Introduction to Homological Algebra

⚡ Key Takeaways

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).

💡 The Core Insight

Non-exactness is information, not noise. When you apply a functor like $-\otimes_R M$ or $\operatorname{Hom}_R(-, M)$ 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?

DefinitionChain Complex and Homology

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

1

\quad \cdots \xrightarrow{d_{n+1}} C_{n+1} \xrightarrow{d_n} C_n \xrightarrow{d_{n-1}} C_{n-1} \xrightarrow{d_{n-2}} \cdots

Derivation hint: 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

2

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

Derivation hint: This is the algebraic shadow of "boundary of a boundary is zero" $\partial^2 = 0$ from simplicial topology. The condition $[2]$ gives $\operatorname{im} d_{n+1} \subseteq \ker d_n$ , so the $n$ -th homology object is well-defined:

3

\quad H_n(C_\bullet) = \ker d_n / \operatorname{im} d_{n+1} = Z_n / B_n,

Derivation hint: Elements of $Z_n = \ker d_n$ are <strong>cycles</strong>, elements of $B_n = \operatorname{im} d_{n+1}$ are <strong>boundaries</strong>; homology is cycles modulo boundaries. Here:

$\mathcal{A}$ — 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.
$C_\bullet$ 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 $H^n$ .

DefinitionChain Map and Quasi-Isomorphism

A chain map $f_\bullet : C_\bullet \to D_\bullet$ 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 ( $f \simeq g$ ) if there exist maps $s_n : C_n \to D_{n+1}$ with

4

\quad f_n - g_n = d_{n+1}^D s_n + s_{n-1} d_n^C.

Derivation hint: 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.

TheoremLong Exact Sequence in Homology

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

5

\quad \cdots \to H_n(A) \xrightarrow{f_*} H_n(B) \xrightarrow{g_*} H_n(C) \xrightarrow{\partial} H_{n-1}(A) \to \cdots

Derivation hint: Build $\partial$ by the "diagram chase" of the Snake Lemma — lift a cycle in $C_n$ to $B_n$ , push through $d$ , and pull back to $A_{n-1}$ .

TheoremDerived Functors

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

6

\quad L_n F(A) = H_n\big(F(P_\bullet)\big),

Derivation hint: Independence of the resolution follows from the Comparison Theorem: any two projective resolutions are chain homotopy equivalent, and $F$ preserves homotopies. with $L_0 F \cong F$ (since $F$ is right exact). Dually, right derived functors $R^n F$ use injective resolutions of a left-exact $F$ . The fundamental special cases:

7

\quad \operatorname{Tor}_n^R(M, N) = L_n(- \otimes_R N)(M), \qquad \operatorname{Ext}^n_R(M, N) = R^n \operatorname{Hom}_R(M, -)(N).

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

ProofThe 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 $0 \to A_\bullet \xrightarrow{f} B_\bullet \xrightarrow{g} C_\bullet \to 0$ 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 $d^B b \in \ker g_{n-1} = \operatorname{im} f_{n-1}$ .

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 $\operatorname{im} f_n$ ; 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, $g_* f_* = (gf)_* = 0$ , and a class in $\ker g_*$ lifts back along $f$ by the construction above. $\blacksquare$

∎

ProofIndependence of Resolution (Comparison Theorem)

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

Step 1. Let $P_\bullet \to A$ and $Q_\bullet \to A$ be two projective resolutions. We claim there is a chain map $\varphi : P_\bullet \to Q_\bullet$ lifting $\operatorname{id}_A$ .

Step 2. Build $\varphi$ inductively. Since $P_0$ is projective and $Q_0 \to A$ is surjective, the map $P_0 \to A$ lifts to $\varphi_0 : P_0 \to Q_0$ . Continuing degree by degree, using projectivity of $P_n$ and exactness of $Q_\bullet$ , produces all $\varphi_n$ .

Step 3. Any two such lifts $\varphi, \varphi'$ are chain homotopic — same inductive construction applied to the difference, which lifts the zero map.

Step 4. By symmetry there is $\psi : Q_\bullet \to P_\bullet$ , and $\psi\varphi \simeq \operatorname{id}_{P}$ , $\varphi\psi \simeq \operatorname{id}_Q$ . So $P_\bullet$ and $Q_\bullet$ are chain homotopy equivalent.

Step 5. Additive functors preserve homotopies (apply $F$ to equation $[4]$ ). Hence $F(P_\bullet) \simeq F(Q_\bullet)$ , and homotopic complexes have isomorphic homology. Therefore $H_n(F(P_\bullet)) \cong H_n(F(Q_\bullet))$ . $\blacksquare$

∎

mathematician · 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.

🧠 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. $\operatorname{Tor}_1(\mathbb{Z}/n, \mathbb{Z}/m) = \mathbb{Z}/\gcd(n,m)$ literally measures the shared torsion. $\operatorname{Ext}^1(A, B)$ classifies the ways $B$ can be "extended" by $A$ — the secret life of short exact sequences.

Figure

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

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ExampleComputing Tor over ℤ

Given: Compute $\operatorname{Tor}_n^{\mathbb{Z}}(\mathbb{Z}/6, \mathbb{Z}/4)$ for all $n$ .

