1 Statement of the Snake Lemma

1.1 Preliminaries: abelian categories and exactness

The Snake Lemma is formulated in the setting where kernels and cokernels exist and behave well with respect to exact sequences. In practice, one works either with abelian categories (where exactness is defined abstractly via images and kernels) or with module categories (for which the same notion can be expressed using homomorphisms of abelian groups).

A sequence \[ A\xrightarrow{f}B\xrightarrow{g}C \] is exact at \(B\) if \(\ker(g)=\operatorname{im}(f)\). In an abelian category, this condition is equivalent to demanding that \(g\circ f=0\) and that every element in \(\ker(g)\) arises from some element in \(A\).

Exact rows in the Snake Lemma are sequences that are exact at the relevant objects: typically, one has kernels matching images at the middle terms, along with compatibility to form a diagram.

1.2 The standard commutative diagram with two exact rows

A typical Snake Lemma diagram involves two short exact-style “exact rows” connected by vertical morphisms. Concretely, one considers a commutative diagram of the form \[ \begin{array}{ccccccccc} A'&\xrightarrow{}&A&\xrightarrow{}&A''&\xrightarrow{}&0\\ \downarrow && \downarrow && \downarrow \\ B'&\xrightarrow{}&B&\xrightarrow{}&B''&\xrightarrow{}&0\\ && \downarrow && \downarrow && \\ && C&& C'' && \end{array} \] in the usual presentations, often specialized to the case where the two horizontal rows are exact sequences \[ A'\to A\to A''\quad\text{and}\quad B'\to B\to B'' \] with vertical maps \(A'\to B'\), \(A\to B\), \(A''\to B''\) making the whole rectangle commute. The essential feature is having exactness in the rows and commutativity so that kernels and cokernels can be related consistently.

When stated in its standard form, one uses a diagram with two exact rows \[ 0\to A'\to A\to A''\to 0,\qquad 0\to B'\to B\to B''\to 0 \] or a slightly less restrictive version in which only the middle exactness is needed for the construction, depending on the variant being used.

1.3 Construction of the connecting morphism

The Snake Lemma produces a morphism, often called the connecting morphism, that links the cokernel of one vertical map to the kernel of the next.

Assume the diagram has vertical maps \(A'\to B'\), \(A\to B\), and \(A''\to B''\) between the exact rows. One defines:

  • a map from \(\operatorname{coker}(A'\to B')\) to \(\ker(A''\to B'')\), and then
  • additional maps between kernels/cokernels that extend this into a long exact sequence.

The construction is typically diagram-chasing:

  1. Start with an element (or morphism class) representing a cokernel element on one side.
  2. Lift it along a surjection in the top exact row to a suitable element in the middle object.
  3. Apply the relevant vertical map to land in the bottom object.
  4. Use exactness to show the result lands in an appropriate kernel.
  5. Check that different choices of lifts differ by elements that map to zero, ensuring the connecting morphism is well-defined.

In an abelian category, the same idea is implemented using universal properties of kernels and cokernels rather than element-wise arguments.

1.4 Exactness properties and the resulting long exact sequence

The output of the lemma is an exact sequence (the “snake”) in which consecutive morphisms have images equal to kernels.

In the standard form for a diagram with exact rows, the lemma yields a long exact sequence \[ \ker(A\to B)\to \ker(A''\to B'')\to \operatorname{coker}(A'\to B')\to \operatorname{coker}(A\to B)\to \operatorname{coker}(A''\to B'') \] with the precise arrangement depending on the indexing of objects in the chosen diagram.

A common formulation gives an exact sequence \[ \ker(A\to B)\to \ker(A''\to B'')\xrightarrow{\delta} \operatorname{coker}(A'\to B')\to \operatorname{coker}(A\to B) \] and then continues by mapping to \(\operatorname{coker}(A''\to B'')\) when the diagram includes terminal objects (e.g., zeros) arranged appropriately.

The defining feature is that the connecting morphism \(\delta\) makes the entire sequence exact at each term. This is the mechanism by which the lemma connects the failure of surjectivity (a cokernel) on one part of the diagram with the failure of injectivity (a kernel) on the next.

