In homological algebra, more particularly in algebraic Topology and cohomology of group, a spectral continuation is a succession of differential modules ( E _N , D _N ) such as

E _{N +1} = H ( E _N ) = ker D _N /im D _N

is the homology of E _N .

There are several manners to obtain such a continuation in practice. Historically, since 1950, the arguments of the spectral continuations were a powerful tool for research, in particular in the theory of homotopy.

General explanation

A manner of visualizing what occurs in a spectral continuation is the use of the metaphor of the block-note . E ₁ being the first datasheet, the sheet E ₂ derives from it by a definite process; and so on for the sheets E ₃, E ₄… The end result of calculation would be the last sheet of the scratch pad.

In practice E _i contains certain data of gradation, often same two. Each sheet is then transformed into table, arranged in lines and column, with an abelian group in each cell. Each sheet has also " différentielles" , which acts since each cell of the sheet on another cell. The definite process is then a means of calculating the state of each cell on the following sheet starting from the current sheet, while following the differentials.

To be rigorous, the element E _N should indicate two indices:

E _N ^{p, Q}

with the differentials D _N ^{p, Q} acting since E _N ^{p, Q} on a E _N ^{p+a, q+b} , with has and B depending only on N.

Filtrations

Spectral continuations often appear during calculations of Filtration of a Module E ₀. A filtration

$A\_\; \{-\; 2\}\; =\; A\_\; \{-\; 1\}\; =\; A\_0\; \backslash \; supset\; A\_1\; \backslash \; supset\; A\_2\; \backslash \; supset\; \backslash \; ldots$

of a module a short exact continuation induces

$0\; \backslash \; to\; has\; \backslash \; hookrightarrow\; has\; \backslash \; to\; B\; \backslash \; to\; 0,$

with B , the quotient J of has by its image under the inclusion I , and whose differentials is induced by that of has. Either has ₁ = H ( has ) and B ₁ = H ( B ); a exact series long,

$\backslash ldots\; \backslash to\; A\_1\; \backslash to\; A\_1\; \backslash to\; B\_1\; \backslash to\; A\_1\; \backslash to\; \backslash ldots$

is given by the lemma of the snake . If we call the posted charts I ₁, J ₁, and K ₁, and if has ₂ = i₁A ₁ and B ₂ = ker j₁k₁ /im j₁k₁ , one can show that

$\backslash ldots\; \backslash to\; A\_2\; \backslash to\; A\_2\; \backslash to\; B\_2\; \backslash to\; A\_2\; \backslash to\; \backslash ldots$

is another exact continuation.

Examples

Some notorious spectral continuations:

• spectral Continuation of Leray-Greenhouse

• spectral Continuation of Hochschild-Greenhouse
• spectral Continuation of Adams
• spectral Continuation of Atiyah-Hirzebruch
• spectral Continuation of Adams-Novikov
• spectral Continuation of chromatic Grothendieck
• spectral Continuation
• spectral Continuation of Eilenberg-Moore
• spectral Continuation of Bockstein

References

• has User' S Guide to Spectral Sequences by John McCleary

• Cohomology Operations and Applications in Homotopy Theory , by Robert Mosher and Martin Tangora
• Séminaire Henri Cartan of the National university, 1950/1951. Cohomology of the groups, spectral continuation, beams

Random links:

Aimargues | Charles Ives | Pierre Klossowski | Léo Marion | Carl Hårleman