- Lemma 41.53.5
- Lemma 41.53.6
- Lemma 41.53.7
- Lemma 41.53.8
- Lemma 41.53.9
- Remark 41.53.10: Variant for immersions
Section 41.54: Calculating some classes
- Lemma 41.54.1
- Lemma 41.54.2
- Lemma 41.54.3
- Lemma 41.54.4
Section 41.55: An Adams operator
- Lemma 41.55.1
- Remark 41.55.2
- Remark 41.55.3
- Remark 41.55.4
- Lemma 41.55.5
- Remark 41.55.6
- Remark 41.55.7
- Lemma 41.55.8
- Remark 41.55.9
- Remark 41.55.10
- Remark 41.55.11
Section 41.56: Chow groups and K-groups revisited
- Proposition 41.56.1
Section 41.57: Rational intersection products on regular schemes
- Lemma 41.57.1
Section 41.58: Gysin maps for local complete intersection morphisms
- Lemma 41.58.1
- Lemma 41.58.2
- Lemma 41.58.3
- Definition 41.58.4
- Lemma 41.58.5
- Lemma 41.58.6
- Lemma 41.58.7
- Lemma 41.58.8
- Remark 41.58.9
- Lemma 41.58.10
- Lemma 41.58.11
Section 41.59: Gysin maps for diagonals
- Lemma 41.59.1
- Proposition 41.59.2 reference
Section 41.60: Exterior product
- Lemma 41.60.1
- Lemma 41.60.2
- Lemma 41.60.3
- Remark 41.60.4
- Lemma 41.60.5
Section 41.61: Intersection products
- Lemma 41.61.1
- Lemma 41.61.2
- Lemma 41.61.3
- Lemma 41.61.4
- Lemma 41.61.5
- Lemma 41.61.6
- Lemma 41.61.7
Section 41.62: Exterior product over Dedekind domains
- Lemma 41.62.1
- Lemma 41.62.2
- Lemma 41.62.3
- Remark 41.62.4
- Lemma 41.62.5
Section 41.63: Intersection products over Dedekind domains
- Lemma 41.63.1
- Lemma 41.63.2
Section 41.64: Todd classes
Section 41.65: Grothendieck-Riemann-Roch
Section 41.66: Appendix A: Alternative approach to key lemma
- Subsection 41.66.1: Determinants of finite length modules
- Definition 41.66.2
- Lemma 41.66.3
- Lemma 41.66.4
- Lemma 41.66.5
  - Equation 41.66.5.1
- Lemma 41.66.6
- Lemma 41.66.7
- Lemma 41.66.8
- Remark 41.66.9
- Lemma 41.66.10
- Example 41.66.11
- Subsection 41.66.12: Periodic complexes and determinants
- Definition 41.66.13
- Remark 41.66.14
- Lemma 41.66.15
- Lemma 41.66.16
- Lemma 41.66.17
- Lemma 41.66.18
- Example 41.66.19
- Example 41.66.20
- Example 41.66.21
- Example 41.66.22
- Example 41.66.23
- Lemma 41.66.24
- Lemma 41.66.25
- Subsection 41.66.26: Symbols
- Lemma 41.66.27
- Lemma 41.66.28
- Definition 41.66.29
- Lemma 41.66.30
- Definition 41.66.31
- Lemma 41.66.32
- Lemma 41.66.33
- Lemma 41.66.34
- Lemma 41.66.35
- Lemma 41.66.36
- Lemma 41.66.37
- Lemma 41.66.38
- Lemma 41.66.39
- Subsection 41.66.40: Lengths and determinants
- Lemma 41.66.41
- Lemma 41.66.42
- Proposition 41.66.43
- Lemma 41.66.44
- Subsection 41.66.45: Application to the key lemma
- Lemma 41.66.46: Key Lemma reference
Section 41.67: Appendix B: Alternative approaches
- Subsection 41.67.1: Rational equivalence and K-groups
- Lemma 41.67.2
- Lemma 41.67.3
- Subsection 41.67.4: Cartier divisors and K-groups
- Lemma 41.67.5
- Lemma 41.67.6
- Lemma 41.67.7