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Splitting lemma

In homological algebra, the splitting lemma is a fundamental theorem that characterizes split short exact sequences in an abelian category, stating that for a short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0, the sequence splits—meaning B \cong A \oplus C compatibly with i and p—if and only if there exists either a retraction \pi: B \to A such that \pi \circ i = \mathrm{id}_A or a section s: C \to B such that p \circ s = \mathrm{id}_C.^[1] This equivalence holds in any abelian category and is proven by constructing the complementary maps and isomorphisms from the given inverse.^[2] The lemma originates in the study of module categories over rings, where it implies that a submodule N of a module M is a direct summand if and only if the quotient map M \to M/N admits a section, providing a concrete criterion for decomposability.^[2] In broader contexts, such as chain complexes, split exact sequences simplify homological computations by reducing them to direct sums.^[3] The lemma underscores the role of additive structure in homological algebra, highlighting how seemingly rigid exact sequences can decompose under mild splitting conditions.

Fundamentals

Short Exact Sequences

In an abelian category \mathcal{A}, a short exact sequence is a sequence of objects and morphisms of the form

$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0

such that the sequence is exact at each object: the map f: A \to B is a kernel of g (equivalently, \ker g = \im f), the map g: B \to C is a cokernel of f (equivalently, \coker f = C), and the maps to and from the zero object are the zero morphisms.^[4] This implies that f is a monomorphism (injective in the categorical sense) and g is an epimorphism (surjective in the categorical sense).^[4] Exactness at B specifically requires that the image of f coincides with the kernel of g, ensuring no "loss of information" in the middle term beyond the intended subobject structure.^[5] In concrete abelian categories like modules over a ring, this translates to the standard set-theoretic conditions on kernels and images.^[6] A classic example in the category of abelian groups is the sequence

$0 \to 2\mathbb{Z} \xrightarrow{\cdot 2} \mathbb{Z} \xrightarrow{\mod 2} \mathbb{Z}/2\mathbb{Z} \to 0,

where multiplication by 2 embeds the even integers as a subgroup, and the quotient map yields the cyclic group of order 2; exactness holds because the kernel of the quotient is precisely the image of the embedding.^[7] In the category of vector spaces over a field, consider $0 \to U \to V \to V/U \to 0, where the first map is the inclusion of a subspace U into V and the second is the canonical projection; this is exact since the kernel of the projection is U itself.^[8] Short exact sequences are preserved under isomorphisms in the category: if \sigma is an isomorphism, then applying \sigma to each term yields another short exact sequence.^[4] They also exhibit functoriality: an additive functor between abelian categories preserves exactness of such sequences if it is exact, mapping them to short exact sequences in the target category.^[5]

Splitting Conditions

A split exact sequence is a short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0 in an abelian category that admits a retraction r: B \to A satisfying r \circ i = \mathrm{id}_A, or equivalently, a section s: C \to B satisfying p \circ s = \mathrm{id}_C.^[9]^[10] The retraction provides a right inverse to the inclusion i, ensuring that A embeds as a direct summand in B, while the section serves as a left inverse to the projection p, embedding C similarly.^[9] These maps satisfy the compatibility conditions that make the original sequence exact while allowing a decomposition of the middle term.^[10] Alternative characterizations of splitting include the isomorphism B \cong A \oplus C, where the inclusion and projection correspond to the standard maps into and out of the direct sum.^[9]^[10] In this case, the short exact sequence is equivalent to the canonical split sequence $0 \to A \to A \oplus C \to C \to 0.^[10] The splitting maps can be visualized in a commutative diagram that incorporates both the original sequence and the additional morphisms:

\begin{CD} 0 @>>> A @>i>> B @>p>> C @>>> 0 \\ @. @| @V r V @V s V @| \\ 0 @>>> A @= A @>>> C @>>> 0 \end{CD}

Here, the vertical maps r and s ensure the triangles commute, with r retracting along i and s sectioning along p.^[9]^[10]