2 Variants and Generalizations

2.1 The dual statement (co-snake form)

There is a dual version of the Snake Lemma obtained by reversing arrows. Dualizing exchanges kernels with cokernels and transforms exactness statements accordingly.

If one replaces the original commutative diagram by its opposite (or equivalently reverses all morphisms), the lemma yields a “co-snake” connecting morphism that runs in the opposite direction:

  • cokernels correspond to kernels in the dual statement,
  • and the resulting long exact sequence is reversed in orientation.

This duality is a structural feature of abelian categories: exactness is self-dual, so the lemma has an immediate mirror form.

2.2 The version for short exact sequences

A frequent variant assumes the rows are short exact sequences \[ 0\to A'\to A\to A''\to 0,\qquad 0\to B'\to B\to B''\to 0 \] and then studies the induced behavior of vertical maps.

Under these assumptions, the lemma often gives a connecting morphism \[ \delta:\ker(A''\to B'')\to \operatorname{coker}(A'\to B') \] and an exact chain that begins with \(\ker(A\to B)\) and ends with \(\operatorname{coker}(A''\to B'')\). The additional zeros simplify the bookkeeping and provide a clean long exact sequence without needing partial exactness hypotheses.

2.3 Snake lemma in an arbitrary abelian category

The most robust form holds in any abelian category, not only for modules. The core requirements are:

  • existence of kernels and cokernels,
  • exactness defined in terms of images and kernels (equivalently cokernels),
  • and the usual categorical properties (monomorphisms behave like kernels, epimorphisms behave like cokernels, etc.).

Because the lemma can be proved using universal properties rather than elements, the connecting morphism becomes a canonical morphism in the category. This categorical nature supports its use in more abstract contexts like coherent sheaves, representations, or derived category constructions.

2.4 Relation to kernels, cokernels, and diagram chasing

The Snake Lemma is tightly linked to the behavior of kernels and cokernels under commutative diagrams. The “snake” can be understood as the systematic accounting of how exactness in rows constrains the possible maps between kernels and cokernels.

Diagram chasing offers an operational view: given commutative squares, one tracks how an element in one object must map to zero in another, then uses exactness to lift or descend along the horizontal arrows. Universal properties provide the more conceptual view: kernels and cokernels are characterized by factorization properties, and the connecting morphism arises from factoring certain canonical morphisms through these universal objects.

In either perspective, the lemma functions as a bridge between local information (exactness in rows) and global consequences (a long exact sequence relating kernels/cokernels across the diagram).

3 Proof Approaches

3.1 Diagram chasing proof

In the element-based approach (commonly taught first for module categories), one proves well-definedness and exactness by explicit lift-and-check arguments.

Sketch of the typical method:

  1. Take an element in a kernel or a class in a cokernel that should be related by the connecting morphism.
  2. Use exactness of a horizontal row to lift it to the middle object.
  3. Apply the commutative vertical map to obtain an element in the target object.
  4. Show this new element lies in the desired kernel (or has image in the desired cokernel).
  5. Prove independence of choices by checking that different lifts differ by something mapped to zero due to exactness.
  6. Prove exactness at each stage by verifying that an element maps to zero exactly when it arises from a previous morphism.

This approach is concrete and illuminates why the construction yields a well-defined connecting morphism.

3.2 Proof via universal properties (kernels/cokernels)

A more abstract approach avoids elements entirely. One uses the fact that a kernel is characterized by a universal property: a morphism factors uniquely through the kernel precisely when its composite with the defining arrow is zero. Similarly for cokernels: a morphism factors uniquely through the cokernel when it vanishes on the kernel of the cokernel map.

Under the hypotheses of the diagram, one constructs certain intermediate morphisms between kernels and cokernels by universal factorization. The connecting morphism is obtained by forcing a suitable map into a kernel (or through a cokernel) using exactness.

The advantage of this proof is that it immediately generalizes to any abelian category and clarifies that the connecting morphism is canonical.

