22 Differential Graded Algebra

Section 22.1: Introduction
Section 22.2: Conventions
Section 22.3: Differential graded algebras
- Definition 22.3.1
- Definition 22.3.2
- Definition 22.3.3
- Definition 22.3.4
- Lemma 22.3.5
Section 22.4: Differential graded modules
- Definition 22.4.1
- Lemma 22.4.2
  - Equation 22.4.2.1
- Definition 22.4.3
Section 22.5: The homotopy category
- Definition 22.5.1
- Lemma 22.5.2
- Definition 22.5.3
- Lemma 22.5.4
Section 22.6: Cones
- Definition 22.6.1
- Lemma 22.6.2
Section 22.7: Admissible short exact sequences
- Definition 22.7.1
- Lemma 22.7.2
- Lemma 22.7.3
- Lemma 22.7.4
- Lemma 22.7.5
- Lemma 22.7.6
Section 22.8: Distinguished triangles
- Lemma 22.8.1
  - Equation 22.8.1.1
- Definition 22.8.2
Section 22.9: Cones and distinguished triangles
- Lemma 22.9.1
- Lemma 22.9.2
- Lemma 22.9.3
- Lemma 22.9.4
Section 22.10: The homotopy category is triangulated
- Lemma 22.10.1
- Lemma 22.10.2
- Proposition 22.10.3
Section 22.11: Left modules
- Definition 22.11.1
- Lemma 22.11.2
- Definition 22.11.3
Section 22.12: Tensor product
Section 22.13: Hom complexes and differential graded modules
- Lemma 22.13.1
- Lemma 22.13.2
- Lemma 22.13.3
- Lemma 22.13.4
- Remark 22.13.5
- Remark 22.13.6
- Remark 22.13.7
Section 22.14: Projective modules over algebras
Section 22.15: Projective modules over graded algebras
Section 22.16: Projective modules and differential graded algebras
- Lemma 22.16.1
- Lemma 22.16.2
Section 22.17: Injective modules over algebras
Section 22.18: Injective modules over graded algebras
Section 22.19: Injective modules and differential graded algebras
- Lemma 22.19.1
- Lemma 22.19.2
- Lemma 22.19.3
Section 22.20: P-resolutions
- Lemma 22.20.1
- Lemma 22.20.2
- Lemma 22.20.3
- Lemma 22.20.4
Section 22.21: I-resolutions
- Lemma 22.21.1
- Lemma 22.21.2
- Lemma 22.21.3
- Lemma 22.21.4
Section 22.22: The derived category
- Lemma 22.22.1
- Definition 22.22.2
- Lemma 22.22.3
- Lemma 22.22.4
- Remark 22.22.5
Section 22.23: The canonical delta-functor
- Lemma 22.23.1
- Lemma 22.23.2
Section 22.24: Linear categories
- Definition 22.24.1
- Definition 22.24.2
Section 22.25: Graded categories
- Definition 22.25.1
- Definition 22.25.2
- Definition 22.25.3
- Definition 22.25.4
- Example 22.25.5: Graded category of graded objects
- Example 22.25.6: Graded category of graded modules
- Remark 22.25.7
Section 22.26: Differential graded categories
- Definition 22.26.1
- Definition 22.26.2
- Definition 22.26.3
- Definition 22.26.4
- Lemma 22.26.5
- Example 22.26.6: Differential graded category of complexes
  - Equation 22.26.6.1
- Lemma 22.26.7
- Example 22.26.8: Differential graded category of differential graded modules
- Lemma 22.26.9
- Lemma 22.26.10
Section 22.27: Obtaining triangulated categories
- Lemma 22.27.1
- Situation 22.27.2
- Lemma 22.27.3 slogan
- Lemma 22.27.4
- Lemma 22.27.5
- Lemma 22.27.6
- Lemma 22.27.7
- Lemma 22.27.8
- Lemma 22.27.9
- Lemma 22.27.10
- Lemma 22.27.11
- Lemma 22.27.12
- Lemma 22.27.13
  - Equation 22.27.13.1
- Lemma 22.27.14
- Lemma 22.27.15
- Proposition 22.27.16
- Lemma 22.27.17
Section 22.28: Bimodules
- Definition 22.28.1
- Lemma 22.28.2
- Lemma 22.28.3
- Lemma 22.28.4
- Lemma 22.28.5
Section 22.29: Bimodules and tensor product
- Lemma 22.29.1
- Remark 22.29.2
Section 22.30: Bimodules and internal hom
- Lemma 22.30.1
- Remark 22.30.2
- Lemma 22.30.3
Section 22.31: Derived Hom
- Equation 22.31.0.1
- Lemma 22.31.1
- Lemma 22.31.2
- Lemma 22.31.3
- Lemma 22.31.4
- Lemma 22.31.5
Section 22.32: Variant of derived Hom
- Lemma 22.32.1
Section 22.33: Derived tensor product
- Equation 22.33.0.1
- Lemma 22.33.1
- Lemma 22.33.2
- Lemma 22.33.3
- Lemma 22.33.4
- Lemma 22.33.5
- Example 22.33.6
- Lemma 22.33.7
- Lemma 22.33.8
- Lemma 22.33.9
Section 22.34: Composition of derived tensor products
- Equation 22.34.0.1
- Lemma 22.34.1
  - Equation 22.34.1.1
- Lemma 22.34.2
- Lemma 22.34.3
Section 22.35: Variant of derived tensor product
- Lemma 22.35.1
- Lemma 22.35.2
- Lemma 22.35.3
- Lemma 22.35.4
- Lemma 22.35.5
- Lemma 22.35.6
Section 22.36: Characterizing compact objects
- Remark 22.36.1
- Remark 22.36.2
- Lemma 22.36.3
- Proposition 22.36.4
- Lemma 22.36.5
- Lemma 22.36.6
Section 22.37: Equivalences of derived categories
- Lemma 22.37.1
- Lemma 22.37.2
- Remark 22.37.3
- Lemma 22.37.4
- Remark 22.37.5
- Proposition 22.37.6
- Remark 22.37.7
- Lemma 22.37.8
- Remark 22.37.9
- Remark 22.37.10
Section 22.38: Resolutions of differential graded algebras
- Lemma 22.38.1
- Lemma 22.38.2
- Lemma 22.38.3