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Spectral sequence

In algebraic topology and homological algebra, a spectral sequence is a powerful computational device that approximates the homology or cohomology groups of a chain complex or topological space through a sequence of successively refined pages, each derived as the homology of the previous page's differentials, ultimately converging to the target graded groups associated with a filtration.^[1]^[2] Introduced by French mathematician Jean Leray in 1946 while studying sheaf cohomology for general topological spaces, spectral sequences emerged from his wartime work on avoiding simplicial approximations in favor of more abstract sheaf-based methods.^[3]^[1] The foundational example in algebraic topology is the Serre spectral sequence, developed by Jean-Pierre Serre in his 1951 thesis, which relates the homology (or cohomology) of a fibration F \to X \to B by starting with the E_2 page given by E_2^{p,q} = H_p(B; H_q(F; G)) for coefficients G, and converging to H_{p+q}(X; G) under suitable conditions like a simply connected base.^[2] This tool has proven essential for calculating invariants of complex spaces, such as the cohomology ring of the Eilenberg-MacLane space K(\mathbb{Z}, 2) \cong \mathbb{C}P^\infty, which is \mathbb{Z} with |x| = 2, or detecting torsion in homotopy groups of spheres like \pi_{n+2}(S^n) \cong \mathbb{Z}/2\mathbb{Z}.^[2] Beyond fibrations, spectral sequences arise more generally from filtered chain complexes via exact couples, a construction formalized by William Massey in 1952, enabling applications in diverse areas including stable homotopy theory through the Adams spectral sequence and persistent homology in topological data analysis.^[1]^[4] Their "spectral" name reflects the layered, iterative nature reminiscent of spectra in physics, though they demand careful handling of convergence, differentials of length r on the r-th page, and bigrading to extract precise subquotient information from the limiting E_\infty page.^[3]

History and Motivation

Discovery

Jean Leray introduced spectral sequences in 1946 while studying sheaf cohomology in algebraic topology, developing the concept during his imprisonment as a prisoner of war in Oflag XVII-A to compute sheaf cohomology from filtered complexes associated to sheaves on topological spaces.^[3] His initial announcement appeared in a note to the Comptes Rendus de l'Académie des Sciences, titled "Structure de l'anneau d'homologie d'une représentation," where he outlined the sequence arising from filtrations on spaces.^[5] In his 1950 paper "Vanneaux spectraux et vanneaux filtrés d'homologie d'un espace localement compact et d'une application continue," published in the Journal de Mathématiques Pures et Appliquées, Leray provided a more detailed treatment of hypercohomology and its associated spectral sequence for fiber bundles.^[3] This work built directly on his 1946 ideas, providing a more detailed treatment of the sequences in the context of sheaf theory and topological applications.^[6] Spectral sequences gained early adoption in the 1950s among topologists, notably Henri Cartan, who incorporated them into his seminars at the École Normale Supérieure and suggested key algebraic refinements, such as viewing filtered complexes as the central object.^[7] Their importance was later recognized by William S. Massey, who stated in 1955 that the spectral sequence is "one of the fundamental algebraic structures needed for dealing with topological problems."^[8] Others, including Jean-Pierre Serre and Jean-Louis Koszul, contributed to their refinement during this period for computing cohomology groups in various settings.^[6] A key milestone came with the 1956 book Homological Algebra by Henri Cartan and Samuel Eilenberg, which provided a systematic formalization of spectral sequences in the broader framework of homological algebra, including their construction from filtered complexes and exact couples.^[7] Earlier, Eilenberg and Norman Steenrod's 1952 Foundations of Algebraic Topology had touched on related ideas but did not fully develop spectral sequences. Initial challenges in the theory involved defining convergence, particularly for unbounded filtrations, where early formulations by Leray assumed bounded or complete filtrations to ensure the sequence abuts to a specific target group.^[9] These issues persisted into the 1950s, with Cartan and Eilenberg addressing bounded cases rigorously, while unbounded convergence required later advancements, such as those by J. Michael Boardman in the 1960s.^[10]

Original Motivation

Spectral sequences arose from the need to compute cohomology groups of topological spaces through successive approximations, particularly when direct methods proved intractable. In the context of sheaf cohomology, Jean Leray developed this approach to handle local data on spaces without relying on global simplicial decompositions, using sheaves to localize cohomology and approximate it via subcomplexes associated to closed subsets.^[3] This was essential for spaces where traditional combinatorial techniques, such as simplicial approximations, were cumbersome or inapplicable, allowing cohomology to be built iteratively from finer to coarser levels.^[3] The conceptual foundation draws an analogy to long exact sequences in homology or cohomology, which arise from short exact sequences or pairs of spaces, but extends this to multi-step filtrations. In a filtered complex, a single cohomology group decomposes into contributions from a grid-like structure, where each page refines the approximation by incorporating higher-order differentials, much like how long exact sequences capture boundary maps between successive terms.^[11] This grid enables tracking how elements survive or are killed across filtration levels, providing a systematic way to approximate the ultimate cohomology without resolving the entire unfiltered complex at once.^[11] A key algebraic motivation came from exact couples, introduced as a mechanism to generate these iterative differentials. By starting with an exact triangle of maps between abelian groups, one can derive a sequence of approximating pages without upfront resolution of the full chain complex, facilitating computations in filtered settings.^[12] An early prominent application was Jean-Pierre Serre's spectral sequence for fibrations, which relates the cohomology of the total space to that of the base and fiber, enabling deductions about one from the others in scenarios like principal bundles or loop spaces.^[2]

Formal Definitions

Bigraded Spectral Sequences

A bigraded spectral sequence is an algebraic structure consisting of a sequence of pages \{E_r^{p,q}\}_{r \geq 1}, where each E_r is a bigraded module over a commutative ring (such as \mathbb{Z}), equipped with differentials d_r: E_r^{p,q} \to E_r^{p+r, q-r+1} satisfying d_r^2 = 0. The differential d_r has bidegree (r, 1-r), increasing the total degree p+q by 1. The (r+1)-th page is defined as the homology of the r-th page: E_{r+1}^{p,q} = \ker(d_r^{p,q}) / \im(d_r^{p-r, q+r-1}), where the kernel and image are taken in the respective bidegrees.^[13] In the first-quadrant case, a common setup restricts E_r^{p,q} = 0 for p < 0 or q < 0, ensuring that the differentials eventually vanish for sufficiently large r, leading to stabilization at E_\infty^{p,q}. This page E_\infty is bigraded and arises as the successive homology, with each E_r^{p,q} fitting into short exact sequences relating it to E_{r+1}. The structure allows for iterative computation, where elements surviving to E_\infty represent graded pieces of a target module.^[14] Spectral sequences often converge to an abutment, a graded module H^* equipped with a filtration \{F_p H^n\}_{p \in \mathbb{Z}} such that the associated graded module satisfies \mathrm{gr}_p H^{p+q} \cong E_\infty^{p,q}. Convergence means that the filtration is complete and exhaustive, with E_\infty^{p,q} isomorphic to F_p H^{p+q} / F_{p+1} H^{p+q}. This setup provides a filtered approximation to H^*, enabling the recovery of the target from the limiting page via extension problems.^[13] The differentials exhibit properties essential for multiplicative structures: in cases where the pages form bigraded algebras, the Leibniz rule holds, d_r(ab) = d_r(a)b + (-1)^{p+q} a d_r(b) for a \in E_r^{p,q}, and anticommutativity follows from the odd total degree of d_r, ensuring d_r^2 = 0. This algebraic compatibility preserves products across pages, facilitating computations in cohomology rings. Standard notation emphasizes the bidegrees, with pages visualized as arrays where arrows indicate differential targets, though detailed diagrams are addressed separately.^[14]

