Exact sequence, short five lemma, snake lemma

Intuition [Beginner]

A homomorphism between groups has a kernel (the elements crushed to the identity) and an image (the elements it produces). An exact sequence is a chain of groups connected by homomorphisms where each map's image matches the next map's kernel perfectly. At every link in the chain, the outputs that arrive are precisely the inputs that vanish.

Think of it as a relay race. Runner A hands the baton to Runner B, who hands it to Runner C. Exactness means that Runner B receives exactly the baton that Runner A produced — nothing extra arrives, and nothing is lost.

Why does this concept exist? Exact sequences encode how a group is assembled from simpler pieces, and they provide the language for connecting kernels and cokernels across diagrams of homomorphisms.

Visual [Beginner]

A row of three circles connected by arrows. The left circle is labelled $A$, the middle $B$, the right $C$. The arrow from $A$ to $B$ and the arrow from $B$ to $C$ overlap at $B$: the image of the first arrow fills precisely the kernel of the second.

The overlap region represents exactness at $B$: what comes in equals what goes to zero on the way out.

Worked example [Beginner]

Consider the integers under addition, written $\mathbb{Z}$, and the integers modulo $3$, written $\mathbb{Z}/3\mathbb{Z}$.

Step 1. Define the inclusion map $f$ from $3\mathbb{Z}$ to $\mathbb{Z}$ that sends $3k$ to $3k$. The image of $f$ is the set $\{\dots ,-6,-3,0,3,6,\dots \}$.

Step 2. Define the quotient map $g$ from $\mathbb{Z}$ to $\mathbb{Z}/3\mathbb{Z}$ that sends each integer to its remainder modulo $3$. The kernel of $g$ is the set $\{\dots ,-6,-3,0,3,6,\dots \}$.

Step 3. The image of $f$ equals the kernel of $g$. The sequence $3\mathbb{Z}\to \mathbb{Z}\to \mathbb{Z}/3\mathbb{Z}$ is exact at the middle group $\mathbb{Z}$.

What this tells us: the group $\mathbb{Z}$ is assembled from the subgroup $3\mathbb{Z}$ and the quotient $\mathbb{Z}/3\mathbb{Z}$, with exactness recording how they fit together.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $A$, $B$, $C$ be groups (in this section, abelian groups written additively) and let $f:A\to B$, $g:B\to C$ be group homomorphisms.

Definition (Exact at one point). The pair $A\stackrel{f}{\to }B\stackrel{g}{\to }C$ is exact at $B$ if $\mathrm{im}f=\ker g$.

Definition (Exact sequence). A sequence of groups and homomorphisms $$ \cdots \xrightarrow{f_{n-2}} A_{n-1} \xrightarrow{f_{n-1}} A_n \xrightarrow{f_n} A_{n+1} \xrightarrow{f_{n+1}} \cdots $$ is exact if it is exact at every group $A_n$ in the sequence: $\mathrm{im}{f}_{n-1}=\ker f_n$ for all $n$.

Definition (Short exact sequence). A sequence $0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0$ is short exact if:

• $\mathrm{im}(0\to A)=\ker f$, which forces $\ker f=0$ (so $f$ is injective);
• $\mathrm{im}f=\ker g$ (exactness at $B$);
• $\mathrm{im}g=\ker (C\to 0)=C$ (so $g$ is surjective).

In this case $C\cong B/f(A)\cong B/A$ (identifying $A$ with its image in $B$) ^{[Rotman §2.1]}.

