Introduction to Homological Algebra

Homological algebra is the systematic study of chain complexes and their homology, providing a unified language for measuring the failure of sequences to be exact. Its central computational tools are derived functors (Tor, Ext, and their relatives), which extract "hidden" invariants from objects in abelian categories. It applies anywhere algebra meets topology, geometry, or arithmetic — algebraic topology, commutative algebra, algebraic geometry, group cohomology, and representation theory. Founded in the work of Poincaré, formalized by Eilenberg–Mac Lane, Cartan–Eilenberg (1956), and Grothendieck (1957).

Homological algebra abstracts the formal machinery of homology from algebraic topology and recasts it as a self-contained theory about chain complexes in abelian categories. A chain complex is a sequence of objects connected by maps $d$ satisfying $d^{2} = 0$ ; its homology $H_{n} = ker d_{n} / im d_{n + 1}$ measures how far the complex is from being exact. The subject matters because almost every invariant in modern algebra — torsion phenomena, extensions of modules, obstructions to lifting, sheaf cohomology — turns out to be a derived functor, computed by replacing objects with resolutions and applying a functor. The surprising fact: the homology you compute does not depend on which (projective or injective) resolution you choose — a robustness encoded in the Comparison Theorem and ultimately in the derived category.

The Core Insight

Non-exactness is information, not noise. When you apply a functor like

- \otimes_{R} M

Hom_{R} (-, M)

to a short exact sequence, exactness usually breaks. Rather than discard this failure, homological algebra measures it with a sequence of derived functors. These measurements assemble into long exact sequences that turn local algebraic data into global invariants. The whole discipline is the answer to one question: given a functor that "almost" preserves exactness, what exactly does it lose?

Formal Statement

Chain Complex and Homology

A chain complex $(C_{∙}, d)$ in an abelian category $A$ is a sequence of objects and morphisms

\dots d_{n + 1} C_{n + 1} d_{n} C_{n} d_{n - 1} C_{n - 1} d_{n - 2} \dots

⟳The indices on

d

follow the "homological" convention

d_{n} : C_{n + 1} \to C_{n}

in some texts and

d_{n} : C_{n} \to C_{n - 1}

in others; fix one and the rest is bookkeeping.

subject to the defining condition

d_{n} \circ d_{n + 1} = 0 for all n .

⟳This is the algebraic shadow of "boundary of a boundary is zero"

\partial^{2} = 0

from simplicial topology.

The condition $[2]$ gives $im d_{n + 1} \subseteq ker d_{n}$ , so the $n$ -th homology object is well-defined:

H_{n} (C_{∙}) = ker d_{n} / im d_{n + 1} = Z_{n} / B_{n},

⟳Elements of

Z_{n} = ker d_{n}

are

cycles, elements of $B_{n} = im d_{n + 1}$ are boundaries; homology is cycles modulo boundaries.

Here:

$A$ — an abelian category (e.g. $R$ -Mod, sheaves of abelian groups), where kernels, cokernels, images exist and behave like in modules.
$Z_{n}$ — the object of cycles; $B_{n}$ — the object of boundaries.
$C_{∙}$ is exact at $C_{n}$ iff $H_{n} = 0$ ; exact everywhere iff it is an acyclic complex.

A cochain complex reverses arrows ( $d^{n} : C^{n} \to C^{n + 1}$ ) and gives cohomology $H^{n}$ .

Chain Map and Quasi-Isomorphism

A chain map $f_{∙} : C_{∙} \to D_{∙}$ is a family $f_{n} : C_{n} \to D_{n}$ commuting with differentials: $d_{n}^{D} f_{n} = f_{n - 1} d_{n}^{C}$ . It induces $H_{n} (f) : H_{n} (C) \to H_{n} (D)$ . A chain map is a quasi-isomorphism if every $H_{n} (f)$ is an isomorphism. Two chain maps $f, g$ are chain homotopic ( $f ≃ g$ ) if there exist maps $s_{n} : C_{n} \to D_{n + 1}$ with

f_{n} - g_{n} = d_{n + 1}^{D} s_{n} + s_{n - 1} d_{n}^{C} .

