Exact Sequences and the Snake Lemma

One-line statement: An exact sequence is a sequence of objects and morphisms in an abelian category where the image of each map equals the kernel of the next; the Snake Lemma constructs a natural long exact sequence connecting the kernels and cokernels of three vertically related short exact sequences via a "connecting homomorphism." Key result in words: Given a commutative diagram of three short exact sequences stacked into two rows, the kernels and cokernels of the vertical maps splice together into a single six-term exact sequence, with the magic "snake" connecting map δ threading from the leftmost cokernel back to the rightmost kernel. Where it applies: Homological algebra, algebraic topology (long exact sequences of pairs, Mayer–Vietoris), commutative algebra, sheaf cohomology, representation theory, algebraic K-theory. Who/when: The diagrammatic methods crystallized in the 1940s–1950s (Eilenberg, Mac Lane, Cartan, Steenrod); the Snake Lemma became standard with the rise of homological algebra (Cartan–Eilenberg, 1956) and category theory.

An exact sequence in an abelian category is a (possibly infinite) chain of objects connected by morphisms such that at every interior object the image of the incoming morphism coincides exactly with the kernel of the outgoing one. This single condition encodes injectivity, surjectivity, and the precise "failure of exactness" that homology measures. Exact sequences matter because they convert global structural questions—does this extension split? how do these cohomology groups relate?—into bookkeeping over maps, and because the Snake Lemma provides the universal mechanism for producing the long exact sequences that pervade topology and algebra. The surprising fact is that the connecting homomorphism δ, which appears to "jump" across the diagram from cokernels back to kernels, is canonical, natural, and entirely forced by the diagram—even though its definition requires a non-canonical choice of preimage that miraculously washes out.

The Core Insight

Exactness localizes a global property: instead of asking whether a long composite is zero, you check one balance condition (image = kernel) at each node. The Snake Lemma then says that whenever you have a 3×2 commutative grid of short exact sequences, the obstruction to "kernels mapping onto kernels and cokernels injecting into cokernels" is itself measured by a single connecting map, and the entire defect organizes into one exact sequence. Exactness is the conservation law of homological algebra, and the snake map is the current that flows when conservation is locally violated.

Formal Statement

Exact Sequence

Let $A$ be an abelian category. A sequence of objects and morphisms

\dots ⟶ A_{n + 1} f_{n + 1} A_{n} f_{n} A_{n - 1} ⟶ \dots

⟳This is a diagram in

A

; composability requires

cod (f_{n + 1}) = dom (f_{n}) = A_{n}

is exact at $A_{n}$ if $im f_{n + 1} = ker f_{n}$ as subobjects of $A_{n}$ . The sequence is exact if it is exact at every interior object. A first consequence is that consecutive maps compose to zero:

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

⟳

im f_{n + 1} = ker f_{n}

implies

im f_{n + 1} \subseteq ker f_{n}

, so

f_{n}

kills everything in the image of

f_{n + 1}

; this is necessary but not sufficient for exactness.

Short Exact Sequence

A short exact sequence (SES) is an exact sequence of the form

0 ⟶ A f B g C ⟶ 0.

⟳Exactness at

A

forces

ker f = im (0 \to A) = 0

, so

f

is monic; exactness at

C

forces

im g = ker (C \to 0) = C

, so

g

is epic; exactness at

B

gives

im f = ker g

Equivalently $f$ is a monomorphism, $g$ is an epimorphism, and $g$ induces an isomorphism $B / im f ≅ C$ . We say $B$ is an extension of $C$ by $A$ .

Kernel, Cokernel, Image in an Abelian Category

For a morphism $φ : M \to N$ :

ker φ = eq (φ, 0), coker φ = coeq (φ, 0), im φ = ker (coker φ) .

⟳In

A

all (co)equalizers with zero exist; the image is defined as the kernel of the cokernel, equivalently the smallest subobject through which

φ

factors. In

R

-Mod these reduce to the usual set-theoretic definitions.

The Snake Lemma

Let $A$ be an abelian category and suppose the following diagram commutes with exact rows:

00 \to \to A ↓ a A^{'} i j B ↓ b B^{'} p q C ↓ c C^{'} \to \to 00

⟳Commutativity means

j \circ a = b \circ i

and

q \circ b = c \circ p

; the top and bottom rows are short exact sequences.

