Add Commutative graded differential algebras.

[miguelmarco]

This patch adds basic CDGA's.

They work as follows, first you create the algebra:

sage: A.<x,y,t> = CDGAlgebra(QQ, degrees = (1, 1, 2))
sage: A
Graded Commutative Algebra over Rational Field with generators ('x', 'y', 't') in degrees (1, 1, 2)

Then define the differential on A to construct a differential graded algebra:

sage: sage: B = A.CDGAlgebra({x: x*y , y: x*y })
sage: B
Commutative Differential Graded Algebra over Rational Field with generators ('x', 'y', 't') in degrees (1, 1, 2), differential 
    x --> x*y
    y --> x*y
    t --> 0

Now you can compute things like a basis of each homogeneous part:

sage: B.basis(5)
[y*t^2, x*t^2]

Or the cohomology in each degree:

sage: B.cohomology(4)
Free module generated by {[t^2]} over Rational Field
sage: B.cohomology(9)
Free module generated by {[-x*t^4 + y*t^4]} over Rational Field

Depends on #17224

CC: @vbraun @simon-king-jena @saraedum @tscrim

Component: algebra

Keywords: sd58, sd59, algebras, noncommutative, graded

Author: Miguel Marco, John Palmieri

Branch: 3900c66

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/16508

[tscrim]

comment:42

Okay here's my first big pass of review. I've reworked a bunch of things:

• Differentials are linear, so they are in the category of modules morphisms.
• Differentials now are endomorphisms of a CGAlgebra.
• Implemented coercions between GCAlgebra and CDGAlgebra.
• Added TestSuite checks (which led to #17224).
• I put the over {base_ring} last for CGAlgebra to better follow the rest of Sage.
• Cleaned up the doc.

There will be more changes and improvements coming, but I figured I should post what I have currently. In particular, I'm going to tweaks things around so that the differential specified for CDGAlgebra acts naturally on its elements rather than using the coercions. I'm also going to implement a method to create a differential morphism, in case someone has a use for that independently.

Actually, that makes me ask this question. Should we put all of the chain complex implementations in the differentials? If so, should we have (decorator) class of CDGAlgebra call those functions (which would mainly be a shorthand) or remove that class all together?

---

New commits:

cd59dbc	Merge branch 'u/jhpalmieri/DGA_new' of trac.sagemath.org:sage into public/algebras/cdga-16508
619624e	Merge branch 'u/jhpalmieri/DGA_new' of trac.sagemath.org:sage into public/algebras/cdga-16508
1b95a7e	First pass of reviewer changes: Various minor to moderate fixes.
7dd57c5	Some more minor tweaks.
13a552e	Fix for pickling NC rings with weighted term order.
9aaf591	Merge branch 'public/pickling/term_order_NC_plural-17224' into public/algebras/cdga-16508
aae8f5b	Merge branch 'u/jhpalmieri/DGA_new' of trac.sagemath.org:sage into public/algebras/cdga-16508
7b6f118	Fixed doctests.

[tscrim]

Changed commit from a4d5b24 to 7b6f118

[tscrim]

Dependencies: #17224

[tscrim]

Changed branch from u/jhpalmieri/DGA_new to public/algebras/cdga-16508

[tscrim]

Changed author from mmarco to Miguel Marco, John Palmieri

[jhpalmieri]

comment:43

Replying to @tscrim:

Actually, that makes me ask this question. Should we put all of the chain complex implementations in the differentials? If so, should we have (decorator) class of CDGAlgebra call those functions (which would mainly be a shorthand) or remove that class all together?

What syntax are you imagining? A.differential().cohomology()? Something like this would not be discoverable by tab completion. I think it makes a lot of sense to have a class for commutative DG algebras, and for instances of those, you should be able to call B.cohomology(), etc.

[tscrim]

comment:44

Replying to @jhpalmieri:

What syntax are you imagining? A.differential().cohomology()? Something like this would not be discoverable by tab completion. I think it makes a lot of sense to have a class for commutative DG algebras, and for instances of those, you should be able to call B.cohomology(), etc.