⟳Verify directly that a homotopy forces

H_{n} (f) = H_{n} (g)

by checking that

d^{D} s_{n} + s_{n - 1} d^{C}

kills cycles after passing to homology.

Long Exact Sequence in Homology

A short exact sequence of chain complexes $0 \to A_{∙} f B_{∙} g C_{∙} \to 0$ (exact in each degree) induces a connecting homomorphism $\partial : H_{n} (C) \to H_{n - 1} (A)$ and a long exact sequence

\dots \to H_{n} (A) f_{*} H_{n} (B) g_{*} H_{n} (C) \partial H_{n - 1} (A) \to \dots

⟳Build

\partial

by the "diagram chase" of the Snake Lemma — lift a cycle in

C_{n}

B_{n}

, push through

d

, and pull back to

A_{n - 1}

Derived Functors

Let $F : A \to B$ be a right-exact additive functor between abelian categories with enough projectives. For each object $A$ , choose a projective resolution $P_{∙} \to A$ . The left derived functors are

L_{n} F (A) = H_{n} (F (P_{∙})),

⟳Independence of the resolution follows from the Comparison Theorem: any two projective resolutions are chain homotopy equivalent, and

F

preserves homotopies.

with $L_{0} F ≅ F$ (since $F$ is right exact). Dually, right derived functors $R^{n} F$ use injective resolutions of a left-exact $F$ . The fundamental special cases:

Tor_{n}^{R} (M, N) = L_{n} (- \otimes_{R} N) (M), Ext_{R}^{n} (M, N) = R^{n} Hom_{R} (M, -) (N) .

⟳Tor is balanced — resolving either variable gives the same answer; the same holds for Ext, proved via a double-complex spectral sequence argument.

Why It Holds / Derivation

The Snake Lemma and the Connecting Map

We construct the connecting homomorphism $\partial$ of equation $[5]$ , the engine of the whole theory.

Step 1 — Setup. Given

0 \to A_{∙} f B_{∙} g C_{∙} \to 0

exact in each degree, take a homology class

[c] \in H_{n} (C)

with

c \in Z_{n} (C)

, so

d^{C} c = 0

. Step 2 — Lift. Since

g_{n} : B_{n} \to C_{n}

is surjective, choose

b \in B_{n}

with

g_{n} (b) = c

. Step 3 — Push. Apply the differential:

d^{B} b \in B_{n - 1}

. Then

g_{n - 1} (d^{B} b) = d^{C} g_{n} (b) = d^{C} c = 0

, so

d^{B} b \in ker g_{n - 1} = im f_{n - 1}

. Step 4 — Pull back. By exactness and injectivity of

f_{n - 1}

, there is a unique

a \in A_{n - 1}

with

f_{n - 1} (a) = d^{B} b

. One checks

d^{A} a = 0

(push

a

forward, use that

f

is a chain map and

d^{2} = 0

), so

a \in Z_{n - 1} (A)

. Step 5 — Define. Set

\partial [c] = [a] \in H_{n - 1} (A)

. Step 6 — Well-defined. Different choices of

b

in Step 2 differ by an element of

im f_{n}

; tracing through shows

[a]

changes only by a boundary. Changing the representative

c

by a boundary similarly leaves

[a]

unchanged. Step 7 — Exactness. Exactness at each of the three spots (

H_{n} (B)

H_{n} (C)

H_{n - 1} (A)

) is a routine diagram chase. For instance,

g_{*} f_{*} = (g f)_{*} = 0

, and a class in

ker g_{*}

lifts back along

f

by the construction above.

■

Independence of Resolution (Comparison Theorem)

We justify why $L_{n} F (A)$ in $[6]$ does not depend on the chosen projective resolution.

