#6491: [with spkg, needs review] Modular Cohomology Rings of Finite p-Groups
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Reporter: SimonKing | Owner: SimonKing
Type: enhancement | Status: assigned
Priority: major | Milestone: sage-4.1.1
Component: optional packages | Keywords: cohomology ring finite p-group
Reviewer: | Author: Simon King
Merged: |
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Comment(by SimonKing):
Replying to [comment:9 SimonKing]:
> - The most important difference: Apparently CRIME has no completeness
criterion. Section 2.4 states: "A test or series of tests for completion
of the calculation will hopefully be implemented soon. See [CTVZ03] for
the details." So, there is no analogue for our {{{H.make()}}}.
Since people don't know the purpose of the {{{make}}} method, a little
elaboration.
There are two ways of usage:
- {{{H.make(n)}}}: Compute at most out to degree {{{n}}}. This
corresponds to the capabilities of CRIME.
- {{{H.make()}}}: Compute in increasing degree, until (our improved
version of) Benson's criterion tells us that we can stop. This can't be
done in CRIME, but HAP has a completeness criterion.
----
Perhaps it is better to compare HAP (or actually HAPprime), rather than
CRIME, with our package.
HAPprime [http://www.gap-
system.org/~gap/Manuals/pkg/happrime-0.3.2/doc/userguide/manual.pdf] uses
a clever idea: Frst compute the Lyndon-Hoschild-Serre Spectral Sequence
until convergence to find the additive structure of the cohomology ring,
and then compute the cohomology ring itself, based on projective
resolutions, up to and including the maximum necessary generator or
relation degree.
The point is that this completeness criterion is perfect! However, an
application of this criterion involves the computation of spectral
sequences, which is perhaps not the easiest thing to do.
Here is a random collection of features, and how the three packages
compare. Disclaimer: I am rather tired, so, perhaps I do mistakes. Sorry
in advance.
__Complete computations:__ Possible with HAPprime and our package,
impossible with CRIME.
__Induced homomorphisms:__ CRIME can do, we can do, but I didn't find it
addressed in the HAPprime manual.
__Ring theoretic invariants:__ CRIME has nothing. HAPprime has Poincaré
series. We have Poincaré series, depth, dimension, a-invariants, and can
also compute the nil radical.
__Speed:__ I don't know about CRIME. I was told by Paul Smith that
HAPprime needs a few days to compute the cohomology for the groups of
order 64. We need less than 60 minutes.
__Coefficients:__ CRIME can do it not only over GF(p) but with higher-
dimensional modules. We can do it over GF(p), p a prime that must not be
greater than 255. HAPprime can do it over GF(2); Paul Smith says it also
works over GF(p), p>2, but Graham Ellis seems to have some doubts.
Best regards,
Simon
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6491#comment:11>
Sage <http://sagemath.org/>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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