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Implementation of Ore modules #38703

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@xcaruso xcaruso commented Sep 23, 2024

This PR implements modules over Ore polynomial rings.

More precisely, if $A[X;\theta,\partial]$ is a Ore polynomial ring, we propose an implementation of finite free modules $M$ over $A$ equipped with a map $f : M \to M$ such that $f(ax) = \theta(a) f(x) + \partial(a) x$ for all $a \in R$ and $x \in M$.
Such a map is called pseudolinear and it endows M with a structure of module over $A[X;\theta,\partial]$ (the map $f$ corresponding to the multiplication by $X$).

This PR includes:

  • an implementation of the category of Ore modules
  • an implementation of Ore modules, their submodules and their quotients (with an option to give chosen names to elements in a distinguished basis)
  • a constructor to create quotients of the form $A[X;\theta,\partial] / A[X;\theta,\partial]P$
  • an implementation of morphisms between Ore modules, including methods for computing kernels, cokernels, images and coimages

This is the second step (after PR #38650) towards the implemetation of Anderson motives.

📝 Checklist

  • The title is concise and informative.
  • The description explains in detail what this PR is about.
  • I have linked a relevant issue or discussion.
  • I have created tests covering the changes.
  • I have updated the documentation and checked the documentation preview.

⌛ Dependencies

#38650: pseudomorphisms

@xcaruso xcaruso requested a review from kryzar September 23, 2024 16:36
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kryzar commented Sep 24, 2024

Quick question: why is it necessary that $M$ is free and finite (or just free with finite rank?)?

Also: love that you're creating these intermediate facilities that can be relevant in their own right, instead of just putting all of this in the implementation of Anderson motives.

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xcaruso commented Sep 24, 2024

Quick question: why is it necessary that M is free and finite (or just free with finite rank?)?

Well, it is not for the definition. But for the implementation, it is really useful as we want to represent the map $f$ by its matrix.

@fchapoton fchapoton self-assigned this Feb 12, 2025
@xcaruso xcaruso added the sd128 tickets of Sage Days 128 Le Teich label Feb 14, 2025
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Here are some minor syntax mistakes in the doc.

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Fix 11 E302 "expected 2 blank lines, found " errors of pycodestyle-minimal

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Fix another E302 "expected 2 blank lines, found " error of pycodestyle-minimal (sorry, blame the TGV's WiFi)

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