Step 1. Let

P_{∙} \to A

and

Q_{∙} \to A

be two projective resolutions. We claim there is a chain map

φ : P_{∙} \to Q_{∙}

lifting

id_{A}

. Step 2. Build

φ

inductively. Since

P_{0}

is projective and

Q_{0} \to A

is surjective, the map

P_{0} \to A

lifts to

φ_{0} : P_{0} \to Q_{0}

. Continuing degree by degree, using projectivity of

P_{n}

and exactness of

Q_{∙}

, produces all

φ_{n}

. Step 3. Any two such lifts

φ, φ^{'}

are chain homotopic — same inductive construction applied to the difference, which lifts the zero map. Step 4. By symmetry there is

ψ : Q_{∙} \to P_{∙}

, and

ψ φ ≃ id_{P}

φ ψ ≃ id_{Q}

. So

P_{∙}

and

Q_{∙}

are chain homotopy equivalent. Step 5. Additive functors preserve homotopies (apply

F

to equation

[4]

). Hence

F (P_{∙}) ≃ F (Q_{∙})

, and homotopic complexes have isomorphic homology. Therefore

H_{n} (F (P_{∙})) ≅ H_{n} (F (Q_{∙}))

■

Samuel Eilenberg & Saunders Mac Lane | 1940s–1950s

Eilenberg (1913–1998) and Mac Lane (1909–2005) transformed scattered topological computations into a structural theory. Their joint work on group cohomology, the universal coefficient theorem, and the bar construction crystallized the idea that functors and natural transformations are the right organizing concepts — leading directly to category theory (1945). Eilenberg's collaboration with Henri Cartan produced Homological Algebra (1956), the book that named and founded the field. Mac Lane's Homology (1963) became its standard graduate text. Both insisted that the formalism of $\partial^{2} = 0$ deserved to be studied in its own right, independent of any geometric realization.

Building Intuition

Think of a chain complex as a machine that almost cancels itself out. Each differential $d$ sends information "down" one level, and $d^{2} = 0$ guarantees that what gets sent down twice vanishes completely. Homology asks: what is sitting in the kernel that wasn't already pushed there from above? That leftover is a genuine, coordinate-free invariant.

For Tor and Ext specifically: a module $M$ may be "rigid" (free/projective) or "flexible" (full of torsion and relations). When you tensor or hom against $M$ , the rigid part survives intact while the flexible part generates discrepancies. Derived functors are precisely the catalogue of those discrepancies, organized by how many resolution steps you need to uncover them. $Tor_{1} (Z / n, Z / m) = Z / g cd (n, m)$ literally measures the shared torsion. $Ext^{1} (A, B)$ classifies the ways $B$ can be "extended" by $A$ — the secret life of short exact sequences.

Short exact sequence of complexes producing the long exact homology sequence via the connecting map ∂

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Worked Examples

Computing Tor over ℤ

Given: Compute

Tor_{n}^{Z} (Z /6, Z /4)

for all

n

. Step 1 — Resolve $Z /6$ . A free resolution:

0 \to Z \times 6 Z \to Z /6 \to 0.

P_{1} = Z

P_{0} = Z

d_{1} = \times 6

, and

P_{n} = 0

for

n \geq 2

. Step 2 — Apply $- \otimes_{Z} Z /4$ . Delete the augmentation and tensor:

0 \to Z /4 \times 6 Z /4 \to 0.

Note

\times 6 \equiv \times 2 (mod 4)

. Step 3 — Take homology.

$Tor_{0} = coker (\times 2 : Z /4 \to Z /4) = Z /4/2 (Z /4) = Z /2$ . (Equals $Z /6 \otimes Z /4 = Z / g cd (6, 4)$ . ✓)
$Tor_{1} = ker (\times 2 : Z /4 \to Z /4) = {0, 2} ≅ Z /2$ .
$Tor_{n} = 0$ for $n \geq 2$ (resolution has length 1).

Boxed answer:

Tor_{0} ≅ Z /2, Tor_{1} ≅ Z /2 = Z / g cd (6, 4), Tor_{\geq 2} = 0.

Consistent with the general formula

Tor_{1}^{Z} (Z / a, Z / b) ≅ Z / g cd (a, b)

Computing Ext and Classifying Extensions

Given: Compute

Ext_{Z}^{n} (Z /2, Z)

and interpret

Ext^{1}

. Step 1 — Resolve $Z /2$ .

0 \to Z \times 2 Z \to Z /2 \to 0.