Counterexamples to common slips [Intermediate+]

• Injective plus surjective does not suffice. The sequence $0\to \mathbb{Z}\stackrel{\times 2}{\to }\mathbb{Z}\stackrel{\text{mod }2}{\to }\mathbb{Z}/2\mathbb{Z}\to 0$ is not short exact: the image of multiplication by $2$ is $2\mathbb{Z}$, but the kernel of reduction mod $2$ is all even integers, so exactness at the middle holds, but the issue is that the map to $\mathbb{Z}/2\mathbb{Z}$ must have kernel exactly $\mathrm{im}(\times 2)$. In this case it does, so this is actually short exact. A genuine failure: $0\to \mathbb{Z}\stackrel{\times 2}{\to }\mathbb{Z}\stackrel{\times 3}{\to }\mathbb{Z}\to 0$ is not short exact because $\mathrm{im}(\times 2)=2\mathbb{Z}\ne 3\mathbb{Z}=\ker (\times 3)$.
• Exactness at one point does not imply exactness everywhere. A map $A\stackrel{0}{\to }B\stackrel{0}{\to }C$ is exact at $B$ (since $\mathrm{im}(0)=\{0\}=\ker (0)$), but the sequence $0\to A\stackrel{0}{\to }B\stackrel{0}{\to }C\to 0$ is not exact at $A$ or $C$ unless $A=0$ and $C=0$.
• The middle group need not be the direct sum. For a short exact sequence $0\to A\to B\to C\to 0$, the group $B$ need not be isomorphic to $A\oplus C$. The sequence $0\to \mathbb{Z}\stackrel{\times 2}{\to }\mathbb{Z}\stackrel{\text{mod }2}{\to }\mathbb{Z}/2\mathbb{Z}\to 0$ is short exact but $\mathbb{Z}\ncong \mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$.

Key theorem with proof [Intermediate+]

Theorem (Snake Lemma). Consider a commutative diagram of abelian groups with exact rows: $$ \begin{CD} 0 @>>> A @>{f}>> B @>{g}>> C @>>> 0 \ @. @V{\alpha}VV @V{\beta}VV @V{\gamma}VV @. \ 0 @>>> A' @>{f'}>> B' @>{g'}>> C' @>>> 0 \end{CD} $$ There is a long exact sequence $$ 0 \to \ker \alpha \xrightarrow{f_*} \ker \beta \xrightarrow{g_*} \ker \gamma \xrightarrow{\delta} \operatorname{coker} \alpha \xrightarrow{f'*} \operatorname{coker} \beta \xrightarrow{g'} \operatorname{coker} \gamma \to 0 $$ where $f_$and$g_*$aretherestrictionsof$f$and$g$tothekernels,$f'*$and$g'*$arethemapsinducedoncokernels,and$\delta$ is the connecting homomorphism.

Proof. The proof proceeds by a diagram chase. Write $i_A:\ker \alpha \hookrightarrow A$, $i_B:\ker \beta \hookrightarrow B$, $i_C:\ker \gamma \hookrightarrow C$ for the inclusions, and ${\pi }_{A^{\prime }}:A^{\prime }\to A^{\prime }/\mathrm{im}\alpha$, ${\pi }_{B^{\prime }}:B^{\prime }\to B^{\prime }/\mathrm{im}\beta$, ${\pi }_{C^{\prime }}:C^{\prime }\to C^{\prime }/\mathrm{im}\gamma$ for the projections.

Definition of $\delta$. Let $c\in \ker \gamma$. Since $g$ is surjective, choose $b\in B$ with $g(b)=c$. Then $\gamma (g(b))=\gamma (c)=0$, so $g^{\prime }(\beta (b))=0$. By exactness of the bottom row, $\beta (b)\in \ker g^{\prime }=\mathrm{im}{f}^{\prime }$, so there exists $a^{\prime }\in A^{\prime }$ with $f^{\prime }(a^{\prime })=\beta (b)$. Define $\delta (c)=[a^{\prime }]\in A^{\prime }/\mathrm{im}\alpha =\mathrm{coker}\alpha$.

