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 . Visualization 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. An object can be seen as the checkered page of a book. On these sheets, we will take p to be the horizontal direction and q to be the vertical direction. At each lattice point we have the object . Now turning to the next page means taking homology, that is the page is a subquotient of the page. The total degree n = p + q 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. The differentials change their direction with each turn with respect to r. The red arrows demonstrate the case of a first quadrant sequence (see example below), where only the objects of the first quadrant are non-zero. While turning pages, either the domain or the codomain of all the differentials become zero. Properties Categorical properties The set of cohomological spectral sequences form a category: a morphism of spectral sequences is by definition a collection of maps which are compatible with the differentials, i.e. , and with the given isomorphisms between the cohomology of the rth step and the (r+1)th sheets of E and E' , respectively: . In the bigraded case, they should also respect the graduation: Multiplicative structure A cup product gives a ring structure to a cohomology group, turning it into a cohomology ring. Thus, it is natural to consider a spectral sequence with a ring structure as well. Let be a spectral sequence of cohomological type. We say it has multiplicative structure if (i) are (doubly graded) differential graded algebras and (ii) the multiplication on is induced by that on via passage to cohomology. A typical example is the cohomological Serre spectral sequence for a fibration , when the coefficient group is a ring R. It has the multiplicative structure induced by the cup products of fibre and base on the -page. However, in general