Homological algebra — chain complexes, exact sequences, and derived functors

Intuition Beginner

Imagine a shape built from points, line segments, and triangles glued along their edges. A boundary operator records, for each piece, the pieces on its rim: the boundary of a segment is its two endpoints, and the boundary of a triangle is its three edges. Going around a triangle and adding up its three edges, each inner edge cancels against a neighbour pointing the other way. So the boundary of a boundary is zero.

The reverse can fail. A closed loop of segments has no boundary — it returns to its start — yet no triangle may be present to fill it in. That gap, a closed loop with no filling, is a hole. Homology counts these holes. Exactness is the ideal situation where every closed loop does have a filling; homology measures exactly how far a chain falls short of being exact.

The same pattern — boundary of a boundary is zero, and we measure the leftover — reappears across all of higher algebra. In the algebraic setting it becomes the toolkit called homological algebra, and the holes become obstructions to solving equations that no amount of clever rearrangement can remove.

Visual Beginner

A chain complex is a row of modules connected by boundary operators, each one handing the next what it could not kill.

The shaded overlap — what the incoming operator reaches but the outgoing operator does not catch — is the homology. Vanishing homology means the row is exact; non-vanishing homology is the obstruction the algebra must remember.

Piece	Boundary operator sends it to
triangle (face)	its three edges, with orientation signs
segment (edge)	the difference of its two endpoints
point (vertex)	zero

Worked example Beginner

Take a triangle with vertices $v_0,v_1,v_2$, edges $e_{01},e_{12},e_{20}$, and a single face $f$. The boundary operator sends the face to its three oriented edges, and each edge to the difference of its endpoints.

Step 1. The boundary of the face is $e_{01}+e_{12}+e_{20}$.

Step 2. Apply the boundary operator again, to each edge. Edge $e_{01}$ becomes $v_1-v_0$, edge $e_{12}$ becomes $v_2-v_1$, and edge $e_{20}$ becomes $v_0-v_2$.

Step 3. Add these: $(v_1-v_0)+(v_2-v_1)+(v_0-v_2)$. The $v_1$ terms cancel, the $v_2$ terms cancel, and the $v_0$ terms cancel. The total is $0$.

Boundary of a boundary is zero. Now delete the face but keep the three edges. The loop $e_{01}+e_{12}+e_{20}$ still has boundary zero — it closes — yet no face remains to fill it. That unfilled loop is one homology class: one hole.

Check your understanding Beginner

Formal definition Intermediate+

A chain complex $(C_{\bullet },d)$ over a ring $R$ is a sequence of $R$-modules and maps

$$\cdots ⟶C_{n+1}\stackrel{d_{n+1}}{\to }{C}_n\stackrel{d_n}{\to }{C}_{n-1}⟶\cdots$$

with $d_n\circ d_{n+1}=0$ for every $n$ ^{[Weibel Ch. 1]}. The maps $d_n$ are differentials or boundary operators. The condition $d^2=0$ forces $\mathrm{im}{d}_{n+1}\subseteq \ker d_n$, so the quotient below is defined.

The $n$-th homology of $C_{\bullet }$ is

$$H_n(C_{\bullet })=\frac{\ker d_n}{\mathrm{im}{d}_{n+1}}=\frac{Z_n(C)}{B_n(C)},$$

where $Z_n=\ker d_n$ are the $n$-cycles and $B_n=\mathrm{im}{d}_{n+1}$ are the $n$-boundaries. A complex is exact at $C_n$ when $H_n=0$, equivalently $\mathrm{im}{d}_{n+1}=\ker d_n$. A cochain complex has differentials that raise degree, $d^n:C^n\to C^{n+1}$; its cohomology is $H^n=\ker d^n/\mathrm{im}{d}^{n-1}$.

A chain map $f_{\bullet }:C_{\bullet }\to D_{\bullet }$ is a family of module maps commuting with the differentials ($d_n^D{f}_n=f_{n-1}{d}_n^C$). Chain maps induce maps $H_n(f):H_n(C)\to H_n(D)$. Two chain maps $f,g$ are chain homotopic ($f\simeq g$) when there are maps $s_n:C_n\to D_{n+1}$ with $f_n-g_n=d_{n+1}^D{s}_n+s_{n-1}{d}_n^C$; chain-homotopic maps induce the same map on homology.

