1 Definition and Exactness
A long exact sequence is an infinite sequence of algebraic objects and homomorphisms in which each map fits exactly between its neighbors. The central requirement is that the image of one map equals the kernel of the next at every position. This condition expresses a precise balance: elements that can be continued forward are exactly those that came from the previous stage.
Long exact sequences appear throughout homological algebra because they organize information extracted from chain complexes, short exact sequences, and related constructions. They often convert a difficult problem into a chain of smaller ones, with connecting homomorphisms carrying the remaining information from one stage to the next.
1.1 Exactness at a point in a sequence
Exactness at a single object means that every element entering that object from the left is exactly one that exits to the right, and no others survive to the right. In symbols, if the maps are \(A \to B \to C\), exactness at \(B\) means \(\mathrm{im}(A \to B)=\ker(B \to C)\).
This local condition is the building block for the entire theory. It measures how much of the structure of one term is determined by its neighbors.
1.2 Exactness of a long sequence
A long exact sequence extends indefinitely in both directions or in one direction, depending on context. Exactness must hold at every internal term, so the sequence is not merely a list of objects but a tightly linked chain.
Because the sequence may extend through many degrees, it can encode global information across a family of related invariants. This is one reason long exact sequences are so useful in topology, algebra, and geometry.
1.3 Kernels, images, and consecutive maps
The kernel of a map consists of elements sent to zero, while the image is the set of outputs actually reached. Exactness identifies these two notions at adjacent positions in the sequence. As a result, the failure of one map to be injective is measured by the image of the preceding map, and the failure of a map to be surjective is recorded by the kernel of the following map.
This interplay makes exact sequences a natural language for tracking obstructions and extensions.
1.4 Coherent indexing conventions
Long exact sequences are typically indexed by integers, often degrees in homology or cohomology. The indexing convention matters because the direction of the maps and the grading shifts determine where the connecting homomorphism lands.
Different authors may shift indices or write the sequence in slightly different orientations. Despite these variations, the underlying exactness condition remains the same.
2 Sources of Long Exact Sequences
Long exact sequences usually arise from a short exact sequence of objects together with a homological construction. The most common sources are chain complexes, the snake lemma, and the mapping cone. In each case, a short local piece of structure gives rise to a longer sequence of invariants.
2.1 From short exact sequences of complexes
A short exact sequence of chain complexes can be viewed degree by degree. When the sequence is exact in each degree and the differentials are compatible, one can pass to homology and obtain a long exact sequence.
The resulting sequence links the homology groups of the three complexes, showing how the homology of one term controls or is controlled by the others.
2.1.1 The derived connecting morphism
The connecting morphism is the map that joins one stage of the sequence to a later stage after taking homology. It is obtained by lifting a cycle, applying a differential, and then projecting to the appropriate quotient or homology group.
This map is the key mechanism that turns a short exact sequence of complexes into a long exact sequence of homology groups.
2.2 From the snake lemma
The snake lemma is a diagram-chasing result that starts with a commutative diagram with exact rows. It produces an exact sequence of kernels and cokernels, together with a connecting map that links them.
Although the statement is elementary in appearance, it is one of the fundamental sources of long exact sequences in algebra.
2.2.1 Extracting a long exact sequence from a diagram
When the snake lemma is applied repeatedly across degrees or layers of a diagram, the resulting kernel-cokernel sequence can be organized into a long exact sequence. This is especially useful when studying maps between complexes or families of abelian groups.
The method highlights how exactness in a diagram can propagate into a larger algebraic pattern.
2.3 From the mapping cone construction
The mapping cone packages a map of chain complexes into a new complex whose homology records the effect of the original map. The associated cone sequence yields a long exact sequence in homology.
This construction is central in derived and homotopical settings because it encodes the failure of a map to be a quasi-isomorphism.
2.3.1 Distinct sign and grading conventions
Different texts define the mapping cone with different sign choices in the differential. These conventions affect the precise formula for the connecting map and the indexing of the sequence.