$\delta$ is well-defined. Suppose $b_1,b_2\in B$ both satisfy $g(b_1)=g(b_2)=c$. Then $g(b_1-b_2)=0$, so $b_1-b_2\in \ker g=\mathrm{im}f$ by exactness of the top row. Write $b_1-b_2=f(a)$ for some $a\in A$. The corresponding $a_1^{\prime },a_2^{\prime }$ satisfy $f^{\prime }(a_1^{\prime })=\beta (b_1)$ and $f^{\prime }(a_2^{\prime })=\beta (b_2)$, so $f^{\prime }(a_1^{\prime }-a_2^{\prime })=\beta (b_1-b_2)=\beta (f(a))=f^{\prime }(\alpha (a))$. Since $f^{\prime }$ is injective, $a_1^{\prime }-a_2^{\prime }=\alpha (a)\in \mathrm{im}\alpha$, hence $[a_1^{\prime }]=[a_2^{\prime }]$ in $\mathrm{coker}\alpha$.

$\delta$ is a homomorphism. For $c_1,c_2\in \ker \gamma$, choose lifts $b_1,b_2$. Then $b_1+b_2$ lifts $c_1+c_2$, so $\delta (c_1+c_2)=[a_1^{\prime }+a_2^{\prime }]=[a_1^{\prime }]+[a_2^{\prime }]=\delta (c_1)+\delta (c_2)$.

Exactness at $\ker \beta$. The map $f_{\ast }$ sends $\ker \alpha$ to $\ker \beta$: for $a\in \ker \alpha$, $\beta (f(a))=f^{\prime }(\alpha (a))=f^{\prime }(0)=0$. For exactness, $\mathrm{im}{f}_{\ast }\subseteq \ker g_{\ast }$: $g_{\ast }(f_{\ast }(a))=g(f(a))=0$ by exactness of the top row. For $\ker g_{\ast }\subseteq \mathrm{im}{f}_{\ast }$: if $b\in \ker \beta$ with $g(b)=0$, then $b\in \ker g=\mathrm{im}f$, say $b=f(a)$. Then $0=\beta (b)=\beta (f(a))=f^{\prime }(\alpha (a))$, and injectivity of $f^{\prime }$ gives $\alpha (a)=0$, so $a\in \ker \alpha$.

Exactness at $\ker \gamma$. For $b\in \ker \beta$, $\delta (g(b))$ is constructed using $b$ itself as the lift. Since $\beta (b)=0$, the $a^{\prime }$ satisfying $f^{\prime }(a^{\prime })=\beta (b)$ is $a^{\prime }=0$, so $\delta (g(b))=[0]=0$. Conversely, if $\delta (c)=0$, write $[a^{\prime }]=0$, meaning $a^{\prime }=\alpha (a)$ for some $a\in A$. The lift $b$ of $c$ satisfies $f^{\prime }(a^{\prime })=\beta (b)$, so $\beta (b)=f^{\prime }(\alpha (a))=\beta (f(a))$, giving $\beta (b-f(a))=0$. Thus $b-f(a)\in \ker \beta$ and $g(b-f(a))=g(b)-0=c$, so $c=g_{\ast }(b-f(a))\in \mathrm{im}{g}_{\ast }$.

Exactness at $\mathrm{coker}\alpha$. The map $f_{\ast }^{\prime }$ sends $[a^{\prime }]$ to $[f^{\prime }(a^{\prime })]$. For $c\in \ker \gamma$, $\delta (c)=[a^{\prime }]$ where $f^{\prime }(a^{\prime })=\beta (b)$, so $f_{\ast }^{\prime }([a^{\prime }])=[\beta (b)]=0$ in $\mathrm{coker}\beta$, giving $\mathrm{im}\delta \subseteq \ker f_{\ast }^{\prime }$. Conversely, if $[a^{\prime }]\in \ker f_{\ast }^{\prime }$ then $f^{\prime }(a^{\prime })\in \mathrm{im}\beta$, say $f^{\prime }(a^{\prime })=\beta (b)$. Then $g^{\prime }(\beta (b))=g^{\prime }(f^{\prime }(a^{\prime }))=0$, so $\gamma (g(b))=0$, meaning $g(b)\in \ker \gamma$. Now $\delta (g(b))=[a^{\prime }]$ by construction.

