Introduction to spectral sequences
نویسنده
چکیده
The words “spectral sequence” strike fear into the hearts of many hardened mathematicians. These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful. This is an unfinished handout for my algebraic topology class. In particular I did not have time to reproduce here the little spectral sequence diagrams showing where all the arrows go. For a comprehensive introduction to spectral sequences, see [3]. For more nice explanations of spectral sequences, see [1] and [2]. Finally, the original paper [4] is a good read. A short exact sequence of chain complexes gives rise to a long exact sequence in homology, which is a fundamental tool for computing homology in a number of situations. There is a natural generalization of a short exact sequence of chain complexes, called a “filtered chain complex”. Associated to a chain complex with a filtration is an algebraic gadget generalizing the long exact sequence, which is called a spectral sequence, and which can help compute the homology of the chain complex. 1 The long exact sequence in homology We begin by reviewing the long exact sequence in homology associated to a short exact sequence of chain complexes, from a point of view which naturally generalizes to spectral sequences. Consider a chain complex C∗ with a subcomplex F0C∗. We now have a short exact sequence of chain complexes 0 −→ F0C∗ −→ C∗ −→ C∗/F0C∗ −→ 0. A fundamental lemma in homological algebra asserts that there is then a long exact sequence in homology · · · −→ Hi(F0C∗) −→ Hi(C∗) −→ Hi(C∗/F0C∗) δ −→ Hi−1(F0C∗) −→ · · · .
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