What is a Spectral Sequence?

Abstract

“A spectral sequence is an algebraic object, like an exact sequence, but more complicated.” J. F Adams In Chapter 1 we restricted our examples of spectral sequences to the first quadrant and to bigraded vector spaces over a field in order to focus on the computational features of these objects. In this chapter we treat some deeper structural features including the settings in which spectral sequences arise. In order to establish a foundation of sufficient breadth, we remove the restrictions of Chapter 1 and consider (ℤ × ℤ) -bigraded modules over R , a commutative ring with unity. It is possible to treat spectral sequences in the more general setting of abelian categories (the reader is referred to the thorough treatments in [Eilenberg-Moore62], [Eckmann-Hilton66], [Lubkin80], and [Weibel96]). The approach here supports most of the topological applications we want to consider. In this chapter we present two examples that arise in purely algebraic contexts—the spectral sequence of a double complex and the Klinneth spectral sequence that generalizes the ordinary Kunneth Theorem (Theorem 2.12). For completeness we have included a discussion of basic homological algebra. This provides a foundation for the generalizations that appear in later chapters. Definitions and basic properties We begin by generalizing our First Definition and identifying the basic components of a spectral sequence. Definition 2.1. A differential bigraded module over a ring R, is a collection of R-modules , { E p,q }, where p and q are integers, together with an R-linear mapping, d : E *,* → E *,*, the differential , of bidegree ( s , 1– s ) or(– s , s –1), for some integer s, and satisfying d o d = 0.