Exactness at $\mathrm{coker}\beta$. The map $g_{\ast }^{\prime }$ sends $[b^{\prime }]$ to $[g^{\prime }(b^{\prime })]$. For $[a^{\prime }]\in \mathrm{coker}\alpha$, $g_{\ast }^{\prime }(f_{\ast }^{\prime }([a^{\prime }]))=[g^{\prime }(f^{\prime }(a^{\prime }))]=[0]$. Conversely, if $[b^{\prime }]\in \ker g_{\ast }^{\prime }$, then $g^{\prime }(b^{\prime })\in \mathrm{im}\gamma$, say $g^{\prime }(b^{\prime })=\gamma (c)$. Choose $b\in B$ with $g(b)=c$. Then $g^{\prime }(b^{\prime }-\beta (b))=0$, so $b^{\prime }-\beta (b)=f^{\prime }(a^{\prime })$ for some $a^{\prime }$ by exactness of the bottom row. Hence $[b^{\prime }]=[b^{\prime }-\beta (b)]=[f^{\prime }(a^{\prime })]=f_{\ast }^{\prime }([a^{\prime }])$.

Exactness at $\mathrm{coker}\gamma$. Since $g^{\prime }$ is surjective, $g_{\ast }^{\prime }$ is surjective on cokernels. $\square$

Bridge. The snake lemma builds toward the long exact sequence in homology 01.02.12, where each connecting homomorphism in a long exact sequence is constructed by the same diagram-chase mechanism. This result appears again in 01.02.10 (tensor product of modules), where the snake lemma yields the long exact Tor sequence. The foundational reason is that exactness is a local condition (checked at one group) that propagates globally across a diagram, and this is exactly the mechanism that makes homological algebra work: the central insight is that kernels and cokernels are connected by a systematic homomorphism, the bridge is between the zero-dimensional data (kernels) and the quotient data (cokernels), and the pattern generalises to any abelian category.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Five Lemma). Consider a commutative diagram of abelian groups with exact rows: $$ \begin{CD} A_1 @>>> A_2 @>>> A_3 @>>> A_4 @>>> A_5 \ @V{\alpha_1}VV @V{\alpha_2}VV @V{\alpha_3}VV @V{\alpha_4}VV @V{\alpha_5}VV \ B_1 @>>> B_2 @>>> B_3 @>>> B_4 @>>> B_5 \end{CD} $$ If ${\alpha }_1$ is surjective, ${\alpha }_5$ is injective, and ${\alpha }_2,{\alpha }_4$ are isomorphisms, then ${\alpha }_3$ is an isomorphism.

The proof is a diagram chase in two directions: injectivity uses ${\alpha }_4$ and ${\alpha }_5$, surjectivity uses ${\alpha }_1$ and ${\alpha }_2$ ^{[Weibel §1.3]}. The Short Five Lemma is the special case where the outer groups $A_1,A_5,B_1,B_5$ are all zero.

Theorem 2 (Splitting Lemma, full version). For a short exact sequence $0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0$ of abelian groups, the following are equivalent:

• There exists $r:B\to A$ with $r\circ f={\mathrm{id}}_A$ (a retraction).
• There exists $s:C\to B$ with $g\circ s={\mathrm{id}}_C$ (a section).
• $B\cong A\oplus C$ with $f$ the inclusion into the first factor and $g$ the projection onto the second.

When any (hence all) of these hold, the sequence is called split ^{[Rotman §2.1]}.

Theorem 3 (3x3 Lemma / Nine Lemma). Given a commutative diagram of three rows and three columns of abelian groups, if all three columns and the top two rows are short exact, then the bottom row is also short exact. Equivalently, if all three columns and the bottom two rows are short exact, then the top row is short exact.

This follows from repeated applications of the snake lemma to pairs of adjacent rows ^{[Weibel §1.3]}.

