Hi Sverre,
I ran into similar problems in my "yacop" package
(https://github.com/cnassau/yacop-sage), which also deals with graded
modules over the Steenrod algebra.
I think when I began, Sage didn't even have its own category of graded
things, so I ended up inventing my own category of "Yacop graded
objects". I later had a look at what Sage is offering and it seemed to
lack some important features that I need, so I have stuck with my own
inventions ever since.
One missing thing is inspection of the grading module: if your code gets
a tri-graded, potentially very big module it's desirable to provide a
way to locate the non-zero tridegree's efficiently. My solution was that
a graded module can advertise the "bounding box" for (a subset of) its
elements, so that the non-zero pieces can be located quickly.
More importantly, once you're dealing with graded modules you might want
to operate *on* the grading by taking a suspension or truncation of the
module. I implemented this by providing two new functors "supension" and
"truncation".
My solution is probably not very robust and well-tested, but you could
take a look at my code for inspiration. Here is a sample session, using
the docker image based on Sage 8.2:
~$ docker run -e DISPLAY=$DISPLAY -v /tmp/.X11-unix:/tmp/.X11-unix
--rm -it cnassau/yacop-sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 8.2, Release Date: 2018-05-05 │
│ Type "notebook()" for the browser-based notebook interface. │
│ Type "help()" for help. │
└────────────────────────────────────────────────────────────────────┘
┌────────────────────────────────────────────────────────────────────┐
│ Imported package Yacop (version 2.0) │
└────────────────────────────────────────────────────────────────────┘
sage: from yacop.modules.dickson import DicksonAlgebra
sage: D=DicksonAlgebra(2,3)
sage: D
Dickson algebra D(3) for prime 2
sage: for g in D.graded_basis(tmax=10):
....: print g, g.t, g.degree()
....:
d4 4 region(e = 0, s = 0, t = 4)
1 0 region(e = 0, s = 0, t = 0)
d4*d6 10 region(e = 0, s = 0, t = 10)
d6 6 region(e = 0, s = 0, t = 6)
d7 7 region(e = 0, s = 0, t = 7)
d4**2 8 region(e = 0, s = 0, t = 8)
Here's how you take suspensions and truncations:
sage: suspension(D,t=8)
suspension (8,0,0) of Dickson algebra D(3) for prime 2
sage: from yacop.modules.functors import truncation
sage: truncation(D,tmax=10)
truncation to region(-Infinity < t <= 10) of Dickson algebra D(3)
for prime 2
Finally, to come back to your original question, here is the degree of zero:
sage: D.zero().degree()
---------------------------------------------------------------------------
ValueError Traceback (most recent
call last)
...
ValueError: degree of zero is undefined
I think if you end up needed a different zero in each grading, you're
doing something wrong. After all, your code can always know what the
expected degree of an element (eg. some Sq(R)*whatever) is; you probably
should just not rely on the idea that that degree can be recovered from
the product.
Best,
Christian
On 18.07.20 11:31, Sverre Lunøe-Nielsen wrote:
Dear list,
I have been involved in preparing a package by M. Catanzaro and R.
Bruner lately, which implements finitely presented modules over the
mod `p` Steenrod algebra.
We have encountered a conflict regarding how to present graded
objects, and I am writing to the list to get other people's opinion on
how to proceed on this matter.
Briefly, the issue is that the Steenrod algebra allows inhomogeneous
elements and our graded modules do not. Thus, the Steenrod algebra
has a single zero element with no well defined degree, while our
modules could potentially have one zero element for each degree.
My wish is to allow degreewise zero elements in our graded modules, so
that x.degree() would return an integer for every element x. But
because the unique zero in the Steenrod algebra has no well defined
degree, I am forced to let degree() treat all zero elements in our
modules the same way and return ``None``.
A more precise description of the issue is found in the Sphinx note below.
My questions to the list are: Has similar issues been discussed and/or
resolved before? And more specificly: What acceptable changes could
be made to the Steenrod algebra package to achieve what I want?
Regards,
Sverre Lunøe-Nielsen
.. NOTE::
Our implementation treats a graded module as the disjoint union,
rather than a
direct sum, of vectorspaces of homogeneous elements. Elements are
therefore
always homogeneous, which also implies that sums between elements of
different
degrees are not allowed. This also means that acting by an inhomogeneous
element of the Steenrod algebra makes no sense.
In this setting there is no single zero element, but rather a zero for
every
degree. It follows that, in theory, all elements, including the zero
elements,
have a well defined degree.
This way of representing a graded object differs from the way the graded
Steenrod algebra is represented by :class:`sage.algebras.steenrod`
where inhomogeneous
elements are allowed and there is only a single zero element.
Consequently,
this zero element has no well defined degree.
Thus, because of the module action, we are forced to follow the same
convention
when it comes to the degree of zero elements in a module: The method
:meth:`sage.modules.finitely_presented_over_the_steenrod_algebra.module.fp_element.FP_Element.degree'
returns the value ``None`` for zero elements.
An example which highlights this problem is the following::
sage: F = FPA_Module([0], SteenrodAlgebra(p=2)) # The free
module on a single generator in degree 0.
sage: g = F.generator(0)
sage: x1 = Sq(1)*g
sage: x2 = Sq(1)*x1
Clearly, the code implementing the module action has all the
information it needs
to conclude that the element ``x2`` is the zero element in the second
degree.
However, because of the module action, we cannot distinguish it from
the element::
sage: x2_ = (Sq(1) * Sq(1))*g
The latter is equal to the action of the zero element of the Steenrod
algebra on `g`, but since the zero element has no degree in the
Steenrod algebra,
the module class cannot deduce what degree the zero element `x2_`
should belong
to.
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