Appendix B. Some homological algebra
FC13 — Matsumura, Commutative Ring Theory.
This chapter contains 15 definitions, 11 theorems, 1 corollary and 1 construction.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: complex, homology and cohomology
A chain complex K has d²=0 and homology H_n=ker d_n/im d_{n+1}; a cochain complex has cohomology H^n=ker d^n/im d^{n−1}. Exactness means all homology vanishes.
Definition: morphism of complexes
A morphism f:K→K′ is a degreewise family commuting with differentials and therefore induces maps on homology.
Definition: chain homotopy and homotopy equivalence
Morphisms f,g are homotopic if f−g=dh+hd. Homotopic maps induce the same homology map; homotopy-equivalent complexes have isomorphic homology.
Theorem/construction: long exact homology sequence
A short exact sequence of complexes 0→K′→K→K″→0 has connecting maps H_n(K″)→H_{n−1}(K′) and yields the standard long exact homology sequence.
Definition: double complex and total complex
A double complex has commuting horizontal and vertical differentials squaring to zero. Its total complex uses total degree p+q and differential d=d′+(−1)^p d″.
Theorem: Theorem B1
For a first-quadrant double complex whose vertical homology vanishes above degree zero, total homology is the homology of the degree-zero vertical homology complex; with the symmetric horizontal vanishing, both edge complexes compute the same total homology.
Theorem: Theorem B2
The cohomological analogue of Theorem B1 identifies total cohomology with the appropriate edge cohomology under the corresponding first-quadrant acyclicity hypotheses.
Definition: projective module
P is projective if maps P→N lift across every surjection M→N. Projective modules are exactly direct summands of free modules.
Definition: injective module
I is injective if maps from submodules extend across injections, the arrow-dual lifting condition to projectivity.
Theorem: Theorem B3 (Baer criterion)
An A-module I is injective iff every A-linear map from an ideal a⊆A to I extends to a map A→I.
Definition/construction: projective resolution
Every module M has a projective resolution ⋯→P_1→P_0→M→0 built by successive projective surjections; its positive homology vanishes and H_0≅M.
Corollary: finite free resolutions exist termwise for finite modules over Noetherian rings
If A is Noetherian and M finite, one may choose a projective resolution whose terms are finite free modules.
Definition/construction: injective resolution
Every module M embeds into an injective resolution 0→M→Q0→Q1→⋯ with H^0≅M and H^n=0 for n>0.
Theorem: comparison theorem for resolutions
A module map M→N lifts to a chain map between projective resolutions; the lift is unique up to homotopy. Consequently any two projective resolutions are homotopy equivalent, and dually for injective resolutions.
Definition: Tor functors
For projective resolutions P of M and Q of N, Tor_n^A(M,N) is the common homology of P⊗N, M⊗Q, and the associated double complex; it is independent of the chosen resolutions.
Theorem: basic properties of Tor
Tor_0(M,N)=M⊗N; Tor_n vanishes for n>0 when one entry is flat; Tor is symmetric and covariant in both entries, sends short exact sequences to long exact sequences, and commutes with direct limits in the second entry.
Definition: Ext functors
Using a projective resolution of M or injective resolution of N, Ext_A^n(M,N) is the common cohomology computed from Hom(P,N) or Hom(M,Q); it is independent of resolutions.
Theorem: basic properties of Ext
Ext^0(M,N)=Hom(M,N); higher Ext vanishes for projective first or injective second arguments; Ext is contravariant in the first and covariant in the second, with long exact sequences in either variable; projectivity/injectivity are characterized by Ext^1 vanishing against all modules.
Definition: projective and injective dimension
projdim M is the least length of a projective resolution, or ∞; injdim is defined dually. Dimension zero is equivalent to projectivity/injectivity.
Theorem: Ext characterization of homological dimension
projdim M≤d iff Ext_A^{d+1}(M,N)=0 for every N; dually injdim N≤d iff Ext_A^{d+1}(M,N)=0 for every M.
Definition/theorem: derived functors
Right derived functors of a left-exact functor are obtained from resolutions and are characterized by degree-zero recovery, vanishing on projectives/injectives as appropriate, and natural long exact sequences; left derived functors are defined dually for right-exact functors.
Definition: essential extension
An extension M⊆L is essential if every nonzero submodule of L meets M nontrivially; equivalently every nonzero x∈L has a nonzero scalar multiple in M.
Theorem: Theorem B4
An A-module M is injective iff it has no proper essential extension.
Definition/construction: injective hull
An injective hull E(M) is an injective essential extension of M. It exists by taking a maximal essential extension inside an injective overmodule and is unique up to an isomorphism fixing M.
Definition/construction: minimal injective resolution
Iteratively taking injective hulls of successive cokernels gives the minimal injective resolution of a module.
Theorem: five lemma
In a commutative diagram of exact five-term rows, the standard injectivity and surjectivity hypotheses on the surrounding vertical maps imply injectivity or surjectivity of the middle map.
Theorem: snake lemma
A commutative diagram with exact rows 0→A→B→C→0 and 0→A′→B′→C′→0 induces the standard exact sequence from kernels through cokernels with a connecting map Ker γ→Coker α.
Construction: tensor product of complexes
For complexes K,L, (K⊗L)n=⊕{p+q=n}K_p⊗L_q with d(x⊗y)=dx⊗y+(−1)^p x⊗dy. There are graded-symmetry and associativity isomorphisms; finite iterated tensor products are therefore well-defined.