Split exact sequence

In homological algebra, a split exact sequence is a short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0 of objects and morphisms in an abelian category that admits a splitting, meaning there exists a morphism s: C \to B such that p \circ s = \mathrm{id}_C, or equivalently, a retraction r: B \to A such that r \circ i = \mathrm{id}_A.^[1] These conditions are equivalent by the splitting lemma, which implies that B is isomorphic to the direct sum A \oplus C.^[2] Split exact sequences are characterized by the existence of such complementary morphisms that decompose the middle term into a direct sum, preserving the exactness at each position.^[3] In particular, every short exact sequence of vector spaces over a field splits, as do those involving free modules over a ring (assuming the axiom of choice), due to the projective nature of free modules.^[3] More generally, a short exact sequence splits if the domain of the injection is injective or the codomain of the surjection is projective in the category.^[3] Unlike arbitrary exact sequences, split exact sequences are preserved by any additive functor between abelian categories, making them particularly useful in computations involving homology or cohomology.^[4] They play a central role in understanding extensions of modules or sheaves, where the non-split cases give rise to extension groups like \mathrm{Ext}^1(C, A), while split ones correspond to the trivial element in these groups.^[1] In the context of chain complexes, a long exact sequence is split if it is chain homotopy equivalent to the direct sum of its components.^[3]

Definition

Formal definition

In an abelian category \mathcal{A}, a short exact sequence is a sequence of morphisms $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0 such that i is a monomorphism, p is an epimorphism, and the image of i equals the kernel of p. Such a sequence is said to split (or be a split exact sequence) if there exists a morphism s: C \to B, called a section or right splitting, such that p \circ s = \mathrm{id}_C. Equivalently, the sequence splits if there exists a morphism r: B \to A, called a retraction or left splitting, such that r \circ i = \mathrm{id}_A. In abelian categories, the existence of a left splitting is equivalent to the existence of a right splitting, and in either case the middle object B is isomorphic to the direct sum A \oplus C.

Splitting conditions

A short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0 in an abelian category splits if and only if there exists a homomorphism s: C \to B, called a section or right splitting, such that p \circ s = \mathrm{id}_C.^[5] Equivalently, the sequence splits if there exists a homomorphism r: B \to A, called a retraction or left splitting, such that r \circ i = \mathrm{id}_A.^[5] In abelian categories, such as the category of modules over a ring, the existence of a left splitting is equivalent to the existence of a right splitting, as established by the splitting lemma.^[5] This equivalence follows from diagram chasing or the five lemma applied to the relevant commutative diagrams involving the identity maps on A and C. The presence of either splitting induces a direct sum decomposition of B as B = \mathrm{im}(i) \oplus \mathrm{im}(s), where \mathrm{im}(i) = \ker(p) and \mathrm{im}(s) \cong C.^[5] In non-abelian categories, such as the category of groups, left and right splittings are not necessarily equivalent; a right splitting corresponds to a semidirect product, while a left splitting implies a direct product, but the focus here remains on the abelian case where they coincide.^[6]

Equivalent characterizations

Homological conditions

In homological algebra, short exact sequences of the form $0 \to A \to B \to C \to 0 in an abelian category are classified up to equivalence by the extension group \operatorname{Ext}^1(C, A), which parametrizes the possible "twistings" between A and C in the middle term B.^[7] This group arises from the derived functor of the Hom functor and captures obstructions to splitting.^[8] The set of equivalence classes of such extensions forms an abelian group under the Baer sum operation, which combines two extensions via a pushout along the codiagonal A \oplus A \to A followed by a pullback along the diagonal C \to C \oplus C, yielding a natural group structure on \operatorname{Ext}^1(C, A).^[8] The zero element in this group corresponds precisely to the trivial extension, which is the direct sum sequence $0 \to A \to A \oplus C \to C \to 0 equipped with the standard inclusions and projections.^[7] A short exact sequence splits if and only if its equivalence class in \operatorname{Ext}^1(C, A) is the zero element.^[9] To see this, note that the Baer sum equips the extension classes with inverses: for a nonzero class [E], its inverse [-E] is obtained by a similar pushout-pullback construction, ensuring that only the trivial class satisfies [E] + [-E] = 0, which manifests as the split extension isomorphic to the direct sum.^[8] This characterization highlights how vanishing of the extension class removes the homological obstruction to retraction.

