Homological Algebra: Introduction

Applying a functor to an exact sequence sometimes breaks exactness. How badly? That is measured by Tor and Ext. This 'algebra of defects' turned out to be a universal language: topology (homology of spaces), algebraic geometry (sheaves and their cohomology), number theory (motives), mathematical physics (BRST). It is all the same language of chain complexes.

• Algebraic topology: homology groups H_n(X) are topological invariants of spaces; Tor measures 'torsion' in homology
• Algebraic geometry: derived category of coherent sheaves D^b(X) is the main invariant of a variety; Riemann-Roch theorem computes Ext between sheaves

Предварительные знания

• Rings: Addition and Multiplication

Exact Sequences

BERT's 12 transformer layers form a chain complex with 768×768 boundary maps , homological invariants explain why deeper models extract richer feature hierarchies. A sequence of R-modules and homomorphisms: ... → A →^f B →^g C → ... is called **exact at B** if im(f) = ker(g). A sequence is exact if it is exact at every term. **Short exact sequence:** 0 → A →^f B →^g C → 0 Means: f is injective, g is surjective, im(f) = ker(g). We say B is an **extension** of C by A. Examples: - 0 → Z →×2 Z → Z/2Z → 0 (multiplication by 2, projection mod 2) - 0 → nZ → Z → Z/nZ → 0

**The long exact sequence of cohomology** is the main computational tool. If 0 → A → B → C → 0 is a short exact sequence, applying a functor (e.g., Hom(M, ·)) may break exactness. The defect is measured by a long exact sequence: ... → Hom(M,A) → Hom(M,B) → Hom(M,C) → Ext¹(M,A) → Ext¹(M,B) → ...

Chain Complexes and Homology

A **chain complex** (C•, d) is a sequence of modules and homomorphisms: ... → Cₙ₊₁ →^{dₙ₊₁} Cₙ →^{dₙ} Cₙ₋₁ → ... satisfying d∘d = 0, i.e., dₙ ∘ dₙ₊₁ = 0 (the composition of two consecutive differentials is zero). The **n-th homology:** Hₙ(C•) = ker(dₙ) / im(dₙ₊₁) Homology measures the 'defect of exactness': the complex is exact (a resolution) if and only if all homology groups vanish.

**de Rham's theorem (again):** The de Rham complex (Ω⁰ →^d Ω¹ →^d Ω² → ...) is a chain complex (d∘d = 0)! Its homology groups are the de Rham cohomology H^k(M). By de Rham's theorem they are isomorphic to topological cohomology. Thus the entire exterior algebra is one big chain complex.

Derived Functors: Tor and Ext

The functor ⊗_R: Mod_R → Ab does not preserve exactness - it is **right exact**: if 0 → A → B → C → 0 is exact, then A⊗M → B⊗M → C⊗M → 0 is exact, but 0 → A⊗M may fail. **Tor_n(C, M)** - the n-th derived functor of ⊗_M. Measures how far ⊗ is from left exactness: - Tor₀(C, M) = C ⊗ M - If M or C is flat, then Tor_n = 0 for n ≥ 1 - Long exact sequence of Tor: ... → Tor₁(C,M) → A⊗M → B⊗M → C⊗M → 0 Similarly, Hom(·, M) is left exact, and its derived functors are **Ext^n(·, M)**.

**Origin of the terminology:** - **Tor** from 'Torsion': Tor₁(Z/nZ, M) 'sees' the n-torsion in M - **Ext** from 'Extension': Ext¹(A, B) classifies extensions B → ? → A Derived functors are the central tool of modern mathematics: in algebraic geometry (coherent sheaves, Riemann-Roch theorem), in number theory (motives, étale cohomology), in theoretical physics (BRST quantization).

Key Ideas

• Exact sequence: im(f) = ker(g) at every term
• 0 → A → B → C → 0 (SES): A is the kernel, C is the cokernel, B is an extension
• Chain complex: d∘d = 0; homology H_n = ker(dₙ)/im(dₙ₊₁)
• Hₙ = 0 ⟺ exactness at the n-th term
• Tor_n(M, N) - derived functor of ⊗; Tor₁ measures torsion
• Ext^n(M, N) - derived functor of Hom; Ext¹ classifies extensions

Further Directions

Homological algebra is the foundation of derived categories and spectral sequences. These allow computing invariants in algebraic geometry (Hochschild cohomology, de Rham homology) and number theory (p-adic cohomology, motives).

• Tensor Products — Tor - derived functor of ⊗; Tor₁ = kernel where ⊗ fails left exactness
• Exterior Algebra — The de Rham complex (Ω*, d) is the prime example of a chain complex

Вопросы для размышления

• Prove that for a field k every k-module (vector space) is flat, i.e., Tor₁(V, W) = 0 for any k-vector spaces V, W.
• Compute Ext¹(Z/nZ, Z) over Z. Interpret the answer in terms of extensions.
• Universal Coefficient Theorem: H_n(X; Z/pZ) has a 'correction term' Tor₁(H_{n-1}(X), Z/pZ). For which space X is this correction term nontrivial?

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