Then there is a natural connecting homomorphism $δ : ker c \to coker a$ making the following sequence exact:

0 \to ker a \overset{ˉ}{i} ker b \overset{p}{ˉ} ker c δ coker a \overset{ˉ}{j} coker b \overset{q}{ˉ} coker c \to 0.

⟳The barred maps are induced by restriction (on kernels) and passage to quotients (on cokernels); the leftmost

0

requires

a

's row to begin with

0

, and the rightmost

0

requires

c

's row to end with

0

. If the top row is not assumed to start with

0

, the leftmost

0

may be dropped; symmetrically on the right.

Why It Holds / Derivation

Construction of δ and Exactness — Diagram Chase in R-Mod

We work in $R$ -Mod; the general abelian-category case follows by the Freyd–Mitchell embedding theorem, which embeds any small abelian category exactly and fully faithfully into $R$ -Mod for some ring $R$ , so that element-wise chases are legitimate.

Step 1 — The induced maps exist. If

x \in ker a \subseteq A

, then

b (i (x)) = j (a (x)) = j (0) = 0

, so

i (x) \in ker b

; define

\overset{ˉ}{i} = i ∣_{k e r a}

. Likewise

\overset{p}{ˉ} = p ∣_{k e r b}

. Dually,

j

sends

im a

into

im b

(since

j (a (x)) = b (i (x))

), so

j

descends to

\overset{ˉ}{j} : coker a \to coker b

; similarly

\overset{q}{ˉ}

. Step 2 — Define the connecting map δ. Take

z \in ker c \subseteq C

. Choose

b^{'} \in B

with

p (b^{'}) = z

(possible since

p

is surjective). Now

q (b (b^{'})) = c (p (b^{'})) = c (z) = 0,

b (b^{'}) \in ker q = im j

. Since

j

is injective, there is a unique

a^{'} \in A^{'}

with

j (a^{'}) = b (b^{'})

. Define

δ (z) = a^{'} + im a \in coker a, j (a^{'}) = b (b^{'}), p (b^{'}) = z .

⟳This is the "snake": go up via

p^{- 1}

, across via

b

, then back via

j^{- 1}

. The chase descends one row and crosses the diagram, hence the name.

Step 3 — δ is well-defined. Suppose

b^{''}

is another lift of

z

, so

p (b^{''}) = z = p (b^{'})

, hence

b^{''} - b^{'} \in ker p = im i

, say

b^{''} - b^{'} = i (x)

. Then

b (b^{''}) - b (b^{'}) = b (i (x)) = j (a (x)),

so the corresponding

a^{''}

satisfies

j (a^{''}) = j (a^{'}) + j (a (x))

, giving

a^{''} = a^{'} + a (x)

by injectivity of

j

. Thus

a^{''}

and

a^{'}

differ by

a (x) \in im a

, so

δ (z)

is independent of the lift. Linearity of

δ

follows since lifts can be chosen additively. Step 4 — Exactness at $ker b$ . Clearly

\overset{p}{ˉ} \circ \overset{ˉ}{i} = 0

(it is

p \circ i

restricted, and

p \circ i = 0

). Conversely if

y \in ker b

with

\overset{p}{ˉ} (y) = p (y) = 0

, then

y \in ker p = im i

, so

y = i (x)

. Then

j (a (x)) = b (i (x)) = b (y) = 0

, and

j

injective gives

a (x) = 0

, i.e.

x \in ker a

. Hence

y = \overset{ˉ}{i} (x) \in im \overset{ˉ}{i}

. Step 5 — Exactness at $ker c$ .

δ \circ \overset{p}{ˉ} = 0

: if

z = p (y)

with

y \in ker b

, use

b^{'} = y

as the lift; then

b (b^{'}) = b (y) = 0

, so

a^{'} = 0

and

δ (z) = 0

. Conversely if

δ (z) = 0

, then in Step 2 we may take

a^{'} \in im a

, say

a^{'} = a (x)

. Set

\tilde{b} = b^{'} - i (x)

. Then

p (\tilde{b}) = p (b^{'}) = z

and

b (\tilde{b}) = b (b^{'}) - b (i (x)) = j (a^{'}) - j (a (x)) = 0,

\tilde{b} \in ker b

and

\overset{p}{ˉ} (\tilde{b}) = z

. Hence

z \in im \overset{p}{ˉ}

. Step 6 — Exactness at $coker a$ .

