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Spectral Sequence


A spectral sequence is a tool of homological algebra that has many applications in algebra, algebraic geometry, and algebraic topology. Roughly speaking, a spectral sequence is a system for keeping track of collections of exact sequences that have maps between them.

There are many definitions of spectral sequences and many slight variations that are useful for certain purposes. The most common type is a "first quadrant cohomological spectral sequence," which is a collection of Abelian groups E_r^(p,q) where p, q, and r are integers, with p and q nonnegative and r>a for some positive integer a, usually 2. The groups E_r^(p,q) come equipped with maps

 d_r^(p,q):E_r^(p,q)->E_r^(p+r,q-r+1)
(1)

such that

 d_r^(p+r,p-r+1) degreesd_r^(p,q)=0.
(2)

There is the further restriction that

 E_(r+1)^(p,q)=Ker(d_r^(p,q))/Im(d_r^(p-r,p+r-1)).
(3)

The maps d_r^(p,q) are called boundary maps.

SpectralSequence

A spectral sequence may be visualized as a sequence of grids, one for each value of r. The ps and qs denote positions on the grid, where p is the x-coordinate and q is the y-coordinate. The diagram above shows this for r=2.

The entire collection of groups E_r^(p,q) together with their boundary maps is referred to as the E_r-term. The groups E_(r+1)^(p,q) are completely determined by the E_r-term

For any given value of p and q, the group E_r^(p,q) eventually stabilizes, because there are only a finite number of nonzero boundary maps that either start or end at this position. This stable value is referred to a E_infty^(p,q). Stable values allow the convergence of a spectral sequence to be defined. In particular, the spectral sequence E_r^(p,q) converges to groups H^n, written

 E_r^(p,q)=>H^(p+q)
(4)

if there is a filtration

 0=H_(n+1)^n subset= H_n^n subset= H_(n-1)^n subset= ... subset= H_2^n subset= H_1^n subset= H_0^n=H^n
(5)

such that the successive quotients are equal to the E_infty terms, i.e.,

 E_infty^(p,n-p)=H_p^n/H_(p+1)^n.
(6)

The Serre spectral sequence is used to compute the cohomology groups of the spaces in a fibration. Suppose

 F->E->B
(7)

is a fibration (for example, the Hopf map S^1->S^3->S^2). Then there is a spectral sequence with E_2-term

 E_2^(p,q)=H^p(B;H^q(F;Z)).
(8)

The sequence converges to H^(p+q)(E;Z). (Here, the H^* denotes ordinary cohomology and is unrelated to the H^n's above.) This allows one to compute the cohomology of one of three spaces E, F, and B from the cohomology of the other two.

There are other examples of spectral sequences. The Leray-Serre spectral sequence is used to compute the hypercohomology of complexes of sheaves. The Grothendieck spectral sequence is used to compute the derived functors of the composition of two functors from the derived functors of the two original functors. The Leray-Serre spectral sequence is a special case of the Grothendieck spectral sequence. Finally, the Adams spectral sequence is used to compute the higher homotopy groups of spheres.


See also

Exact Sequence, Homological Algebra

This entry contributed by John Renze

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References

Spanier, E. H. Algebraic Topology. New York: McGraw-Hill, pp. 185-186, 1966.Weibel, C. An Introduction to Homological Algebra. Cambridge, England: Cambridge University Press, 1995.Weston, T. "The Inflation-Restriction Sequence: An Introduction to Spectral Sequences." http://www.math.mcgill.ca/goren/SeminaronCohomology/infres.pdf.

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Spectral Sequence

Cite this as:

Renze, John. "Spectral Sequence." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SpectralSequence.html

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