A chain map $f:C_{\bullet }\to D_{\bullet }$ is a quasi-isomorphism when every induced map $H_n(f):H_n(C)\to H_n(D)$ is an isomorphism. Quasi-isomorphism is weaker than isomorphism of complexes: two distinct complexes can carry the same homology and still differ degree by degree. The passage to the derived category in 04.03.11 formally inverts all quasi-isomorphisms, so that a quasi-isomorphism becomes a genuine isomorphism there; this is the move that lets derived functors be computed from any convenient resolution of the same object.

A sequence $A\stackrel{f}{\to }B\stackrel{g}{\to }C$ is exact at $B$ when $\mathrm{im}f=\ker g$. A short exact sequence is

$$0⟶A^{\prime }\stackrel{i}{\to }A\stackrel{p}{\to }{A}^{\prime \prime }⟶0,$$

exact at every term: $i$ is injective, $p$ is surjective, and $\mathrm{im}i=\ker p$. It splits when $A\cong A^{\prime }\oplus A^{\prime \prime }$ compatibly with $i$ and $p$; equivalently there is a section $s:A^{\prime \prime }\to A$ with $p\circ s=\mathrm{id}$, or a retraction $r:A\to A^{\prime }$ with $r\circ i=\mathrm{id}$.

An $R$-module $P$ is projective when every diagram $P\to B\leftarrow C$ with $B\twoheadrightarrow C$ admits a lift $P\to C$; equivalently ${\mathrm{Hom}}_R(P,-)$ is exact. Dually, $I$ is injective when ${\mathrm{Hom}}_R(-,I)$ is exact, i.e. every map into a submodule extends. A category has enough projectives when every module is a quotient of a projective; $R\text{-Mod}$ always has enough projectives (every module is a quotient of a free module). A projective resolution of $M$ is an exact sequence

$$\cdots ⟶P_2⟶P_1⟶P_0⟶M⟶0$$

with each $P_i$ projective ^{[Cartan-Eilenberg Ch. VI]}. Injective resolutions are defined dually.

A covariant functor $F$ is right exact when it preserves cokernels; $-{\otimes }_{R}N$ is right exact. It is left exact when it preserves kernels; ${\mathrm{Hom}}_R(M,-)$ is left exact. The derived functors $L_{n}F$ (for right exact $F$, via projective resolutions) and $R^{n}F$ (for left exact $F$, via injective resolutions) measure the failure of $F$ to be exact, and the two workhorses are

$${\mathrm{Tor}}_n^R(M,N)=L_n(-{\otimes }_{R}N)(M),{\mathrm{Ext}}_R^n(M,N)=R^n{\mathrm{Hom}}_R(M,-)(N).$$

Counterexamples to common slips

• Not every complex is exact. For $0\to \mathbb{Z}\stackrel{\times 2}{\to }\mathbb{Z}\to 0$ concentrated in degrees $1,0$, the cokernel of $\times 2$ is $\mathbb{Z}/2$, so $H_0\cong \mathbb{Z}/2$ — homology detects the obstruction, it does not vanish.
• Tensor is not left exact. From $0\to \mathbb{Z}\stackrel{\times n}{\to }\mathbb{Z}$, tensoring with $\mathbb{Z}/n$ gives $\mathbb{Z}/n\stackrel{\times n}{\to }\mathbb{Z}/n$, the zero map, so injectivity is lost. ${\mathrm{Tor}}_1^{\mathbb{Z}}(\mathbb{Z}/n,\mathbb{Z}/n)\cong \mathbb{Z}/n$ records this failure.
• Projective need not be free. Over many rings (rings of integers in number fields, or the tangent bundle of a sphere via Swan's theorem) projective modules exist that admit no basis; over a PID, projective coincides with free, which is the simple case.

Key theorem with proof Intermediate+

Theorem (Snake lemma). Given a commutative diagram of $R$-modules with exact rows

$$\begin{array}{ccccccc}& \ker a& \to & \ker b& \to & \ker c& \\ & \downarrow & & \downarrow & & \downarrow & \\ 0\to & A^{\prime }& \stackrel{i}{\to }& A& \stackrel{p}{\to }& A^{\prime \prime }& \to 0\\ & a\downarrow & & b\downarrow & & c\downarrow & \\ 0\to & B^{\prime }& \stackrel{j}{\to }& B& \stackrel{q}{\to }& B^{\prime \prime }& \to 0\\ & \downarrow & & \downarrow & & \downarrow & \\ & \mathrm{coker}a& \to & \mathrm{coker}b& \to & \mathrm{coker}c& \end{array}$$

there is an exact sequence

$$\ker a\to \ker b\to \ker c\stackrel{\delta }{\to }\mathrm{coker}a\to \mathrm{coker}b\to \mathrm{coker}c,$$

in which the connecting homomorphism $\delta$ sends $x\in \ker c\subseteq A^{\prime \prime }$ to the class of $j^{-1}(b(p^{-1}(x)))$ in $\mathrm{coker}a$. The proof is a diagram chase checking exactness at each of the six positions ^{[Weibel Ch. 1]}.

