In the chapters that follow, we will consider, in detail, the algebra of spectral sequences and furthermore, how this formalism can be applied to a topological problem. The user, however, needs to get acquainted with the manipulation of these gadgets without the formidable issue of their origins. This chapter is something of a tool kit, filled with computation techniques that may be employed by the user in the application of spectral sequences to algebraic and topological problems. We take a loosely axiomatic stance and argue from definitions, with most spectral sequences in mind. As in the case of long exact sequences or homology theory, this viewpoint still makes for a substantial enterprise. The techniques developed in this chapter, though elementary, will appear again and again in what follows. The user, facing a computation in later chapters, will profit by returning to this collection of tools and tricks.

“There is a spectral sequence …”

Let us begin with a basic goal: We want to compute H* where H* is a graded R-module or a graded κ-vector space or a graded κ-algebra or … This H* may be the homology or cohomology of some space or some other graded algebraic invariant associated to a space or perhaps an invariant of some algebraic object like a group, a ring or a module; in any case, H* is often difficult to obtain. In order to proceed, we introduce some helpful conditions.