Step 2 — Apply $Hom_{Z} (-, Z)$ . This contravariant functor turns the resolution into a cochain complex:

0 \to Hom (Z, Z) (\times 2)^{*} Hom (Z, Z) \to 0,

i.e.

0 \to Z \times 2 Z \to 0

(precomposition with

\times 2

is multiplication by 2). Step 3 — Cohomology.

$Ext^{0} = ker (\times 2 : Z \to Z) = 0 = Hom (Z /2, Z)$ . ✓
$Ext^{1} = coker (\times 2 : Z \to Z) = Z /2$ .
$Ext^{n} = 0$ for $n \geq 2$ .

Step 4 — Interpret.

Ext^{1} (Z /2, Z) ≅ Z /2

classifies extensions

0 \to Z \to E \to Z /2 \to 0

up to equivalence. The two classes:

the split extension $E = Z \oplus Z /2$ (zero class),
the non-split extension $E = Z$ with $Z \times 2 Z ↠ Z /2$ (nonzero class).

Boxed answer:

Ext^{0} = 0, Ext^{1} ≅ Z /2 (2 extension classes), Ext^{\geq 2} = 0.

Common Misconceptions

"Tor and Ext vanish for nice modules, so they're rarely needed." False emphasis.

Tor_{n} (M, N) = 0

for all

n > 0

exactly characterizes flatness of

M

, and

Ext^{n} (M, -) = 0

for

n > 0

characterizes projectivity — these vanishing statements are the content. The derived functors detect precisely when the friendly behavior fails, and that failure is the invariant of interest (e.g. global dimension, depth, singularities). "The connecting homomorphism is canonical and you can ignore signs." The map

\partial

is canonical, but sign conventions in the total complex and in shifting complexes (

C [1]_{n} = C_{n - 1}

, often with a sign on the differential) are a notorious source of errors. Fix conventions early. "A quasi-isomorphism is invertible as a chain map." No — quasi-isomorphisms are not generally invertible at the chain level. You must pass to the derived category

D (A)

, where they are formally inverted, to treat them as isomorphisms. Right derived functors of an already-exact functor need not vanish above degree 0 unless the functor is exact. Consider

Hom_{Z} (Z /2, -)

. One might guess that since

Z /2

is "small," higher Ext vanishes. But

Ext_{Z}^{1} (Z /2, Z) = Z /2 \neq = 0

, computed above. The functor

Hom (Z /2, -)

is only left exact, not exact, so its higher derived functors are genuinely nonzero.

Conversely, flat does not imply projective: $Q$ is a flat $Z$ -module (all $Tor_{n}^{Z} (Q, -) = 0$ for $n > 0$ ) but $Q$ is not projective over $Z$ — it is not a summand of any free abelian group. So vanishing of Tor and projectivity are different conditions; the slogan "flat = locally free = projective" requires finiteness and Noetherian hypotheses.

Connections

Homological algebra is the connective tissue of modern mathematics:

Singular Homology and Simplicial Homology are the topological origin; the chain complex of singular chains is the prototype for $[1]$ .
Abelian Categories provide the axiomatic stage; Grothendieck's "Tôhoku" paper extended derived functors from $R$ -Mod to any abelian category with enough injectives, enabling Sheaf Cohomology.
Spectral Sequences generalize the long exact sequence $[5]$ to filtered complexes and double complexes, computing the cohomology of composite functors via the Grothendieck spectral sequence.
Group Cohomology is $Ext_{Z G}^{n} (Z, M)$ , recovering $H^{1}$ (crossed homomorphisms), $H^{2}$ (group extensions), and obstruction theory.
Derived Categories and Triangulated Categories are the modern home, where quasi-isomorphisms become isomorphisms and the long exact sequence becomes a distinguished triangle.
Commutative Algebra uses the Koszul complex, depth, and global dimension (Auslander–Buchsbaum) all phrased homologically.