To see how $\delta$ is built, chase an element $x\in \ker c$ through the diagram. Lift it along the surjection $p$ to an element $\tilde{x}\in A$ with $p(\tilde{x})=x$. Apply $b$ and then $q$: since $q(b(\tilde{x}))=c(p(\tilde{x}))=c(x)=0$, exactness of the bottom row places $b(\tilde{x})$ in $\ker q=\mathrm{im}j$, so a unique $y\in B^{\prime }$ has $j(y)=b(\tilde{x})$. The class of $y$ in $\mathrm{coker}a$ is independent of the lift $\tilde{x}$, and that class is $\delta (x)$. This zig-zag — lift right, push down, pull back left — is the prototype of every connecting homomorphism in homological algebra, including the boundary map of the long exact sequence in homology and the Bockstein operations of 04.03.06.

The snake lemma is the algebraic engine behind the long exact sequence in homology: a short exact sequence of complexes $0\to C^{\prime }\to C\to C^{\prime \prime }\to 0$ yields, by applying the snake lemma degree by degree, the long exact sequence

$$\cdots \to H_n(C^{\prime })\to H_n(C)\to H_n(C^{\prime \prime })\stackrel{\delta }{\to }{H}_{n-1}(C^{\prime })\to \cdots$$

Theorem (Five lemma). Given a commutative diagram with exact rows

$$\begin{array}{ccccccccc}{A}_1& \to & A_2& \to & A_3& \to & A_4& \to & A_5\\ f_1\downarrow & & f_2\downarrow & & f_3\downarrow & & f_4\downarrow & & f_5\downarrow \\ B_1& \to & B_2& \to & B_3& \to & B_4& \to & B_5\end{array}$$

if $f_1,f_2,f_4,f_5$ are isomorphisms, then $f_3$ is an isomorphism.

Proof. Injectivity of $f_3$. Let $a_3\in A_3$ with $f_3(a_3)=0$. Then $f_4{g}_3(a_3)=g_3^{\prime }{f}_3(a_3)=0$ by commutativity, and since $f_4$ is injective, $g_3(a_3)=0$. Exactness of the top row gives $a_3=g_2(a_2)$ for some $a_2\in A_2$. Commutativity yields $g_2^{\prime }{f}_2(a_2)=f_3{g}_2(a_2)=f_3(a_3)=0$, so $f_2(a_2)=\ker g_2^{\prime }=\mathrm{im}{g}_1^{\prime }=g_1^{\prime }(b_1)$ for some $b_1$. Since $f_1$ is surjective, $b_1=f_1(a_1)$, and $f_2{g}_1(a_1)=g_1^{\prime }{f}_1(a_1)=g_1^{\prime }(b_1)=f_2(a_2)$. Injectivity of $f_2$ gives $a_2=g_1(a_1)$, whence $a_3=g_2(a_2)=g_2{g}_1(a_1)=0$. So $f_3$ is injective.

Surjectivity of $f_3$. Let $b_3\in B_3$. Since $f_5$ is injective and $g_4^{\prime }{g}_3^{\prime }(b_3)=0$, exactness gives $g_3^{\prime }(b_3)=g_4^{\prime }(b_4)$ for some $b_4$ by first using $f_5{g}_4^{\prime }(b_3)$: $f_5{g}_4^{\prime }(b_3)=g_4^{\prime }{g}_3^{\prime }(b_3)=0$, hence $g_4^{\prime }(b_3)=0$ as $f_5$ injective. Thus $b_3=g_2^{\prime }(b_2)$ for some $b_2\in B_2$. Surjectivity of $f_2$ gives $b_2=f_2(a_2)$. Set $a_3=g_2(a_2)$. Then $f_3(a_3)-b_3=f_3{g}_2(a_2)-g_2^{\prime }(b_2)=g_2^{\prime }{f}_2(a_2)-g_2^{\prime }(b_2)=0$, so $f_3$ hits $b_3$. Hence $f_3$ is surjective, and an isomorphism. $\square$