Statement and Equivalences

The Lemma in Abelian Categories

In an abelian category \mathcal{A}, the splitting lemma asserts that a short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0splits if and only ifB \cong A \oplus C. This equivalence captures the notion that the middle object B$ decomposes as a direct sum of the kernel and cokernel objects when the sequence admits a splitting morphism. The splitting condition is equivalent to the existence of a retraction r: B \to A such that r \circ i = \mathrm{id}_A, or dually, a section s: C \to B such that p \circ s = \mathrm{id}_C. Furthermore, the sequence splits precisely when its extension class in the Ext group \mathrm{Ext}^1_{\mathcal{A}}(C, A) vanishes. This formulation applies broadly in contexts where abelian categories arise, such as the category of modules over a ring, the category of abelian groups, and the category of sheaves of abelian groups on a topological space. In these settings, direct sums and retractions provide concrete realizations of the abstract splitting.

Equivalent Formulations

The splitting lemma admits several equivalent formulations in the context of abelian categories, each providing a different perspective on when a short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0 decomposes into a direct sum. One such rephrasing emphasizes the existence of retraction or section maps: the sequence splits if and only if there exists a morphism s: C \to B such that p \circ s = \mathrm{id}_C (a section), or dually, a retraction r: B \to A such that r \circ i = \mathrm{id}_A. These conditions ensure that B is isomorphic to the direct sum A \oplus C, with i and s (or p and r) serving as the inclusion and projection maps, respectively.^[11] The splitting lemma relates to properties of projective and injective objects: a short exact sequence splits if the codomain C is projective, meaning that the surjection p: B \to C admits a section s: C \to B, or dually, if the domain A is injective, ensuring the existence of a retraction r: B \to A. This perspective highlights the lemma's role in homological algebra, where projectivity or injectivity guarantees the triviality of extensions represented by the sequence. For instance, in the category of modules over a ring, free modules being projective implies that any surjection onto a free module splits, yielding a direct sum decomposition.^[11] The splitting lemma also connects to broader diagram-chasing techniques, such as the snake lemma, which constructs long exact sequences from commutative diagrams of short exact sequences. In this framework, splitting properties propagate through such diagrams: if one row splits and the vertical maps on the ends are isomorphisms, then the middle row splits as well, preserving the direct sum structure across the diagram. This propagation is particularly useful in computing derived functors or analyzing extensions in abelian categories.^[11] Historically, the splitting lemma emerged as a foundational result in homological algebra, originating in the work of Henri Cartan and Samuel Eilenberg, where it served as a key tool for studying group extensions and cohomology theories. Their 1956 treatise formalized these equivalences within the emerging framework of derived categories and functors, influencing subsequent developments in algebraic topology and beyond.^[12]

Proofs

Direction from Direct Sum to Splitting

In an abelian category, the splitting lemma asserts that for a short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0, the existence of an isomorphism \phi: B \to A \oplus C implies the sequence splits. To establish this direction, construct an explicit section r: C \to B satisfying p \circ r = \mathrm{id}_C. The direct sum A \oplus C comes equipped with the standard inclusion \iota_A: A \to A \oplus C given by a \mapsto (a, 0) and projection \pi_C: A \oplus C \to C given by (a, c) \mapsto c. The given maps align via the isomorphism as i = \phi^{-1} \circ \iota_A and p = \pi_C \circ \phi. Define r: C \to B as the composite r = \phi^{-1} \circ j, where j: C \to A \oplus C is the inclusion c \mapsto (0, c). Then,

p \circ r = (\pi_C \circ \phi) \circ (\phi^{-1} \circ j) = \pi_C \circ j = \mathrm{id}_C,

verifying that r is a section. The zero morphisms and kernel properties of the abelian category ensure the exactness is preserved under this decomposition, as the image of i remains the kernel of p. This construction relies on the existence of direct sums and isomorphisms in the category, confirming the section without further diagram chasing beyond the composites. For instance, in the category of abelian groups, the sequence $0 \to \mathbb{Z} \xrightarrow{i} \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \xrightarrow{p} \mathbb{Z}/2\mathbb{Z} \to 0 with i(n) = (n, 0) and p(m, \overline{k}) = \overline{k} is exact and splits, as \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} via the identity, yielding the section r(\overline{k}) = (0, \overline{k}).