\overset{ˉ}{j} \circ δ = 0

: with notation of Step 2,

\overset{ˉ}{j} (δ (z)) = j (a^{'}) + im b = b (b^{'}) + im b = 0

. Conversely if

\overset{ˉ}{j} (a^{'} + im a) = 0

, then

j (a^{'}) \in im b

, say

j (a^{'}) = b (b^{'})

. Put

z = p (b^{'})

. Then

c (z) = c (p (b^{'})) = q (b (b^{'})) = q (j (a^{'})) = 0

since

q \circ j = 0

. So

z \in ker c

and by construction

δ (z) = a^{'} + im a

. Step 7 — Exactness at $coker b$ . Mirror of Step 4 by duality (replace each module by its role in the cokernel sequence; the argument is the formal dual of Step 4).

\overset{q}{ˉ} \circ \overset{ˉ}{j} = 0

from

q \circ j = 0

, and a preimage chase recovers exactness. Step 8 — End terms.

\overset{ˉ}{i}

is injective:

\overset{ˉ}{i} (x) = 0 \Rightarrow i (x) = 0 \Rightarrow x = 0

since

i

injective.

\overset{q}{ˉ}

is surjective: given

c^{'} + im c \in coker c

, since

q

is surjective pick

b^{'}

with

q (b^{'}) = c^{'}

; then

\overset{q}{ˉ} (b^{'} + im b) = c^{'} + im c

■

The Snake: the connecting map δ threads from ker c down across to coker a

Samuel Eilenberg & Saunders Mac Lane | 1909–1998 / 1909–2005

Eilenberg and Mac Lane founded category theory in their 1945 paper "General Theory of Natural Equivalences," precisely to make rigorous the notion of a natural transformation—exactly the kind of canonicity the connecting homomorphism δ exhibits. Eilenberg, together with Steenrod, axiomatized homology (1952) around the long exact sequence of a pair; with Cartan he wrote Homological Algebra (1956), the text in which diagram-chasing tools like the Snake Lemma and the Five Lemma became the working language of the subject. Mac Lane's Homology (1963) and Categories for the Working Mathematician (1971) standardized the abelian-category framework in which the Snake Lemma lives in full generality. Their insistence that the connecting map be functorial in the diagram—not merely existent—is what elevates the Snake Lemma from a computational trick to a structural theorem.

Building Intuition

Think of a short exact sequence $0 \to A \to B \to C \to 0$ as a statement that $B$ is built from a "bottom piece" $A$ and a "top piece" $C$ , with $A$ sitting inside $B$ and $C$ being what's left over. Exactness is conservation: nothing is created or destroyed as you pass along the maps; whatever dies (gets sent to zero) by $f_{n}$ is exactly what was born (lay in the image) under $f_{n + 1}$ .

The Snake Lemma addresses a comparison: you have two such decompositions, the top row and bottom row, linked by vertical "comparison maps" $a, b, c$ . You'd love for kernels to map onto kernels and cokernels into cokernels cleanly, giving two separate short exact sequences. They almost do—but there's a gap, and the snake map δ is precisely the bridge that closes it. Operationally δ is a relay race: start with an element $z$ of $ker c$ , run it backwards up the right column to $B$ , run it down through the middle map $b$ to $B^{'}$ , then run it backwards up the left column to $A^{'}$ , and report the result modulo $im a$ . The non-canonical "backwards" steps introduce ambiguity, but Step 3 of the proof shows the ambiguity lands exactly in $im a$ , which is why we quotient by it. The miracle isn't that δ is defined—it's that this S-shaped chase produces an exact sequence with no leftover error terms.

Worked Examples

Multiplication by n on Z/m, and the connecting map

Consider, for fixed integers $m, n \geq 1$ , the diagram of $Z$ -modules with rows the SES $0 \to Z m Z \to Z / m \to 0$ , vertical maps "multiply by $n$ ":

00 \to \to Z \cdot n ↓ Z \cdot m \cdot m Z \cdot n ↓ Z π π Z / m \cdot n ↓ Z / m \to \to 00

⟳Commutativity is

m \cdot n = n \cdot m

on the left square and

n \cdot π = π \cdot n

on the right; both hold since multiplication in

Z

is commutative.

Given.

a = b = c = (\cdot n)

. We compute the six-term sequence [6]. Steps.