Cohomological Spectral Sequences

In the cohomological setting, spectral sequences arise primarily from filtered cochain complexes, providing a systematic way to compute the cohomology groups of the total complex through successive approximations. Given a cochain complex C^* equipped with a decreasing filtration F^p C^n satisfying standard conditions (such as completeness and exhaustiveness), the associated spectral sequence is bigraded with pages E_r^{p,q} for r \geq 1, where p denotes the filtration degree and q the complementary degree, such that the total degree is p + q. This indexing ensures that the spectral sequence converges to the cohomology of the total complex, specifically abutting to \mathrm{Gr}_p H^{p+q}(C^*), the graded pieces of the induced filtration on H^*(C^*).^[15] The differentials on each page are defined as d_r: E_r^{p,q} \to E_r^{p+r, q-r+1}, which increase the total degree by 1 (since (p+r) + (q-r+1) = p+q+1) and satisfy d_r^2 = 0, allowing the next page E_{r+1} to be the cohomology of E_r with respect to d_r. This orientation reflects the cohomological nature, where differentials raise the degree, in contrast to the homological convention. In diagrammatic representations, these differentials correspond to arrows pointing "up and to the right" on the (p,q)-plane, emphasizing the progression along the filtration.^[15] For a filtered cochain complex C^*, the first page is given by

E_1^{p,q} = H^{p+q}(\mathrm{gr}_p C^*),

where \mathrm{gr}_p C^* = F^p C^* / F^{p+1} C^* is the associated graded complex, and the differential d_1 is induced by the original differential on C^* modulo the filtration. Subsequent pages are obtained iteratively, with d_r incorporating higher-order terms from the filtration structure. This construction, formalized in the context of exact categories, enables the decomposition of complex cohomology computations into manageable graded pieces.^[15]

Homological Spectral Sequences

Homological spectral sequences arise in the context of filtered chain complexes in homological algebra, providing a systematic way to compute the homology groups of the total complex through successive approximations.^[16] In this setup, the spectral sequence is bigraded with terms

E_r_{p,q}

, where p denotes the filtration degree and q the complementary degree, such that the total degree is n = p + q. The sequence converges to the homology H_n(X) of the object X, with the E_\infty page abutting to a graded pieces of a filtration on H_n(X).^[17] The differentials

d_r: E_r_{p,q} \to E_r_{p-r, q+r-1}

on the r-th page decrease the total degree by 1, reflecting the homological nature where boundaries lower the degree. These maps satisfy d_r^2 = 0 and have bidegree ( -r, r-1 ), ensuring compatibility with the chain complex structure. The next page is obtained as the homology of the previous one:

E_{r+1}_{p,q} = H(E_r_{p,q}, d_r) = \ker d_r / \operatorname{im} d_r

, with induced maps from previous differentials. Under suitable boundedness conditions on the filtration, the spectral sequence converges, meaning

\lim_{r \to \infty} E_r_{p,q} \cong E_\infty_{p,q} \cong \operatorname{gr}_p H_{p+q}(X)

.^[16]^[17] A prototypical construction begins with a filtered chain complex (C_*, \partial) where F_p C_n \subseteq F_{p+1} C_n and \partial: C_n \to C_{n-1}. The E_0 page is the associated graded:

E_0_{p,q} = \operatorname{gr}_p C_{p+q} = F_p C_{p+q} / F_{p-1} C_{p+q}

, equipped with the vertical differential d_0 induced by \partial, which maps

E_0_{p,q} \to E_0_{p, q-1}

. Taking homology with respect to d_0 yields

E_1_{p,q} = H_{p+q}(\operatorname{gr}_p C_*)

, where the differential d_1 is horizontal, induced by the component of \partial connecting different filtration levels. Subsequent pages are built iteratively via the general homology construction. This framework is particularly useful for computing homology in algebraic topology and homological algebra, paralleling cohomological spectral sequences in the opposite category.^[17]^[16]

Visualization and Notation

Standard Notation

In spectral sequences, the pages are typically denoted using bidegrees (p, q), where the r-th page consists of groups E_r^{p,q} for cohomological spectral sequences or E_{p,q}^r for homological ones, reflecting the filtration and complementary degrees.^[18]^[2] The differentials on the r-th page, denoted d_r, are maps of bidegree (r, 1-r) in the cohomological convention (i.e., d_r: E_r^{p,q} \to E_r^{p+r, q-r+1}) or (-r, r-1) in the homological convention (i.e., d_r: E_{p,q}^r \to E_{p-r, q+r-1}^r), with higher differentials vanishing on previous images and kernels.^[18]^[19] The sequence of pages converges to a limiting page E_\infty^{p,q}, whose terms are isomorphic to the associated graded pieces of the abutment.^[2] Convergence is indicated by abutment notation such as E_r^{p,q} \Rightarrow H^{p+q}(X) for cohomological sequences or E_{p,q}^r \Rightarrow H_{p+q}(X) for homological ones, meaning the E_\infty terms provide a filtration whose graded quotients match those of the target homology or cohomology groups.^[18] In cases of complete convergence under suitable conditions (e.g., finite filtrations), this yields E_\infty^{p,q} \cong \mathrm{gr}^{p} H^{p+q}(X), where \mathrm{gr}^p denotes the p-th graded piece.^[19]^[2] For multiplicative spectral sequences, which carry compatible ring or algebra structures (often induced from cup products), the differentials satisfy a signed Leibniz rule: d_r(xy) = d_r(x) y + (-1)^{|x|} x d_r(y), where |x| is the total degree of x, ensuring the structure is preserved across pages.^[19]^[2] This convention aligns with the bidegrees, typically taking the total degree as p+q in cohomological settings.^[18] Common abbreviations include "SS" for spectral sequence and "gr" for associated graded object, used throughout the literature to streamline discussions of filtrations and quotients.^[18] These notations, rooted in the bigraded structure of the underlying filtered complexes, facilitate precise descriptions across algebraic topology and homological algebra.^[2]

Diagrammatic Representations

Spectral sequences are commonly visualized on a two-dimensional grid, with the horizontal axis labeled by the filtration degree p increasing to the right and the vertical axis labeled by the complementary degree q increasing upward. Each intersection (p, q) hosts the abelian group E_r^{p,q} on the r-th page of the spectral sequence. The anti-diagonals, where the total degree n = p + q is fixed, align with the graded pieces that converge to the homology or cohomology groups in degree n. This layout facilitates tracking how terms evolve across pages and contribute to the final abutment.^[2] Differentials d_r are represented as directed arrows on the grid. In the cohomological convention, an arrow from E_r^{p,q} points to E_r^{p+r,q-r+1}, spanning r squares rightward along the p-axis and $1-r squares upward (downward for r > 1) along the q-axis. In the homological convention, the directions reverse, with arrows spanning r squares leftward and r-1 squares upward. Terms unaffected by any differential, which persist to the E_\infty page, are often marked distinctly, such as by encircling or boxing them to emphasize their role in the associated graded structure of the target groups.^[2] For spectral sequences arising from bounded filtrations, such as first-quadrant sequences where terms vanish for p < 0 or q < 0, diagrams are restricted to the nonnegative quadrant, simplifying visualization and computation. In contrast, unbounded or multiply filtered cases may require the full plane, though vanishing conditions often confine nonzero terms to specific regions. This distinction aids in deciding the diagram's scope, with first-quadrant representations sufficing for many applications in algebraic topology.^[2] Computational diagrams incorporate visual aids to monitor the progression of differentials. Regions corresponding to kernels of d_r (potential cycles) and images of incoming d_r (boundaries) may be shaded to distinguish killed terms from survivors across pages. The E_\infty terms, representing the permanent cycles modulo boundaries, are frequently boxed or highlighted to directly map to the filtration quotients in the abutment, streamlining the extraction of extension information. These techniques, rooted in standard notational practices, enhance readability and error-checking in manual calculations.^[2]

