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Exact sequence

In homological algebra, an exact sequence is a sequence of objects in an abelian category together with morphisms between consecutive objects such that the image of each morphism equals the kernel of the next morphism.^[1] This condition captures a precise notion of "no information loss" or "tight fitting" between the structures in the sequence, making exact sequences a foundational tool for studying relationships among algebraic objects like modules, groups, and sheaves.^[1] Exact sequences arise naturally in the study of chain complexes, where a complex is exact if its homology groups vanish everywhere, meaning it serves as a resolution without higher obstructions.^[2] They enable the computation of derived functors, such as Ext and Tor groups, which measure deviations from exactness in functor categories.^[3] In applications, exact sequences appear in algebraic topology to relate homology groups of spaces via long exact sequences induced by fibrations or pairs of subspaces.^[3] The concept of exact sequences originated in the early 20th century amid developments in algebraic topology and group cohomology.^[3] It first appeared explicitly in Witold Hurewicz's 1941 work on cohomology sequences for topological spaces.^[3] The term "exact sequence" was coined in 1947 by John L. Kelley and Everett Pitcher in their studies of homology.^[3] Samuel Eilenberg and Saunders Mac Lane further developed the idea in 1942–1945 while formalizing homology for group extensions.^[3] The modern framework was solidified in 1956 by Henri Cartan and Samuel Eilenberg in their seminal book Homological Algebra, which unified various homology theories using abelian categories and derived functors.^[3] Particularly notable are short exact sequences of the form $0 \to A \to B \to C \to 0, which imply that the morphism from A to B is injective, the morphism from B to C is surjective, and C is isomorphic to the quotient B/A.^[1] Such sequences describe extensions, where B is built from C by adjoining A as a kernel subgroup.^[1] Long exact sequences, often infinite in one or both directions, arise from applying functors to short exact sequences and preserve exactness under certain conditions, facilitating computations in cohomology and homological invariants across mathematics.^[2]

Definitions

Basic notion of exactness

In homological algebra, the basic notion of exactness concerns sequences of mathematical objects connected by morphisms, where the structure ensures a precise relationship between subspaces or subobjects defined by kernels and images. Specifically, an exact sequence is a sequence in an abelian category where the image of each morphism coincides exactly with the kernel of the subsequent morphism, capturing a form of "no loss or overlap" in the mapping process.^[4] Formally, consider a sequence of objects A_i and morphisms f_i: A_i \to A_{i+1} in an abelian category \mathcal{A}, written as \dots \to A_{n-1} \xrightarrow{f_{n-1}} A_n \xrightarrow{f_n} A_{n+1} \to \dots. The sequence is exact at A_n if \operatorname{im}(f_{n-1}) = \ker(f_n), meaning every element in the image of the incoming map is precisely the set of elements mapped to zero by the outgoing map.^[4] This condition holds locally at each object, and a sequence is fully exact if it satisfies exactness at every position. Trivial cases illustrate this: the sequence $0 \to A \xrightarrow{\operatorname{id}_A} A is exact at A since the image of the zero map from $0 is \{0\}, which equals the kernel of the identity map; similarly, A \xrightarrow{\operatorname{id}_A} A \to 0 is exact at A as the image of the identity is all of A, matching the kernel of the map to $0.^[4] The concept presupposes a setting where kernels and images are well-defined, such as an abelian category or the category of modules over a ring, where every morphism admits a kernel and a cokernel, and images are normal monomorphisms.^[5] This framework generalizes linear algebra, where the rank-nullity theorem states that for a linear map f: [V](/page/V.) \to [W](/page/W) between vector spaces, \dim V = \dim \ker f + \dim \operatorname{[im](/page/IM)} f; exact sequences extend this by chaining maps such that the "deficiency" at one step (the kernel) is fully accounted for by the previous map's range, ensuring dimensional additivity across the sequence without gaps.^[6] For instance, in vector spaces, a short exact sequence $0 \to U \to V \to W \to 0 implies \dim V = \dim U + \dim W via repeated application of rank-nullity.^[7] The notion of exactness was formalized in the development of homological algebra by Henri Cartan and Samuel Eilenberg in their seminal 1956 monograph, where it serves as a foundational tool for studying homology and derived functors.^[3] Short exact sequences represent a special finite case of this general condition, often terminating with zero objects to indicate injectivity and surjectivity at the ends.^[4]