Although the formulas differ, the exactness statement and the conceptual role of the cone remain unchanged.
3 Long Exact Sequence in Homology and Variants
In homology, long exact sequences arise naturally from short exact sequences of chain complexes or from pairs of spaces and subspaces. They relate cycles, boundaries, and homology classes across adjacent degrees.
These sequences are among the most important computational tools in algebraic topology and homological algebra.
3.1 Short exact sequence of chain complexes
A short exact sequence of chain complexes is exact in each degree, with maps commuting with the differentials. This setup allows the homology functor to produce a long exact sequence.
The sequence connects the homology groups of the subcomplex, the ambient complex, and the quotient complex.
3.1.1 Induced long exact sequence on homology
Applying homology to a short exact sequence of complexes gives exactness on the level of homology groups, except that a connecting homomorphism is needed to bridge the degree gap. That map comes from the algebraic behavior of cycles lifted through the short exact sequence.
The resulting long exact sequence is a standard tool for computing unknown homology groups from known ones.
3.2 Degree shifts and grading behavior
Homology groups are graded, so a long exact sequence must respect degree changes. The connecting map usually lowers degree by one, or equivalently shifts by one in the opposite direction, depending on convention.
These shifts are not incidental: they reflect the way boundaries in one degree relate to cycles in the next.
3.2.1 Suspension and its effect on connecting maps
Suspension changes grading by raising or lowering degrees in a systematic way. In a long exact sequence, suspension often alters the position of the connecting morphism while preserving exactness.
This is important when comparing different chain models that represent the same underlying object.
3.3 Relative homology viewpoint
Relative homology compares a space with a subspace and measures features present in the larger space but not in the smaller one. The long exact sequence of a pair is one of the most familiar examples of the theory.
It provides a bridge between absolute and relative invariants.
3.3.1 How relative groups fit into the long sequence
Relative groups appear between the homology of the subspace and the homology of the ambient space. The long exact sequence shows how information passes from the subspace to the pair and then back to the space.
This arrangement makes it possible to compute one group from the others when enough data are known.
4 Long Exact Sequence in Cohomology
Cohomology gives a contravariant analogue of homology, and it also admits long exact sequences. These arise from short exact sequences of cochain complexes or from pairs and triples in topological settings.
The cohomological version often reverses arrows relative to homology while retaining the same exactness principle.
4.1 Cohomology from cochain complexes
Cochain complexes use differentials that raise degree rather than lower it. A short exact sequence of cochain complexes therefore yields a long exact sequence in cohomology after passing to cocycles and coboundaries.
The structure is parallel to homology, but the grading direction is reversed.
4.2 Induced maps and contravariant behavior
Cohomology is contravariant, so a map of objects typically induces a map in the opposite direction on cohomology groups. This reversal influences the orientation of the long exact sequence.
Despite the change in direction, exactness still expresses the same image-kernel relationship.
4.3 Connecting homomorphisms in cohomology
The cohomological connecting homomorphism transfers information from one degree to the next, usually increasing degree by one. It is defined by lifting a cocycle, applying the differential, and interpreting the result in the appropriate quotient.
This map is essential for making the cohomology sequence exact across the gap created by the short exact sequence of cochain complexes.
4.4 Comparison with homology sequences
Homology and cohomology long exact sequences are formally similar but differ in variance, grading, and the direction of induced maps. The homological sequence typically runs with degree-lowering connecting maps, while the cohomological version usually raises degree.
Both are manifestations of the same deeper principle: exactness survives passage to derived invariants.
5 Functoriality and Natural Transformations
Long exact sequences are not only exact; they are often natural with respect to maps between underlying short exact sequences or diagrams. This functorial behavior makes them robust and transferable across contexts.
Naturality ensures that constructions commute with morphisms in a controlled way.