Theorem 4 (Long exact sequence from short exact sequences of complexes). If $0\to A_{\bullet }\to B_{\bullet }\to C_{\bullet }\to 0$ is a short exact sequence of chain complexes, there is a long exact sequence in homology: $$ \cdots \to H_n(A) \to H_n(B) \to H_n(C) \xrightarrow{\delta} H_{n-1}(A) \to H_{n-1}(B) \to \cdots $$ The connecting homomorphism $\delta$ is constructed by the same diagram chase as in the snake lemma. In fact, the snake lemma is the special case where each complex has at most two nonzero terms ^{[Weibel §1.3]}.

Theorem 5 (Horseshoe Lemma). Given a short exact sequence $0\to A\to B\to C\to 0$ of modules and projective resolutions $P_{\bullet }\to A$ and $Q_{\bullet }\to C$, there exists a projective resolution $R_{\bullet }\to B$ with $0\to P_n\to R_n\to Q_n\to 0$ exact for all $n$, fitting the original resolutions into a short exact sequence of complexes.

This identifies projective resolutions as behaving well under extensions, which is the foundational input for constructing derived functors ^{[Weibel §2.2]}.

Theorem 6 (Yoneda Ext and exact sequences). The set ${\mathrm{Ext}}^1(C,A)$ of equivalence classes of extensions $0\to A\to B\to C\to 0$ is in bijection with the derived functor ${\mathrm{Ext}}^1(C,A)$. A short exact sequence splits if and only if the corresponding extension class in ${\mathrm{Ext}}^1(C,A)$ is zero.

This is the bridge between the concrete classification of extensions and the abstract homological machinery. The group law on ${\mathrm{Ext}}^1$ corresponds to the Baer sum of extensions ^{[Weibel §3.4]}.

Synthesis. Exact sequences are the foundational reason that homological algebra unifies computation across modules, chain complexes, and abelian categories. The central insight is that a short exact sequence encodes an extension problem — how $B$ is assembled from $A$ and $C$ — and the snake lemma transforms that local extension data into global connectivity between kernels and cokernels. Putting these together with the Five Lemma and the Nine Lemma, the bridge is between local exactness at individual groups and global exactness of the entire diagram, and this is exactly the machinery that makes derived functors computable. The pattern generalises from abelian groups to modules over any ring, to sheaves on a topological space, and to objects in any abelian category, identifying the homological invariants of a mathematical object with the algebraic data of its exact sequences.

Full proof set [Master]

Proposition 1 (Five Lemma). In the five-by-five diagram of Theorem 1, if ${\alpha }_1$ is surjective, ${\alpha }_2$ and ${\alpha }_4$ are isomorphisms, and ${\alpha }_5$ is injective, then ${\alpha }_3$ is an isomorphism.

Proof. Label the horizontal maps in the top row as $d_i:A_i\to A_{i+1}$ and in the bottom row as $e_i:B_i\to B_{i+1}$.

Injectivity of ${\alpha }_3$. Let $a_3\in \ker {\alpha }_3$. Then $e_3({\alpha }_3(a_3))=0$, so ${\alpha }_4(d_3(a_3))=0$ by commutativity. Since ${\alpha }_4$ is injective, $d_3(a_3)=0$. By exactness, $a_3=d_2(a_2)$ for some $a_2\in A_2$. Now ${\alpha }_3(a_3)=0$ gives $e_2({\alpha }_2(a_2))=0$, so ${\alpha }_2(a_2)\in \ker e_2=\mathrm{im}{e}_1$. Write ${\alpha }_2(a_2)=e_1(b_1)$ for some $b_1\in B_1$. Since ${\alpha }_1$ is surjective, $b_1={\alpha }_1(a_1)$ for some $a_1\in A_1$. Then ${\alpha }_2(a_2)=e_1({\alpha }_1(a_1))={\alpha }_2(d_1(a_1))$, and injectivity of ${\alpha }_2$ gives $a_2=d_1(a_1)$. Hence $a_3=d_2(a_2)=d_2(d_1(a_1))=0$ by exactness at $A_2$.