Short exact sequences

A short exact sequence is a finite sequence of the form $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0, where f and g are homomorphisms between modules (or objects in an abelian category) such that the sequence is exact at each term: the map f is injective, g is surjective, and \operatorname{im} f = \ker g.^[1] This exactness implies that A is isomorphic to a submodule of B, and C is isomorphic to the quotient module B / A.^[1] Such a sequence can be interpreted as describing B as an extension of C by A. In the category of modules over a ring, the isomorphism classes of these extensions are classified by the first Ext group \operatorname{Ext}^1(C, A), which parametrizes the possible ways to "glue" A onto C to form B.^[8] A short exact sequence splits if there exists a homomorphism h: C \to B (a section) such that g \circ h = \operatorname{id}_C, or equivalently a retraction B \to A making the diagram commute with the identity on A; in either case, this is equivalent to B \cong A \oplus C as modules.^[1] However, not all short exact sequences split; for example, the sequence $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{\bmod 2} \mathbb{Z}/2\mathbb{Z} \to 0 is short exact but does not split, as \mathbb{Z} is not isomorphic to \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}.^[1] Short exact sequences of chain complexes give rise to long exact sequences in homology groups through connecting homomorphisms that link the homology of the terms.^[1]

Long exact sequences

In homological algebra, a long exact sequence is an infinite sequence of objects A_n and morphisms f_n: A_n \to A_{n+1} in an abelian category, extending bidirectionally as \cdots \to A_{n-1} \xrightarrow{f_{n-1}} A_n \xrightarrow{f_n} A_{n+1} \to \cdots, such that it is exact at every A_n, meaning \operatorname{im}(f_{n-1}) = \ker(f_n) for all n \in \mathbb{Z}.^[9] This condition implies that the homology groups of the associated chain complex vanish everywhere.^[10] Such sequences typically arise from the functoriality of derived functors applied to short exact sequences. Consider a short exact sequence $0 \to A \to B \to C \to 0 of objects in an abelian category with enough injectives, and let F be a left exact covariant functor. The right derived functors R^i F (for i \geq 0) then yield a long exact sequence

\cdots \to R^{i-1} F(C) \to R^i F(A) \to R^i F(B) \to R^i F(C) \xrightarrow{\delta^i} R^{i+1} F(A) \to \cdots,

where the sequence begins with $0 \to R^0 F(A) \to R^0 F(B) \to R^0 F(C) since F preserves the exactness on the left. This construction relies on injective resolutions to compute the derived functors, ensuring the long exactness through the naturality of the functor and the long exact sequence property of cohomology.^[1] The connecting homomorphism, or boundary map, \delta^i: R^i F(C) \to R^{i+1} F(A), is a key component that links consecutive derived functor groups, with exactness at R^i F(C) holding via \operatorname{im}(R^i F(B) \to R^i F(C)) = \ker(\delta^i). This map is induced by lifting the morphisms in the short exact sequence through the resolutions and is natural in the objects A, B, C. A representative example occurs in cohomology from a short exact sequence of chain complexes $0 \to \mathcal{A}_\bullet \to \mathcal{B}_\bullet \to \mathcal{C}_\bullet \to 0. Applying the cohomology functor H^n(-) produces a long exact sequence

\cdots \to H^n(\mathcal{A}_\bullet) \to H^n(\mathcal{B}_\bullet) \to H^n(\mathcal{C}_\bullet) \xrightarrow{\delta^n} H^{n+1}(\mathcal{A}_\bullet) \to \cdots,

where the connecting map \delta^n arises from the snake lemma applied degreewise, sending a cohomology class in \mathcal{C}_\bullet to one in \mathcal{A}_{\bullet+1} via a zigzag of boundaries and lifts.^[1] This sequence connects the cohomology groups across the complexes, facilitating computations in algebraic topology and beyond. In non-abelian settings, such as semi-abelian categories, long exact sequences are generalized using regular epimorphisms and normal monomorphisms for exactness, with left exact functors preserving kernels and right exact functors preserving cokernels; recent category-theoretic developments emphasize these distinctions to extend homological tools beyond abelian structures.^[11]^[10]

