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.

exact sequence | homological algebra