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/03f3aa20-f8b7-4da3-a01f-c2a7e8d0c752n%40googlegroups.com.