Direction from Splitting to Direct Sum

In an abelian category, consider the short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{q} C \to 0. Suppose it splits via a section s: C \to B such that q \circ s = \mathrm{id}_C. This condition implies the existence of a retraction \pi: B \to A such that \pi \circ i = \mathrm{id}_A and i \circ \pi + s \circ q = \mathrm{id}_B, making B the biproduct of A and C with i and s as the canonical inclusions and \pi and q as the canonical projections.^[4] To construct the explicit isomorphism, define the morphism \psi: A \oplus C \to B by

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

for all a \in A and c \in C. This map is well-defined since the direct sum A \oplus C is the biproduct in the abelian category, and the formula respects the universal property of the coproduct combined with the abelian structure.^[2] To verify that \psi is an isomorphism, first check injectivity. Suppose \psi(a, c) = 0. Then i(a) + s(c) = 0, so applying q yields q(i(a)) + q(s(c)) = 0 + c = c = 0. Thus, i(a) = 0, and since i is the kernel of q (by exactness), a = 0. Hence, \ker \psi = 0, so \psi is monic.^[4] For surjectivity, take any b \in B and set c = q(b). Then b - s(c) \in \ker q = \mathrm{im} i, so there exists a unique a \in A such that i(a) = b - s(c). It follows that b = i(a) + s(c) = \psi(a, c). The uniqueness of a ensures \psi is bijective. Alternatively, the retraction \pi can be defined by \pi(b) = a where i(a) = b - s(q(b)), confirming \psi \circ (\mathrm{id}_A, \pi) = \mathrm{id}_B and the inverse structure via the biproduct projections.^[2]^[13] This construction relies on the exactness of the sequence to identify \ker q = \mathrm{im} i and the splitting to ensure the images of i and s are complementary subobjects whose direct sum is B. In the biproduct formulation, the splitting maps satisfy the required commuting diagrams for A \oplus C to be isomorphic to B.^[4]

Direction from Retraction to Splitting

Consider a short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0 in an abelian category, where a retraction r: B \to A exists such that r \circ i = \mathrm{id}_A. This retraction ensures that \ker p = \mathrm{im} i and that r provides a complement to \mathrm{im} i within B. To construct a section s: C \to B such that p \circ s = \mathrm{id}_C, select any b \in B with p(b) = c for c \in C (possible by surjectivity of p). Define

s(c) = b - i(r(b)).

This definition is independent of the choice of b, as any other lift b' with p(b') = c satisfies b' - b = i(a) for some a \in A, and substituting yields the same s(c). Moreover,

p(s(c)) = p(b - i(r(b))) = p(b) - p(i(r(b))) = c - 0 = c,

since \mathrm{im} i \subseteq \ker p. The existence of such a section s completes the splitting of the sequence, establishing that B \cong A \oplus C. In abelian categories, the notions of retraction and section are symmetric due to the additive structure, ensuring that the existence of one implies the other via dual constructions.

Extensions and Limitations

Generalization to Abelian Categories

In an abelian category \mathcal{A}, the splitting lemma asserts that for a short exact sequence $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0, the following conditions are equivalent: there exists a section s: C \to B such that p \circ s = \mathrm{id}_C; there exists a retraction r: B \to A such that r \circ i = \mathrm{id}_A; and B is isomorphic to the biproduct A \oplus C via the canonical inclusion and projection morphisms.^[4] This formulation replaces the direct sums of abelian groups with biproducts, which coincide in categories like modules but are defined abstractly via universal properties in general abelian categories.^[14] The proof adapts the element-free arguments from the abelian group case by relying solely on the existence and universal properties of kernels, cokernels, and the zero morphism in \mathcal{A}. For instance, given a retraction r: B \to A, the idempotent endomorphism e = i \circ r on B induces a decomposition B \cong \ker(e) \oplus \mathrm{im}(e) as biproducts, where \mathrm{im}(e) \cong A and \ker(e) \cong C, using the abelian category axioms to ensure the kernels and images behave appropriately.^[14] Conversely, from the biproduct isomorphism, the canonical projections and inclusions provide the retraction and section directly. This categorical approach avoids explicit choice of elements and holds without additional additivity assumptions beyond the abelian structure.^[4] A key difference from the abelian groups setting is the absence of reliance on concrete group operations or the axiom of choice for constructing splittings; instead, all constructions use commutative diagrams and exactness properties intrinsic to \mathcal{A}. For example, in the category of sheaves of abelian groups on a topological space, which is abelian but lacks global elements in general, the splitting lemma ensures that split exact sequences correspond to biproducts of sheaves, facilitating computations in sheaf cohomology without descending to stalks.

