#6491: [with spkg, needs review] Modular Cohomology Rings of Finite p-Groups
-------------------------------+--------------------------------------------
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: |
-------------------------------+--------------------------------------------
Comment(by SimonKing):
Replying to [comment:8 wdj]:
> In that case, I think it should be made clear in the installation docs
that to install the package you need to have run sage -i database_gap*.
What do you mean by "installation docs"?
[[http://sage.math.washington.edu/home/SimonKing/Cohomology/#installation]]
? Yes, you are right. I will add a few words later today.
> I'm trying to see how this compares with GAP's CRIME package. It seems
to me that you using a completely different way of representing the
cohomology rings. Is that correct? If not, is there a simple example using
your package that can also be used with CRIME to compare their outputs?
> For example, the dihedral group of order 8. It seems that CRIME wants to
truncate the ring at a given degree and you don't (but maybe I'm wrong
here?), so I don't see how to easily compare them.
I must admit that I was not aware of CRIME, although Marcus Bishop
currently is in Galway (so am I, since very recently). I only knew HAP
(from Galway, too), that can also compute cohomology.
Looking at the CRIME manual, I think it compares with our package as
follows:
- We work with coefficients GF(p); CRIME allows more general modules.
- Apparently, when doing {{{C:=CohomologyObject( G )}}} in CRIME, you get
a not-yet-completed cohomology ring; this is similar to
{{{H=CohomologyRing(G)}}} or {{{H=CohomologyRing(q,m)}}}, if G or
SmallGroup(q,m) are not yet in our data base.
- Both CRIME and our package rely on the construction of minimal
projective resolutions. We use David Green's Gröbner basis approach, while
probably CRIME uses linear algebra.
- Indeed, CRIME works with truncated rings. For example,
{{{CohomologyGenerators( C, n )}}} computes the ring out to degree {{{n}}}
and returns the generators that it found. With our package, one could do
the same, doing {{{H.make(n); H.gens()}}}. Our {{{H.make(n)}}} is similar
to {{{CohomologyRing(G,n)}}}.
- {{{CohomologyGenerators( C, n )}}} only yields the ''degrees'' of the
generators. {{{H.gens()}}} yields the generators themselves, and you can
actually do computations with them. Actually, from the documentation, I
get the impression that CRIME does provide (truncated) Cohomology Rings,
but it does not provide ''elements'' of Cohomology Rings.
- 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()}}}.
Also my impression is that we provide more things to do with a cohomology
ring:
We implemented the computation of some ring theoretic invariants: Depth,
Poincaré series, a-invariants. Also we are able to compute the nil
radical. For these operations, we make extensive use of the fact that we
have a cohomology ring: We use Benson's completeness criterion, which
involves the computation of a filter regular system of parameters. These
parameters allow for an easy computation of depth and Poincare series, and
with a slightly extended machinery we also get the a-invariants. The nil
radical is obtained by studying the restrictions to the maximal elementary
abelian subgroups (which we do anyway when we choose our generators!).
My impression from the CRIME doc is that the computation would be much
more clumsy:
* Compute the ring structure C (Problem 1: There is no completeness
criterion)
* Create a quotient of a polynomial ring P/I that has this ring
structure.
* Problem 2: On the way you loose the information that C was a cohomology
ring. Now you just have an abstract ring presentation P/I, and then the
computation of the nil radical or the Poincare series becomes much harder.
Back to your question:
> I'm trying to see how this compares with GAP's CRIME package. It seems
to me that you using a completely different way of representing the
cohomology rings. Is that correct? If not, is there a simple example using
your package that can also be used with CRIME to compare their outputs?
Note that our favourite example (cohomology of the Dihedral Group of order
8) also appears in the CRIME documentation. Of course, we get essentially
the same result -- but we can prove that our result is complete and not
just a truncated version of the cohomology ring.
What you could always do: Make a complete computation with our package,
and use it to get the right truncation degree for CRIME, hence:
{{{
sage: G = your favourit p-group
sage: H = CohomologyRing.user_db(G)
sage: H.make()
sage: n=H.knownDeg
}}}
The last line means that n tells in what degree we drove our computation
before realising that it was complete. So, you could now use CRIME:
{{{
C := CohomologyRing(G,n);
}}}
and get the ring structure.
Then you can compare with our result. Of course, in general the relations
will look different. But: The number of generators sorted by degree and
the number of relations sorted by degree should be the same with CRIME and
our package. And, if you are able to compute the Poincaré series with
CRIME, again the result should be the same.
Actually this is how we compared our computational findings with the
results of Jon F. Carlson.
> BTW, I am testing the innstallation on an older amd64 ubuntu 8.10
machine. Seems to be going okay there too.
Good news!
Cheers,
Simon
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6491#comment:9>
Sage <http://sagemath.org/>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--~--~---------~--~----~------------~-------~--~----~
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To post to this group, send email to [email protected]
To unsubscribe from this group, send email to
[email protected]
For more options, visit this group at
http://groups.google.com/group/sage-trac?hl=en
-~----------~----~----~----~------~----~------~--~---