Real-World Applications

Concrete, named uses across mathematics and beyond:

Sheaf cohomology in algebraic geometry — $H^{i} (X, F)$ computes global obstructions (Serre duality, Riemann–Roch, vanishing theorems of Kodaira and Grothendieck).
The Universal Coefficient Theorem — relates homology with different coefficients via $Tor$ and $Ext$ , essential for computing $H_{*} (X; Z / p)$ in topology.
Group extensions and Galois cohomology — $H^{2} (G, A)$ classifies central extensions; Galois cohomology underpins class field theory and the Brauer group of a field.
Hochschild and cyclic (co)homology — invariants of associative algebras driving deformation theory and noncommutative geometry (Connes), with applications to index theory.
TDA / persistent homology — the chain-complex formalism, with homology over a field tracked across a filtration, powers topological data analysis used in materials science, neuroscience, and shape recognition.
Resolutions in computational algebra — minimal free resolutions (computed in Macaulay2, Singular) determine Betti numbers, regularity, and syzygies of ideals in polynomial rings.
String theory and topological field theory — BRST cohomology and the Batalin–Vilkovisky formalism encode physical state spaces as homology of a differential, directly using $d^{2} = 0$ .

Historical Development

1895–1904 — Henri Poincaré introduces homology of manifolds via cycles and boundaries in Analysis Situs, establishing Betti numbers and torsion coefficients.
1925 — Emmy Noether reframes homology in terms of abelian groups and homomorphisms, replacing Poincaré's numerical invariants with structural ones — the conceptual birth of algebraic homology.
1935 — Hassler Whitney and others develop cohomology; the cup product gives it ring structure.
1942–1947 — Eilenberg and Mac Lane develop group cohomology, the bar resolution, and introduce categories, functors, natural transformations (1945).
1948 — Hopf, Eckmann, and others clarify the relation between group homology and topology of $K (G, 1)$ spaces.
1956 — Cartan and Eilenberg publish Homological Algebra, defining derived functors, Tor, Ext, and balancing — the founding text.
1957 — Grothendieck's Tôhoku paper ("Sur quelques points d'algèbre homologique") axiomatizes abelian categories, establishes enough injectives, and generalizes derived functors to sheaf cohomology.
1960s — Jean-Louis Verdier (in his thesis under Grothendieck) constructs derived and triangulated categories, the modern framework.
1980s–2000s — Persistent homology (Edelsbrunner, Carlsson) and derived algebraic geometry (Lurie, Toën–Vezzosi) extend the reach into data science and higher category theory.

Subtleties & Limits

Where the classical theory strains and active questions remain:

Existence of resolutions: Not every abelian category has enough projectives (sheaves on a space typically do not), forcing reliance on injectives or flat/flasque resolutions and careful tracking of which derived functors even make sense.
Higher structures: Ordinary homology forgets the homotopy coherent data. The passage from the homotopy category to genuine $\infty$ -categories (Lurie's Higher Algebra) is needed to make derived tensor products and composition strictly functorial — many "obvious" diagrams commute only up to higher homotopy.
Computability: Computing $Ext$ and $Tor$ over general non-Noetherian or infinite-dimensional rings can be undecidable or wildly hard; finite-dimensional algebra representation type (tame vs. wild) governs tractability.
Homological conjectures in commutative algebra: The constellation around the Auslander–Reiten conjecture, the (now-resolved by André via perfectoid methods, 2016) Direct Summand Conjecture, and the still-open Serre multiplicity conjectures (positivity of intersection multiplicities) remain deep.
Derived equivalences and the structure of $D (A)$ : Classifying when two rings/varieties are derived equivalent (tilting theory, Bondal–Orlov, the Homological Mirror Symmetry program of Kontsevich) is a vast open frontier connecting algebra, geometry, and physics.

Introduction to Homological Algebra

The Core Insight

Formal Statement

Chain Complex and Homology

Chain Map and Quasi-Isomorphism

Long Exact Sequence in Homology

Derived Functors

Why It Holds / Derivation

The Snake Lemma and the Connecting Map

Independence of Resolution (Comparison Theorem)

Samuel Eilenberg & Saunders Mac Lane | 1940s–1950s

Building Intuition

Short exact sequence of complexes producing the long exact homology sequence via the connecting map ∂

Worked Examples

Computing Tor over ℤ

Computing Ext and Classifying Extensions

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads