Mathematical Field Notes

Archive for the ‘Algebra’ Category

Some simple spectral sequences

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I keep finding myself trying to explain how the very simplest spectral sequences arise (spectral sequence to compute the cohomology of a cone or an iterated cone), so I have taken the time to TeX the explanation into a sequence of guided exercises:

Some simple spectral sequences

This is all very formal and diagram-chasy. One of the off-putting things about spectral sequences is all the indices; in these exercises I have suppressed gradings and concentrated on the very simplest cases to avoid overcomplicating the notation. Once you’ve seen how the proof goes, you should go and look in Bott-Tu or McCleary for some actual examples and computations.

Please let me know of any errors in the exercises!

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Written by Jonny Evans

April 28, 2015 at 3:10 pm

TikZ code for the octahedral axiom

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If anyone finds it useful, I’ve created a LaTeX command for drawing the octahedral axiom (requires the TikZ package) based on this example of Stefan Kottwitz. You can easily edit the code to add labels to arrows (they would go in the empty brackets in the lines that say “edge”) or add these as extra arguments to the function.

Include the following code in the head of your LaTeX document:


\usepackage{tikz}
\usetikzlibrary{arrows}

\newcommand{\Octa}[6]{\begin{center}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm,
  thick]

  \node (1) {$#1$};
  \node (4) [below right of=1] {$#4$};
  \node (6) [below right of=4] {$#6$};
  \node (2) [above right of=4] {$#2$};
  \node (5) [above right of=6] {$#5$};
  \node (3) [above right of=5] {$#3$};


  \path
    (1) edge [out=25,in=155] node [right] {} (3)
        edge node[right] {} (2)
    (2) edge node [right] {} (3)
        edge node[right] {} (4)
    (3) edge node [right] {} (5)
        edge [out=270,in=0] node [right] {} (6)
    (4) edge node [right] {} (6)
    (6) edge node [right] {} (5);
  \path[dashed]
    (4) edge node[right] {} (1)
    (5) edge node[right] {} (4)
        edge node[right] {} (2)
    (6) edge [out=180,in=270] node [right] {} (1);
\end{tikzpicture}\end{center}}

Then the command \Octa{X}{Y}{Z}{A}{B}{C} will produce a diagram like this:

The octahedral axiom for taking the cone of a composition.

The octahedral axiom for taking the cone of a composition.

Written by Jonny Evans

March 1, 2013 at 11:37 am