Abstract algebra of chain complexes, exact sequences, derived functors and spectral sequences. Provides computational invariants (Tor, Ext) for modules over rings; foundational for algebraic topology, algebraic geometry, and representation…
homological-algebra
Chain complex
Sequence of abelian groups (or modules) Cₙ with boundary maps ∂ₙ: Cₙ → Cₙ₋₁ satisfying ∂ₙ ∘ ∂ₙ₊₁ = 0. The primitive object of homological…
Cochain complex and cohomology
Dual of a chain complex: differentials dⁿ: Cⁿ → Cⁿ⁺¹ with dⁿ⁺¹ dⁿ = 0. Cohomology Hⁿ = ker dⁿ / im dⁿ⁻¹ carries a contravariant functorial…
Tor functor
Left-derived functors of the tensor product: Torₙ^R(A, B) = Hₙ(P• ⊗_R B) for a projective resolution P• → A. Measures torsion: Tor₁^ℤ(A,…
Ext functor
Right-derived functors of Hom: Extⁿ_R(A,B) = Hⁿ(Hom_R(P•, B)). Ext¹ classifies extensions 0 → B → E → A → 0; Ext^n classifies n-fold Yoneda…
Projective resolution
Exact sequence … → P₁ → P₀ → A → 0 with each Pᵢ projective. Unique up to chain-homotopy, so derived functors depend only on A. Injective…
Snake lemma
For a commutative diagram of two short exact sequences joined by vertical maps f,g,h, there is a natural connecting map δ: ker h → coker f…
Long exact sequence in (co)homology
Every short exact sequence of chain complexes 0 → A• → B• → C• → 0 induces a long exact sequence … → Hₙ(A) → Hₙ(B) → Hₙ(C) → Hₙ₋₁(A) → ……
Spectral sequence
Book of pages (E_r^{p,q}, d_r) with d_r² = 0 and E_{r+1} = H(E_r, d_r), converging to a filtered graded object E_∞. Key tool for computing…
Derived functor
For a left-exact functor F on an abelian category with enough injectives, the right-derived functors R^n F(A) = H^n(F(I•)) for injective…
Grothendieck spectral sequence
Given composable left-exact functors F: A → B, G: B → C with F carrying injectives of A to G-acyclics, there is a spectral sequence R^p G ∘…
Five lemma
In a commutative diagram of two exact rows A₁→A₂→A₃→A₄→A₅ and B₁→B₂→B₃→B₄→B₅, if the outer vertical maps are suitably iso/epi/mono, the…
Injective resolution
Dual to projective resolution: an exact sequence 0 → M → I⁰ → I¹ → ⋯ with each I^k injective. Every object in an abelian category with…
Flat module
An R-module M is flat iff the tensor functor −⊗ᵣM preserves short exact sequences (equivalently, Tor₁^R(−, M) vanishes). Over ℤ: flat ⇔…
Künneth formula
Homology of a product: over a field, H_*(X×Y) = H_*(X) ⊗ H_*(Y). Over ℤ an extra Tor-correction appears. Gives H*(Tⁿ) = Λⁿ (exterior…
Universal coefficient theorem (cohomology)
Relates cohomology with coefficients in G to integral homology via a natural split short exact sequence involving Hom and Ext. Splits but…
Mayer-Vietoris sequence
For an open cover X = U ∪ V, a long exact sequence computes H_*(X) from H_*(U), H_*(V), H_*(U∩V). Iteration on a CW-cover provides a…
Hom-tensor adjunction
Natural isomorphism making ⊗ left-adjoint to Hom. The reason Tor is a left-derived functor of ⊗ and Ext is a right-derived functor of Hom.…
Spectral sequence convergence
Convergence criterion for a cohomological spectral sequence: first-quadrant sequences (E_r^{p,q} = 0 off p, q ≥ 0) stabilise at each slot…
Horseshoe lemma
Given a short exact sequence 0 → A' → A → A'' → 0 and projective resolutions P'_• → A' and P''_• → A'', the horseshoe lemma constructs a…
Zig-zag (connecting homomorphism) lemma
A short exact sequence of chain complexes 0 → A → B → C → 0 induces a natural long exact sequence in homology, with connecting map ∂:…
Euler characteristic of a chain complex
For a bounded chain complex of finitely-generated abelian groups (or finite-dim vector spaces), the alternating sum of ranks of the chains…
Projective dimension
The projective dimension pd_R(M) of a left R-module M is the minimal length of a projective resolution, or ∞ if no finite one exists. …
Global dimension of a ring
The global dimension gldim(R) is the supremum of projective dimensions over all left R-modules; equivalently the supremum of injective…
Baer's criterion for injectivity
A left R-module Q is injective iff every R-homomorphism from a left ideal a ⊆ R to Q extends to a homomorphism R → Q. Baer 1940. …
Koszul complex
For x₁, …, xₙ in a commutative ring R, the Koszul complex K_•(x; R) has K_p = Λ^p(R^n) with differential contracting by the vector (x₁, …,…
Cartan–Eilenberg resolution
Every bounded-below complex C_• admits a Cartan–Eilenberg resolution: a double complex P_{p,q} of projectives with exact rows (each row is…