5.1 Naturality of induced maps
When a morphism acts on the objects producing a long exact sequence, the resulting maps on homology or cohomology are compatible with the sequence. This compatibility means that the sequence is preserved under the induced transformations.
Naturality is essential for comparing different computations and for proving that a construction is independent of choices.
5.2 Compatibility with composition
If one map factors through another, the induced maps on the long exact sequence also compose accordingly. This property follows from the functorial nature of the homological construction.
As a result, complex transformations can be analyzed step by step without losing track of exactness.
5.3 Morphisms of short exact sequences
A morphism between short exact sequences is a commutative diagram respecting exact rows. Such a diagram induces a morphism between the corresponding long exact sequences.
This principle allows one to transport information from one exact sequence to another, often simplifying calculations or comparisons.
5.4 Commuting diagrams and induced exact sequences
Commutative diagrams are the natural setting for exact-sequence arguments. When the relevant squares commute, the induced maps on kernels, cokernels, or homology groups fit together into a larger exact diagram.
This organized viewpoint makes it easier to verify exactness and identify the role of each connecting map.
6 Computations Using Exactness
Long exact sequences are powerful because they permit indirect computation. If several adjacent terms are known, exactness often determines the remaining one or narrows it down to a small set of possibilities.
Their utility lies in converting global calculations into local algebraic steps.
6.1 Diagram chasing techniques
Diagram chasing is the method of following elements through commuting diagrams to prove exactness or identify maps. It is a standard technique for working with long exact sequences.
Although elementary in spirit, it is often the most effective way to establish detailed properties of connecting homomorphisms.
6.2 Five-lemma-style reasoning and generalizations
The five lemma and related results use exactness in a diagram to conclude that one map is an isomorphism when nearby maps already are. Variants such as the four lemma and the zig-zag lemma extend this approach.
These arguments are widely used to deduce isomorphisms without computing every group directly.
6.3 Splitting criteria and consequences of exactness
Exact sequences can sometimes split, meaning that the middle object decomposes as a direct sum of the neighboring ones. Splitting often simplifies a long exact sequence into shorter pieces or reveals hidden structure.
Conversely, failure to split may indicate the presence of nontrivial extension data.
6.4 Applications to determining unknown invariants
When most terms in a long exact sequence are known, exactness can determine the remaining ones up to isomorphism or narrow bounds. This is especially useful in algebraic topology, where homology or cohomology groups of complicated spaces may be computed from simpler subspaces.
The sequence thus serves as a computational bridge between familiar and unfamiliar invariants.
7 Boundary and Connecting Homomorphisms
The connecting homomorphism is the distinctive feature that turns a short exact sequence into a long one. It measures the obstruction to lifting a class from one stage to another and often carries geometric or algebraic meaning beyond mere bookkeeping.
Understanding this map is central to using exact sequences effectively.
7.1 Construction of the connecting map
To define the connecting map, one typically chooses a representative cycle or cocycle, lifts it to the middle object, and applies the differential or analogous structure. The result is then interpreted in the preceding or following homology group.
Although the construction depends on choices, the resulting homomorphism is well defined on equivalence classes.
7.2 Sign and convention issues
The sign of the connecting morphism can vary with grading conventions, cone definitions, or the order in which maps are written. These differences are usually harmless if treated consistently.
Careful attention to signs is important when comparing formulas from different sources.
7.3 Relation to chain-level representatives
The connecting map is often computed using explicit representatives at the chain level. A cycle that becomes a boundary after lifting produces a class in the neighboring degree.
This chain-level viewpoint clarifies why exactness holds and how the map reflects hidden algebraic structure.
7.4 Exactness verification near the connecting morphism
Exactness near the connecting map requires checking that the image of one map equals the kernel of the next on both sides of the boundary. This step is usually the most delicate part of the proof.
Once verified, it ties together the entire long exact sequence and confirms that no information is lost between adjacent degrees.