3.3 Categorical proof using kernels of induced maps

Another categorical proof organizes the argument around induced morphisms on kernels and images.

One begins by considering the induced map between kernels arising from the commutative diagram, such as the map \[ \ker(A\to B)\to \ker(A''\to B'') \] coming from restriction of the vertical morphism to kernels. Then one analyzes how images relate to kernels using exactness of the rows. From there, one identifies which quotient object must map to which other quotient object, producing the connecting morphism as a map between a cokernel and a kernel.

This method emphasizes the “functorial” nature of kernels and cokernels under induced maps, making the structure of the long exact sequence feel inevitable.

3.4 Handling non-split cases and boundary maps

The Snake Lemma does not require the diagram to split. Its connecting morphism captures the obstruction to splitting in a controlled way.

In many worked examples, one sees that even when short exact sequences fail to split, the Snake Lemma still provides meaningful information through the boundary map. Proofs must therefore address that:

  • lifts are not unique in a strict sense,
  • but their differences land in subobjects that get annihilated in the passage to kernels/cokernels.

The boundary map (connecting morphism) is precisely the mechanism ensuring that non-splitting phenomena are reflected as nontrivial elements of the long exact sequence rather than forcing the whole diagram to trivialize.

4 Applications

4.1 Computing kernels and cokernels in exact diagrams

A direct application is computational. Suppose one has a commutative diagram where rows are exact and some vertical maps are known to be injective or surjective. Then the Snake Lemma can be used to deduce properties of the remaining vertical map.

For example:

  • if two of the vertical maps are injective, then the kernel term for the third vertical map can often be identified with the image of a related map;
  • if a vertical map is surjective, corresponding cokernel terms vanish, shortening the long exact sequence.

This yields a systematic way to compute kernels and cokernels of composite morphisms or to relate them across a diagram without constructing them from scratch.

4.2 Long exact sequences in homology and cohomology

The most celebrated application is in homological algebra: the Snake Lemma underlies the construction of long exact sequences in homology and cohomology.

Typically, one takes a short exact sequence of chain complexes \[ 0\to C^\bullet \to D^\bullet \to E^\bullet \to 0 \] and applies the Snake Lemma degreewise to obtain a long exact sequence in homology: \[ \cdots \to H^n(C^\bullet)\to H^n(D^\bullet)\to H^n(E^\bullet)\xrightarrow{\delta} H^{n+1}(C^\bullet)\to \cdots \] The connecting morphisms in such long exact sequences are precisely the boundary maps produced by Snake-lemma-style reasoning, though implemented in the language of cycles, boundaries, and exactness within the category of abelian groups.

4.3 Applications to derived functors

Derived functors such as \(\operatorname{Ext}\) and \(\operatorname{Tor}\) rely on the existence of long exact sequences induced by short exact sequences of modules or complexes.

Snake Lemma logic is part of the standard toolkit for proving that derived functors are “delta-functors,” meaning they convert short exact sequences into long exact sequences. While the full derived-functor construction involves resolutions and additional machinery, the connecting morphism ultimately traces back to the same kernel/cokernel interface governed by exact diagrams.

4.4 Use in establishing functorial exactness statements

When studying a functor between abelian categories, one often seeks to understand whether it preserves exactness, at least partially. Exactness properties can be checked using diagrams and the Snake Lemma.

For instance, if one knows a functor is left exact or right exact, the lemma helps interpret how the failure of full exactness manifests as connecting morphisms in associated long exact sequences. This leads to concise proofs that certain derived or correction functors measure non-exactness.

5 Computational and Practical Tools

5.1 How to identify the connecting morphism

In concrete diagrams, identifying the connecting morphism \(\delta\) is often the main step. Practically, one:

  1. Determines which objects in the diagram correspond to the domain kernel and codomain cokernel.
  2. Uses exactness in the horizontal rows to express “a preimage exists” or “a map lands in a kernel.”
  3. Follows the commutative square/cube relations to obtain the induced map between those kernel/cokernel terms.

