1 Definitions and exactness

1.1 Sequences of objects and morphisms

In algebra, an exact sequence is typically displayed as a finite or infinite “chain” of objects with arrows between them. In a setting such as groups, modules, or more general categories, one considers a sequence \[ \cdots \to A_{n-1} \xrightarrow{f_{n-1}} A_n \xrightarrow{f_n} A_{n+1} \to \cdots \] where each arrow is a morphism in the ambient category. The objects \(A_n\) may be modules, abelian groups, vector spaces, or others, and the arrows are structure-preserving maps.

1.2 Image–kernel condition

The defining property of exactness is the alignment of “what maps to zero” with “what comes from earlier in the chain.” Concretely, at each stage \(A_n\), one requires that \[ \operatorname{Im}(f_{n-1})=\ker(f_n). \] Thus, an element (or element-like object) in \(A_n\) lies in the kernel of the next map precisely when it originated as the image of the previous map. This condition enforces that no nontrivial information is lost between consecutive steps beyond what is intended by the vanishing of maps.

1.3 Exactness at a position vs exactness of a sequence

Exactness can be discussed “at a single position” or for the entire sequence. A sequence is exact at \(A_n\) when \(\operatorname{Im}(f_{n-1})=\ker(f_n)\) holds at that particular object. The sequence is exact if the equality holds at every relevant object in the chain (or at every object except possibly endpoints, depending on the convention used).

At ends of finite sequences, the meaning of \(\ker\) and \(\operatorname{Im}\) may involve either zero objects or specified initial/terminal morphisms. In categorical settings, one often formulates exactness using the universal properties of kernels and images (or their categorical analogues), but the intuition remains the same: “the next kernel is exactly the previous image.”

1.4 Short exactness and split short exactness

A short exact sequence has the form \[ 0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0, \] with maps \(i\) and \(p\) such that exactness holds at \(A\), \(B\), and \(C\). Typically this means:

  • \(i\) is injective (so \(\ker i=0\)),
  • \(p\circ i=0\),
  • the image of \(i\) equals the kernel of \(p\),
  • \(p\) is surjective (so \(\operatorname{coker} i\cong C\)).

A short exact sequence is split if the middle object decomposes as a direct sum in a compatible way. One common formulation is that there exists a morphism \(s:C\to B\) with \(p\circ s=\mathrm{id}_C\) (a right inverse for \(p\), i.e., a section). Equivalently, there exists a morphism \(r:B\to A\) with \(r\circ i=\mathrm{id}_A\) (a left inverse for \(i\), i.e., a retraction). In that case, \(B\) is isomorphic to \(A\oplus C\).

1.5 Split monomorphisms and split epimorphisms

A morphism \(i:A\to B\) is a split monomorphism if it has a left inverse \(r:B\to A\) with \(r\circ i=\mathrm{id}_A\). Such a map is automatically injective in algebraic contexts where kernels exist. Dually, a morphism \(p:B\to C\) is a split epimorphism if it admits a right inverse \(s:C\to B\) with \(p\circ s=\mathrm{id}_C\), hence is surjective in suitable settings. In short exact sequences, splitness corresponds precisely to whether the relevant mono or epi has this inverse property, guaranteeing a direct-sum decomposition.

2 Types of exact sequences

2.1 Short exact sequences

Short exact sequences are the basic building blocks for many constructions in module theory and homological algebra. They encode an object \(B\) as an extension of \(C\) by \(A\): \(A\) sits inside \(B\), and \(C\) is the corresponding quotient.

2.1.1 Interpretation via extensions

In an extension \[ 0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0, \] the subobject \(i(A)\) represents the part of \(B\) isomorphic to \(A\), while the quotient \(B/i(A)\) recovers \(C\). Two extensions can be considered equivalent when there is an isomorphism between the middle terms commuting with the maps from \(A\) and to \(C\). This viewpoint is central in classification problems, where one studies all possible ways \(C\) can be “glued” onto \(A\) to form \(B\).