General Properties

Categorical Aspects

Spectral sequences exhibit functoriality when constructed from filtered chain complexes in an abelian category, where a morphism of filtered complexes induces a morphism of the associated spectral sequences. Specifically, if f: (C_\bullet, F) \to (C'_\bullet, F') is a chain map preserving the filtrations, it maps the E^r_{p,q}-page of the spectral sequence of C_\bullet to that of C'_\bullet for each r, commuting with the differentials d^r. This structure allows spectral sequences to be viewed as functors from the category of filtered complexes (with filtration-preserving morphisms) to a category of bigraded objects equipped with differentials of increasing length.^[20]^[21] The naturality of differentials in spectral sequences arises from the compatibility of induced maps with the boundary operators on each page. Under a filtration-preserving chain map f, the map f^r: E^r \to E'^r satisfies f^r \circ d^r = d'^r \circ f^r, ensuring that the homology of subsequent pages is preserved. This natural transformation property holds because the differentials are derived from the original chain map via the filtration quotients, maintaining exactness in the abelian category setting. For instance, in the exact couple construction, morphisms between couples induce compatible maps on the derived pages, preserving the exact triangles formed by the structure maps.^[22]^[20] Categorical constructions of spectral sequences rely on exact couples within abelian categories, where an exact couple consists of objects A, E and morphisms i: E \to A, j: A \to E, k: A \to E satisfying exactness conditions such as \operatorname{im} i = \ker j and \operatorname{im} k = \ker i. The first page E^1 is the homology of E^0 = E under d^0 = j \circ i, and higher pages arise from derived couples, yielding a spectral sequence in the category. These constructions are compatible with short exact sequences: if $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 is exact with induced filtrations, a long exact sequence of spectral sequences emerges, often via the five-lemma applied pagewise. Morphisms of exact couples are functorial, inducing maps on the spectral sequences that respect the abelian category structure.^[22]^[21]^[20] In triangulated categories, spectral sequences manifest as derived functors, capturing the universal properties of compositions of exact functors. For functors F: \mathcal{A} \to \mathcal{B} and G: \mathcal{B} \to \mathcal{C} between abelian categories with enough injectives, the derived functors RF and RG extend to the derived categories D(\mathcal{A}) and D(\mathcal{B}), which are triangulated. The composition R(GF) fits into a distinguished triangle with RG \circ RF, inducing a spectral sequence E^2_{p,q} = R^p G (R^q F (A)) \Rightarrow R^{p+q} (GF)(A). This Grothendieck spectral sequence encodes the universal property that derived functors preserve exact triangles and homotopy equivalences, providing a categorical framework for computing higher derived objects without explicit resolutions.^[23]^[20]

Multiplicative Structures

Multiplicative spectral sequences arise when the pages E_r of a spectral sequence carry additional algebraic structure, specifically as graded-commutative rings equipped with differentials that act as derivations. In such a setup, each E_r is a bigraded ring with a product operation that is bilinear and graded-commutative, meaning the product \mu: E_r^{p,q} \otimes E_r^{p',q'} \to E_r^{p+p', q+q'} satisfies \mu(x \otimes y) = (-1)^{(p+q)(p'+q')} \mu(y \otimes x) for homogeneous elements x \in E_r^{p,q} and y \in E_r^{p',q'}, and the differentials d_r: E_r^{p,q} \to E_r^{p-r, q+r-1} are derivations with respect to this product, obeying the Leibniz rule d_r(xy) = d_r(x)y + (-1)^{p+q} x d_r(y). This structure ensures that the multiplicative properties are preserved across pages, as the homology functor defining E_{r+1} = H(E_r, d_r) is compatible with the ring operations.^[14]^[2] The primary construction of multiplicative spectral sequences originates from filtered differential graded algebras (DGAs), where a DGA (A, d, \mu) equipped with a decreasing filtration F induces a spectral sequence whose pages E_r inherit DGA structures. Specifically, the filtration on A leads to associated graded pieces that form DGAs, with the product on A descending to a product on the E_1 or E_2 page via the quotient maps, and subsequent differentials remaining derivations due to the Leibniz property in the underlying DGA. For instance, in cohomological spectral sequences derived from filtered cochain complexes of algebras, the E_2 page often carries the induced product, as seen in the equation for cup products \cup: E_2^{p,q} \otimes E_2^{p',q'} \to E_2^{p+p', q+q'}, which is compatible with the differentials and extends to higher pages when the filtration is multiplicative. This construction is functorial in the category of filtered DGAs, briefly aligning with the categorical aspects of spectral sequences.^[14]^[2] A prominent example occurs in cohomology spectral sequences, such as the Serre spectral sequence for a fibration, where cup products in the cohomology of the total space induce a multiplicative structure on the E_2 page, given by E_2^{p,q} \cong H^p(B; \mathcal{H}^q(F; \mathbb{Z})) with the product twisted by a sign factor (-1)^{qs} to account for the bigrading. Here, the differentials act as derivations, preserving the ring structure up to E_\infty. Another key illustration involves actions of the Steenrod algebra in mod p cohomology spectral sequences, where the E_r pages become modules over the Steenrod algebra \mathcal{A}_p, a graded Hopf algebra generated by Steenrod operations like squares or powers; these actions commute with the differentials, enhancing the multiplicative framework to detect unstable homotopy groups or cohomology rings of Eilenberg-MacLane spaces.^[2]^[14]

Key Constructions

Exact Couple Construction

An exact couple consists of abelian groups D and E, together with homomorphisms i: D \to D, j: D \to E, k: E \to D satisfying the exactness conditions \operatorname{im}(i) = \ker(j), \operatorname{im}(j) = \ker(k), and \operatorname{im}(k) = \ker(i).^[14] This structure, introduced by William S. Massey, forms the basis for deriving a spectral sequence through iterative application of homology constructions.^[24] The differential on the initial page is defined as d_0 = j \circ k: E \to E. Exactness implies k \circ j = 0, so d_0 \circ d_0 = j \circ k \circ j \circ k = j \circ (k \circ j) \circ k = j \circ 0 \circ k = 0, confirming d_0 is a differential.^[14] The zeroth page of the spectral sequence is thus E_0 = E equipped with d_0. The first page is obtained as the homology E_1 = H(E_0, d_0) = \ker(d_0)/\operatorname{im}(d_0).^[25] To derive higher pages, construct the first derived exact couple (D_1, E_1; i_1, j_1, k_1) where D_1 = \operatorname{im}(i) \subseteq D, i_1 is the restriction of i to D_1, j_1: D_1 \to E_1 sends an element i(x) \in D_1 to the homology class [j(x)] \in E_1, and k_1: E_1 \to D_1 is the map induced by k on \ker(d_0) that factors through \operatorname{im}(d_0).^[14] This derived couple satisfies the exactness conditions by the five-lemma applied to the commutative diagram relating the original and derived structures.^[25] The differential on the first page is then d_1 = j_1 \circ k_1: E_1 \to E_1, and E_2 = H(E_1, d_1). Iterating this derivation yields the r-th page E_r = H(E_{r-1}, d_{r-1}), where d_r = j_r \circ k_r is the differential of bidegree (-r, r-1) in homological grading (or (r, 1-r) in cohomological grading), with bigrading compatible with the degrees of i_r (bidegree (1, -1)), j_r (bidegree (0, 0)), and k_r (bidegree (-1, 0)).^[14] In general, for an exact couple (A, A', B; \alpha, \beta, \gamma) with \alpha: A \to A', \beta: A' \to B, \gamma: B \to A and exactness \operatorname{im} \alpha = \ker \beta, \operatorname{im} \beta = \ker \gamma (extending to \operatorname{im} \gamma = \ker \alpha for closure), the identification i = \alpha, j = \beta \circ \alpha^{-1} on images (where invertible on relevant subgroups), and k = \gamma yields E_1 \cong \ker(\beta \circ \alpha \circ \gamma)/\operatorname{im}(\beta \circ \alpha \circ \gamma), with d_1 induced by the action of \alpha on this homology via boundary connections in the derived triangle.^[26] This axiomatic approach unifies all major spectral sequence constructions by producing exact couples from underlying data, such as long exact sequences in filtered or double complexes, while also accommodating cases without explicit filtrations, such as certain sheaf cohomology computations.^[14] The iterative homology process ensures convergence under suitable boundedness conditions on the bigradings, yielding E_\infty terms that fit into exact sequences relating to the abutment.^[25]