8 Structural Properties and Reformulations
Long exact sequences admit several equivalent descriptions in more advanced language. They can be viewed through derived functors, exact couples, or homotopical constructions, each emphasizing a different aspect of the same phenomenon.
These reformulations are useful because they reveal why exact sequences arise so persistently.
8.1 Translating between homological and derived language
In derived settings, long exact sequences often come from exact triangles or derived functor formalism. The homological sequence then appears as the shadow of a more abstract categorical structure.
This translation clarifies how exactness fits into modern algebraic frameworks.
8.2 Exact couples and spectral-sequence motivation
Exact couples are algebraic data from which spectral sequences are generated. They package several exact relationships in a compact form, and repeated derivation leads to successive pages of a spectral sequence.
Long exact sequences can be seen as one of the simplest manifestations of this broader pattern.
8.3 Persistence across chain homotopies
Chain homotopic maps induce the same map in homology or cohomology. As a result, long exact sequences built from such maps are stable under chain homotopy.
This invariance explains why the sequences depend on homotopy type rather than on the precise chain-level model.
8.4 Equivalent formulations via derived functors
Derived functors such as Ext and Tor often come equipped with long exact sequences when applied to short exact sequences. These are another expression of the same exactness principle.
The derived-functor viewpoint generalizes many familiar examples and situates long exact sequences within a larger categorical theory.
9 Examples
Concrete examples show how long exact sequences are used in practice. They also reveal the role of the connecting map, which is often invisible in abstract formulations.
The examples below illustrate typical computations rather than exhaustive classifications.
9.1 A basic example from a short exact sequence
Consider a short exact sequence of chain complexes with two known homology groups and one unknown group. The long exact sequence links these groups, and exactness may force the unknown group to vanish or to match a neighboring group.
Such examples are common in introductory calculations because they demonstrate the method with minimal overhead.
9.2 An example using relative constructions
For a pair consisting of a space and a subspace, the long exact sequence of the pair relates the homology of the subspace, the ambient space, and the relative groups. If two of these are easy to compute, the third often follows.
This approach is particularly effective when the subspace is simpler than the whole space.
9.3 A computation illustrating boundary maps
A boundary map may send a class represented by a cycle in one degree to a class one degree lower or higher, depending on context. Computing it explicitly shows how a geometric feature becomes an algebraic obstruction.
Such computations are often the point where abstract exactness becomes concrete.
9.4 Interpreting results in terms of invariants
After the sequence has been used to determine a group or map, the result is interpreted as an invariant of the original object. The exact sequence then provides not only a computation but also a structural explanation for why the invariant has that form.
This perspective is one of the main reasons long exact sequences are so influential in algebraic topology and homological algebra.
10 Common Mistakes and Best Practices
Working with long exact sequences requires care with indices, directions, and exactness conditions. Small notation errors can lead to incorrect conclusions, especially when multiple grading shifts are involved.
A disciplined approach to conventions helps avoid these problems.
10.1 Indexing and degree-shift errors
One frequent mistake is placing the connecting map in the wrong degree. Since homology and cohomology sequences shift degrees differently, it is easy to lose track of where a class belongs.
Checking the grading at each step prevents such errors.
10.2 Confusing kernels with images
Exactness says that a kernel equals an image, but these are not the same object in every context. Confusing the two can lead to incorrect deductions about injectivity or surjectivity.
It is best to state explicitly which map is being considered at each stage.
10.3 Ignoring direction and contravariant behavior
Homology and cohomology do not behave identically with respect to maps. Forgetting whether the theory is covariant or contravariant can reverse arrows and invalidate a calculation.
Keeping track of the variance of the functor is therefore essential.
10.4 Diagram orientation and sign conventions
Different diagram orientations and sign conventions may change the written form of a long exact sequence without changing its meaning. Problems arise when formulas from different sources are combined without adjustment.
Consistent notation, careful reference to the chosen convention, and a check against exactness help ensure correct results.