Homological Algebra

Job: measure failure of exactness via derived functors; organize chain complexes into derived and triangulated categories used across algebra, geometry, and topology.

Core

Homological algebra upgrades diagram chasing into functors \(R^iF\), \(L_iF\) and, further, into morphisms in derived categories where quasi-isomorphisms are inverted. The same formalism computes \(\operatorname{Ext}\), \(\operatorname{Tor}\), sheaf cohomology, and group cohomology.

Layer Content
Abelian categories kernels, cokernels, exact sequences; enough projectives/injectives
Derived functors \(\operatorname{Ext}\), \(\operatorname{Tor}\); resolutions
Derived categories complexes, \(D(\mathcal{A})\), triangulated structure, \(t\)-structures

Reading map

Page Focus
Foundations abelian categories, projective/injective resolutions, Ext/Tor
Derived categories \(K(\mathcal{A})\), quasi-isos, \(D^b(\mathcal{A})\), triangles, \(t\)-structures pointer

Dependency order

  1. Modules and exact sequences — modules.
  2. Categories / functors — category theory.
  3. Derived functors on modules and sheaves.
  4. Derived categories as localization; then applications (geometry, reps).

Interfaces

Area Contact
Abstract algebra \(\operatorname{Ext}^1\) classifies extensions; \(\operatorname{Tor}\) and torsion
Algebraic geometry \(H^i(X,\mathcal{F})=R^i\Gamma\); derived pushforwards; dualizing complexes
Algebraic topology singular chains; universal coefficient / Künneth via Tor/Ext
Representation theory group cohomology \(H^*(G,M)=\operatorname{Ext}^*_{\mathbb{Z}[G]}(\mathbb{Z},M)\)
Category theory abelian/triangulated axioms; adjunctions and derived functors

Notation (paper-standard)

  • \(0\to A\to B\to C\to 0\) exact; \(\operatorname{pd}\), \(\operatorname{id}\), \(\operatorname{gl.dim}\)
  • \(\operatorname{Ext}^i_R(M,N)\), \(\operatorname{Tor}_i^R(M,N)\), \(R^iF\), \(LF\)
  • \(\operatorname{Ch}(\mathcal{A})\), \(K(\mathcal{A})\), \(D(\mathcal{A})\), \(D^b(\mathcal{A})\), \(A\to B\to C\to A[1]\)

See it

Grothendieck’s six functors. For schemes (or spaces), \(f_*,f^*,f_!,f^!,\otimes,\operatorname{RHom}\) act on derived categories of sheaves with adjunctions and base-change morphisms. Classical LES and spectral sequences become formal shadows of distinguished triangles. Reading modern algebraic geometry or geometric representation theory without \(D^b\) is not realistic.

Universal coefficient theorem. \(H_*(X;\mathbb{Z})\) and \(\operatorname{Hom}/\operatorname{Ext}\) with coefficients \(G\) determine \(H^*(X;G)\) up to non-canonical splitting — pure homological algebra applied to singular chains.

Check

Why is \(\operatorname{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/n,\mathbb{Z})\cong\mathbb{Z}/n\) and what extension does a generator represent?

Resolve \(\mathbb{Z}/n\) by \(0\to\mathbb{Z}\xrightarrow{\times n}\mathbb{Z}\to\mathbb{Z}/n\to 0\); apply \(\operatorname{Hom}(-,\mathbb{Z})\): \(\operatorname{Ext}^1\cong\mathbb{Z}/n\mathbb{Z}\). Generator \(\leftrightarrow\) nonsplit extension \(0\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}/n\to 0\) (mult. by \(n\)) up to the usual identification with \(0\to\mathbb{Z}\xrightarrow{\times n}\mathbb{Z}\to\mathbb{Z}/n\to 0\) viewed as extension class — equivalently pushout constructions of \(\mathbb{Z}\xrightarrow{n}\mathbb{Z}\).

If \(F\) is exact, what are \(R^iF\)?

\(R^iF=0\) for \(i>0\) and \(R^0F\cong F\) (on the abelian category, under standard setup with enough injectives).

Pitfalls

  • Left vs right derived: \(\otimes\) is right exact \(\leadsto L_i\); \(\operatorname{Hom}\) left exact \(\leadsto R^i\) — directions reverse with variance.
  • Bounded vs unbounded derived categories need different existence results for resolutions (Spaltenstein, etc.).
  • Spectral sequences converge under hyp.; \(E_2\) page alone does not determine the abutment without differentials vanishing.

Refs

  • Weibel, An Introduction to Homological Algebra
  • Gelfand–Manin, Methods of Homological Algebra
  • Hartshorne, Residues and Duality (derived categories in geometry)
  • Stacks Project, Homological Algebra / Derived Categories

Next

Foundations · Derived categories · Modules · Category theory · Algebraic geometry cohomology · Algebraic topology