Bridge. This diagram-chase lemma builds toward 04.03.01 (sheaf cohomology), where the long exact sequence is the engine that computes obstructions on spaces, and appears again in 04.03.06 (derived functors) as the mechanism by which a short exact sequence of sheaves yields a long exact sequence of cohomology groups. The foundational reason diagram chases recur is that exactness is a pointwise condition that nevertheless propagates globally along the arrows; this is exactly the pattern that makes homological algebra the universal language for measuring obstructions, and the bridge is that every derived functor is a systematic device for extending a half-exact functor to a fully exact one by adjoining these connecting maps.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none is recorded for this unit. Mathlib already carries the surrounding infrastructure — CategoryTheory.Abelian.ShortComplex for a short exact sequence, HomologicalComplex for $(C_{\bullet },d)$, the homotopy category, and the statement of the snake lemma — but the project has not yet wired the elementary definitions here (chain complex over a fixed ring, $H_n=\ker d_n/\mathrm{im}{d}_{n+1}$, $\mathrm{Tor}$ and $\mathrm{Ext}$ over $\mathbb{Z}$) to the downstream sheaf-cohomology 04.03.01 and derived-category 04.03.11 consumers. That wiring is the open task captured in Mathlib gap analysis; until it lands, no Lean module is declared.

Advanced results Master

The defining achievement of homological algebra is that every half-exact functor extends, via resolutions, to a sequence of derived functors connected by long exact sequences. Concretely, a left-exact covariant functor $F:\mathcal{A}\to \mathcal{B}$ between abelian categories, with $\mathcal{A}$ having enough injectives, gives right derived functors $R^{n}F$ defined by $R^{n}F(A)=H^n(F(I^{\bullet }))$ for any injective resolution $A\to I^{\bullet }$; the result is independent of the resolution chosen, because injective resolutions are unique up to chain homotopy ^{[Grothendieck Tohoku]}. A short exact sequence $0\to A^{\prime }\to A\to A^{\prime \prime }\to 0$ then yields the long exact sequence

$$0\to F(A^{\prime })\to F(A)\to F(A^{\prime \prime })\to R^{1}F(A^{\prime })\to R^{1}F(A)\to R^{1}F(A^{\prime \prime })\to R^{2}F(A^{\prime })\to \cdots$$

so $R^{1}F$ precisely repairs the failure of $F$ to be right exact. The dual construction with projective resolutions gives the left derived functors $L_{n}F$ of a right-exact functor ^{[Cartan-Eilenberg Ch. V]}.

The two universal examples are $\mathrm{Ext}$ and $\mathrm{Tor}$. Over any ring, ${\mathrm{Ext}}_R^n(M,N)$ may be computed either from a projective resolution of $M$ (applying ${\mathrm{Hom}}_R(-,N)$) or from an injective resolution of $N$ (applying ${\mathrm{Hom}}_R(M,-)$); the two agree — this is the balance of $\mathrm{Ext}$, and $\mathrm{Tor}$ satisfies the analogous symmetry ${\mathrm{Tor}}_n^R(M,N)\cong {\mathrm{Tor}}_n^R(N,M)$ when $R$ is commutative ^{[Weibel §2.7]}. In the lowest degree, ${\mathrm{Ext}}_R^1(M,N)$ classifies extensions of $M$ by $N$ up to the Yoneda equivalence relation: the split extension is the zero class, and every non-split short exact sequence $0\to N\to E\to M\to 0$ is a non-zero element of ${\mathrm{Ext}}^1$.

Three structural facts make the theory usable. First, ${\mathrm{Tor}}_0^R(M,N)\cong M{\otimes }_{R}N$ and ${\mathrm{Ext}}_R^0(M,N)\cong {\mathrm{Hom}}_R(M,N)$, so the derived functors literally extend the original ones. Second, if $M$ has a projective resolution of length $\ell$, then $L_{n}F(M)=0$ for $n>\ell$; over a PID every module has a length-$1$ resolution, so only $L_0$ and $L_1$ survive, which is why computations over $\mathbb{Z}$ reduce to a single map (Exercises 7 and 8). Third, a functor is exact precisely when all its higher derived functors vanish, restoring the clean case in which short exact sequences are preserved.

