Spectral Sequence - Formal Definition

Formal Definition

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:

1. For all integers r ≥ r₀, an object E_r, called a sheet (as in a sheet of paper), or sometimes a page or a term,
2. Endomorphisms d_r : E_r → E_r satisfying d_r d_r = 0, called boundary maps or differentials,
3. Isomorphisms of E_r+1 with H(E_r), the homology of E_r with respect to d_r.

Usually the isomorphisms between E_r+1 and H(E_r) are suppressed, and we write equalities instead. Sometimes E_r+1 is called the derived object of E_r.

The most elementary example is a chain complex C_•. An object C_• in an abelian category of chain complexes comes with a differential d. Let r₀ = 0, and let E₀ be C_•. This forces E₁ to be the complex H(C_•): At the i'th location this is the i'th homology group of C_•. The only natural differential on this new complex is the zero map, so we let d₁ = 0. This forces E₂ to equal E₁, and again our only natural differential is the zero map. Putting the zero differential on all the rest of our sheets gives a spectral sequence whose terms are:

• E₀ = C_•
• E_r = 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. Consequently we can get no more information at later steps. Usually, to get useful information from later sheets, we need extra structure on the E_r.

In the ungraded situation described above, r₀ is irrelevant, but in practice most spectral sequences occur in the category of doubly graded modules over a ring R (or doubly graded sheaves of modules over a sheaf of rings). In this case, each sheet is a doubly graded module, so it decomposes as a direct sum of terms with one term for each possible bidegree. The boundary map is defined as the direct sum of boundary maps on each of the terms of the sheet. Their degree depends on r and is fixed by convention. For a homological spectral sequence, the terms are written and the differentials have bidegree (-r,r-1). For a cohomological spectral sequence, the terms are written and the differentials have bidegree (r, 1 − r). (These choices of bidegree occur naturally in practice; see the example of a double complex below.) Depending upon the spectral sequence, the boundary map on the first sheet can have a degree which corresponds to r = 0, r = 1, or r = 2. For example, for the spectral sequence of a filtered complex, described below, r₀ = 0, but for the Grothendieck spectral sequence, r₀ = 2. Usually r₀ is zero, one, or two.

A morphism of spectral sequences E → E' is by definition a collection of maps f_r : E_r → E'_r which are compatible with the differentials and with the given isomorphisms between cohomology of the r-th step and the (r + 1)-st sheets of E and E', respectively. The category of spectral sequences is an abelian category.