Isomorphism to direct sums

A fundamental characterization of split exact sequences in abelian categories is that they correspond precisely to direct sum decompositions of the middle term. Specifically, for a short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0 that splits via a section s: C \to B satisfying p \circ s = \mathrm{id}_C, the module B is isomorphic to the direct sum A \oplus C.^[1] This isomorphism arises categorically from the splitting data, providing an explicit equivalence between the sequence and the standard direct sum presentation $0 \to A \xrightarrow{\mathrm{in}_A} A \oplus C \xrightarrow{\mathrm{pr}_C} C \to 0, where \mathrm{in}_A and \mathrm{pr}_C denote the inclusion and projection maps, respectively.^[5] The isomorphism \phi: A \oplus C \to B is constructed explicitly as

\phi(a, c) = i(a) + s(c)

for all a \in A and c \in C. To verify that \phi is an isomorphism, first note that it is a category morphism commuting with the sequence maps: \phi \circ \mathrm{in}_A = i and p \circ \phi = \mathrm{pr}_C. For injectivity, suppose \phi(a, c) = 0; then i(a) = -s(c), so applying p yields $0 = p(s(c)) = c, hence c = 0 and i(a) = 0, implying a = 0 since i is injective. For surjectivity, take any b \in B; let c = p(b), so b - s(c) \in \ker p = \mathrm{im} i, say b - s(c) = i(a) for some a \in A, whence \phi(a, c) = b. Finally, the inverse map \psi: B \to A \oplus C is given by \psi(b) = (r(b), p(b)), where r: B \to A is the retraction satisfying r \circ i = \mathrm{id}_A (which exists uniquely from the section s); direct computation confirms \phi \circ \psi = \mathrm{id}_B and \psi \circ \phi = \mathrm{id}_{A \oplus C}.^[1]^[5] This equivalence underscores the "trivial" nature of split exact sequences, distinguishing them from nonsplit extensions where no such direct sum decomposition exists.^[5]

Properties

Existence of complements

In a split exact sequence of modules $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0, the image \operatorname{im}(i) is a direct summand of B, meaning there exists a submodule D \subseteq B such that B = \operatorname{im}(i) \oplus D.^[5] This property arises because the existence of a splitting map s: C \to B with p \circ s = \operatorname{id}_C ensures that D = \operatorname{im}(s) serves as the complement, satisfying \operatorname{im}(i) \cap \operatorname{im}(s) = 0 and \operatorname{im}(i) + \operatorname{im}(s) = B.^[5] Equivalently, since \ker(p) = \operatorname{im}(i), the kernel admits a complement in B that is isomorphic to C, as \operatorname{im}(s) \cong C via s.^[5] In the category of modules over a ring R, this decomposition implies that B \cong A \oplus C as R-modules, where the isomorphism is induced by the splitting.^[5] When A, B, and C are R-modules of finite length, the length function is additive over the short exact sequence: \operatorname{length}_R(B) = \operatorname{length}_R(A) + \operatorname{length}_R(C). This holds for any short exact sequence of finite-length modules, and thus in particular for split ones.^[10]