Properties

Five lemma

The five lemma is a diagram-chasing lemma in homological algebra that establishes isomorphisms in commutative diagrams of exact sequences within abelian categories. Consider the following commutative diagram, where the rows are exact sequences:

\begin{CD} A_1 @>>> A_2 @>>> A_3 @>>> A_4 @>>> A_5 \\ @V{\alpha_1}VV @V{\alpha_2}VV @V{\alpha_3}VV @V{\alpha_4}VV @V{\alpha_5}VV \\ B_1 @>>> B_2 @>>> B_3 @>>> B_4 @>>> B_5 \end{CD}

If the vertical maps \alpha_1, \alpha_2, \alpha_4, \alpha_5 are isomorphisms, then the middle map \alpha_3: A_3 \to B_3 is also an isomorphism.^[12] To prove this, first show that \alpha_3 is a monomorphism (injective). Suppose x \in \ker \alpha_3. By exactness at A_3, x is in the image of the map A_2 \to A_3. Let y \in A_2 such that the map sends y to x. Commutativity implies \alpha_2(y) maps to \alpha_3(x) = [0](/page/0) in B_3, so \alpha_2(y) \in \ker(B_2 \to B_3). Exactness at B_3 means this kernel is the image of B_1 \to B_2. Tracing back via the isomorphisms \alpha_2 and \alpha_1, which preserve kernels and images, yields y = [0](/page/0), hence x = [0](/page/0). Thus, \ker \alpha_3 = [0](/page/0).^[12] For surjectivity (epimorphism), a dual argument using cokernels shows \operatorname{coker} \alpha_3 = 0. Consider an element in B_3; by exactness, it lifts from B_4 or traces via images. Using the isomorphisms at A_4, A_5 and commutativity, one constructs a preimage in A_3 via the exactness at A_3 and the surjectivity induced by \alpha_4^{-1}. Duality in abelian categories (reversing arrows and swapping kernels with cokernels) confirms \alpha_3 is an epimorphism, hence an isomorphism.^[12] Generalizations include the short five lemma, which applies to diagrams of five-term exact sequences where vertical maps at the ends are isomorphisms, implying the middle vertical map is an isomorphism under suitable conditions on the adjacent maps (e.g., the first vertical map being an epimorphism and the fourth a monomorphism). Conversely, if the vertical maps at the ends (\alpha_1 and \alpha_5) are isomorphisms and the adjacent ones (\alpha_2, \alpha_4) satisfy appropriate mono/epi conditions, the middle map \alpha_3 is an isomorphism. These variants facilitate diagram chasing in broader contexts.^[13] The five lemma plays a key role in proving the uniqueness (up to natural isomorphism) of derived functors in abelian categories, by applying it to the long exact sequences arising from projective or injective resolutions.^[14]

Snake lemma

The snake lemma provides a method to construct a long exact sequence from a commutative diagram in the category of abelian groups (or modules over a ring) featuring two short exact rows connected by vertical homomorphisms. Consider the following commutative diagram, where the horizontal maps form short exact sequences:

\begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0 \\ @VVV @V{\phi}VV @V{\psi}VV @V{\xi}VV @VVV \\ 0 @>>> A' @>>f'> B' @>>g'> C' @>>> 0 \end{CD}

Here, the vertical maps \phi: A \to A', \psi: B \to B', and \xi: C \to C' render the squares commutative. The snake lemma asserts the existence of a connecting homomorphism \delta: \ker \xi \to \coker \phi such that the following sequence is exact:

$0 \to \ker \phi \to \ker \psi \to \ker \xi \xrightarrow{\delta} \coker \phi \to \coker \psi \to \coker \xi \to 0.

^[15]^[16] The connecting homomorphism \delta is defined via diagram chasing. For an element x \in \ker \xi \subseteq C, lift x to an element y \in B such that g(y) = x (possible since the top row is exact at B, so g is surjective). By commutativity of the right square, \xi(g(y)) = g'( \psi(y) ), so g'(\psi(y)) = \xi(x) = 0. Thus, \psi(y) \in \ker g' \subseteq B'. By exactness of the bottom row at B', there exists z \in A' such that f'(z) = \psi(y). Define \delta(x) = z + \im \phi in \coker \phi = A' / \im \phi. Well-definedness follows from showing independence of choices via further chasing, and \delta is a homomorphism. Exactness at each term is verified by chasing elements through the diagram to show injections, surjections, and that images equal kernels.^[17]^[18] This lemma is fundamental in homological algebra for generating long exact sequences that relate kernels and cokernels across related exact sequences, enabling computations in derived functors and cohomology.^[15]^[16]

