Variations on a theme: the cohomology of the Steenrod algebra, providing the E_2 page of the Adams spectral sequence.
This program computes Ext_A(M,F_p), where
- A is the Steenrod algebra or its subalgebra A(n),
- M is an A-module, either one of a few standard examples, or else specified in Bruner's MDF format,
- p is any prime.
A couple of other miscellaneous computations are included, such as the E_2 page of the Cartan-Eilenberg spectral sequence.
Output takes the form of a scrollable chart, with products by the first three Hopf elements drawn. For a laugh, there's also a 3D viewer for the trigraded computations.
How to use
Running the program requires only the file
resolution.jar, available at
For trigraded computations in the 2D viewer, there are controls to limit the visible range for the third grading; these are also jointly controlled by the PgUp and PgDn keys on your keyboard, in case you want to quickly step through cross-sections.
For trigraded computations in the 3D viewer, you can drag the mouse to rotate, Shift+drag to pan the view, and either scroll or Ctrl+drag to zoom.
If there are features you'd like to see in this program, please get in touch!
The program can load modules specified in the `module definition format' of Bob Bruner: http://www.math.wayne.edu/~rrb/cohom/modfmt.html
The format changes slightly for odd primes, since we need to include coefficients in the action. The format for the action lines is now:
g r k c_1 g_1 c_2 g_2 ... c_k g_k
to indicate that Sq^r of generator g equals c_1 g_1 + c_2 g_2 + ... + c_k g_k. Choice of integer representative (mod p) of the coefficients doesn't matter.
So e.g. S/p as a module over the mod-p Steenrod algebra is given as:
2 0 1 0 1 1 1 1
for any odd p. (Usually, of course, the same module file won't work for all odd primes.)
This repository includes shell scripts
make for compiling,
run for running, and
prof for profiling. Currently they're a bit specialized to my machine, in particular referring to a Java 6 runtime
rt.jar in a subfolder, but it shouldn't be too hard to get it built if you're into Java. Feel free to contact me.
Most of the features in this program were developed during a period of collaboration with Michael Andrews, and so he has guided its direction to a large extent. Thanks also to Mark Behrens for feature suggestions.