# Constructing Spectral Sequences

While teaching Homological Algebra, I had the occasion to give an introduction to spectral sequences. The approach I took was similar to that of McCleary:

McCleary,
*A User's Guide to Spectral Sequences*.

Cambridge studies in advanced mathematics, 58. 2001.

I gave a fast description of what a spectral sequence is, and then focused on various techniques by working through illustrative examples over the next week. After we had done several examples, I wanted to go back and show my students the construction of a spectral seqeunce from a filtered chain complex. My first attempt at this turned out to be confusing and poorly presented—I believe I effectively communicated how easy it is to get lost in indexing and subquotients of subquotients!

To make a better second attempt, I looked for an outline and illustration of the construction. Not finding one, I produced the following picture. I made scans of three stages in the drawing, so that I could make slides showing successive layers of the picture. I then filled in notes explaining the picture, and decided to share them here.

## Slides and notes

I hope you enjoy the slides and notes, but let me apologize in
advance: *These are not written
for novice users*. They focus exclusively on the construction of
a spectral sequence, assuming that the reader is familiar with the
goals of this construction. There are many good resources which
describe *how* to use spectral sequences, and which give
motivation for *why* they are useful. These notes do not
attempt to address either of those important questions.

Our goal is to give a visual aid to complement the standard construction of a spectral sequence for the homology of a filtered chain complex. The notation follows §5.4 of Weibel:

Weibel,
*An introduction to homological algebra*.

Cambridge studies in advanced mathematics, 38. 1994.

As Weibel does, we drop the complementary degree *q* for
readability.

- Slides present the illustration in three layers, with more information on each successive layer.
- Notes to accompany the slides, including explanatory remarks not written on the slides. The notes contain a larger version of the final illustration, but leave out the earlier stages for ease of printing.

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