Nine lemma

The nine lemma, also known as the 3×3 lemma, is a key result in homological algebra that establishes isomorphisms in commutative diagrams of exact sequences within abelian categories. It extends the five lemma to three rows, concluding that vertical maps in the middle column are isomorphisms when those in the outer columns are. Consider a commutative diagram consisting of three exact rows:

\begin{array}{ccccccc} 0 & \to & A_1 & \to & B_1 & \to & C_1 & \to & 0 \\ & & \downarrow^f & & \downarrow^g & & \downarrow^h & \\ 0 & \to & A_2 & \to & B_2 & \to & C_2 & \to & 0 \\ & & \downarrow^{f'} & & \downarrow^{g'} & & \downarrow^{h'} & \\ 0 & \to & A_3 & \to & B_3 & \to & C_3 & \to & 0 \end{array}

where each row is a short exact sequence. The nine lemma states that if the vertical maps in the left and right columns are isomorphisms (i.e., f: A_1 \to A_2 and f': A_2 \to A_3 are isomorphisms; h: C_1 \to C_2 and h': C_2 \to C_3 are isomorphisms), then the vertical maps in the middle column (g: B_1 \to B_2 and g': B_2 \to B_3) are also isomorphisms. A dual version holds by reversing arrows and interchanging kernels and cokernels.^[19]^[20] The proof applies the five lemma to successive pairs of rows. For the first two rows, consider the commutative diagram including the zero maps:

\begin{CD} 0 @>>> A_1 @>>> B_1 @>>> C_1 @>>> 0 \\ @VVV @V{f}VV @V{g}VV @V{h}VV @VVV \\ 0 @>>> A_2 @>>> B_2 @>>> C_2 @>>> 0 \end{CD}

Since the vertical maps on the ends (identities on 0, f, h) are isomorphisms and the rows are exact, the five lemma implies g is an isomorphism. Similarly, applying the five lemma to the bottom two rows yields that g' is an isomorphism. This establishes the result for the middle column.^[19] The nine lemma plays a crucial role in establishing the exactness properties of derived functors such as \operatorname{Hom} and \operatorname{Ext}. For instance, when applying \operatorname{Hom}(A, -) to a short exact sequence, the resulting long exact sequence in \operatorname{Ext} groups relies on the lemma to verify that the induced maps preserve exactness in multi-row diagrams arising from resolutions, ensuring the functor is left exact and that higher derived functors behave as expected. Similarly, in proofs of the exactness of \operatorname{Ext}^n(A, -), the lemma composes multiple applications of the five lemma to handle the grid structures from projective or injective resolutions.^[19]

Examples

Exact sequences of abelian groups

In the category of abelian groups, exact sequences often arise from subgroup inclusions and quotient constructions, providing insight into the structure of cyclic and torsion groups via modular arithmetic. A fundamental short exact sequence illustrates the relationship between the integers \mathbb{Z}, its even subgroup $2\mathbb{Z}, and the cyclic group of order 2. Specifically, the sequence $0 \to 2\mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0 is exact, where the map $2\mathbb{Z} \to \mathbb{Z} is the inclusion (or equivalently, multiplication by 2 from \mathbb{Z} \to \mathbb{Z}), and \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} is the canonical projection reducing modulo 2. This sequence is short exact because the inclusion is injective (kernel is 0), the projection is surjective (cokernel is 0), and the image of the first map equals the kernel of the second (even integers are precisely those congruent to 0 modulo 2). Unlike split exact sequences in vector spaces, this extension does not split, as there is no subgroup of \mathbb{Z} isomorphic to \mathbb{Z}/2\mathbb{Z} that complements $2\mathbb{Z}, highlighting non-trivial extensions in infinite abelian groups. To compute the exactness explicitly, consider an element k \in \ker(\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}); this means k \equiv 0 \pmod{2}, so k = 2m for some m \in \mathbb{Z}, which lies in the image of multiplication by 2. Conversely, the image consists of even integers, whose kernel under the projection is trivial since only 0 maps to 0 in \mathbb{Z}/2\mathbb{Z}. This example extends the basic notion of integers modulo 2 by embedding it in the broader group-theoretic context of cyclic groups, where \mathbb{Z}/2\mathbb{Z} serves as a prototypical finite abelian group. Another key example involves the torsion subgroup of an abelian group A, captured by a long exact sequence derived from tensoring with \mathbb{Z}/n\mathbb{Z}. For a fixed integer n > 0, the sequence $0 \to A \to A \xrightarrow{\times n} A \to A/nA \to 0 is exact, where A = \{a \in A \mid na = 0\} is the n-torsion subgroup, the first map is inclusion, \times n is multiplication by n, and A/nA = A \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z} is the cokernel of multiplication by n (quotient A / nA). Exactness holds at A by definition of the torsion kernel, at the first A because elements annihilated by n are precisely the torsion elements, and at A/nA since the cokernel of multiplication by n matches the tensor product structure. This sequence generalizes the short exact case for n=2, revealing how tensoring with cyclic groups detects torsion in arbitrary abelian groups, such as when A = \mathbb{Z}^\infty (direct sum of infinitely many \mathbb{Z}), where

A{{grok:render&&&type=render_inline_citation&&&citation_id=2&&&citation_type=wikipedia}} \cong (\mathbb{Z}/2\mathbb{Z})^\infty

Exact sequences in vector spaces

In the category of vector spaces over a field, an exact sequence consists of vector spaces and linear maps such that the image of each map equals the kernel of the next, providing a framework to study subspaces and quotients via linear algebra tools like the rank-nullity theorem.^[21] A canonical example is the short exact sequence arising from a subspace U of a finite-dimensional vector space V:

$0 \to U \xrightarrow{i} V \xrightarrow{p} V/U \to 0,

where i: U \to V is the inclusion map and p: V \to V/U is the canonical projection. This sequence is exact at U because i is injective (its kernel is zero), exact at V because \ker p = U = \im i, and exact at V/U because p is surjective (its image is all of V/U).^[22] By the rank-nullity theorem applied to p, the dimensions satisfy \dim V = \dim \ker p + \dim \im p = \dim U + \dim (V/U).^[21] Every such short exact sequence of finite-dimensional vector spaces splits, meaning there exists a linear map s: V/U \to V (a section) such that p \circ s = \id_{V/U}, yielding an isomorphism V \cong U \oplus s(V/U).^[22] To see this, select a basis \{u_1, \dots, u_m\} for U (where m = \dim U), extend it to a basis \{u_1, \dots, u_m, v_{m+1}, \dots, v_n\} for V (where n = \dim V), and define s on the induced basis for V/U by sending the class of v_j (for j > m) to v_j; this s is linear and satisfies the section property.^[22] Applying the covariant Hom functor \Hom(A, -) (for a fixed vector space A) to the short exact sequence $0 \to U \to V \to W \to 0 produces a long exact sequence

$0 \to \Hom(A, U) \to \Hom(A, V) \to \Hom(A, W) \to \Ext^1(A, U) \to \Ext^1(A, V) \to \Ext^1(A, W) \to 0,

but since vector spaces over a field are free (hence projective) modules, all higher Ext groups vanish (\Ext^1(A, U) = 0), so the long exact sequence reduces to a pair of short exact sequences, and \Hom(A, -) is exact.^[23] In contrast to modules over a general ring, where non-trivial extensions can occur (with \Ext^1 \neq 0), finite-dimensional vector spaces over a field admit no such extensions, reflecting their semisimple nature.^[24]

Exact sequences from chain complexes

A chain complex (C_\bullet, \partial) in an abelian category, such as the category of modules over a ring R, consists of objects C_n and morphisms \partial_n: C_n \to C_{n-1} satisfying \partial_{n-1} \circ \partial_n = 0 for all n. This complex is exact if the induced homology groups vanish, that is, H_n(C_\bullet) = \ker \partial_n / \operatorname{im} \partial_{n+1} = 0 for every integer n, or equivalently, if \operatorname{im} \partial_{n+1} = \ker \partial_n at every degree.^[25] Exactness thus means that the complex has no nontrivial cycles modulo boundaries, capturing a form of "global balance" in the differentials. Exact sequences emerge prominently from resolutions in homological algebra, where one approximates a module by a sequence of simpler modules. A projective resolution of an R-module M is an exact sequence \cdots \to P_2 \to P_1 \to P_0 \to M \to 0, in which each P_i (for i \geq 0) is a projective R-module; finite-length versions, such as the short exact sequence $0 \to P_1 \to P_0 \to M \to 0, are common when the projective dimension of M is low.^[25] Here, exactness holds at each P_i (with \operatorname{im} \partial_{i+1} = \ker \partial_i) and at M (where the map P_0 \to M is surjective with kernel equal to the image from P_1), ensuring the resolution faithfully reconstructs M without cohomological obstructions. Such resolutions extend the notion of exactness from short sequences of modules (like intersections and direct sums) to infinite chains, providing tools for computing derived functors. The augmentation map \varepsilon: P_0 \to M in a projective resolution forms the augmented chain complex \cdots \to P_1 \to P_0 \xrightarrow{\varepsilon} M \to 0, which is exact if and only if the unaugmented complex P_\bullet is acyclic (i.e., H_n(P_\bullet) = 0 for all n > 0) and \varepsilon is surjective with kernel \operatorname{im}(P_1 \to P_0).^[25] This exactness of the augmented complex confirms the resolution's acyclicity, a key property for applications like deriving Ext and Tor groups, where the resolution serves as a "free" replacement for M.

Applications

Computing homology groups

In algebraic topology and homological algebra, exact sequences provide a powerful framework for computing homology groups by relating the homology of different chain complexes or topological spaces. A key application arises from a short exact sequence of chain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0, where the maps are chain maps compatible with the differentials. This induces a long exact sequence in homology:

\cdots \to H_n(A_\bullet) \to H_n(B_\bullet) \to H_n(C_\bullet) \xrightarrow{\partial} H_{n-1}(A_\bullet) \to H_{n-1}(B_\bullet) \to H_{n-1}(C_\bullet) \to \cdots

The connecting homomorphism \partial: H_n(C_\bullet) \to H_{n-1}(A_\bullet) is defined by lifting cycles in C_\bullet to boundaries in B_\bullet and then projecting to A_\bullet, ensuring exactness at each term through diagram chasing or the snake lemma. This sequence allows computation of unknown homology groups when some are known, such as determining relative homology H_n(X, A) from the absolute homologies H_n(X) and H_n(A).^[26] A prominent example is the Mayer-Vietoris sequence, which computes the homology of a space X = U \cup V decomposed into open subspaces U and V with path-connected intersection U \cap V, assuming suitable excision conditions hold. The sequence is:

\cdots \to H_n(U \cap V) \xrightarrow{(i_*, j_*)} H_n(U) \oplus H_n(V) \xrightarrow{k_* - l_*} H_n(X) \xrightarrow{\partial} H_{n-1}(U \cap V) \to \cdots

Here, i_* and j_* are induced by inclusions into U and V, while k_* - l_* subtracts the inclusions into X. This derives from the short exact sequence of singular chain complexes $0 \to C_\bullet(U \cap V) \to C_\bullet(U) \oplus C_\bullet(V) \to C_\bullet(X) \to 0, enabling inductive calculations for spaces like spheres or tori. For instance, decomposing the 2-sphere S^2 into two open hemispheres yields H_2(S^2) \cong \mathbb{Z} and H_n(S^2) = 0 for n \neq 2, confirming known results via exactness.^[26] Exact sequences also preserve the Euler characteristic, a topological invariant defined as \chi(X) = \sum_{n \geq 0} (-1)^n \operatorname{rank} H_n(X; \mathbb{Z}) for spaces with finitely generated homology. In a short exact sequence of chain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 with finite-dimensional homology over a field (or free abelian ranks), the induced long exact sequence implies \chi(B_\bullet) = \chi(A_\bullet) + \chi(C_\bullet), as the alternating sum of ranks in the long sequence vanishes due to exactness. This additivity facilitates quick verifications of computations, such as confirming \chi(S^1 \vee S^2) = 0 from the wedge sum formula \chi(X \vee Y) = \chi(X) + \chi(Y) - 1.^[26]

de Rham cohomology

In de Rham cohomology, exact sequences arise naturally from the de Rham complex of differential forms on a smooth manifold M. The de Rham complex is the chain complex

$0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \cdots \xrightarrow{d} \Omega^n(M) \to 0,

where \Omega^k(M) denotes the space of smooth k-forms on M and d is the exterior derivative, satisfying d^2 = 0. The cohomology groups of this complex, denoted H^k_{dR}(M), capture topological invariants of M. A key exactness property holds when M is contractible: the Poincaré lemma states that every closed form (i.e., d\omega = 0) is exact (i.e., \omega = d\eta for some form \eta), making the complex exact at each \Omega^k(M) for k \geq 1. For manifolds with boundary, relative de Rham cohomology is defined using the subcomplex of forms vanishing on the boundary \partial M. This leads to a long exact sequence in cohomology:

\cdots \to H^k_{dR}(M, \partial M) \to H^k_{dR}(M) \to H^k_{dR}(\partial M) \to H^{k+1}_{dR}(M, \partial M) \to \cdots,

which relates the absolute and relative cohomologies via the connecting homomorphism \partial: H^k_{dR}(\partial M) \to H^{k+1}_{dR}(M, \partial M), analogous to the algebraic case for pairs of spaces.^[27] This sequence is exact, providing a tool to compute relative invariants from absolute ones, such as in the study of manifolds with prescribed boundary conditions. On compact oriented Riemannian manifolds, the Hodge theorem establishes a profound connection between exact sequences and harmonic forms. Specifically, every de Rham cohomology class [ \omega ] \in H^k_{dR}(M) has a unique harmonic representative \alpha, where \Delta \alpha = 0 and \Delta = d\delta + \delta d is the Laplace-Beltrami operator with \delta the codifferential. On compact Kähler manifolds, this implies a Hodge decomposition H^k_{dR}(M, \mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(M), where H^{p,q}(M) is the space of harmonic (p,q)-forms, orthogonal under the L^2 inner product. This decomposition reveals exactness relations through the identification of cohomology with harmonic forms, facilitating computations and linking differential geometry to algebraic topology.

Spectral sequences

Spectral sequences provide a systematic way to compute the homology or cohomology of a filtered chain complex by organizing successive approximations into a sequence of pages, each comprising a long exact sequence of groups with induced differentials. Formally, given a filtered chain complex (C_\bullet, F), where the filtration F is an exhaustive and decreasing sequence of subcomplexes F^p C_\bullet \subseteq C_\bullet with F^p C_\bullet / F^{p+1} C_\bullet forming the associated graded pieces, the spectral sequence \{E_r^{p,q}, d_r\} arises such that each page E_r is a first-quadrant bigraded module equipped with a differential d_r: E_r^{p,q} \to E_r^{p+r, q-r+1} satisfying \ker d_r / \operatorname{im} d_r = E_{r+1}^{p,q}, and the sequence converges to the graded pieces \operatorname{gr} H_{p+q}(C_\bullet) \cong E_\infty^{p,q}. This structure generalizes long exact sequences by iterating refinements, allowing computation of homology groups that are otherwise intractable directly.^[28]^[29] The construction of a spectral sequence typically originates from an exact couple, a diagram consisting of maps \alpha: A \to A, f: A \to E, and g: E \to A forming a long exact sequence \cdots \to A \xrightarrow{\alpha} A \xrightarrow{f} E \xrightarrow{g} A \to \cdots, where A and E are often bigraded. Deriving successive exact couples via homology of the previous page yields the spectral sequence pages, with E_1 as the homology of the associated graded complex and higher differentials capturing interactions across filtration levels. In the context of short exact sequences of filtered complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0, the long exact sequence in homology induces an exact couple on the associated graded modules, producing a spectral sequence that refines the direct long exact sequence into iterative exact sequences on each page. This framework, introduced by William Massey, underpins many spectral sequences in algebraic topology and homological algebra.^[30]^[31] A prominent example is the Serre spectral sequence, which computes the cohomology of a Serre fibration F \to E \to B with fiber F, total space E, and simply connected base B. It takes the form E_2^{p,q} = H^p(B; \mathcal{H}^q(F; \mathbb{Z})) \Rightarrow H^{p+q}(E; \mathbb{Z}), where the E_2 page arises from the long exact sequence in cohomology of the pair (E, F), and higher differentials d_r are exact sequence maps of bidegree (r, 1-r). This spectral sequence, developed by Jean-Pierre Serre, converges strongly under finiteness conditions on the spaces, providing exact sequences at each page that approximate the cohomology of E via that of B and F; for instance, in the Hopf fibration S^1 \to S^3 \to S^2, it collapses at E_2 to yield the known cohomology ring. The method highlights how exact sequences enable multi-step computations in fibrations, a cornerstone of modern algebraic topology since the 1950s.^[32]

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