Surjectivity of ${\alpha }_3$. Let $b_3\in B_3$. Then $e_3(b_3)\in B_4$. Since ${\alpha }_4$ is surjective, $e_3(b_3)={\alpha }_4(a_4)$ for some $a_4\in A_4$. Then ${\alpha }_5(d_4(a_4))=e_4({\alpha }_4(a_4))=e_4(e_3(b_3))=0$. Since ${\alpha }_5$ is injective, $d_4(a_4)=0$. By exactness, $a_4=d_3(a_3^{\prime })$ for some $a_3^{\prime }\in A_3$. Now ${\alpha }_4(a_4)={\alpha }_4(d_3(a_3^{\prime }))=e_3({\alpha }_3(a_3^{\prime }))$, so $e_3(b_3-{\alpha }_3(a_3^{\prime }))=0$. By exactness, $b_3-{\alpha }_3(a_3^{\prime })=e_2(b_2)$ for some $b_2\in B_2$. Since ${\alpha }_2$ is surjective, $b_2={\alpha }_2(a_2)$ for some $a_2\in A_2$. Hence $b_3={\alpha }_3(a_3^{\prime })+e_2({\alpha }_2(a_2))={\alpha }_3(a_3^{\prime })+{\alpha }_3(d_2(a_2))={\alpha }_3(a_3^{\prime }+d_2(a_2))$. $\square$

Proposition 2 (Long exact sequence from short exact sequence of complexes). Given a short exact sequence of chain complexes $0\to A_{\bullet }\stackrel{f}{\to }{B}_{\bullet }\stackrel{g}{\to }{C}_{\bullet }\to 0$, there is a long exact sequence $$ \cdots \to H_n(A) \xrightarrow{f_*} H_n(B) \xrightarrow{g_*} H_n(C) \xrightarrow{\delta} H_{n-1}(A) \to \cdots $$

Proof. The connecting homomorphism $\delta :H_n(C)\to H_{n-1}(A)$ is constructed by a diagram chase. Let $[c_n]\in H_n(C)$, so $c_n\in C_n$ with $\partial c_n=0$. Since $g_n$ is surjective, choose $b_n\in B_n$ with $g_n(b_n)=c_n$. Then $g_{n-1}(\partial b_n)=\partial g_n(b_n)=\partial c_n=0$, so $\partial b_n\in \ker g_{n-1}=\mathrm{im}{f}_{n-1}$. Write $\partial b_n=f_{n-1}(a_{n-1})$ for some $a_{n-1}\in A_{n-1}$. Define $\delta ([c_n])=[a_{n-1}]$.

Well-defined: $\partial a_{n-1}=0$ because $f_{n-2}(\partial a_{n-1})=\partial f_{n-1}(a_{n-1})={\partial }^2{b}_n=0$ and $f_{n-2}$ is injective. A different lift $b_n^{\prime }$ gives $b_n-b_n^{\prime }=f_n(a_n)$ for some $a_n$, and then $\partial b_n^{\prime }=\partial b_n-\partial f_n(a_n)=f_{n-1}(a_{n-1})-f_{n-1}(\partial a_n)=f_{n-1}(a_{n-1}-\partial a_n)$, so $[a_{n-1}]=[a_{n-1}-\partial a_n]$ in homology. Changing $c_n$ by a boundary $\partial c_{n+1}$ similarly changes $[a_{n-1}]$ by a boundary.

Exactness at $H_n(C)$: if $[c_n]=g_{\ast }([b_n])$, then the lift of $c_n$ can be chosen as $b_n$ itself, giving $\partial b_n\in \mathrm{im}\partial \cap B_{n-1}$, so $a_{n-1}=f_{n-1}^{-1}(\partial b_n)$ is a boundary in $A_{n-1}$, hence $\delta ([c_n])=0$. Conversely, if $\delta ([c_n])=0$, then $a_{n-1}=\partial a_n$ for some $a_n\in A_n$, so $\partial b_n=f_{n-1}(\partial a_n)=\partial (f_n(a_n))$, giving $b_n-f_n(a_n)\in \ker \partial$, and $g_n(b_n-f_n(a_n))=c_n$, so $[c_n]=g_{\ast }([b_n-f_n(a_n)])$.

