Mathematics.

homological methods

Homological Algebra

Abstract Algebra II120 minDifficulty9 out of 10

You should know: rings, modules

Overview

Homological algebra is the branch of mathematics that uses algebraic techniques — particularly chain complexes, exact sequences, and derived functors — to study algebraic structures. It arose from algebraic topology (computing homology groups of spaces) and now pervades algebra, geometry, and representation theory. Its core insight is that many algebraic properties can be measured by the failure of certain functors to be exact.

Intuition

If you try to reverse the arrow of a functor (like Hom or tensor), it sometimes fails to be exact — it might not preserve all short exact sequences. Homological algebra quantifies exactly how much it fails using derived functors (Ext, Tor). The homology groups are like 'error terms' that measure obstruction.

Formal Definition

Definition

A central object is the chain complex: a sequence of modules connected by boundary maps whose composition is zero. The homology of the complex measures the 'gap' between the image of one boundary map and the kernel of the next.

⋯→Cn+1→∂n+1Cn→∂nCn−1→⋯ ,∂n∘∂n+1=0\cdots \to C_{n+1} \xrightarrow{\partial_{n+1}} C_n \xrightarrow{\partial_n} C_{n-1} \to \cdots, \quad \partial_n \circ \partial_{n+1} = 0
Chain complex
Hn(C∙)=ker⁡(∂n)/im(∂n+1)H_n(C_\bullet) = \ker(\partial_n) / \text{im}(\partial_{n+1})
Homology group
0→A→fB→gC→0 exact⇔f injective,  g surjective,  ker⁡g=imf0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 \text{ exact} \Leftrightarrow f \text{ injective}, \; g \text{ surjective}, \; \ker g = \text{im} f
Short exact sequence

Worked Examples

  1. A split short exact sequence has a section \(s: C \to B\) with \(g \circ s = \text{id}_C\) (right split) or a retraction \(r: B \to A\) with \(r \circ f = \text{id}_A\) (left split).

    g∘s=idCg \circ s = \text{id}_C

  2. Define \(\phi: A \oplus C \to B\) by \(\phi(a,c) = f(a) + s(c)\). Check injectivity: if \(f(a) + s(c) = 0\), apply \(g\) to get \(c = 0\), then \(f(a) = 0\) so \(a=0\).

    ϕ(a,c)=f(a)+s(c)\phi(a,c) = f(a) + s(c)

  3. Surjectivity: for any \(b \in B\), let \(c = g(b)\) and \(a = r(b - s(c))\); then \(\phi(a,c) = b\).

    b=f(r(b−s(g(b))))+s(g(b))=ϕ(r(b−s(g(b))), g(b))b = f(r(b - s(g(b)))) + s(g(b)) = \phi(r(b-s(g(b))),\, g(b))

Answer: The splitting gives an isomorphism \(B \cong A \oplus C\).

Practice Problems

Difficulty 7/10

Define a projective module and explain how it relates to the exactness of \(\text{Hom}(P,-)\).

Difficulty 8/10

Define an injective module and state how it relates to the exactness of \(\text{Hom}(-,I)\).

Difficulty 9/10

State the snake lemma and describe one of its applications.

Common Mistakes

Common Mistake

Free modules and projective modules are the same thing.

All free modules are projective, but the converse fails in general. Over a PID (like \(\mathbb{Z}\)), projective implies free. But over other rings (e.g., \(\mathbb{Z}/6\mathbb{Z}\)), there exist non-free projectives.

Common Mistake

A short exact sequence always splits.

Splitting requires additional structure. The sequence \(0 \to \mathbb{Z} \xrightarrow{\cdot 2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0\) does not split (\(\mathbb{Z}\) has no \(\mathbb{Z}/2\mathbb{Z}\) submodule of the right form).

Quiz

Homology \(H_n(C_\bullet)\) of a chain complex is defined as:
A module \(P\) is projective if and only if:
The snake lemma produces which type of exact sequence?

Summary

  • Homological algebra studies chain complexes and their homology, measuring the failure of exactness.
  • Short exact sequences are the fundamental building blocks; their behavior under functors is central.
  • Projective modules make \(\text{Hom}(P,-)\) exact; injective modules make \(\text{Hom}(-,I)\) exact.
  • The snake lemma and long exact sequence in homology are core tools.
  • Derived functors (Ext, Tor) quantify the failure of Hom and tensor to be exact.

References

  1. WebsiteWikipedia — Homological algebra