Filtered Complex Construction

A filtered complex in homological algebra consists of a chain complex (C_*, d) together with an increasing filtration F_p C_*, meaning \cdots \subset F_{p-1} C_* \subset F_p C_* \subset F_{p+1} C_* \subset \cdots, where each F_p C_n is a subcomplex of C_n and the differential d maps F_p C_n into F_{p+1} C_{n-1}. To align with the homological convention used elsewhere, we adopt a decreasing filtration convention here: \cdots \supset F_{p+1} C_* \supset F_p C_* \supset F_{p-1} C_* \supset \cdots, with d: F_p C_n \to F_p C_{n-1}. The associated graded object is defined as \mathrm{gr}_p C_n = F_p C_n / F_{p+1} C_n, which inherits an induced differential d_0 from d, making (\mathrm{gr}_p C_*, d_0) a chain complex graded by p.^[14]^[17] The E_0-page of the spectral sequence is given by E_0^{p,q} = \mathrm{gr}_p C_{p+q}, with the differential d_0: E_0^{p,q} \to E_0^{p,q-1} acting vertically within each graded piece \mathrm{gr}_p C_*.^[14] The homology of this graded complex yields the E_1-page: E_1^{p,q} = H_{p+q}(\mathrm{gr}_p C_*, d_0), where cycles and boundaries are taken modulo the filtration. The d_1-differential on E_1 is induced by the connecting homomorphisms from the long exact sequences associated to the short exact sequences $0 \to \mathrm{gr}_p C_* \to F_p C_* / F_{p+2} C_* \to \mathrm{gr}_{p+1} C_* \to 0, resulting in a map d_1: E_1^{p,q} \to E_1^{p-1,q}.^[17] Higher differentials d_r: E_r^{p,q} \to E_r^{p-r, q+r-1} for r \geq 2 arise from the original differential d modulo the lower filtration levels, capturing the failure of d to preserve the filtration strictly.^[27] These differentials satisfy d_r^2 = 0, allowing the formation of subsequent pages E_{r+1}^{p,q} = H_{p+q}(E_r^{*,*}, d_r), with the sequence abutting to the graded pieces of the homology of the original complex. Under suitable completeness and boundedness conditions on the filtration—such as the filtration being Hausdorff and exhaustive—the spectral sequence converges in the sense that F_p H_n(C_*) / F_{p+1} H_n(C_*) \cong E_\infty^{p, n-p}.^[14]

Double Complex Construction

A double complex consists of a bigraded collection of abelian groups (C_{p,q})_{p,q \in \mathbb{Z}}, together with horizontal differentials d_h: C_{p,q} \to C_{p-1,q} of bidegree (-1,0) and vertical differentials d_v: C_{p,q} \to C_{p,q-1} of bidegree (0,-1), satisfying d_h^2 = 0, d_v^2 = 0, and d_h d_v + d_v d_h = 0.^[20]^[9] This anticommutation relation ensures compatibility between the two differentials, allowing the structure to model situations with two commuting or anticommuting operations, such as in derived functors or chain complexes from products of spaces.^[20] The total complex associated to the double complex is the single-graded chain complex \mathrm{Tot}(C) with \mathrm{Tot}(C)_n = \bigoplus_{p+q=n} C_{p,q} and differential d = d_h + d_v, which squares to zero by the properties of d_h and d_v.^[20]^[9] This total complex captures the overall homology that the spectral sequences aim to approximate through successive refinements. To construct spectral sequences, one imposes filtrations on \mathrm{Tot}(C). The column filtration is the decreasing filtration F_p \mathrm{Tot}(C) = \bigoplus_{i \geq p} \bigoplus_{q \in \mathbb{Z}} C_{i,q}, where each graded piece \mathrm{gr}_p \mathrm{Tot}(C) = F_p / F_{p+1} \cong \bigoplus_q C_{p,q} is identified with the p-th column.^[20] Alternatively, the row filtration is F^q \mathrm{Tot}(C) = \bigoplus_{j \geq q} \bigoplus_{p \in \mathbb{Z}} C_{p,j}, with graded pieces corresponding to rows.^[20] These filtrations extend the single-filtration setup by incorporating the bidegree structure, yielding dual spectral sequences from the two orthogonal directions.^[9] For the column filtration, the spectral sequence begins with the E_0-page E_0^{p,q} = C_{p,q} and differential d_0 = d_v, the vertical differential preserving the filtration index p.^[20] The E_1-page is then the vertical homology

E_1^{p,q} = H_q(C_{p,\bullet}, d_v) = \frac{\ker(d_v: C_{p,q} \to C_{p,q-1})}{\operatorname{im}(d_v: C_{p,q+1} \to C_{p,q})},

with the d_1-differential induced by d_h on this homology.^[20]^[9] This d_1: E_1^{p,q} \to E_1^{p-1,q} has bidegree (-1,0), reflecting the horizontal direction. The row filtration yields a second spectral sequence, where the E_0-page has differential d_0 = d_h (horizontal, preserving rows), and the E_1-page is the horizontal homology E_1^{p,q} = H_p(C_{\bullet,q}, d_h), with d_1 induced by d_v of bidegree (0,-1).^[20]^[9] Under suitable boundedness conditions, such as the double complex being first-quadrant (i.e., C_{p,q} = 0 for p < 0 or q < 0), both spectral sequences converge to the homology of the total complex: E_\infty^{p,q} \cong F_p H_{p+q}(\mathrm{Tot}(C)) / F_{p+1} H_{p+q}(\mathrm{Tot}(C)) for the column filtration, and analogously for the row filtration.^[20]^[9] This dual convergence provides complementary approximations, often with the E_2-page of one relating to the E_1-page of the other via derived functor interpretations.^[20]

Convergence and Completion

Abutment and Filtrations

In spectral sequences arising from filtered chain complexes, the abutment refers to the graded object associated with the target homology groups that the sequence converges to. Specifically, the infinity page E_\infty^{p,q} consists of the graded pieces of a filtration on the homology H_{p+q}(C_\bullet), satisfying the isomorphism

E_\infty^{p,q} \cong \frac{F_p H_{p+q}(C_\bullet)}{F_{p+1} H_{p+q}(C_\bullet)},

where F_\bullet H_n(C_\bullet) denotes the induced filtration on the homology.^[21] This structure allows the total homology group H_n(C_\bullet) to be reconstructed as a successive extension problem over the abelian groups E_\infty^{p, n-p} for p \in \mathbb{Z}, though solving these extensions generally requires additional information beyond the spectral sequence itself.^[28] The filtration on the homology groups is induced from the filtration on the underlying chain complex C_\bullet. For a decreasing filtration F_p C_\bullet \supseteq F_{p+1} C_\bullet, the p-th level is defined as

F_p H_n(C_\bullet) = \operatorname{im}\bigl( H_n(F_p C_\bullet) \to H_n(C_\bullet) \bigr),

where the map is the one induced by the inclusion F_p C_\bullet \hookrightarrow C_\bullet.^[21] This filtration is exhaustive, meaning \bigcup_p F_p H_n(C_\bullet) = H_n(C_\bullet), provided the original filtration on C_\bullet is exhaustive.^[28] It is Hausdorff, meaning \bigcap_p F_p H_n(C_\bullet) = 0, if the filtration on C_\bullet satisfies a boundedness condition in each degree, ensuring the intersection of the kernels stabilizes appropriately.^[29] Convergence theorems establish when the pages E_r approach the abutment. For a filtered complex that is bounded below—meaning, for each total degree n, F_p C_n = C_n for all p <<0 sufficiently negative—the spectral sequence exhibits strong convergence to the infinity page:

E_r^{p,q} \Rightarrow E_\infty^{p,q},

with the differentials d_r eventually vanishing in each bidegree after finitely many steps.^[21] This bounded-below hypothesis ensures the process terminates, yielding a finite filtration on each H_n(C_\bullet).^[28] For the spectral sequence to be relevant to the abutment, meaning the derived limit terms vanish and the convergence reflects the full homology filtration, the induced filtration on H_\bullet(C_\bullet) must be complete in the sense that the completion \widehat{H_n(C_\bullet)} = \varprojlim F_p H_n(C_\bullet) equals H_n(C_\bullet).^[30] The key relevancy condition requires that the \lim^1 term in the derived inverse limit,

R^1 \lim_p \bigl( F_p H_n(C_\bullet)/F_{p+1} H_n(C_\bullet) \bigr) = 0,

vanishes, which holds for Hausdorff and complete filtrations under the Mittag-Leffler condition on the transition maps.^[30] This ensures strong convergence to the abutment without conditional pathologies.^[21]

Degeneration Phenomena

In spectral sequences arising from filtered complexes, degeneration occurs at the E_r-page if all differentials d_s = 0 for s \geq r, implying that E_r \cong E_\infty and the abutment decomposes as a direct sum of the associated graded pieces of E_r.^[20] This collapse simplifies computations by stabilizing the sequence early, allowing the homology or cohomology to be read directly from the E_r-terms without further differentials.^[9] A prominent example is E_2-degeneration, as occurs in the spectral sequence arising from a short exact sequence of chain complexes, which induces a two-step filtration on the middle term. In this case, the E_2 page is the abutment, and the only possible non-trivial differentials are the d_2 maps, yielding a long exact sequence in homology analogous to that from the snake lemma.^[20]^[9] For filtered complexes where the first differential d_1 = 0, the E_2-page simplifies to E_2^{p,q} \cong H^{p+q}(F_p C / F_{p+1} C), the homology of the associated graded pieces of the filtration. This situation often arises in bounded filtrations and connects to exact sequences like the Wang sequence in fiber bundle cohomology.^[20] Such degeneration highlights how the initial page's structure directly informs the abutment without intermediate computations.^[9] In double complexes, E_1-degeneration—where one of the vertical or horizontal spectral sequences collapses at E_1—implies the commutativity of derived functors such as Tor and Ext. For example, if one complex, say Q_\bullet, is flat, then the spectral sequence for the double complex P_\bullet \otimes Q_\bullet degenerates at E_1, yielding the Künneth isomorphism H_*(P \otimes Q) \cong H_*(P) \otimes H_*(Q), which relates to computing Tor groups via tensor products of resolutions.^[20] This relies on flatness ensuring vanishing higher Tor terms and no interfering differentials.^[9]

Core Examples

First-Quadrant Computations

In first-quadrant spectral sequences, the terms E_r^{p,q} are zero unless p \geq 0 and q \geq 0, with the additional property of finite support along each anti-diagonal p + q = n, which ensures strong convergence to the abutment graded by the filtration.^[2] A canonical setting for such sequences is the cohomological Serre spectral sequence arising from a Serre fibration F \to E \to B over a path-connected base B, where F is path-connected and \pi_1(B) acts trivially on H^*(F; \mathbb{Z}). In this case, the E_2-page is E_2^{p,q} = H^p(B; H^q(F; \mathbb{Z})), with differentials d_r: E_r^{p,q} \to E_r^{p+r, q - r + 1}, converging to H^{p+q}(E; \mathbb{Z}).^[2] These sequences are particularly tractable for sphere bundles, such as the unit sphere bundle S^{k-1} \to ES(V) \to B of an oriented real vector bundle V of rank k over B. Here, H^q(S^{k-1}; \mathbb{Z}) = \mathbb{Z} for q = 0 and q = k-1, and vanishes otherwise, so the E_2-page consists of two rows: E_2^{p,0} \cong H^p(B; \mathbb{Z}) and E_2^{p,k-1} \cong H^p(B; \mathbb{Z}).^[2] The differentials d_r for r < k vanish, as their targets lie outside the nonzero rows or the quadrant. The first nontrivial differential is the transgression d_k: E_k^{0,k-1} \to E_k^{k,0}, which sends the generator of H^{k-1}(S^{k-1}; \mathbb{Z}) to the Euler class e(V) \in H^k(B; \mathbb{Z}).^[2] If e(V) generates H^k(B; \mathbb{Z}), this differential is an isomorphism on the relevant terms, killing the generator in the top row at p = 0 and the bottom row generator at q = 0, p = k. Subsequent differentials may act on the surviving terms, but for simply connected B with cells in dimensions not congruent to k modulo the period, the sequence often stabilizes at E_{k+1}, with E_\infty^{p,q} = 0 for most low-degree terms along the affected diagonals.^[2] A concrete illustration is the Hopf fibration S^1 \to S^3 \to S^2, the unit circle bundle associated to the tautological complex line bundle over \mathbb{CP}^1 \cong S^2, computed with \mathbb{Z}/2-coefficients to avoid torsion issues. The E_2-page has E_2^{p,0} = H^p(S^2; \mathbb{Z}/2) = \mathbb{Z}/2 for p = 0, 2 and zero otherwise, while E_2^{p,1} = \mathbb{Z}/2 for p = 0, 2 (since H^1(S^1; \mathbb{Z}/2) = \mathbb{Z}/2). The differential d_2: E_2^{0,1} \to E_2^{2,0} is an isomorphism, sending the generator \sigma \in H^1(S^1; \mathbb{Z}/2) to the generator u \in H^2(S^2; \mathbb{Z}/2). Thus, on E_3, the terms E_3^{0,1} = 0 and E_3^{2,0} = 0, leaving E_3^{0,0} = \mathbb{Z}/2 and E_3^{2,1} = \mathbb{Z}/2; higher differentials vanish due to the finite support. The sequence stabilizes at E_3 = E_\infty.^[2] The E_\infty-page provides the associated graded \mathrm{gr} H^*(S^3; \mathbb{Z}/2), with E_\infty^{0,0} contributing to H^0(S^3; \mathbb{Z}/2) \cong \mathbb{Z}/2 and E_\infty^{2,1} to H^3(S^3; \mathbb{Z}/2) \cong \mathbb{Z}/2, while intermediate degrees vanish, matching the known cohomology. Recovering the ungraded abutment requires resolving short exact sequences from the filtration, such as $0 \to F^2 H^3(S^3; \mathbb{Z}/2) \to H^3(S^3; \mathbb{Z}/2) \to E_\infty^{2,1} \to 0; in this case, the extensions are trivial, yielding the direct sum structure.^[2]

Bounded Support Cases

In spectral sequences arising from filtered chain complexes with bounded support in the horizontal direction—specifically, where the E_2 page is nonzero only in two adjacent columns, such as p=0 and p=1—the structure simplifies dramatically due to the absence of higher columns. This setup commonly occurs when constructing a spectral sequence from a short exact sequence of chain complexes $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0, where the filtration on B^\bulletis defined byF^0 B^\bullet = \operatorname{im}(A^\bullet \to B^\bullet)andF^1 B^\bullet = B^\bullet. In this case, the E_2page consists ofE_2^{0,q} \cong H_q(A^\bullet)in the first column andE_2^{1,q} \cong H_q(C^\bullet)$ in the second column, with all other terms vanishing.^[20] The differentials on the E_2 page are d_2: E_2^{p,q} \to E_2^{p+2,q-1}, so from the p=0 column, any d_2 would target the nonexistent p=2 column, rendering all d_2 zero. Higher differentials d_r for r \geq 3 similarly map to columns p \geq 2 or p \leq -1, which are also zero, causing the spectral sequence to stabilize immediately at E_2 = E_\infty. The abutment H_n(B^\bullet) then admits a filtration with exactly two nonzero graded pieces: \operatorname{gr}_0 H_n(B^\bullet) \cong E_\infty^{0,n} \cong H_n(A^\bullet) and \operatorname{gr}_1 H_n(B^\bullet) \cong E_\infty^{1,n-1} \cong H_{n-1}(C^\bullet). This gives a short exact sequence $0 \to \operatorname{gr}_1 H_n(B^\bullet) \to H_n(B^\bullet) \to \operatorname{gr}_0 H_n(B^\bullet) \to 0$ from the filtration.^[20] Connecting these short exact sequences across degrees via the boundary maps from the long exact sequence of homology (arising from the snake lemma applied to the filtered complex) produces the full long exact sequence

