page_title: Homological algebra
section_title: Spectral sequence
hierarchical_section_title: Homological algebra / Standard tools / Spectral sequence
caption_attribution_description: Corrected spectral sequence diagram illustration - by saaska -
context_page_description: Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.
  The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.
  From its very origins, homological algebra has played an enormous role in algebraic topology.
context_section_description: Fix an abelian category, such as a category of modules over a ring.  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 sheet of paper), or sometimes a page or a term,
  Endomorphisms dᵣ : Eᵣ → Eᵣ satisfying dᵣ o dᵣ = 0, called boundary maps or differentials,
  Isomorphisms of Eᵣ₊₁ with H(Eᵣ), the homology of Eᵣ with respect to dᵣ.
  A doubly graded spectral sequence has a tremendous amount of data to keep track of, but there is a common visualization technique which makes the structure of the spectral sequence clearer.  We have three indices, r, p, and q.  For each r, imagine that we have a sheet of graph paper.  On this sheet, we will take p to be the horizontal direction and q to be the vertical direction.  At each lattice point we have the object .
  It is very common for n = p + q to be another natural index in the spectral sequence. n runs diagonally, northwest to southeast, across each sheet.  In the homological case, the differentials have bidegree (−r, r − 1), so they decrease n by one.  In the cohomological case, n is increased by one.  When r is zero, the differential moves objects one space down or up.  This is similar to the differential on a chain complex.  When r is one, the differential moves objects one space to the left or right.  When r is two, the differential moves objects just like a knight's move in chess.  For higher r, the differential acts like a generalized knight's move.
caption: The E2 sheet of a cohomological spectral sequence
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