The connecting morphism is not an arbitrary construction: it is forced by the demand that the resulting sequence be exact at each stage, which mirrors the defining lift-and-quotient process in the proof.

5.2 Typical commutative diagram patterns

Several recurring patterns appear in applications:

  • a 3×3 commutative diagram where both horizontal rows are exact and the vertical maps are arbitrary;
  • diagrams where one vertical morphism is injective or surjective, causing terms in the long exact sequence to vanish;
  • diagrams arising from short exact sequences of complexes, handled degreewise.

Recognizing these patterns helps one deploy the lemma quickly rather than re-deriving the connecting map from scratch.

5.3 Worked example template

A common workflow for applying the lemma looks like this:

  1. Write the exact rows explicitly and label the vertical maps.
  2. Verify commutativity.
  3. Determine the kernels and cokernels that appear at each end of the exact sequence.
  4. Construct \(\delta\) by choosing a representative in the kernel/cokernel term, lifting it across an exact row, and mapping it via the vertical morphism.
  5. Confirm that the resulting maps form an exact sequence by checking that images match kernels, usually using the “if and only if” derived from exactness in the rows.

Even without computing specific numbers, this template ensures correct placement of each term and correct direction of \(\delta\).

5.4 Common pitfalls in applying exactness hypotheses

Misapplication often stems from subtle bookkeeping:

  • using exactness at the wrong spot (exactness at an outer term is often automatic in short exact sequences with zeros, but not in general);
  • assuming a map is surjective or injective without verifying it from the diagram;
  • forgetting that commutativity must hold for the construction to be well-defined;
  • mixing up kernel-versus-cokernel terms in dualized versions.

Careful attention to which horizontal sequences are exact, and at which objects, prevents these errors.

6 Relationship to Other Results

6.1 Five lemma and nine lemma connections

The Five Lemma is a classic result in commutative diagrams that gives isomorphism criteria when maps are monomorphisms/epimorphisms and certain composites are zero. The Snake Lemma complements it by translating exactness conditions into a long exact sequence of kernels and cokernels.

The Nine Lemma (often used in 3×3 diagram settings) deals with when induced morphisms between kernels/images are isomorphisms under exactness assumptions. In many arguments, one reduces to kernel/cokernel comparisons that ultimately resemble the Snake Lemma’s mechanism.

Thus, while the Five Lemma focuses on isomorphisms between objects, the Snake Lemma focuses on exactness structures and connecting morphisms, providing finer-grained information.

6.2 Comparison with the long exact sequence of a pair conceptual level

In topology and related contexts, long exact sequences of pairs relate homology of a space, a subspace, and their quotient. Conceptually, these sequences are constructed by fitting chain complexes into short exact sequences and then extracting connecting morphisms.

The Snake Lemma contributes the abstract “machine” behind such connecting morphisms: it turns exactness of rows in a diagram (here, sequences of cycles and boundaries) into boundary maps in a long exact sequence. While the topological setting uses additional structures, the core logic mirrors the Snake Lemma framework.

The Horseshoe Lemma provides a method to build resolutions from resolutions of subobjects and quotients in a short exact sequence. While its statement concerns constructing objects (resolutions), not directly kernels/cokernels in a single diagram, it connects to the Snake Lemma through the broader theme: short exact sequences induce structured long exact sequences after applying derived constructions.

In other words, the Horseshoe Lemma helps create the inputs needed for derived functor computations, and the Snake Lemma supplies the exact-sequence outputs that such computations rely on.

6.4 Compatibility with functor composition and naturality

An important conceptual property is that the Snake Lemma behaves well under functoriality. Given a morphism of diagrams (a compatible change of vertical maps), the induced long exact sequences commute appropriately. This expresses a naturality principle for the connecting morphism: it depends only on the diagram up to the structure enforced by commutativity and exactness.

Moreover, when composing functors, exactness properties and induced connecting morphisms can be tracked using the same universal constructions that define kernels and cokernels. As a result, the Snake Lemma’s boundary maps can be made compatible with broader categorical frameworks, such as derived functors and spectral sequence machinery.