Step 1 — Resolve $\mathbb{Z}/6$ . A free resolution: $0 \to \mathbb{Z} \xrightarrow{\times 6} \mathbb{Z} \to \mathbb{Z}/6 \to 0.$ So $P_1 = \mathbb{Z}$ , $P_0 = \mathbb{Z}$ , $d_1 = \times 6$ , and $P_n = 0$ for $n \geq 2$ .

Step 2 — Apply $- \otimes_{\mathbb{Z}} \mathbb{Z}/4$ . Delete the augmentation and tensor: $0 \to \mathbb{Z}/4 \xrightarrow{\times 6} \mathbb{Z}/4 \to 0.$ Note $\times 6 \equiv \times 2 \pmod 4$ .

Step 3 — Take homology.

$\operatorname{Tor}_0 = \operatorname{coker}(\times 2 : \mathbb{Z}/4 \to \mathbb{Z}/4) = \mathbb{Z}/4 / 2(\mathbb{Z}/4) = \mathbb{Z}/2$ . (Equals $\mathbb{Z}/6 \otimes \mathbb{Z}/4 = \mathbb{Z}/\gcd(6,4)$ . ✓)
$\operatorname{Tor}_1 = \ker(\times 2 : \mathbb{Z}/4 \to \mathbb{Z}/4) = \{0, 2\} \cong \mathbb{Z}/2$ .
$\operatorname{Tor}_n = 0$ for $n \geq 2$ (resolution has length 1).

Boxed answer:

\boxed{\operatorname{Tor}_0 \cong \mathbb{Z}/2,\quad \operatorname{Tor}_1 \cong \mathbb{Z}/2 = \mathbb{Z}/\gcd(6,4),\quad \operatorname{Tor}_{\geq 2} = 0.}

Consistent with the general formula $\operatorname{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/a, \mathbb{Z}/b) \cong \mathbb{Z}/\gcd(a,b)$ .

ExampleComputing Ext and Classifying Extensions

Given: Compute $\operatorname{Ext}^n_{\mathbb{Z}}(\mathbb{Z}/2, \mathbb{Z})$ and interpret $\operatorname{Ext}^1$ .

Step 1 — Resolve $\mathbb{Z}/2$ . $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2 \to 0.$

Step 2 — Apply $\operatorname{Hom}_{\mathbb{Z}}(-, \mathbb{Z})$ . This contravariant functor turns the resolution into a cochain complex: $0 \to \operatorname{Hom}(\mathbb{Z}, \mathbb{Z}) \xrightarrow{(\times 2)^*} \operatorname{Hom}(\mathbb{Z}, \mathbb{Z}) \to 0,$ i.e. $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to 0$ (precomposition with $\times 2$ is multiplication by 2).

Step 3 — Cohomology.

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

Step 4 — Interpret. $\operatorname{Ext}^1(\mathbb{Z}/2, \mathbb{Z}) \cong \mathbb{Z}/2$ classifies extensions $0 \to \mathbb{Z} \to E \to \mathbb{Z}/2 \to 0$ up to equivalence. The two classes:

the split extension $E = \mathbb{Z} \oplus \mathbb{Z}/2$ (zero class),
the non-split extension $E = \mathbb{Z}$ with $\mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \twoheadrightarrow \mathbb{Z}/2$ (nonzero class).

Boxed answer:

\boxed{\operatorname{Ext}^0 = 0,\quad \operatorname{Ext}^1 \cong \mathbb{Z}/2 \;(\text{2 extension classes}),\quad \operatorname{Ext}^{\geq 2} = 0.}

⚠️ Common mistake

"Tor and Ext vanish for nice modules, so they're rarely needed." False emphasis. $\operatorname{Tor}_n(M,N)=0$ for all $n>0$ exactly characterizes flatness of $M$ , and $\operatorname{Ext}^n(M,-)=0$ 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 $D(\mathcal{A})$ , where they are formally inverted, to treat them as isomorphisms.

Counterexample

Right derived functors of an already-exact functor need not vanish above degree 0 unless the functor is exact. Consider $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}/2, -)$ . One might guess that since $\mathbb{Z}/2$ is "small," higher Ext vanishes. But $\operatorname{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/2, \mathbb{Z}) = \mathbb{Z}/2 \neq 0$ , computed above. The functor $\operatorname{Hom}(\mathbb{Z}/2,-)$ is only left exact, not exact, so its higher derived functors are genuinely nonzero. Conversely, flat does not imply projective: $\mathbb{Q}$ is a flat $\mathbb{Z}$ -module (all $\operatorname{Tor}_n^{\mathbb{Z}}(\mathbb{Q},-) = 0$ for $n>0$ ) but $\mathbb{Q}$ is not projective over $\mathbb{Z}$ — 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.

connection

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 $\operatorname{Ext}^n_{\mathbb{Z}G}(\mathbb{Z}, M)$ , 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.

🔬 Where this appears

⏳ History

- 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.

open-problem

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 $\operatorname{Ext}$ and $\operatorname{Tor}$ 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 $D(\mathcal{A})$ : 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.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also