Finitely presented graded modules over graded connected algebras

[jhpalmieri]

This is a precursor to #30680, laying out the framework for finitely presented modules over graded connected algebras. #30680 will focus on the case of the Steenrod algebra, with specific applications in mind.

CC: @sverre320 @sagetrac-kvanwoerden @jhpalmieri @tscrim @rrbruner @cnassau

Component: algebra

Author: Bob Bruner, Michael Catanzaro, Sverre Lunøe-Nielsen, Koen van Woerden, John Palmieri, Travis Scrimshaw

Branch/Commit: a1a9467

Reviewer: John Palmieri, Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/32505

[jhpalmieri]

comment:71

This now allows for the possibility of the terms in the resolution being instances of FreeGradedModuleMorphism (except for the initial map, which has codomain the original module). I've also added a (slow) method which will compute the top dimension of a finite-dimensional algebra, for use in determining when to stop computing the resolution. I've also added a computation of a free resolution of a trivial module over an exterior algebra.

[jhpalmieri]

comment:72

So maybe the choice for resolution was because the initial map has codomain M which is likely not free. With the new as_free=True option, the later maps are all converted to type FreeGradedModuleMorphism, and you could just omit the 0th term to get an honest (graded) free resolution out of it.

Maybe now we're done?

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

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

6c1ab8d

trac 32505: more fixes

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

Changed commit from 11b3725 to 6c1ab8d

[jhpalmieri]

comment:74

Fixed a few more things: the free_graded_module method for algebras with basis wasn't handling the names argument correctly, and a few doctests were failing. I changed some of the documentation, too.

[tscrim]

comment:75

I was just running into that problem in comment:74 and about to fix it. Thanks. I will be pushing a bunch of changes shortly too.

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

Changed commit from 6c1ab8d to 5074464

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

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

72e455e	We do not require the base algebra's base ring to be a field.
fcce24d	Merge branch 'public/modules/free_graded_modules-32505' of git://trac.sagemath.org/sage into public/modules/free_graded_modules-32505
2e0518f	Using free modules where possible and improved compatibility.
6cf6d14	Merge branch 'public/modules/free_graded_modules-32505' of git://trac.sagemath.org/sage into public/modules/free_graded_modules-32505
5074464	Some last fixes and removing redundancy.

[tscrim]

comment:77

I took this a bit further and made everything in the resolution a FreeGradedModule. As I suspected, there were some compatibility issues to sort out, but I took care of all of the ones that came up in doctests (and a few others I saw along the way). There might be a few others (which will show up as missing methods), but we can fix them as we come across them.

Towards this, I made the free module morphism/homspace a subclass of the respective non-free version. This required some slight workaround, but the code is much cleaner IMO because there are no duplicated methods and the subclassing makes mathematical sense.

I tweaked the documentation to say "free module" basically everywhere we use "vector space" because we are not using the field properties AFAICS.

As of right now, I am happy with the code. Please check my changes. If they are good, then go ahead and set a positive review.

[jhpalmieri]

comment:78

I saw that you replaced PlusInfinity() with infinity, so I made the same change a few other places. I also removed a "todo" about dealing with finite dimensionality rather than just finiteness, handling it the same way we did elsewhere in the code. If someone gets motivated (on another ticket), they can try to handle the case where self.base_ring().dimension() is not defined.

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

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

ffe7179

trac 32505: replace "PlusInfinity()" with "infinity"

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

Changed commit from 5074464 to ffe7179

[jhpalmieri]

Changed author from Bob Bruner, Michael Catanzaro, Sverre Lunøe-Nielsen, Koen van Woerden, John Palmieri to Bob Bruner, Michael Catanzaro, Sverre Lunøe-Nielsen, Koen van Woerden, John Palmieri, Travis Scrimshaw

[tscrim]

comment:81

Replying to @jhpalmieri:

I saw that you replaced PlusInfinity() with infinity, so I made the same change a few other places.

Indeed, this is a micro-optimization since they are the same object, but infinity is already created (and nailed in memory).

I also removed a "todo" about dealing with finite dimensionality rather than just finiteness, handling it the same way we did elsewhere in the code.

Thank you. I had made that note for myself, but I forgot about it.

If someone gets motivated (on another ticket), they can try to handle the case where self.base_ring().dimension() is not defined.

I concur.

Thank you.

[sverre320]

comment:82

Good question. Comments from the original authors? Why not just use free modules in resolution?

In the original version, the free module class was not intended for public use.

[vbraun]

comment:83

[sagemath_doc_html-none] [modules  ] The inventory files are in local/share/doc/sage/inventory/en/reference/modules.
[sagemath_doc_html-none] Error building the documentation.
[sagemath_doc_html-none] Traceback (most recent call last):
[sagemath_doc_html-none]   File "/home/release/Sage/local/var/lib/sage/venv-python3.9.9/lib/python3.9/runpy.py", line 197, in _run_module_as_main
[sagemath_doc_html-none]     return _run_code(code, main_globals, None,
[sagemath_doc_html-none]   File "/home/release/Sage/local/var/lib/sage/venv-python3.9.9/lib/python3.9/runpy.py", line 87, in _run_code
[sagemath_doc_html-none]     exec(code, run_globals)
[sagemath_doc_html-none]   File "/home/release/Sage/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage_docbuild/__main__.py", line 2, in <module>
[sagemath_doc_html-none]     main()
[sagemath_doc_html-none]   File "/home/release/Sage/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage_docbuild/__init__.py", line 1731, in main
[sagemath_doc_html-none]     builder()
[sagemath_doc_html-none]   File "/home/release/Sage/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage_docbuild/__init__.py", line 776, in _wrapper
[sagemath_doc_html-none]     getattr(DocBuilder, build_type)(self, *args, **kwds)
[sagemath_doc_html-none]   File "/home/release/Sage/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage_docbuild/__init__.py", line 136, in f
[sagemath_doc_html-none]     runsphinx()
[sagemath_doc_html-none]   File "/home/release/Sage/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage_docbuild/sphinxbuild.py", line 323, in runsphinx
[sagemath_doc_html-none]     sys.stderr.raise_errors()
[sagemath_doc_html-none]   File "/home/release/Sage/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage_docbuild/sphinxbuild.py", line 258, in raise_errors
[sagemath_doc_html-none]     raise OSError(self._error)
[sagemath_doc_html-none] OSError: /home/release/Sage/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/modules/fp_graded/free_homspace.py:docstring of sage.modules.fp_graded.free_homspace.FreeGradedModuleHomspace.identity:4: WARNING: Literal block expected; none found.
[sagemath_doc_html-none] 
[sagemath_doc_html-none]     Note: incremental documentation builds sometimes cause spurious
[sagemath_doc_html-none]     error messages. To be certain that these are real errors, run
[sagemath_doc_html-none]     "make doc-clean doc-uninstall" first and try again.

[jhpalmieri]

comment:84

Travis, was your intention to delete the identity method from free_homspace? Perhaps also remove zero?

[tscrim]

comment:85

Replying to @jhpalmieri:

Travis, was your intention to delete the identity method from free_homspace? Perhaps also remove zero?

Yes it was. They are now inherited as they had (effectively) the same code as the not-necessarily-free homspace.

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

Changed commit from ffe7179 to a1a9467

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

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

a1a9467

trac 32505: remove redundant "zero" and "identity" methods

[tscrim]

comment:88

Thank you.

[jhpalmieri]

comment:89

Followup in #33275.

[vbraun]

Changed branch from public/modules/free_graded_modules-32505 to a1a9467