Exactness at $H_n(B)$ and $H_{n-1}(A)$ follow by similar diagram chases. $\square$

Proposition 3 (Snake Lemma implies the Short Five Lemma). Given the diagram of the Short Five Lemma with $\alpha$ and $\gamma$ isomorphisms, $\beta$ is an isomorphism.

Proof. Apply the snake lemma to the diagram. The kernel-cokernel sequence is $$ 0 \to \ker\alpha \to \ker\beta \to \ker\gamma \xrightarrow{\delta} \operatorname{coker}\alpha \to \operatorname{coker}\beta \to \operatorname{coker}\gamma \to 0. $$ Since $\alpha$ is an isomorphism, $\ker \alpha =0$ and $\mathrm{coker}\alpha =0$. Since $\gamma$ is an isomorphism, $\ker \gamma =0$ and $\mathrm{coker}\gamma =0$. The sequence collapses to $0\to \ker \beta \to 0$ and $0\to \mathrm{coker}\beta \to 0$, giving $\ker \beta =0$ and $\mathrm{coker}\beta =0$, so $\beta$ is an isomorphism. $\square$

Connections [Master]

• Subgroup, coset, quotient group, isomorphism theorems 01.02.02. The First Isomorphism Theorem $G/\ker \phi \cong \mathrm{im}\phi$ is the prototypical exact sequence: $0\to \ker \phi \to G\to \mathrm{im}\phi \to 0$. The three isomorphism theorems developed in the subgroup-coset unit are the group-theoretic precursors of the exactness conditions formalised here. The foundational reason that kernels and images encode structural information appears in both units.

• Tensor product of modules 01.02.10. The snake lemma applied to the functor $M{\otimes }_R-$ produces the long exact Tor sequence $0\to {\mathrm{Tor}}_1(M,C)\to M\otimes A\to M\otimes B\to M\otimes C\to 0$ whenever $0\to A\to B\to C\to 0$ is exact and $M$ is flat. The exactness properties of the tensor product are measured by the vanishing of Tor groups, and this is exactly the bridge between the homological machinery of the present unit and the module constructions of the tensor product unit.

• Group action, orbit-stabiliser, class equation 01.02.03. The orbit-stabiliser exact sequence $1\to \mathrm{Stab}(x)\to G\to \mathrm{Orb}(x)\to 1$ for a transitive group action (in the category of sets, not groups) previews the exact-sequence viewpoint on group actions. The stabiliser as kernel of the action map and the orbit as image build toward the exact-sequence formalism. The class equation decomposes a group into conjugacy classes whose sizes are indices of centralisers, and these indices are the orders of cokernels in the action maps.

Historical & philosophical context [Master]

Exact sequences originated in the algebraic topology of the 1940s. Eilenberg and Steenrod introduced the exactness axioms for homology theories in their 1952 Foundations of Algebraic Topology ^{[Eilenberg-Steenrod 1952]}, establishing the long exact sequence of a pair as a fundamental axiom. The snake lemma in its general form appeared in Cartan and Eilenberg's 1956 Homological Algebra ^{[Cartan-Eilenberg 1956]}, which systematised the diagram-chase techniques that had been circulating informally in the work of Kelley and Pitcher (1948) and in Eilenberg's earlier papers on spectral sequences.

The five lemma is sometimes attributed to Jans (1964 Mem. AMS 4), though the result was used implicitly throughout Cartan-Eilenberg 1956 and in Grothendieck's 1957 Tôhoku paper. Grothendieck's Tôhoku memoir reoriented homological algebra around abelian categories and derived functors, making the snake lemma and the five lemma into formal consequences of the abelian-category axioms ^{[Weibel §1.3]}. Weibel's 1994 An Introduction to Homological Algebra remains the canonical modern exposition of these results and their generalisations.

Bibliography [Master]

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