On Saturday, July 18, 2020 at 2:31:01 AM UTC-7, 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. >
In my experience, algebraic topologists often think of graded objects as disjoint unions, and you can often get away with this, but really they're not — they're direct sums. I think you should use Sage's categories framework, graded modules with basis or whatever, to set these up. In any case where the degree matters, you should first test whether an element is zero (in which case it won't have a degree) and then perhaps whether it is homogeneous. If not, you can raise an error (to keep someone from multiplying a module element by Sq(1) + Sq(2), for example). If it is homogeneous, you can proceed the way you want. -- John -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/d007dcb0-e85c-433f-8ad2-2935abbdceabo%40googlegroups.com.