2.1.1.1 Pushouts and pullbacks (motivating properties)

Short exact sequences behave functorially with respect to certain universal constructions.

  • A pullback along a morphism \(A\to A'\) modifies the left end: given

\[ 0 \to A \to B \to C \to 0 \] and a map \(A\to A'\), one can form a new object that fits into a diagram where the updated sequence still has exactness at the appropriate positions. Pullbacks are often used to compare extensions obtained by changing the embedded subobject.

  • A pushout along a morphism \(C'\to C\) modifies the right end: starting from the same extension, one can form a new middle object so that the quotient changes from \(C\) to \(C'\) while maintaining exactness. Pushouts help relate extensions under morphisms of the quotient.

These constructions are “motivating properties” because they demonstrate how exact sequences can be transported and combined while preserving the essential image–kernel relationship.

2.2 Long exact sequences

Long exact sequences extend the short exact template to many consecutive terms: \[ \cdots \to A_{n-1} \to A_n \to A_{n+1} \to \cdots \] Exactness at each intermediate object organizes how information moves through a sequence of morphisms. Such sequences frequently arise from applying functors to short exact sequences and then passing to derived invariants. A canonical example is the long exact sequence in homology associated with a short exact sequence of chain complexes or a short exact sequence of coefficient objects.

2.3 Exact sequences in module categories

In the category of modules over a ring \(R\), exactness has a concrete interpretation in terms of elements and linear maps. Kernels and images exist, and the exactness condition \(\operatorname{Im}(f_{n-1})=\ker(f_n)\) can be checked by element computations. Many standard results—such as the characterization of submodules via kernels and of quotients via cokernels—are special cases of general exactness principles.

Because module categories are abelian, exact sequences can also be treated cleanly with categorical language: monomorphisms are kernels and epimorphisms are cokernels, which makes many proofs systematic rather than ad hoc.

2.4 Exact sequences in abelian categories (scope and intuition)

In an abelian category, one has a robust notion of kernel, cokernel, and image-like factorization. Exactness can be defined without referring to elements, relying on the existence and universal properties of kernels and cokernels. The image–kernel condition translates into the statement that each morphism in the sequence is, up to equivalence, the kernel of the next and the cokernel of the previous.

This broadened scope is important: many arguments from modules and vector spaces carry over verbatim to other abelian contexts. The intuition remains the same—exact sequences track how successive maps eliminate precisely the previous contributions and nothing more.

3 Morphisms between exact sequences

3.1 Commutative diagrams and chain maps

Morphisms between sequences are encoded by commutative diagrams. If one has two chains of objects \[ A_\bullet:\ \cdots \to A_{n-1}\xrightarrow{f_{n-1}}A_n\xrightarrow{f_n}A_{n+1}\to\cdots \] and \[ B_\bullet:\ \cdots \to B_{n-1}\xrightarrow{g_{n-1}}B_n\xrightarrow{g_n}B_{n+1}\to\cdots, \] a chain map (or more generally a morphism of complexes, in the common specialization) is a collection of morphisms \(\phi_n:A_n\to B_n\) such that all squares commute: \[ \phi_{n+1}\circ f_n = g_n\circ \phi_n. \] When these diagrams commute, they ensure that the maps respect the exactness structure “in transit,” allowing comparison of kernels and images along the chain.

3.2 Morphisms of short exact sequences

A morphism of short exact sequences is a commutative diagram where each row is short exact: \[ \begin{array}{ccccccccc} 0 & \to & A & \to & B & \to & C & \to & 0\\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & A' & \to & B' & \to & C' & \to & 0. \end{array} \] Such morphisms preserve the boundary structure: the maps from \(A\) to \(B\) and from \(B\) to \(C\) intertwine with the corresponding maps in the primed sequence. They are the setting where one can often deduce injectivity/surjectivity properties of some vertical arrows from others, using the exactness of the rows.

