**Spectral Sequences**

In homological algebra and algebraic topology, a **spectral sequence** is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.

Read more about Spectral Sequences: Discovery and Motivation, Formal Definition, Exact Couples, Visualization, Convergence, Degeneration, and Abutment, Further Examples

### Other articles related to "spectral sequences, spectral sequence, sequence, sequences":

Spectral Sequence - Further Examples

... Some notable

... Some notable

**spectral sequences**are Adams**spectral sequence**in stable homotopy theory Adams–Novikov**spectral sequence**, a generalization to extraordinary cohomology theories ... Atiyah–Hirzebruch**spectral sequence**of an extraordinary cohomology theory Bar**spectral sequence**for the homology of the classifying space of a group ... Barratt**spectral sequence**converging to the homotopy of the initial space of a cofibration ...Spectral Sequence - Convergence, Degeneration, and Abutment

... that we began with, the sheets of the

... that we began with, the sheets of the

**spectral sequence**were constant once r was at least 1 ... In that setup it makes sense to take the limit of the**sequence**of sheets Since nothing happens after the zeroth sheet, the limiting sheet E∞ is the same as E1 ... They are one of the most powerful aspects of**spectral sequences**...Spectral Sequence - Exact Couples

... The most powerful technique for the construction of

... The most powerful technique for the construction of

**spectral sequences**is William Massey's method of exact couples ... Exact couples are particularly common in algebraic topology, where there are many**spectral sequences**for which no other construction is known ... In fact, all known**spectral sequences**can be constructed using exact couples ...**Spectral Sequences**- Further Examples

... Some notable

**spectral sequences**are Adams

**spectral sequence**in stable homotopy theory Adams–Novikov

**spectral sequence**, a generalization to extraordinary cohomology theories ... Atiyah–Hirzebruch

**spectral sequence**of an extraordinary cohomology theory Bar

**spectral sequence**for the homology of the classifying space of a group ... Barratt

**spectral sequence**converging to the homotopy of the initial space of a cofibration ...

Spectral Sequence - Formal Definition

... A

... A

**spectral sequence**is a choice of a nonnegative integer r0 and a collection of three**sequences**For all integers r ≥ r0, an object Er, called a sheet (as in a ... Putting the zero differential on all the rest of our sheets gives a**spectral sequence**whose terms are E0 = C• Er = H(C•) for all r ≥ 1 ... The terms of this**spectral sequence**stabilize at the first sheet because its only nontrivial differential was on the zeroth sheet ...### Famous quotes containing the word spectral:

“How does one kill fear, I wonder? How do you shoot a spectre through the heart, slash off its *spectral* head, take it by its *spectral* throat?”

—Joseph Conrad (1857–1924)

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