A second technique, dimension shifting, turns the long exact sequence into a recursive computation tool. Splicing the long exact sequence for $0\to N\to E\to M\to 0$ with a resolution of $M$ expresses $R^{n}F(N)$ in terms of $R^{n-1}F$ of the next syzygy, stepping the degree down by one at the cost of replacing the module by a kernel of the resolution. Iterating reduces any $R^{n}F$ to a computation of $R^{1}F$ on successive syzygies — which is why a projective resolution of length $\ell$ kills $R^{n}F$ for $n>\ell$, and why finitely generated modules over a regular local ring have bounded cohomological dimension.

When the input is not a single short exact sequence but a whole filtered complex, the long exact sequences assemble into a spectral sequence: a page $E_r^{p,q}$ of bigraded modules with differentials $d_r$ of bidegree $(r,1-r)$ converging, under mild boundedness hypotheses, to the associated graded of the desired homology ^{[Weibel Ch. 5]}. The two archetypes are the Grothendieck spectral sequence $R^{p}G\circ R^{q}F\Rightarrow R^{p+q}(G\circ F)$ for composed functors and the hypercohomology spectral sequence that computes sheaf cohomology from an open cover — the latter being the bridge from this unit to the Čech-to-derived-functor comparison in 04.03.03 and the vanishing theorems of 04.03.05.

Synthesis. Derived functors are the load-bearing construction of the whole subject: the snake lemma proved in the Key theorem builds toward 04.03.06 where $\mathrm{Ext}$ and $\mathrm{Tor}$ are computed by resolving either argument, the long exact sequence in homology appears again in 04.03.01 as the sheaf-cohomology sequence governing coherent sheaves on schemes, this is exactly the mechanism that converts a half-exact functor into a fully exact one, ${\mathrm{Ext}}^1$ classifying extensions generalises the split-versus-non-split dichotomy to all modules at once, the projective-versus-injective resolution story is dual to itself so that left and right derived functors are two faces of one construction, and the central insight is that homology measures the precise obstruction to exactness; putting these together, the bridge is that $\mathrm{Tor}$, $\mathrm{Ext}$, sheaf cohomology, and group cohomology are all instances of the single derived-functor construction.

Full proof set Master

Proposition (Existence of projective resolutions). In $R\text{-Mod}$, every module $M$ admits a projective resolution.

Proof. Every module is a quotient of a free module: choose a generating set $\{m_{\alpha }\}$ of $M$ and let $P_0={⨁}_{\alpha }R$ surject onto $M$ by $e_{\alpha }\mapsto m_{\alpha }$; call the kernel $K_0$. Since free modules are projective, $P_0$ is projective. Now repeat: $K_0$ is itself a quotient of a free module $P_1$, giving $P_1\twoheadrightarrow K_0$ with kernel $K_1$. Composing $P_1\twoheadrightarrow K_0\hookrightarrow P_0$ yields $d_1$. Iterating produces an exact sequence $\cdots \to P_2\to P_1\to P_0\to M\to 0$ with each $P_i$ free, hence projective. $\square$

Proposition (${\mathrm{Ext}}^1$ classifies extensions). For $R$-modules $M,N$, the set of equivalence classes of short exact sequences $0\to N\to E\to M\to 0$ (the Yoneda equivalence) is in bijection with ${\mathrm{Ext}}_R^1(M,N)$.

Proof sketch. Fix a projective resolution $P_{\bullet }\to M$ and let $K_0=\ker (P_0\to M)$. Apply ${\mathrm{Hom}}_R(-,N)$ to the truncated resolution $0\to K_0\to P_0\to M\to 0$; since $P_0$ is projective, $\mathrm{Hom}(P_0,N)\to \mathrm{Hom}(K_0,N)\to {\mathrm{Ext}}^1(M,N)\to 0$ is exact, so ${\mathrm{Ext}}^1(M,N)\cong \mathrm{Hom}(K_0,N)/\mathrm{im}\mathrm{Hom}(P_0,N)$. A short exact sequence $0\to N\to E\to M\to 0$ is the pushout of $0\to K_0\to P_0\to M\to 0$ along a map $K_0\to N$; two maps give equivalent extensions exactly when they differ by an element of $\mathrm{im}\mathrm{Hom}(P_0,N)$, which is precisely the quotient. The split extension corresponds to the zero class. $\square$

Connections Master

• Rings and modules 01.03.01. Homological algebra lives inside the abelian category $R\text{-Mod}$ of modules over a ring 01.03.01; every construction here — chain complex, exact sequence, $\mathrm{Tor}$, $\mathrm{Ext}$ — is a machine built on top of the modules and homomorphisms established in that prerequisite unit, and tensor products of modules are the input from which $\mathrm{Tor}$ is derived.