Behavior under exact functors

Exact functors between abelian categories preserve split exact sequences. Specifically, if F: \mathcal{A} \to \mathcal{B} is an exact functor and $0 \to A \to B \to C \to 0 is a split short exact sequence in \mathcal{A}, then $0 \to F(A) \to F(B) \to F(C) \to 0 is a split short exact sequence in \mathcal{B}. This follows because exact functors are additive and thus map the direct sum decomposition B \cong A \oplus C to F(B) \cong F(A) \oplus F(C), preserving the splitting maps.^[11] Additive functors more generally preserve direct sums, and since a short exact sequence splits if and only if it is isomorphic to the direct sum sequence $0 \to A \to A \oplus C \to C \to 0, any additive functor sends split exact sequences to split exact sequences. Exact functors, being a special case of additive functors, inherit this property while also ensuring the preservation of exactness for all short exact sequences.^[11] Non-additive functors, however, may fail to preserve the splitting of exact sequences. For instance, a functor that does not respect direct sums, such as a non-additive covariant functor between categories of modules, can map a split sequence to one that is exact but lacks a splitting. In contrast, common examples like the tensor product functor - \otimes_R M for a non-flat module M over a ring R is additive and right exact but still preserves splitting due to additivity, though it may distort non-split exact sequences.^[12] In representation theory, this preservation is particularly useful under equivalence functors between module categories, which are exact and thus maintain the split structure of sequences, allowing the transfer of decompositions between equivalent representations of algebras or groups.^[13]

Examples

In abelian groups

In the category of abelian groups, a concrete example of a split exact sequence is the short exact sequence $0 \to 2\mathbb{Z} \xrightarrow{i} \mathbb{Z} \xrightarrow{\pi} \mathbb{Z}/2\mathbb{Z} \to 0, where i is the inclusion map and \pi is the canonical projection modulo 2. This sequence splits because there exists a homomorphism s: \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z} such that \pi \circ s = \mathrm{id}_{\mathbb{Z}/2\mathbb{Z}}, defined by s(0 + 2\mathbb{Z}) = 0 and s(1 + 2\mathbb{Z}) = 1. The splitting induces an isomorphism \mathbb{Z} \cong 2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}, where the direct sum decomposition corresponds to even and odd integers.^[5] In contrast, the short exact sequence $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{\pi} \mathbb{Z}/2\mathbb{Z} \to 0, where the first map is multiplication by 2 and \pi is the projection modulo 2, does not split. If it split, there would exist a section s: \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z} with \pi \circ s = \mathrm{[id](/page/ID)}, implying an element of order 2 in \mathbb{Z}, which is impossible since \mathbb{Z} is torsion-free. Consequently, no such direct sum decomposition \mathbb{Z} \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} exists, as it would require \mathbb{Z}/2\mathbb{Z} \cong 0. A similar non-splitting occurs for general n > 1 in $0 \to \mathbb{Z} \xrightarrow{\times n} \mathbb{Z} \xrightarrow{\pi} \mathbb{Z}/n\mathbb{Z} \to 0.^[2] For an infinite example, consider the short exact sequence $0 \to \bigoplus_{k=1}^\infty \mathbb{Z} \xrightarrow{i} \bigoplus_{k=1}^\infty \mathbb{Z} \oplus \mathbb{Q} \xrightarrow{\pi} \mathbb{Q} \to 0, where i includes the direct sum into the first factor and \pi projects onto the second factor. This splits via the obvious section s: \mathbb{Q} \to \bigoplus_{k=1}^\infty \mathbb{Z} \oplus \mathbb{Q} that embeds \mathbb{Q} into the second component, yielding the direct sum decomposition of the middle term. Such sequences illustrate how splitting holds trivially for direct sums in abelian groups, even with infinite direct summands.^[14]