Behavior in Non-Abelian Groups

In the category of groups, which includes non-abelian groups and is not abelian, the splitting lemma does not hold in its full form as in abelian categories. Instead, for a short exact sequence $1 \to N \to G \xrightarrow{\pi} H \to 1 where N is normal in G, the sequence splits if there exists a group homomorphism s: H \to G such that \pi \circ s = \mathrm{id}_H. In this case, G is isomorphic to the semidirect product N \rtimes H, where the structure is determined by a homomorphism \phi: H \to \mathrm{Aut}(N) encoding the action of H on N via conjugation in G.^[15]^[16] This semidirect product isomorphism arises because the splitting homomorphism provides a complement to N in G, with G = Ns and N \cap s(H) = \{e\}, but the elements of s(H) need not commute with those of N unless the action \phi is trivial. When \phi is the trivial homomorphism, the semidirect product reduces to the direct product N \times H, recovering the abelian case equivalence. In non-abelian settings, non-trivial actions are common, leading to structures where G is not a direct product even if the sequence splits.^[15]^[16] Group extensions in this context are classified up to equivalence by the second cohomology group H^2(H, N), assuming N is abelian and equipped with an action of H. The splitting condition corresponds precisely to the trivial element in H^2(H, N), yielding the semidirect product, while isomorphism to the direct product requires both the trivial cohomology class and the trivial action on N. For non-abelian N, the classification is more involved, relying on the theory of crossed modules rather than standard cohomology, but the splitting still yields a semidirect product.^[17]

Counterexamples in Non-Abelian Settings

A prominent counterexample illustrating the failure of the splitting lemma in non-abelian settings arises from the short exact sequence $1 \to [A_3](/page/Alternating_group) \to [S_3](/page/Symmetric_group) \to \mathbb{Z}/2\mathbb{Z} \to 1, where [A_3](/page/Alternating_group) is the alternating group on three letters (isomorphic to \mathbb{Z}/3\mathbb{Z}) and [S_3](/page/Symmetric_group) is the symmetric group on three letters. This sequence admits a splitting, as there exists a homomorphism \sigma: \mathbb{Z}/2\mathbb{Z} \to [S_3](/page/Symmetric_group) (for instance, mapping the generator to a transposition like (1\ 2)) such that the composition with the quotient map is the identity on \mathbb{Z}/2\mathbb{Z}. Consequently, [S_3](/page/Symmetric_group) is isomorphic to the semidirect product [A_3](/page/Alternating_group) \rtimes \mathbb{Z}/2\mathbb{Z}. However, [S_3](/page/Symmetric_group) is not isomorphic to the direct product [A_3](/page/Alternating_group) \oplus \mathbb{Z}/2\mathbb{Z}, which is abelian, because elements from [A_3](/page/Alternating_group) and the image of \sigma do not commute—for example, (1\ 2) conjugates the 3-cycle (1\ 2\ 3) to (1\ 3\ 2). This demonstrates that the presence of a retraction does not guarantee a direct product decomposition in the non-abelian case, as the lack of commutativity introduces non-trivial conjugation actions.^[18] Another key counterexample is provided by the central extension $1 \to \mathbb{Z}/2\mathbb{Z} \to [Q_8](/page/Quaternion_group) \to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \to 1, where [Q_8](/page/Quaternion_group) is the quaternion group of order 8 and \mathbb{Z}/2\mathbb{Z} is the center \{ \pm 1 \}. Here, the quotient is the Klein four-group, but the sequence does not split: there is no homomorphism s: \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \to [Q_8](/page/Quaternion_group) serving as a section, as [Q_8](/page/Quaternion_group) contains no subgroup isomorphic to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} (all its subgroups of order 4 are cyclic, generated by i, j, or k). Thus, [Q_8](/page/Quaternion_group) cannot be expressed as a semidirect product of its center by the quotient, highlighting how non-abelian structure can prevent even the existence of a splitting in central extensions.^[15] These examples underscore the necessity of the abelian condition in the splitting lemma, as non-commutativity allows for semidirect products (partial analogs via non-trivial actions) without yielding direct sums, or blocks splittings altogether due to incompatible subgroup structures.^[18]