• Sheaf cohomology 04.03.01. The global-sections functor $\Gamma (X,-)$ on sheaves of modules is left exact but not exact, and its right derived functors $H^n(X,-)=R^n\Gamma (X,-)$ are sheaf cohomology 04.03.01; the long exact sequence proved here is exactly the sheaf-cohomology long exact sequence that drives algebraic geometry.

• Derived functors and Ext in geometry 04.03.06. The $\mathrm{Ext}$ and derived-functor formalism developed here is specialised to coherent sheaves in 04.03.06, where $\mathrm{Ext}$ sheaves classify extensions of sheaves and underpin Serre duality; the abstract balance of $\mathrm{Ext}$ becomes a concrete geometric tool.

• Derived categories 04.03.11. Passing from individual derived functors to the derived category $\mathbf{D}(\mathcal{A})$ — the localisation of the category of complexes at quasi-isomorphisms 04.03.11 — packages all the long exact sequences here into a single triangulated structure in which $RF$ and $LF$ become exact functors between derived categories.

• Galois and group cohomology 01.02.15. Group cohomology $H^n(G,M)$ is the right derived functor of the $G$-invariants functor $M\mapsto M^G$ on $G$-modules, and specialises to Galois cohomology and Hilbert's theorem 90 in 01.02.15; it is the most direct arithmetic application of the derived-functor machine.

Historical & philosophical context Master

Homological algebra was forged between 1940 and 1957 from three pressures: the need to organise the cohomology theories then proliferating in topology (singular, Čech, sheaf); the algebraic reformulation of extension problems in group theory and ring theory; and Grothendieck's insistence that the subject be re-derived inside an arbitrary abelian category rather than only module categories ^{[Grothendieck 1957]}. The founding synthesis was Cartan and Eilenberg's 1956 Homological Algebra, which unified $\mathrm{Tor}$, $\mathrm{Ext}$, resolutions, and spectral sequences under a single diagrammatic discipline ^{[Cartan-Eilenberg 1956]}.

The 1957 Tohoku paper of Grothendieck then showed that the entire Cartan-Eilenberg machine depended only on the axioms of an abelian category and the existence of enough injectives, which made the theory portable to sheaves of modules and so planted the seed of modern algebraic geometry. Weibel's 1994 textbook and Gelfand-Manin's 2003 Methods of Homological Algebra record the further passage, in the late twentieth century, from derived functors to derived categories — a shift of viewpoint in which the long exact sequences proved here become shadows of a single triangulated structure.

The philosophical point is that homology replaces the unanswerable question "is this functor exact?" with the answerable one "how far does it deviate from exactness, in every degree?" The obstruction, once named and graded, becomes a computable invariant — and that move, from a yes/no failure to a measured invariant, is the move that makes homological algebra the lingua franca of modern geometry, topology, and number theory.

Bibliography Master

@book{Weibel1994,
  author = {Weibel, Charles A.},
  title = {An Introduction to Homological Algebra},
  publisher = {Cambridge University Press},
  series = {Cambridge Studies in Advanced Mathematics},
  volume = {38},
  year = {1994},
}

@book{GelfandManin2003,
  author = {Gelfand, Sergei I. and Manin, Yuri I.},
  title = {Methods of Homological Algebra},
  edition = {2nd},
  publisher = {Springer Monographs in Mathematics},
  year = {2003},
}

@book{CartanEilenberg1956,
  author = {Cartan, Henri and Eilenberg, Samuel},
  title = {Homological Algebra},
  publisher = {Princeton University Press},
  year = {1956},
}

@book{HiltonStammbach1997,
  author = {Hilton, Peter J. and Stammbach, Urs},
  title = {A Course in Homological Algebra},
  edition = {2nd},
  publisher = {Springer GTM 4},
  year = {1997},
}

@article{Grothendieck1957,
  author = {Grothendieck, Alexander},
  title = {Sur quelques points d'alg\`ebre homologique},
  journal = {Tohoku Mathematical Journal},
  volume = {9},
  number = {2},
  pages = {119--221},
  year = {1957},
}

@book{MacLane1963,
  author = {Mac Lane, Saunders},
  title = {Homology},
  publisher = {Springer Classics in Mathematics},
  year = {1963},
}