Alphabeta MathDerived Categories
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1 · Prerequisites
- Abelian Categories
- Adjunctions Units and Counits
- Binary Operations, Monoids, Groups and Subgroups
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Delta Functors and Universality
- Derived Functors
- Exactness and the Member Calculus
- Ext and Balanced Resolutions
- Finite Counting, Factorials and Binomial Coefficients
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Homomorphisms and the Isomorphism Theorems
- Limits and Colimits
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Preadditive and Additive Categories and Biproducts
- Projective and Injective Resolutions
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Subobject Lattices Generators and the Grothendieck Axioms
- Tensor Products of Modules
- The Diagram Lemmas in an Abelian Category
- The ZFC Axioms and the Basic Set Constructions
- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Properties, Representables and the Yoneda Lemma
- Yoneda Extensions and Homological Dimension
2 · Logical flowchart
3 · Items
Definition
Multiplicative system in a category
Definition
Localization of a category at a class of morphisms
Definition
Left roof representing a localized morphism
Definition
Common refinement equivalence of roofs
Lemma
Roof equivalence is an equivalence relation
Lemma
Composition of roofs is well defined
Theorem
The calculus of fractions constructs the localization
Proposition
Quasi isomorphisms contain identities and are closed under composition
Theorem
Two out of three for quasi isomorphisms
Lemma
Quasi isomorphisms admit the roof calculus in the homotopy category
Definition
Derived category of an abelian category
Proposition
The localization functor sends quasi isomorphisms to isomorphisms
Proposition
Cohomology factors through the derived category
Proposition
A complex is zero in the derived category exactly when it is acyclic
Lemma
Addition of roofs makes an additive localization
Lemma
Finite roof squares and composable pairs can be cleared
Lemma
Localized cone triangles satisfy tr one through tr three
Lemma
Localized cone triangles satisfy the octahedral axiom
Theorem
The derived category inherits a triangulated structure
Theorem
The derived category is the verdier quotient by acyclic complexes
Definition
Homotopically projective bounded above complex
Definition
Homotopically injective bounded below complex
Theorem
A bounded above complex of projectives is homotopically projective
Theorem
A bounded below complex of injectives is homotopically injective
Proposition
Morphisms from a homotopically projective complex need no roof
Proposition
Morphisms into a homotopically injective complex need no roof
Definition
Brutal truncation of a complex
Definition
Canonical truncation of a complex
Lemma
Canonical truncation is a complex and has the claimed cohomology
Theorem
Canonical truncations fit a distinguished triangle
Proposition
Bounded derived localizations embed fully faithfully
Lemma
Bounded above complexes admit projective replacements
Lemma
Bounded below complexes admit injective replacements
Theorem
Projective complexes model the bounded above derived category
Theorem
Injective complexes model the bounded below derived category
Theorem
Ext is hom in the derived category
Proposition
Yoneda product is composition in the derived category
Definition
Canonical t structure on a derived category
Theorem
The canonical pair is a t structure
Theorem
The heart of the canonical t structure is equivalent to the original abelian category
Definition
Left total derived functor on the bounded above derived category
Lemma
Left total derived functor is independent up to a unique augmentation-compatible natural isomorphism
Theorem
Existence of the bounded above left total derived functor
Definition
Right total derived functor on the bounded below derived category
Theorem
Existence of the bounded below right total derived functor
Proposition
Total derived functors send distinguished triangles to distinguished triangles
Proposition
Classical derived functors are the cohomology objects of the total derived functor
Lemma
Bounded above flat tensor complexes preserve quasi isomorphisms
Definition
Derived tensor product in the bounded above setting
Proposition
Homology of the derived tensor product is tor
Definition
Derived hom in the bounded setting
Proposition
Cohomology of derived hom is ext
Proposition
Splitting a bounded complex by vanishing higher Ext
False statement
fs-localization-identifies-a-quasi-isomorphism-with-an-identity-morphism.md
False statement
fs-two-roofs-are-equal-whenever-their-right-hand-arrows-are-equal.md
False statement
fs-the-derived-category-is-the-same-category-as-the-homotopy-category.md
False statement
fs-every-complex-of-projectives-is-homotopically-projective.md
False statement
fs-brutal-and-canonical-truncation-are-the-same.md
False statement
fs-an-unbounded-total-derived-functor-exists-from-enough-injectives-alone.md
False statement
fs-a-derived-functor-is-canonical-without-supplied-replacement-data.md
Sources
- 10.3.1–10.3.14, pp. 379–384
- 13.11.1–13.11.6
- 10.3.11, p. 383
- 13.5.5–13.5.6, including all TR1–TR4 proof paragraphs
- Lemma 4.27.10, complete proof including footnote common denominator construction
- Definition 13.6.7, specialized to acyclic complexes
- 6.5.1 and 13.1 (K-injective definition); cochain convention
- 13.19.3–13.19.8; W 10.4.8 for the equivalence
- 13.18.3–13.18.8; W 10.4.8 for the equivalence
- 12.15, all four chain and four cochain truncations
- Remark 13.12.4 and its three triangles
- Lemma 13.15.4, full descending induction
- Lemma 13.15.5, dual to 13.15.4
- 10.4.7 and 10.7.5, pp. 388, 400
- Definitions 13.27.4 (06XT), Lemma 13.27.5 (06XU), and the following composition paragraphs
- Lecture 3, section 3, definition and Main example, pp. 28–29
- Proposition 12.4.1, p. 64
- 10.5.1–10.5.8, pp. 391–393; restrict to supplied replacement data
- 10.6.1–10.6.4 and Exercise 10.6.1, p. 395; elementary finite-diagonal replacement for spectral sequence proof
- 10.7.2–10.7.5 and Exercise 10.7.1, pp. 399–400
- Lemmas 13.27.8–13.27.10
- Lemma 13.4.11
- Boundary check against the licensed construction