$ker (\cdot n : Z \to Z) = 0$ for the left and middle, so $ker a = ker b = 0$ .
$ker (\cdot n : Z / m \to Z / m)$ : elements $x + m Z$ with $n x \equiv 0 (mod m)$ , i.e. $x \equiv 0 (mod m / d)$ where $d = g cd (m, n)$ . This subgroup has order $d$ , so $ker c ≅ Z / d$ .
$coker (\cdot n : Z \to Z) = Z / n$ for left and middle, so $coker a = coker b = Z / n$ .
$coker (\cdot n : Z / m \to Z / m)$ has order $d$ , so $coker c ≅ Z / d$ .
The sequence [6] reads $0 \to 0 \to 0 \to Z / d δ Z / n \overset{ˉ}{j} Z / n \overset{q}{ˉ} Z / d \to 0$ .
Exactness at $ker c$ forces δ injective; trace δ: lift $x + m Z \in ker c$ to $x \in Z$ (middle), apply $\cdot n$ to get $n x$ , then divide by $m$ (left $\cdot m$ ) — but $n x$ need not be divisible by $m$ in $Z$ ... here we use that $x \in ker c$ means $n x \equiv 0 mod m$ , so $n x = m \cdot t$ and $δ (x) = t + n Z$ .

Boxed answer.

0 \to Z / d δ Z / n \to Z / n \to Z / d \to 0, d = g cd (m, n)

This is the algebraic shadow of the Bockstein homomorphism; orders multiply consistently:

d \cdot n = n \cdot d

. (Units: pure group orders, dimensionless.)

Recovering the long exact sequence in homology

Given. A short exact sequence of chain complexes

0 \to A_{∙} f B_{∙} g C_{∙} \to 0

(exact in each degree). We derive the long exact homology sequence. Steps.

For each $n$ , the differential $d$ induces a $3 \times 2$ diagram comparing $A_{n} / im d, B_{n} / im d, C_{n} / im d$ (top) with cycle subgroups $Z_{n - 1} (A), Z_{n - 1} (B), Z_{n - 1} (C)$ (bottom), arising from the maps $\overset{ˉ}{d}$ induced by the differential.
Concretely the rows

0 \to A_{n} / B_{n}^{d} \to B_{n} / \dots \to C_{n} / \dots \to 0, 0 \to Z_{n - 1} A \to Z_{n - 1} B \to Z_{n - 1} C \to 0

are exact, and the vertical maps are induced by

d

; the Snake Lemma applied to this diagram has

ker

giving

H_{n}

and

coker

giving

H_{n - 1}

⟳

ker (\overset{ˉ}{d}) = Z_{n} / B_{n} = H_{n}

and

coker

of the induced

d

on cycles is

Z_{n - 1} / im d = H_{n - 1}

; this is the standard "snake produces homology" identification.

The connecting map δ becomes the homological boundary $\partial : H_{n} (C) \to H_{n - 1} (A)$ : lift a cycle $c \in C_{n}$ to $b \in B_{n}$ , apply $d$ to get $d b \in B_{n - 1}$ which maps to $0$ in $C$ , hence comes from $a \in A_{n - 1}$ ; set $\partial [c] = [a]$ .

Boxed answer.

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

This long exact sequence is the engine behind Mayer–Vietoris and the long exact sequence of a pair in Singular Homology.

Common Misconceptions

" $f_{n} \circ f_{n + 1} = 0$ means the sequence is exact." False. The vanishing of composites only gives

im f_{n + 1} \subseteq ker f_{n}

(a chain complex). Exactness is the much stronger equality

im f_{n + 1} = ker f_{n}

. The discrepancy

ker f_{n} / im f_{n + 1}

is precisely homology; exactness means homology vanishes. "The connecting map δ is canonical so it must be defined without choices." Misleading. The construction of δ genuinely uses a choice of lift (an inverse image under a non-injective surjection). What is canonical is the output, because the ambiguity is exactly quotiented away. Confusing "choice-free definition" with "well-defined result" is a common trap. "You can chase elements in any abelian category." Not directly—a general abelian category has no underlying sets. Element-chasing is justified only via the Freyd–Mitchell embedding (for small categories) or by a more careful diagrammatic/membership-relation formalism (generalized elements). The conclusion holds, but the method needs justification.

A complex that is not exact, and where the Snake hypotheses fail

Take $0 \to Z \cdot 2 Z \cdot 2 Z \to 0$ . Here $(\cdot 2) \circ (\cdot 2) = \cdot 4 \neq = 0$ , so it is not even a complex—exactness is impossible. Even repairing to $Z \cdot 2 Z π Z /2$ , the composite is zero but the leftmost map is not injective onto $ker π = 2 Z$ ... it is here, so this one is exact—illustrating how delicate the equality condition is.