In modules over rings

In the category of modules over a field k, every short exact sequence $0 \to V \to W \to U \to 0 of finite-dimensional vector spaces splits. This is achieved by selecting a basis for U and lifting it to linearly independent elements in W whose images form that basis; the subspace they span is isomorphic to U, and its complement in W is isomorphic to V.^[5] Over the ring of integers \mathbb{Z}, which is a principal ideal domain, not all short exact sequences split, illustrating the dependence on the ring structure. Consider the sequence $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0, where the first map is multiplication by 2 and the second is the canonical projection. This sequence is exact but does not split, as any purported section s: \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z} satisfying the projection composition to the identity would require s(1 \mod 2) to be an element of order 2 in \mathbb{Z}, which does not exist since \mathbb{Z} is torsion-free. A similar non-splitting occurs in $0 \to \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0, where the inclusion sends $1 \mod 2 to $2 \mod 4 and the projection is modulo 2; no section exists because \mathbb{Z}/4\mathbb{Z} lacks an element of order 2 outside the image of the inclusion.^[15] For modules over a principal ideal domain R, a short exact sequence $0 \to A \to B \to C \to 0 splits if and only if its class in \operatorname{Ext}^1_R(C, A) vanishes. Over the polynomial ring k with k a field, consider $0 \to k \xrightarrow{\times x} k \xrightarrow{\mathrm{ev}_0} k \to 0, where the first map is multiplication by x (with image the ideal (x)) and the second is evaluation at 0. As k-vector spaces, this splits by sending the basis element of k to the constant polynomial 1. However, as k-modules, it does not split, since any section s: k \to k would satisfy x \cdot s(1) = s(x \cdot 1) = s(0) = 0, but s(1) must have constant term 1, so x \cdot s(1) has linear term 1, a contradiction.^[16] In general, for a ring R and ideal I, the short exact sequence $0 \to I \to R \to R/I \to 0 (with inclusion and canonical projection) splits if and only if R/I is a projective R-module. Over a PID, ideals are free of rank 1 and thus projective, but the sequence still fails to split unless I = R (i.e., the ideal is the unit ideal), as the cokernel R/I is generally not projective.^[16]

Short exact sequences

A short exact sequence in an abelian category, such as the category of abelian groups or modules over a ring, is a sequence of the form $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 that is exact at each term: the map f is a monomorphism (injective), g is an epimorphism (surjective), and the image of f equals the kernel of g.^[17] This means B can be viewed as an extension of C by A, where A embeds as a kernel subgroup and C appears as a quotient. Unlike split exact sequences, where B \cong A \oplus C, general short exact sequences need not decompose in this direct way, capturing more intricate algebraic structures.^[17] A classic example of a non-split short exact sequence is $0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0, where the first map is the natural inclusion of integers into rationals and the second is the canonical projection onto the quotient. This sequence is exact because \mathbb{Z} is the kernel of the projection and embeds injectively into \mathbb{Q}, but it does not split: there is no subgroup of \mathbb{Q} isomorphic to \mathbb{Q}/\mathbb{Z} that complements \mathbb{Z}.^[18] In general, the equivalence classes of short exact sequences $0 \to A \to B \to C \to 0 (up to congruence via commutative diagrams) are in one-to-one correspondence with elements of the Ext group \operatorname{Ext}^1(C, A), where the split sequences correspond precisely to the zero element in this group.^[17] This classification highlights how non-trivial elements in \operatorname{Ext}^1(C, A) produce non-split extensions. The framework of short exact sequences and their classification via derived functors like Ext was formalized in the foundational text of homological algebra by Cartan and Eilenberg in 1956.^[17]

Projective and injective objects

In an abelian category, an object P is projective if, for every epimorphism f: A \twoheadrightarrow B and every morphism g: P \to B, there exists a morphism h: P \to A such that f \circ h = g.^[19] This lifting property ensures that projective objects interact favorably with surjections, preserving homomorphisms in a way that facilitates decompositions.^[19] A key consequence is that if C is projective in a short exact sequence $0 \to A \to B \to C \to 0, then the sequence splits.^[20] Specifically, the projectivity of C implies the existence of a section to the epimorphism B \to C, making B isomorphic to the direct sum A \oplus C.^[20] This holds equivalently because the functor \Hom(P, -) is exact when P is projective, applying the identity on C to lift through the kernel.^[20] Dually, an object I is injective if, for every monomorphism f: A \hookrightarrow B and every morphism g: A \to I, there exists a morphism h: B \to I such that h \circ f = g.^[21] In a short exact sequence $0 \to I \to A \to B \to 0, if I is injective, the sequence splits, with a retraction from A to I providing the decomposition A \cong I \oplus B.^[22] This dual lifting property against injections guarantees the splitting via the exactness of \Hom(-, I).^[22] In the category of modules over a ring R, free modules—direct sums of copies of R—are projective.^[23] Thus, any short exact sequence $0 \to A \to B \to F \to 0 with F free splits, yielding B \cong A \oplus F.^[23] This property underscores the role of free modules in providing explicit splittings within module categories.^[23]