\cdots \to H_n(A^\bullet) \to H_n(B^\bullet) \to H_n(C^\bullet) \xrightarrow{\partial} H_{n-1}(A^\bullet) \to \cdots,

where \partial is the connecting homomorphism. This degeneration at E_2 exemplifies how bounded support enforces immediate collapse, directly recovering the classical long exact sequence in homology from the short exact sequence of complexes.^[20]^[2] A concrete example is the universal coefficient theorem, where the short exact sequence $0 \to \operatorname{Ext}^1(H_{n-1}(X;\mathbb{Z}), G) \to H^n(X; G) \to \operatorname{Hom}(H_n(X;\mathbb{Z}), G) \to 0of cochain complexes (for coefficients in an abelian groupG) induces a two-column spectral sequence on the E_2page withE_2^{0,q} \cong \operatorname{Hom}(H_q(X;\mathbb{Z}), G)andE_2^{1,q} \cong \operatorname{Ext}^1(H_q(X;\mathbb{Z}), G)$, degenerating to the short exact sequence relating cohomology to homology and Ext groups.^[2] For generalizations to k consecutive nonzero columns (say p=0 to p=k-1), the spectral sequence features nontrivial differentials d_r only for r \leq k, as higher d_r would target columns outside the support. The process terminates after finitely many pages (at most E_k), yielding a filtration on the abutment H_n with k graded pieces \operatorname{gr}_p H_n \cong E_\infty^{p,n-p}, connected by extension problems that can be resolved using the abutment's structure. This finite-column case thus computes homology via a bounded chain of exact sequences, generalizing the two-column long exact sequence to multi-step filtrations.^[20]

Advanced Features

Edge Morphisms

In a cohomological spectral sequence associated to a filtered cochain complex converging to the cohomology groups H^*(X), the edge morphism along the left edge (the vertical axis where p=0) is defined as the map \alpha_r^n: E_r^{0,n} \to H^n(X) for each page r \geq 2. This map is induced by the natural inclusion F^0 C^n \hookrightarrow C^n of the zeroth filtration level into the full cochain group, composed with the connecting homomorphism in cohomology.^[14] Since F^0 C^* = C^* in the standard decreasing filtration convention for cohomology, the inclusion is the identity on cochains, but the map on E_r^{0,n} arises from considering cycles in the associated graded complex at that filtration level that survive to page r.^[2] More generally, edge morphisms exist along both the left and bottom edges of the spectral sequence pages. Along the bottom edge (where q=0), there is a map \beta_r^p: E_r^{p,0} \to F^p H^p(X)/F^{p+1} H^p(X), induced by the quotient map F^p C^* \twoheadrightarrow \mathrm{gr}^p C^* and the subsequent projection to the associated graded piece of the abutment. Along the left edge, the dual map is \gamma_r^n: E_r^{0,n} \to F^0 H^n(X) / F^1 H^n(X), reflecting the kernel of the map to the next graded piece. These maps factor through the stable E_\infty page in the limit, embedding into or projecting onto the graded abutment \mathrm{gr} H^*(X) \cong E_\infty^{p,q}.^[20]^[2] Visually, in the standard diagram of a spectral sequence page, these edge morphisms appear as horizontal arrows from the bottom row E_r^{p,0} to the graded pieces \mathrm{gr}^p H^p(X) and vertical arrows from the left column E_r^{0,q} to \mathrm{gr}^q H^q(X), or their reverses depending on the direction of the filtration. The properties of these morphisms include naturality with respect to morphisms of filtered complexes, meaning they commute with induced maps on the E_r pages and abutments, and compatibility with differentials up to page r-1, as elements on the edges are permanent cycles until hit by higher differentials.^[14]^[2] Transgression maps, which connect interior terms to edges via higher differentials, can be viewed as generalizations of these boundary edge morphisms.^[20]

Transgression Maps

In a homological spectral sequence, the transgression map \tau: E_r^{p,0} \to E_r^{0,p-1} for r > 1 is defined when the intermediate terms in the relevant filtration vanish, specifically as the differential d_r with r = p that connects the bottom edge to the left edge of the E_r-page.^[14] This map arises in situations where lower differentials do not affect the relevant cycles, allowing it to detect non-trivial extensions between the abutment groups.^[2] The construction of the transgression proceeds as a connecting homomorphism in the long exact sequence associated to the short exact sequence of cycles and boundaries in the filtered chain complex: $0 \to Z_r^{p,0} \to C_{p-1} \to B_{r-1}^{0,p-1} \to 0, where Z_r^{p,0} denotes the cycles on the bottom edge and B_{r-1}^{0,p-1} the boundaries on the left edge.^[14] More explicitly, up to sign, \tau = \pi_F \circ d_{p} \circ i_B^{-1}, where i_B: E_\infty^{p,0} \hookrightarrow E_r^{p,0} is the inclusion of the permanent cycles from the base filtration (a monomorphism), d_p is the spectral differential, and \pi_F: E_r^{0,p-1} \twoheadrightarrow E_\infty^{0,p-1} is the projection onto the homology of the fiber filtration (an epimorphism).^[14] This composition ensures that \tau measures how elements on one edge transgress to the opposite edge after surviving prior pages.^[2] In the cohomological Serre spectral sequence for a fibration F \to E \to B with acyclic total space E (i.e., H^*(E) = 0), the transgression identifies \tau: H^q(B) \to H^{q+1}(F) as an isomorphism, arising from the long exact sequence of the fibration where the middle terms vanish.^[31] This map, originally introduced in the context of singular cohomology of fibrations, detects the extent to which cohomology classes in the base map to cohomology classes in the fiber under contractibility assumptions.^[31]

Applications

Topology and Geometry

Spectral sequences play a pivotal role in topology and geometry by facilitating the computation of cohomology groups associated with fibrations, sheaves, and other geometric structures. The Serre spectral sequence, introduced for Serre fibrations, provides a method to relate the cohomology of the total space E to that of the base B and fiber F in a fibration F \to E \to B. Specifically, it converges to H^*(E; \mathbb{Z}) with E_2^{p,q} = H^p(B; \mathcal{H}^q(F; \mathbb{Z})), where \mathcal{H}^q(F; \mathbb{Z}) denotes the local coefficient system induced by the action of \pi_1(B) on H^q(F; \mathbb{Z}). This sequence is particularly effective for simply connected spaces, where local coefficients trivialize, simplifying computations. A classic application is the quaternionic Hopf fibration S^3 \to S^7 \to S^4, where the Serre spectral sequence confirms that H^*(S^7; \mathbb{Z}) = \mathbb{Z} in degrees 0 and 7, and trivial otherwise, leveraging the known cohomologies of S^3 and S^4. The Atiyah-Hirzebruch spectral sequence extends this framework to generalized cohomology theories, computing groups h^n(X) for a space X from the ordinary cohomology H^*(X; \mathbb{Z}) and the coefficients h^*(pt). It abuts to h^*(X) with E_2^{p,q} = H^p(X; h^q(pt)), and its differentials encode obstructions related to k-invariants, which detect the Postnikov tower structure of the Eilenberg-MacLane spaces underlying the theory.^[32] In complex K-theory, for instance, this sequence relates K^*(X) to H^{even}(X; \mathbb{Z}), with differentials arising from the Chern character and Adams operations, enabling computations of K-groups for projective spaces and other manifolds.^[32] In geometric settings involving sheaves, the Leray spectral sequence computes the hypercohomology \mathbb{H}^*(X, \mathcal{F}^\bullet) of a complex of sheaves \mathcal{F}^\bullet on a space X via a cover \{U_i\}, with E_2^{p,q} = H^p(\{U_i\}, \mathcal{H}^q(\mathcal{F}^\bullet)) converging to \mathbb{H}^{p+q}(X, \mathcal{F}^\bullet). This is essential for sheaf cohomology. For a single sheaf \mathcal{F}, the higher cohomology sheaves vanish (\mathcal{H}^q(\mathcal{F}) = 0 for q > 0, \mathcal{H}^0(\mathcal{F}) = \mathcal{F}), so the sequence becomes E_2^{p,0} = \check{H}^p(\{U_i\}; \mathcal{F}) \Rightarrow H^p(X; \mathcal{F}), comparing Čech and derived functor cohomology, and it degenerates to an isomorphism for fine or acyclic covers. Applications include de Rham cohomology on manifolds, where the sequence relates global forms to local computations, providing insights into Hodge structures. The Eilenberg-Moore spectral sequence applies to pullback diagrams in topology, particularly for computing the cohomology of a space P in the fibration \Omega B \to P \to E \times_B E, converging to H^*(P) from the Tor groups \Tor_{H^*(B)}^{H^*(E), H^*(E)}(H^*(E \times_B E), \mathbb{Z}). In manifold geometry, it relates to Koszul complexes, where the fiber product structure models algebraic resolutions; for example, on a smooth manifold M with a group action, the sequence computes invariants of quotient spaces or orbit spaces by resolving the Koszul complex associated to the action, yielding differential forms and equivariant cohomology data.

