Module VIII·Article II·~1 min read
Homological Algebra: Fundamentals
Tensor Algebra and Sylow Theorems (Advanced)
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Chain Complexes
A chain complex is a sequence of abelian groups (or modules) and homomorphisms: ... → Cₙ₊₁ → Cₙ → Cₙ₋₁ → ... with the condition dᵢdᵢ₊₁ = 0 (d² = 0).
Elements of ker dₙ are cycles Zₙ, Im dₙ₊₁ are boundaries Bₙ.
Homology: Hₙ = Zₙ/Bₙ = ker dₙ / Im dₙ₊₁.
Homology measures “how far the complex is from being exact”.
Homology in Topology
Singular homology of a topological space X: we construct a chain complex from singular simplices (continuous images of standard simplices) with a boundary operator.
H₀(X) ≅ ℤ^(number of connected components). H₁(X) — “holes” of dimension 1 (cycles). H₂(X) — “voids”, etc.
Sphere S²: H₀ = ℤ, H₁ = 0, H₂ = ℤ. Torus: H₀ = ℤ, H₁ = ℤ², H₂ = ℤ.
The Künneth Theorem
H*(X × Y) is computed in terms of H*(X) and H*(Y) according to the Künneth formula: H_n(X×Y) ≅ ⊕{p+q=n} H_p(X) ⊗ H_q(Y) ⊕ ⊕{p+q=n-1} Tor(H_p(X), H_q(Y)).
Derived Functors
An exact sequence 0→A→B→C→0 does not always remain exact after applying a functor (for example, Hom or ⊗).
Derived functors (Ext and Tor) measure the “degree of non-exactness”:
Torᵢ(A, B), Extⁱ(A, B) — algebraic invariants connecting algebra with topology.
Homological algebra is the language of modern geometry, number theory, and theoretical physics.
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