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Cartaneilenberg Spectral SequenceEdit

The Cartaneilenberg Spectral Sequence (CSS) is a computational framework in homological algebra and algebraic topology that helps extract the total cohomology of a filtered object by peeling it apart along two compatible filtrations. Built as a variant of the classic Cartan–Eilenberg approach, the CSS formalizes how information from the pieces of a bicomplex or a filtered total complex assembles into the global cohomology. It sits in the same family as spectral sequence techniques used to study extensions, filtrations, and derived functors, and it interacts closely with notions such as filtration, double complex, and total complex.

Unlike a single-filtration construction, the Cartaneilenberg Spectral Sequence leverages a two-dimensional filtration on a total complex Tot(C^{,}) coming from a bicomplex C^{,} with two compatible filtrations. This structure naturally produces a sequence of pages E_r^{p,q} and differentials d_r that converge (under suitable boundedness conditions) to the total cohomology H^{n}(Tot(C^{,})). In practice, the CSS provides a way to organize complicated extension and interaction data between the horizontal and vertical directions of the bicomplex into a computable ladder of successive approximations.

Overview

  • The setting is a bicomplex C^{,} in an abelian category (often with enough projectives or injectives) together with a compatible filtration on the total complex Tot(C^{,}). The filtration yields a chain of subcomplexes whose associated graded pieces feed the successive pages of the sequence.
  • The E_0-page reflects the raw graded pieces of the filtration; passing to cohomology with respect to one differential produces the E_1-page, and further taking cohomology with respect to the remaining differential produces higher pages.
  • The CSS captures two layers of information: the intrinsic cohomology of the individual pieces and the way those pieces glue together across the filtration to form the total cohomology.
  • It recovers and interrelates several classical constructions in spectral sequence theory, and it specializes to familiar sequences when one of the filtrations is trivial or when the bicomplex reduces to a single-filtered complex.

Definition and construction

  • Setup: Start with a bicomplex C^{p,q} in an abelian category, with horizontal differential d_h: C^{p,q} → C^{p+1,q} and vertical differential d_v: C^{p,q} → C^{p,q+1} satisfying d_h^2 = d_v^2 = d_h d_v + d_v d_h = 0. Equip Tot(C^{,}) with a filtration F^p Tot(C^{,}) arising from the p-th column (or from a dual filtration in the other direction).
  • E_0-page: The associated graded pieces E_0^{p,q} are determined by the filtration quotients of Tot(C^{,}).
  • E_1-page: Taking cohomology with respect to the horizontal differential (or vertical, depending on convention) yields E_1^{p,q} ≅ H^q(C^{p,*}, d_h). The differential d_1 is induced by d_v on these horizontal cohomology classes.
  • E_2-page and beyond: The E_2^{p,q} terms come from the cohomology of the E_1-page with respect to the induced differential d_1, i.e., E_2^{p,q} ≅ H^p(H^q(C^{,}, d_h), d_v). Higher differentials d_r: E_r^{p,q} → E_r^{p+r,q−r+1} encode increasingly subtle extension data across the two filtrations.
  • Convergence: Under suitable boundedness or finiteness hypotheses (for example, the filtrations are bounded in each total degree, or the category has finite-length objects), the CSS converges to H^{n}(Tot(C^{,})), meaning the abutment recovers the total cohomology from the stabilized pages.

For readers familiar with Cartan-Eilenberg spectral sequence theory, the Cartaneilenberg Spectral Sequence can be viewed as a two-dimensional extension of that framework: it keeps track of how horizontal and vertical cohomology interact through the filtration, producing a two-way decomposition of the total cohomology.

E_2-page, differentials, and interpretations

  • The E_2-page often provides a tangible handle on the computation because it packages cohomology of cohomology data: E_2^{p,q} reflects how the q-th cohomology of the horizontal slices interacts with the p-th cohomology of the resulting vertical structure.
  • The differentials d_r for r ≥ 2 measure higher-order obstructions to extending local cohomology information across the two filtrations. In representation-theoretic or geometric contexts, these obstructions frequently correspond to extension classes or compatibility conditions across subobjects.
  • When specialized to particular choices of filtration or restricted classes of bicomplexes, the CSS can reduce to more familiar sequences (for instance Lyndon–Hochschild–Serre spectral sequences in suitable group-extension situations, or the standard Cartan–Eilenberg spectral sequence in a one-filtration limit).

Convergence and relationship to other constructions

  • Convergence criteria typically require the bicomplex to be bounded in at least one direction or for the filtrations to be finite in each total degree. Under these hypotheses, the CSS abuts to H^{}(Tot(C^{,*})).
  • The Cartaneilenberg Spectral Sequence is closely related to and often compares with the Leray spectral sequence, the Lyndon spectral sequence, and various filtrations arising in sheaf cohomology and homological algebra. In many situations, CSS provides a convenient scaffold to organize information that would otherwise be scattered across several computations.
  • The CSS also interacts with derived functor machinery, such as the Ext and Tor functors, and with the language of abelian category and exact sequences.

Examples and applications

  • Group cohomology: In contexts involving a short exact sequence of groups or a group extension, a Cartaneilenberg-type construction can be used to compute cohomology by splitting the computation into horizontal and vertical layers and then reassembling via the CSS.
  • Sheaf theory and algebraic geometry: When studying the cohomology of filtered sheaves or bicomplexes arising from hypercoverings, the CSS provides a structured way to handle the interaction between local data and its global glueing.
  • Topology and fibered spaces: For fibrations and their associated spectral sequences, the Cartaneilenberg framework clarifies how base and fiber cohomology pieces contribute to the total cohomology.

Variants and related notions

  • Variants of the CSS can be formulated with different types of filtrations (e.g., increasing vs. decreasing filtrations) or with alternative bicomplex structures that reflect specific algebraic or geometric constraints.
  • Related tools include spectral sequences derived from filtrations in cohomology theories, as well as specialized sequences designed to handle particular classes of extensions, modules, or sheaves.
  • The CSS shares methodological kinship with other two-dimensional filtering techniques, and it often serves as a bridge between purely algebraic computations and geometric or topological interpretations.

See also