Homological Algebra

In homological algebra, spectral sequences provide powerful tools for computing derived functors such as \operatorname{Ext}_R and \operatorname{Tor}_R in module categories over a ring R. The Grothendieck spectral sequence, introduced for the composition of left-exact functors on abelian categories, is central to these computations. Specifically, for left-exact functors G: \mathcal{C} \to \mathcal{D} and F: \mathcal{D} \to \mathcal{E} where G sends injectives in \mathcal{C} to F-acyclics in \mathcal{D}, there arises a first-quadrant spectral sequence

E_2^{p,q} = R^q F (R^p G (A)) \implies R^{p+q} (F \circ G)(A)

for any object A \in \mathcal{C}, assuming the category has enough injectives.^[33] This sequence arises from a Cartan-Eilenberg resolution of the derived functor R^\bullet G(A) and subsequent application of F, yielding a double complex whose total homology is R^\bullet (F \circ G)(A). In the context of modules over R, this applies to computing \operatorname{Ext}_R(M, N) via resolutions of either M or N. For instance, resolving N by an injective resolution I^\bullet \to N and applying \operatorname{Hom}_R(M, -) gives \operatorname{Ext}_R(M, N) = H^\bullet (\operatorname{Hom}_R(M, I^\bullet)), while resolving M by projectives P_\bullet \to M and applying \operatorname{Hom}_R(-, N) yields the same via homology. The Grothendieck spectral sequence relates these approaches in composite functor settings, such as base change over ring extensions, where E_2^{p,q} = \operatorname{Ext}_S^p (M \otimes_R S, \operatorname{Ext}_R^q (S, N)) \implies \operatorname{Ext}_R^{p+q} (M, N) under suitable flatness conditions on S as an R-module.^[34] The Künneth spectral sequence addresses computations involving tensor products of modules or complexes. For R-modules M and N, consider projective resolutions P_\bullet \to M and Q_\bullet \to N; the tensor product complex P_\bullet \otimes_R Q_\bullet forms a double complex whose total homology is \operatorname{Tor}_\bullet^R (M, N). Filtering by columns (degrees in Q_\bullet) induces a spectral sequence

E_2^{p,q} = \operatorname{Tor}_p^R (H_q (Q_\bullet), M) \cong \operatorname{Tor}_p^R (N, H_{-q} (P_\bullet)) \implies \operatorname{Tor}_{p+q}^R (M, N),

converging strongly in the first-quadrant case. This relates the Tor groups of the modules to the homology of the tensor product of their resolutions. If one resolution is acyclic except in degree zero (as for flat modules), the sequence degenerates at E_2, yielding the classical Künneth isomorphism \operatorname{Tor}_\bullet^R (M, N) \cong \bigoplus_{p+q=\bullet} H_p (P_\bullet) \otimes_R H_q (Q_\bullet). More generally, for complexes A^\bullet and B^\bullet, the Künneth spectral sequence computes H_\bullet (A^\bullet \otimes_R B^\bullet) via E_2^{p,q} = \operatorname{Tor}_{-p}^R (H_q (A^\bullet), H_\bullet (B^\bullet)), providing a tool to relate derived tensor products to Tor of homologies.^[7] Change-of-rings theorems use spectral sequences to relate Tor groups over different rings, particularly proving commutativity in tensor product settings. For commutative k-algebras R and S, the tensor product ring R \otimes_k S satisfies a change-of-rings spectral sequence for modules M over R and N over S: resolving M over R and N over S, the double complex P_\bullet \otimes_k Q_\bullet computes \operatorname{Tor}_\bullet^{R \otimes_k S} (M \otimes_k S, R \otimes_k N) via the total complex, with the induced spectral sequence

E_2^{p,q} = \operatorname{Tor}_p^R (M, \operatorname{Tor}_q^S (k, N)) \implies \operatorname{Tor}_{p+q}^{R \otimes_k S} (M, N).

This converges to the Tor groups over the composite ring, and under Tor-dimension bounds (e.g., finite projective dimension), edge maps provide isomorphisms showing commutativity \operatorname{Tor}_\bullet^{R \otimes_k S} (M, N) \cong \operatorname{Tor}_\bullet^R (M, N) \otimes_k S when applicable. The sequence demonstrates how Tor over the tensor product decomposes into iterated Tor over the factors, facilitating computations in algebraic k-theory and derived categories.^[34] A concrete example of spectral sequences in extensions arises from Cartan-Eilenberg resolutions, which extend projective or injective resolutions to compute higher Ext groups classifying extensions. For R-modules M and N, to compute \operatorname{Ext}_R^\bullet (M, N) via a resolution incorporating extensions, construct a Cartan-Eilenberg projective resolution of M: start with a projective resolution P_\bullet \to M, then for each syzygy Z_p = \ker (P_p \to P_{p-1}), attach projectives resolving the Ext sheaves or kernels in a double complex setup. The total complex yields a resolution whose Hom to N gives a double complex, inducing a spectral sequence

E_2^{p,q} = \operatorname{Ext}_R^p (H_q (P_\bullet), N) \implies \operatorname{Ext}_R^{p+q} (M, N).

In the case of extensions, for \operatorname{Ext}_R^1 (M, N) classifying short exact sequences $0 \to N \to E \to M \to 0, higher terms in the resolution capture obstructions to lifting extensions, with the spectral sequence degenerating appropriately for low degrees but providing nontrivial differentials for infinite extensions or in nonprojective cases. This construction, detailed in the theory of derived functors for complexes, exemplifies how spectral sequences resolve extension problems in module categories.^[7]