3.3 The five lemma and its variants (high-level statement)

The five lemma is a standard tool for comparing maps between exact sequences. In typical form, it concerns a commutative diagram with two exact rows of length five (in abelian settings), providing conditions under which the middle vertical morphism is an isomorphism. Variants (such as the four lemma) address cases where one or more end morphisms are injective or surjective, leading to conclusions about the remaining map(s).

At a high level, the lemma formalizes a principle: in a diagram where exactness controls kernels and images, the behavior of maps at the ends forces the behavior in the middle.

4 Fundamental constructions

4.1 Quotients and subobjects in exact sequences

In the abelian/module context, short exact sequences encode quotient-subobject relationships. Exactness at the middle term implies that the subobject corresponding to the image of \(i\) is precisely the kernel of \(p\). Hence \(A\) embeds into \(B\) as \(\ker p\), and \(C\) is realized as the quotient \(B/\ker p\). These identifications allow one to treat \(B\) as assembled from a subobject and a quotient without ambiguity.

4.2 Pullback construction

The pullback of an exact sequence along a map on the left produces a new sequence whose left end is adjusted. In essence, one replaces the subobject \(A\) by another object mapping to it, then takes the fiber product that makes the corresponding square commute. Under appropriate hypotheses (e.g., in abelian categories), pullbacks preserve exactness at the relevant positions. This construction is used to transport extension data and to compare how kernels behave under change of domain.

4.3 Pushout construction

Dually, a pushout modifies the right end. Starting with \(0\to A\to B\to C\to 0\) and a morphism \(C\to C'\), one forms a universal object receiving a map from \(B\) and mapping onto \(C'\). In favorable settings, the resulting diagram yields another exact sequence in which the quotient is the new \(C'\). Pushouts therefore provide a mechanism to transfer extension structure along morphisms of quotients.

4.4 Snake lemma (diagrammatic viewpoint)

The snake lemma relates kernels and cokernels in a commutative diagram with exact rows. Diagrammatically, one considers a grid where two rows are exact and certain vertical morphisms are present. The lemma then produces a long exact sequence of kernels and cokernels, often summarized as a chain \[ \ker(\text{left}) \to \ker(\text{middle}) \to \ker(\text{right}) \to \operatorname{coker}(\text{left}) \to \operatorname{coker}(\text{middle}) \to \operatorname{coker}(\text{right}), \] with appropriate connecting morphisms.

The “snake” terminology comes from the way elements are traced through the diagram: one follows how an element in one kernel maps across the grid, using exactness to locate its preimages, and then returns via cokernel information. This lemma is a key engine for producing derived exact sequences from simpler ones.

5 Homological consequences

5.1 Applying a functor to an exact sequence

Homological methods frequently start by applying a functor \(F\) to an exact sequence and studying what remains exact. In general, functors do not preserve exactness automatically. However, the effect of \(F\) on kernels and images can be constrained by whether the functor is right or left exact, or by whether it is part of a derived framework.

When one applies \(F\) to \[ 0 \to A \to B \to C \to 0, \] one obtains \[ 0 \to F(A) \to F(B) \to F(C) \to 0 \] only under additional conditions. Exactness may fail at the beginning or end: the kernel–image equalities can be distorted depending on the functor’s properties.

5.2 Left and right exactness of functors

A functor is left exact if it preserves kernels; equivalently, it preserves exactness at the left side of short exact sequences. In many conventions, left exactness ensures that \[ 0\to F(A)\to F(B)\to F(C) \] is exact at \(F(A)\) and \(F(B)\). A functor is right exact if it preserves cokernels and thus preserves exactness at the right side: \[ F(A)\to F(B)\to F(C)\to 0 \] is exact at \(F(B)\) and \(F(C)\).

These distinctions are central in homological algebra because they predict where exactness will break and therefore where derived functors must compensate.