For the Snake Lemma's hypotheses, consider dropping exactness of a row. Take rows where the bottom is $0 \to Z \cdot 2 Z \to Z \to 0$ with the wrong right map (say the identity rather than $π$ ), making $im (\cdot 2) \neq = ker (id)$ . Then the bottom is not exact, $coker$ identifications break, and the six-term sequence [6] fails to be exact at $coker a$ : one finds an element of $ker \overset{ˉ}{j}$ not hit by δ. Exact rows are not decoration; they are load-bearing.

Connections

The Snake Lemma is the keystone joining several pillars. It specializes to the long exact sequence in Singular Homology and to the long exact sequence of Derived Functors (e.g. $Ext$ and $Tor$ ): a short exact sequence of modules induces long exact $Tor_{n} (-, M)$ and $Ext^{n} (-, M)$ sequences whose connecting maps are snake maps in disguise. In Sheaf Cohomology, a short exact sequence of sheaves yields the cohomology long exact sequence via the same mechanism applied to injective resolutions. The lemma is the prototype of the connecting morphism in any Triangulated Category, where distinguished triangles axiomatize "snake-like" rotation, and it underlies the Five Lemma and Nine Lemma (the $3 \times 3$ lemma), which are corollaries obtained by composing snake/diagram chases. It also sits at the heart of the Spectral Sequence formalism, where the page-to-page differentials generalize δ.

Real-World Applications

Algebraic topology — long exact sequence of a pair. Computing $H_{n}$ of spaces by relating a space, a subspace, and the quotient; the boundary map δ detects how cycles in the quotient lift to relative chains. Used to compute homology of spheres, projective spaces, CW complexes.
Mayer–Vietoris computations. Decomposing a space into two open sets and assembling cohomology; the connecting map is a snake map for the Čech/singular complex SES. Essential in computing de Rham cohomology of manifolds.
Number theory — Bockstein and Galois cohomology. The mod- $p$ Bockstein (Example 1) and the connecting maps in the long exact sequences of Galois cohomology (e.g. from the Kummer sequence $1 \to μ_{n} \to G_{m} \to G_{m} \to 1$ ) drive computations of $H^{1} (G, μ_{n})$ and the structure of class groups.
Commutative algebra — depth and local cohomology. Snake/long exact sequences in local cohomology $H_{m}^{i}$ relate depth, dimension, and Cohen–Macaulayness of rings and modules.
Representation theory. Long exact $Ext$ sequences classify extensions of representations and compute cohomology of groups and Lie algebras; the connecting map encodes obstruction classes.
Algebraic K-theory. The localization long exact sequence relating $K$ -groups of a ring, a localization, and the relevant quotient category is built from connecting homomorphisms of snake type.
Differential geometry / physics (anomalies). Descent equations and the relation between gauge anomalies in different dimensions are organized by connecting homomorphisms (Stora–Zumino), a physical avatar of δ; characteristic-class transgression is a snake-map phenomenon.

Historical Development

1895–1933. Poincaré's analysis situs and the work of Noether, Hopf, and Vietoris recast "Betti numbers and torsion" into homology groups, making maps between groups—and eventually exact sequences—the natural objects.
1941. Hurewicz and others use exact sequence reasoning implicitly; the term "exact sequence" gestates in the homotopy/homology literature.
1945. Eilenberg and Mac Lane publish "General Theory of Natural Equivalences," founding category theory and giving precise meaning to naturality—the property that makes δ canonical.
1947–1949. Kelley–Pitcher and others give the "exact sequence" its modern name and notation; the long exact sequence of a pair becomes a standard tool.
1952. Eilenberg and Steenrod's Foundations of Algebraic Topology axiomatizes homology, placing the long exact sequence (hence the connecting homomorphism) among the Eilenberg–Steenrod axioms.
1956. Cartan and Eilenberg's Homological Algebra systematizes diagram chasing; the Snake Lemma, Five Lemma, and derived functors $Tor, Ext$ enter the canon, with long exact sequences derived uniformly.
1957. Grothendieck's Tôhoku paper generalizes homological algebra to abelian categories, removing dependence on elements and setting the stage for sheaf cohomology and δ in great generality.
1960s. Freyd (1964