Homotopy Theory

In homotopy theory, spectral sequences provide a powerful framework for computing homotopy groups, particularly in the stable regime where spaces are replaced by spectra. These tools arise naturally from filtrations on spectra or model categories, allowing the approximation of homotopy groups through successive pages of differentials. A key example is the spectral sequence associated to a filtered spectrum, where the filtration induces a sequence whose E_1-page consists of the homotopy groups of the cofibers of the filtration maps, and which converges strongly to the homotopy groups of the total spectrum under suitable completeness conditions, such as when the filtration is exhaustive and complete.^[35]^[36] The Adams spectral sequence stands as a cornerstone for determining stable homotopy groups of spectra. Introduced by J. Frank Adams, it computes the p-primary component of \pi_*^S(X) for a spectrum X, starting from the E_2-page given by \operatorname{Ext}_{A_*}^{s,t}(H_*(X; \mathbb{Z}/p), \mathbb{Z}/p), where A_* is the dual Steenrod algebra and H_* denotes homology with \mathbb{Z}/p-coefficients.^[37] The sequence converges to the p-local stable homotopy groups \pi_*^S(X) \otimes \mathbb{Z}_{(p)}, detecting elements via permanent cycles and extensions, and has been instrumental in resolving the p-primary components of the stable stems \pi_*^S.^[37] For the sphere spectrum, it yields charts of homotopy groups up to high dimensions, with differentials often computed via secondary operations in the Steenrod algebra.^[37] For unstable homotopy groups of spheres, the EHP spectral sequence offers a fibration-based approach. Derived from the filtration S^0 \to \Omega S^1 \to \Omega^2 S^2 \to \cdots \to \Omega^\infty S^\infty, its E^1-page is isomorphic to \pi_i(\Omega^{n+2} S^{2n+1}), relating \pi_{i+n+2}(S^{2n+1}) through differentials involving the Hopf invariant and Whitehead products.^[38] This sequence, developed in the context of composition methods, converges to \pi_*(\Omega^\infty S^\infty), the stable homotopy groups, and was extensively used by Hirosi Toda to compute \pi_n(S^k) for k \leq 19 and n \leq 20, including p-local variants via fibrations like \hat{S}^{2m} \to JS^{2m} \to JS^{2mp}.^[39] Properties such as the vanishing of differentials below the desuspension threshold \rho(n) of the Whitehead square facilitate these calculations.^[38]^[39] In modern contexts, the Bousfield-Kan spectral sequence addresses completions in model categories. For a cosimplicial space X^\bullet, it arises from the totalization \operatorname{Tot}(X^\bullet), with the E_2-page given by the cohomology of the normalized chain complex associated to X^\bullet, converging to the homotopy groups \pi_*(\operatorname{Tot}(X^\bullet)).^[40] This sequence underpins R-completions for a ring R, such as p-completion, by resolving spaces via cosimplicial models and computing the homotopy of the completed totalization.^[40] Extensions to stable \infty-categories, as in derived algebraic geometry, generalize these constructions: filtered objects in a stable \infty-category with a t-structure yield spectral sequences whose E_1-page is the homotopy of cokernels, abutting to the homotopy groups of the colimit, unifying homotopy computations with derived stacks and E_\infty-ring spectra.^[36]^[40]

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The main purpose of this paper is to introduce a new algebraic object int topology. This new algebraic structure is called an exact couple of groups (or (.
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[PDF] Notes on spectral sequence
In practice, we have the differential dr of bidegree (r,1 − r) (for a spec- tral sequence of cohomology type) or (−r, r − 1) (for a spectral sequence of.Missing: d_r E_r^{
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[PDF] A Primer on Spectral Sequences - UChicago Math
Filtered chain complexes give rise to exact couples and therefore to spectral sequences. This is one of the most basic sources of spectral sequences. The ...
[15]
[PDF] HOMOLOGICAL ALGEBRA
The purpose of this book is to present a unified account of these developments and to lay the foundations of a full-fledged theory. The invasion of algebra has ...
[16]
[PDF] Math 754 Chapter II: Spectral Sequences and Applications
Mar 14, 2018 · A spectral sequence is a machine that takes (co)homology of base and fiber and outputs the (co)homology of the total space in a fibration.
[17]
[PDF] Introduction to spectral sequences - UC Berkeley math
Apr 28, 2011 · We now sketch two constructions of the Leray-Serre spectral sequence. The first construction only works in the special case when B is a CW.
[18]
spectral sequence in nLab
### Summary of Spectral Sequence Notation from nLab
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[PDF] Chapter 4 - Spectral sequences and Serre classes
He had to recast the theory to work with singular cohomology, on much more general spaces, but in return he considered only what we now call Serre fibrations.<|control11|><|separator|>
[20]
[PDF] Spectral Sequences
Spectral sequences were invented by Jean Leray, as a prisoner of war during. World War II, in order to compute the homology (or cohomology) of a chain.
[21]
[PDF] Exact Couples and Their Spectral Sequences - arXiv
May 21, 2022 · Abstract. Given a bigraded exact couple of modules over some ring, we determine the meaning of the E8 -terms of its associated spectral ...
[22]
12.21 Spectral sequences: exact couples - Stacks Project
An exact couple is a datum (A, E, \alpha , f, g) where A, E are objects of \mathcal{A} and \alpha , f, g are morphisms as in the following diagram.
[23]
[PDF] The Derived Category
2). 10.5 Derived Functors. There is a category of triangulated categories; a morphism F:K -> K' of triangulated categories is ...
[24]
[PDF] Products in Exact Couples - UiO
Exact couples were introduced by the author in [2]; in that paper it was shown how exact couples may be applied to various problems in algebraic.
[25]
[PDF] 1 Exact couples - Kiran S. Kedlaya
We explain the construction (or rather, one particular construction) of spectral sequences, enough to explain how they are used as part of the computation ...
[26]
[PDF] Lecture 12: Spectral sequences - Duke Mathematics Department
Feb 15, 2015 · A spectral sequence is a sequence of differential groups (En,dn) for n ≥ 2 (or 1 or 0) such that En. ∼. = H(En−1,dn−1). This definition is ...
[27]
12.24 Spectral sequences: filtered complexes - Stacks project
Hence according to Section 12.23 we obtain a spectral sequence (E_ r, d_ r)_{r \geq 0}. In this section we work out the terms of this spectral sequence, and we ...
[28]
[PDF] MAT9580: Spectral Sequences - UiO
▷ The key feature of a homological spectral sequence is that the dr -differential reduces the filtration degree from s to s − r. Page 18. Morphisms of ...<|control11|><|separator|>
[29]
[PDF] The homology of groups. Part II : spectral sequences Contents
J. McCleary, A user's guide to spectral sequences, Cambridge studies in advanced mathematics 58, Cambridge University Press, 1994. • C. A. Weibel, ...
[30]
[PDF] Conditionally Convergent Spectral Sequences Contents Introduction
In §1, we discuss limits and colimits of abelian groups, and in §4, we do the same for spectra. An important special case is §2 on filtered groups, where the ...
[31]
[PDF] Homologie Singuliere Des Espaces Fibres Jean-Pierre Serre The ...
Jun 27, 2007 · Homologie Singuliere Des Espaces Fibres. Jean-Pierre Serre. The Annals of Mathematics, 2nd Ser., Vol. 54, No. 3. (Nov., 1951), pp. 425-505 ...
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[PDF] VECTOR BU1\'DLES A.,\'D BO)IOGEl''EOUS SPACES
Cambridge. Philos. Soc.). This justifies the new theory. In spite of its length, the present paper is by no means a final exposition.
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[PDF] sur qtjelques points d'algebre homologique. - » Tous les membres
I. Contenu du travail. Ce travail a son origine dans une tentative d'exploiter Panalogie formelle entre la theorie de la cohomologie d'un espace.
[34]
[PDF] AN INTRODUCTION TO HOMOLOGICAL ALGEBRA
... Weibel An introduction to homological algebra. 39 W. Bruns and J. Herzog ... Spectral Sequences. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 6 Grouj. 6.1. 6.2.
[35]
spectral sequence of a filtered stable homotopy type in nLab
### Summary of Spectral Sequence of a Filtered Stable Homotopy Type
[36]
[PDF] Derived Algebraic Geometry I: Stable ∞-Categories
Oct 8, 2009 · If C is a stable ∞-category equipped with a t-structure, then every filtered object of C determines a spectral sequence; we will give the ...<|control11|><|separator|>
[37]
[1801.07530] A Guide for Computing Stable Homotopy Groups - arXiv
Jan 23, 2018 · We provide a working guide to the stable homotopy category, to the Steenrod algebra and to computations using the Adams spectral sequence.
[38]
[PDF] The EHP Spectral Sequence
1 The homological spectral sequence as a special case of the homo- topical ... Definition For α ∈ πi(X) and β ∈ πj(X), the Whitehead product [α, β] ...
[39]
https://press.princeton.edu/books/paperback/9780691095868/composition-methods-in-homotopy-groups-of-spheres
[40]
Homotopy Limits, Completions and Localizations - SpringerLink
The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X.