#17096: Implement categories for filtered algebras
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Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.4
Component: categories | Resolution:
Keywords: filtered algebras | Merged in:
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/categories/filtered_algebras-17096|
bb234ec71d1276c6cc14320074b3de6fdc606192
Dependencies: | Stopgaps:
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Comment (by tscrim):
Replying to [comment:4 darij]:
> 1) What does the `_element_constructor_` in `associated_graded.py` do? I
understand that it constructs an element of gr A from an element a of A,
but the way it does this I do not think is correct.
>
> The "right" thing to do is this: For a given pair `(a, n)` where `n` is
a nonnegative integer and `a` is an element of the `n`-th filtered part of
`A`, the residue class of `a` modulo the `(n-1)`-th filtered part of `A`
is an element of the `n`-th graded component of `\gr A`. The `n` is part
of the input; you can try to reconstruct it as the smallest `i` such that
`a` lies in the `i`-th filtered part of `A`, but such a definition will be
ill-behaved.
Hmmm....maybe this shouldn't be a coercion then since if `ab = c1 + c2`
(where `deg(c1) > deg(c2)`), then `G(ab) != G(c1) + G(c2)`. The
implementation of `_element_constructor_` is definitely the right
conversion however (the natural vector space isomorphism). So I'll remove
the coercion part but leave the `_element_constructor_`.
> 2) I think you need some requirements on the basis of a
FilteredModulesWithBasis for your code to work. I would guess you want the
basis to be a sequence `(B_0, B_1, B_2, ...)` of sets such that for every
`n`, the union `B_0 \cup B_1 \cup ...\cup B_n` is a basis of the `n`-th
filtered component. Is it what you want?
No, the basis does not have to be ordered with respect to the degree. I
also don't see where in the code this is used (I just moved it over from
the graded modules, so I may not have looked hard enough).
--
Ticket URL: <http://trac.sagemath.org/ticket/17096#comment:5>
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