5.3 Derived functors and the role of exactness

Derived functors quantify the failure of a functor to be fully exact. For instance, if a functor is left exact but not right exact, its right-derived functors measure how far surjectivity or cokernel exactness fails. Similarly, left-derived functors measure failures of kernel exactness for right exact functors.

Exact sequences serve as the input data that derived functors process: the long exact sequences in homology or cohomology emerge from applying derived constructions and using the fact that derived functors transform short exact sequences into long exact sequences, with connecting morphisms capturing the obstruction data.

5.4 Relationship to Ext and Tor (conceptual overview)

In module theory, Ext and Tor are derived constructions that classify extension and measure tensor-related homological defects.

  • Ext can be interpreted as measuring extension classes: higher \(\operatorname{Ext}\) groups track increasingly indirect ways in which one module can fail to lift across exact sequences. Informally, \(\operatorname{Ext}^1\) corresponds to equivalence classes of extensions, while higher degrees arise from iterated resolutions.
  • Tor measures failure of tensor product to preserve exactness, especially exactness with respect to one variable. Tensoring a short exact sequence typically yields only right or left exactness; the derived functors quantify the missing exactness and relate to homology of tensor products.

Together, Ext and Tor connect the abstract definition of exactness with concrete computations of homological invariants.

6 Examples and standard computations

6.1 Exactness from kernel and image computations

In practice, exactness is often verified by direct calculation of kernels and images. Given maps \(A_{n-1}\xrightarrow{f_{n-1}}A_n\xrightarrow{f_n}A_{n+1}\), one computes \(\ker(f_n)\) and \(\operatorname{Im}(f_{n-1})\), then checks equality. This approach is particularly common for modules and vector spaces where linear algebra provides explicit descriptions.

A useful diagnostic is to check two inclusions:

  1. \(\operatorname{Im}(f_{n-1})\subseteq \ker(f_n)\), often ensured by the relation \(f_n\circ f_{n-1}=0\).
  2. \(\ker(f_n)\subseteq \operatorname{Im}(f_{n-1})\), which is the substantive part.

6.2 The sequence 0 → A → B → C → 0

For the short exact sequence \[ 0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0, \] exactness is verified by showing:

  • \(i\) is injective, so elements of \(A\) embed into \(B\) without collapsing;
  • \(p\) is surjective, so every element of \(C\) comes from some element of \(B\);
  • the image of \(i\) equals the kernel of \(p\), ensuring that the only elements mapped to zero in \(C\) are precisely those coming from \(A\).

Under these conditions, \(C\) is canonically isomorphic to the quotient \(B/i(A)\), and \(A\) is isomorphic to \(\ker p\).

6.3 Constructing exact sequences from presentations

Exact sequences frequently arise from presentations of algebraic objects. For modules, one may describe a module \(M\) as a quotient of a free module by relations: \[ R^{(S)} \xrightarrow{\;\;\phi\;\;} R^{(T)} \to M \to 0, \] where \(R^{(S)}\) and \(R^{(T)}\) are free modules on chosen generating sets. By completing the sequence with a kernel at the left, one obtains a short exact or longer exact sequence reflecting both generators and relations. This is a standard method to turn explicit construction data into the language of exactness.

6.4 Standard examples in modules and vector spaces

In vector spaces, exact sequences can be interpreted through dimension counts. For instance, for a sequence \[ 0\to A\xrightarrow{i}B\xrightarrow{p}C\to 0, \] exactness implies \(\dim B=\dim A+\dim C\) and identifies \(A\) with \(\ker p\) and \(C\) with \(B/\ker p\). Many computations reduce to elementary linear algebra: rank–nullity gives the kernel dimension, and the image dimension aligns with it.

More broadly, in module categories, standard examples include sequences derived from homomorphisms between free modules, sequences defining syzygies (relations among relations), and sequences associated with tensor products or hom functors, all of which serve as templates for understanding how exactness behaves under common algebraic operations.