Oh, that is very nice. Thank you for the solution to my issue! I would still classify the workaround as Very Clever, and not a solution that an average user would come up with on their own. Perhaps this could be stated more clearly in the documention. On Sunday, August 18, 2024 at 7:35:21 PM UTC+9 mmarco wrote:
> Yes, It allows to use bigradings and it works as you would expect. We have > used that approach in some computations we have done. You can see some > examples in the documentation here: > > https://doc.sagemath.org/html/en/reference/algebras/sage/algebras/commutative_dga.html#sage.algebras.commutative_dga.DifferentialGCAlgebra_multigraded > > > > El domingo, 18 de agosto de 2024 a las 7:27:20 UTC+2, Benjamin McMillan > escribió: > >> Ah, it would make it harder to compute cohomology. I haven't run into >> that problem because I am not working in a context where you would want to >> do that. >> (To set my context: I am working with differential forms, but without >> writing everything out in coordinates, as one might do on a Lie group with >> left invariant forms. The vector spaces are indeed infinite dimensional, >> and this is why one typically does not compute de Rham cohomology directly, >> rather using the de Rham theorem and a more computable version of homology.) >> >> Bigrading is a very interesting solution. I take it that the differential >> should have degree (1,0). >> Does the cdg_algebra package as implemented allow you to do bigradings >> like this to do calculations on cdga with "degree 0" terms? >> >> On Sunday, August 18, 2024 at 12:27:31 AM UTC+9 mmarco wrote: >> >>> The problem with that approach is that, if there are degree zero >>> generators, the homogeneous part of each degree becomes an infinite >>> dimensional vector space. And hence, we can't compute a matrix representing >>> the differential operator in a given degree (which is needed to compute >>> cohomology). >>> >>> One way to workaround this problem is to use a bigraded CDGA: for the >>> generators that usually would be of degree i>0, use the degree (i,0). For >>> the generators of degree 0, use the degree (0,2) (we want a 2 so they are >>> considered even). This way, we can still have finite dimensional spaces in >>> each bigrade (and hence we can compute the bigraded cohomology), but the >>> algebra structure would be the one you expect, and if you ignore the second >>> index in the grading, you get the grading you expect. >>> El sábado, 17 de agosto de 2024 a las 10:27:20 UTC+2, Benjamin McMillan >>> escribió: >>> >>>> I would like to propose a simple but large improvement to the >>>> commutative_dga package. >>>> In short, one currently cannot use the package to create graded >>>> commutative algebras that include non-closed degree 0 terms. >>>> (For example, this exclude the package from being used for the algebra >>>> of differential forms on a manifold, because any non-constant function is >>>> non-closed.) >>>> >>>> For my purposes, this can be fixed easily, changing 1 line of code. >>>> However, I am unclear if this breaks parts of the module that I don't use. >>>> >>>> As I understand it, the only reason that degree 0 doesn't work is in >>>> the constructor for a new CDGA. >>>> If you pass a degree 0 generator, then the call in the constructor to >>>> create a g_algebra (line 1010) uses a TermOrder weighted by degree. But >>>> the >>>> TermOrder package assumes a polynomial ring with generators of positive >>>> weight, and so throws an error when you pass weight = degree = 0 >>>> generators. >>>> This can be fixed by using weight = degree + 1, but I worry that this >>>> might break some non-obvious assumption elsewhere. (It will also maybe >>>> mean >>>> that your monomials are ordered slightly differently than expected.) >>>> >>>> I can submit a pull request, but I read in the developer guide that it >>>> is best to start a discussion here first. >>> >>> -- 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/67259cd1-5df0-4bc7-9c4a-6cd061e3a769n%40googlegroups.com.
