Linear_Algebra_Done_Spectral

This monograph develops the complete theory of spectral sequences using only linear algebra over a field. No topology, no category theory, no homological algebra beyond what is explicitly constructed from vector spaces and linear maps. The central object is a filtered differential space: a finite-dimensional vector space V equipped with a linear endomorphism d satisfying d2 = 0 and a flag of d-invariant subspaces 0 = F−1V ⊆ F0V ⊆ ⋯ ⊆ FsV = V. The spectral sequence {Er, dr}r≥0 is defined as the output of a single algorithm — iterated rank-nullity applied to the restrictions of d across the flag — and every theorem about spectral sequences is proved as a theorem about dimensions, ranks, and nullities of linear maps. Chapters 1–10 develop the theory from scratch: nilpotent endomorphisms, the homology dimension formula, flags and graded spaces, the spectral sequence algorithm, exact couples, convergence, products, double complexes, and resolutions. Chapter 11 — the Plug-In Principle — shows how to recover every named spectral sequence in the literature (Serre, Adams, Eilenberg–Moore, May, Grothendieck, Atiyah–Hirzebruch, Bockstein, Lyndon–Hochschild–Serre, and others) by choosing a specific input (V, d, F). The topology enters only in Chapter 11, and only as the source of the input data. The entire content of McCleary's A User's Guide to Spectral Sequences is subsumed by Chapters 1–10.