every , , the homology of with respect to . Usually the isomorphisms are suppressed and we write instead. An object is called sheet (as in a sheet of paper), or sometimes a page or a term; an endomorphism is called boundary map or differential. Sometimes is called the derived object of . Bigraded spectral sequence In reality spectral sequences mostly occur in the category of doubly graded modules over a ring R (or doubly graded sheaves of modules over a sheaf of rings), i.e. every sheet is a bigraded R-module So in this case a cohomological spectral sequence is a sequence of bigraded R-modules and for every module the direct sum of endomorphisms of bidegree , such that for every it holds that: , . The notation used here is called complementary degree. Some authors write instead, where is the total degree. 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. In the ungraded situation described above, r0 is irrelevant. Homological spectral sequence Mostly the objects we are talking about are chain complexes, that occur with descending (like above) or ascending order. In the latter case, by replacing with and with (bidegree ), one receives the definition of a homological spectral sequence analogously to the cohomological case. Spectral sequence from a chain complex The most elementary example in the ungraded situation is a chain complex C•. An object C• in an abelian category of chain complexes naturally 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 to equal , 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• Er = H(C•) for all r ≥ 1. The terms of this spectral sequence stabilize at the first sheet because its only nontrivial