That's no more tab completable than A.CDGAlgebra().cohomology(). The semantic would be d = A.differential({some: data}). However I'm not saying that removing the CDGAlgebra class is the best or right thing to do based on my current data/ideas. Yet I'm thinking of putting the actually implementations in the differential class and having CDGAlgebra's corresponding methods call the corresponding differential's method in case we want to remove the class. IMO those methods fits naturally with the differential as well.

[jhpalmieri]

comment:45

From my point of view, doing B=A.CDGAlgebra(...) and then manipulating B feels like the natural thing: I view the differential graded algebra as the central object of study here. Doing instead d=A.differential(...) and then manipulating d would not be the obvious thing to do at all.

So as long as in this situation, B.cohomology(...) (etc.) works, I don't think I care about how it's implemented.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

fb16080	Merge branch 'develop' into public/algebras/cdga-16508
b8328e6	Second pass of reworking things around by adding functionality.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 7b6f118 to b8328e6

[tscrim]

comment:47

Replying to @jhpalmieri:

From my point of view, doing B=A.CDGAlgebra(...) and then manipulating B feels like the natural thing: I view the differential graded algebra as the central object of study here. Doing instead d=A.differential(...) and then manipulating d would not be the obvious thing to do at all.

So as long as in this situation, B.cohomology(...) (etc.) works, I don't think I care about how it's implemented.

Now we can do both. I've also made the other changes I stated. There's one last thing, there is the method CDGAlgebra and the class CDGAlgebra, and if things are reordered in the module, you get failures from a name conflict. So the question is do we want to rename the method and fully spell that out to commutative_differential_graded_algebra and/or the shorter cdg_algebra, or do we leave it as it is? I'm strongly in favor of renaming the method to something else.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from b8328e6 to d732dbe

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

d732dbe

Commutative DGAs: rename some classes

[jhpalmieri]

comment:49

I've made a few changes: maybe most crucially but also most trivially, there was a missing EXAMPLES:: that was causing docbuilding to fail.

Then I renamed some classes. The one I feel most strongly about is DifferentialCGA: to me, this sounds like a "differential commutative graded algebra", not a "differential defined on a commutative graded algebra". So I just renamed it to Differential. I changed CDGAlgebra to DifferentialGCAlgebra. (I toyed with GCAlgebraWithDifferential, but didn't like it as much. I don't care that much, since these class names won't be that visible to most users.)

I also added a bit to the docstrings for the two CDGAlgebra methods.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from d732dbe to 497d9d6

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

497d9d6

Merge branch 'public/algebras/cdga-16508' of trac.sagemath.org:sage into public/algebras/cdga-16508

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 497d9d6 to 3900c66

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

3900c66

Renamed CDGAlgebra -> cdg_algebra.

[tscrim]

comment:52

Replying to @jhpalmieri:

I've made a few changes: maybe most crucially but also most trivially, there was a missing EXAMPLES:: that was causing docbuilding to fail.

Whoops.

Then I renamed some classes. The one I feel most strongly about is DifferentialCGA: to me, this sounds like a "differential commutative graded algebra", not a "differential defined on a commutative graded algebra". So I just renamed it to Differential. I changed CDGAlgebra to DifferentialGCAlgebra. (I toyed with GCAlgebraWithDifferential, but didn't like it as much. I don't care that much, since these class names won't be that visible to most users.)

I'm happy with this name change (it gets the association from the module name). However I've rename the method CDGAlgebra to cdg_algbra match conventions.

I also added a bit to the docstrings for the two CDGAlgebra methods.

Looks good. So if you're happy with my changes, then I think we can set a positive review.

[jhpalmieri]

comment:53

Travis, should you be listed as an author in the file commutative_dga.py?

[tscrim]

comment:54

I don't consider myself as an author since I mostly just reorganized code and did touch-ups.

[vbraun]

Changed branch from public/algebras/cdga-16508 to 3900c66

[jhpalmieri]

comment:57

See #21845 as a followup: cohomology algebra generators.

[jhpalmieri]

Changed commit from 3900c66 to none

[fchapoton]

Changed keywords from sd58, sd59, algebras, nonconmutative, graded to sd58, sd59, algebras, noncommutative, graded