Applications

In Homological Algebra

The splitting lemma plays a fundamental role in homological algebra by facilitating the analysis of long exact sequences arising from short exact sequences of modules. When a short exact sequence $0 \to A \to B \to C \to 0 splits, the induced long exact sequences in the derived functors \operatorname{Ext}^* and \operatorname{Tor}_* decompose into direct sums, simplifying computations of these groups. For instance, the long exact sequence \cdots \to \operatorname{Ext}^n(A, M) \to \operatorname{Ext}^n(B, M) \to \operatorname{Ext}^n(C, M) \to \operatorname{Ext}^{n+1}(A, M) \to \cdots becomes a collection of short exact sequences $0 \to \operatorname{Ext}^n(A, M) \to \operatorname{Ext}^n(B, M) \to \operatorname{Ext}^n(C, M) \to 0 when B \cong A \oplus C, allowing direct calculation of extension groups without boundary maps.^[19] In the context of projective resolutions, the splitting lemma ensures that if a module is projective, its trivial resolution $0 \to P \to P \to 0 splits, which aids in homology calculations by confirming that higher derived functors vanish: \operatorname{Ext}^i(P, M) = 0 for i > 0. More generally, splitting properties in resolutions of complexes enable the decomposition of chain complexes into direct summands, streamlining the evaluation of Tor and Ext through tensor products or Hom applications. This is particularly useful in constructing minimal resolutions or verifying exactness in derived categories.^[20] Historically, the splitting lemma was instrumental in the foundational work of Cartan and Eilenberg on classifying module extensions. In their development of homological algebra, split extensions correspond precisely to the trivial elements in \operatorname{Ext}^1(C, A), providing a cohomological classification where non-split extensions are parameterized by this group. This framework, introduced in their 1956 treatise, revolutionized the study of exact sequences and derived functors by linking algebraic decompositions to global homological invariants.^[20]

In Module Theory

In module theory, the splitting lemma provides a fundamental criterion for when a short exact sequence of modules $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 over a ring R splits, meaning B \cong A \oplus C as R-modules, if and only if there exists a homomorphism s: C \to B such that g \circ s = \mathrm{id}_C (a section) or a retraction r: B \to A with r \circ f = \mathrm{id}_A.^[2] This equivalence holds in the category of R-modules, which is an abelian category, allowing the lemma to characterize direct sum decompositions directly.^[21] The Krull–Schmidt theorem, which asserts the uniqueness (up to isomorphism and permutation) of decompositions of certain modules into indecomposable summands, relies on the splitting lemma to ensure that endomorphisms of indecomposables are either zero or isomorphisms, facilitating the lifting of idempotents and thus stable decompositions.^[22] Specifically, for modules of finite length over any ring, or more generally for modules where endomorphism rings are semiperfect, the splitting property guarantees that any two such decompositions are equivalent, providing a complete invariant for module structure in these cases.^[23] A concrete application arises over principal ideal domains (PIDs), where the structure theorem decomposes every finitely generated module M as a direct sum M \cong F \oplus T, with F free (torsion-free) and T torsion; the splitting lemma ensures this decomposition by providing sections for the inclusion of the torsion submodule.^[24] For example, over \mathbb{Z}, the integers, any finitely generated abelian group splits into its free part (isomorphic to \mathbb{Z}^r) and torsion part (a direct sum of cyclic groups of prime power order), allowing explicit classification via invariant factors or elementary divisors.^[25] The splitting lemma also serves as a detection tool for direct summands: a submodule A \subseteq B is a direct summand if and only if the short exact sequence $0 \to A \to B \to B/A \to 0 admits a splitting, which can be verified through the existence of complementary projections or by invariants such as the rank function in free modules, where equality of ranks implies potential splitting under additional conditions like projectivity.^[2] This criterion is particularly useful in identifying projective or free summands, as projective modules split off from extensions when the quotient is also projective.^[26]

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