Allow partial result in minimal_model, and translate OS-algebra to cdga #35782

miguelmarco · 2023-06-16T16:44:27Z

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The title is concise, informative, and self-explanatory.
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 accordingly.

tscrim

Why is the method to return the OS algebra as differential algebra and not just a CGA? It seems like that would be the more natural thing to have as the user could then easily enough add the differential.

src/sage/algebras/commutative_dga.py

src/sage/algebras/orlik_solomon.py

miguelmarco · 2023-06-18T14:23:13Z

Why is the method to return the OS algebra as differential algebra and not just a CGA? It seems like that would be the more natural thing to have as the user could then easily enough add the differential.

The OS algebra is the cohomology ring of the complement of the arrangement, so i guess it makes sense to consider it as a differential algebra.

But as you said, since it is so easy to add the differential structure, it is almost equivalent to return it just as gca.

Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

tscrim · 2023-06-19T14:15:44Z

Why is the method to return the OS algebra as differential algebra and not just a CGA? It seems like that would be the more natural thing to have as the user could then easily enough add the differential.

The OS algebra is the cohomology ring of the complement of the arrangement, so i guess it makes sense to consider it as a differential algebra.

Ah, I see. We have modded out by the image in the original chain complex, so the differentials all become 0.

But as you said, since it is so easy to add the differential structure, it is almost equivalent to return it just as gca.

Perhaps we can have both methods, but with the gdca simply calling the gca() and adding the differential structure?

miguelmarco · 2023-06-19T17:30:43Z

Perhaps we can have both methods, but with the gdca simply calling the gca() and adding the differential structure?

Yes, that makes sense.

tscrim

Thanks. Just a few other essentially trivial things.

src/sage/algebras/commutative_dga.py

src/sage/algebras/orlik_solomon.py

Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

miguelmarco · 2023-06-22T08:04:53Z

So, is it ready for positive review?

tscrim

Sorry, just a few more trivial things I noticed while looking at the compiled doc. One formatting thing, and then breaking up the long output lines (might even be a bit better to be slightly shorter per line too; I didn't check closely).

I've approved the PR, so once you push the changes, you can switch the status to positive review.

tscrim · 2023-06-22T12:18:50Z

src/sage/algebras/commutative_dga.py

+          on the `partial_result` flag.
+
+        - ``partial_result``  -- boolean (default: `False`); wether to return
+          the partial result if the `max_iterations` limit is reached.


Suggested change

on the `partial_result` flag.

- ``partial_result`` -- boolean (default: `False`); wether to return

the partial result if the `max_iterations` limit is reached.

on the ``partial_result`` flag.

- ``partial_result`` -- boolean (default: ``False``); wether to return

the partial result if the ``max_iterations`` limit is reached

tscrim · 2023-06-22T12:19:19Z

src/sage/algebras/commutative_dga.py

+            ValueError: could not cover all relations in max iterations in degree 2
+            sage: S.minimal_model(partial_result=True)
+            Commutative Differential Graded Algebra morphism:
+              From: Commutative Differential Graded Algebra with generators ('x1_0', 'x1_1', 'x1_2', 'y1_0', 'y1_1', 'y1_2') in degrees (1, 1, 1, 1, 1, 1) over Rational Field with differential:


Suggested change

From: Commutative Differential Graded Algebra with generators ('x1_0', 'x1_1', 'x1_2', 'y1_0', 'y1_1', 'y1_2') in degrees (1, 1, 1, 1, 1, 1) over Rational Field with differential:

From: Commutative Differential Graded Algebra with generators

('x1_0', 'x1_1', 'x1_2', 'y1_0', 'y1_1', 'y1_2') in degrees (1, 1, 1, 1, 1, 1)

over Rational Field with differential:

tscrim · 2023-06-22T12:19:32Z

src/sage/algebras/commutative_dga.py

+               y1_0 --> x1_0*x1_1 - x1_0*x1_2 + x1_1*x1_2
+               y1_1 --> x1_0*y1_0 - x1_2*y1_0
+               y1_2 --> x1_1*y1_0 - x1_2*y1_0
+              To:   Commutative Differential Graded Algebra with generators ('a', 'b', 'c') in degrees (1, 1, 1) with relations [a*b - a*c + b*c] over Rational Field with differential:


Suggested change

To: Commutative Differential Graded Algebra with generators ('a', 'b', 'c') in degrees (1, 1, 1) with relations [a*b - a*c + b*c] over Rational Field with differential:

To: Commutative Differential Graded Algebra with generators ('a', 'b', 'c')

in degrees (1, 1, 1) with relations [a*b - a*c + b*c] over Rational Field with differential:

tscrim · 2023-06-22T12:20:41Z

src/sage/algebras/orlik_solomon.py

+            Graded Commutative Algebra with generators ('e0', 'e1', 'e2', 'e3', 'e4', 'e5', 'e6') in degrees (1, 1, 1, 1, 1, 1, 1) with relations [e1*e2 - e1*e3 + e2*e3, e0*e2 - e0*e4 + e2*e4, e0*e1 - e0*e5 + e1*e5, e3*e4 - e3*e5 + e4*e5, e0*e3 - e0*e6 + e3*e6, e1*e4 - e1*e6 + e4*e6, e2*e5 - e2*e6 + e5*e6] over Rational Field
+
+        """


Suggested change

Graded Commutative Algebra with generators ('e0', 'e1', 'e2', 'e3', 'e4', 'e5', 'e6') in degrees (1, 1, 1, 1, 1, 1, 1) with relations [e1*e2 - e1*e3 + e2*e3, e0*e2 - e0*e4 + e2*e4, e0*e1 - e0*e5 + e1*e5, e3*e4 - e3*e5 + e4*e5, e0*e3 - e0*e6 + e3*e6, e1*e4 - e1*e6 + e4*e6, e2*e5 - e2*e6 + e5*e6] over Rational Field

"""

Graded Commutative Algebra with generators ('e0', 'e1', 'e2', 'e3', 'e4', 'e5', 'e6')

in degrees (1, 1, 1, 1, 1, 1, 1) with relations

[e1*e2 - e1*e3 + e2*e3, e0*e2 - e0*e4 + e2*e4, e0*e1 - e0*e5 + e1*e5,

e3*e4 - e3*e5 + e4*e5, e0*e3 - e0*e6 + e3*e6, e1*e4 - e1*e6 + e4*e6,

e2*e5 - e2*e6 + e5*e6] over Rational Field

"""

tscrim · 2023-06-22T12:21:14Z

src/sage/algebras/orlik_solomon.py

+            sage: H = hyperplane_arrangements.braid(3)
+            sage: O = H.orlik_solomon_algebra(QQ)
+            sage: O.as_cdga()
+            Commutative Differential Graded Algebra with generators ('e0', 'e1', 'e2') in degrees (1, 1, 1) with relations [e0*e1 - e0*e2 + e1*e2] over Rational Field with differential:


Suggested change

Commutative Differential Graded Algebra with generators ('e0', 'e1', 'e2') in degrees (1, 1, 1) with relations [e0*e1 - e0*e2 + e1*e2] over Rational Field with differential:

Commutative Differential Graded Algebra with generators ('e0', 'e1', 'e2') in degrees

(1, 1, 1) with relations [e0*e1 - e0*e2 + e1*e2] over Rational Field with differential:

miguelmarco · 2023-06-22T16:28:01Z

Sorry, I just realized that the computations where only valid for the case of line arrangements (that is, we only cared about rank 2 relations). I changed the method to add missing relations.

I think it is correct now, but another pair of eyes would help.

tscrim · 2023-06-23T08:54:18Z

Good catch. Is there some way we can easily/quickly test that the result matches, say, (partially) computing the (graded) degrees?

miguelmarco · 2023-06-23T11:25:02Z

We can check if the Betti numbers match the coefficients of the poincare polynomial. For instance:

sage: H = hyperplane_arrangements.Catalan(4,QQ)
sage: O = H.cone().orlik_solomon_algebra(QQ)
sage: B = O.as_cdga()
sage: H.cone().poincare_polynomial()
210*x^4 + 317*x^3 + 125*x^2 + 19*x + 1
sage: [len(B.basis(i)) for i in range(5)]
[1, 19, 125, 317, 210]

sage: H = hyperplane_arrangements.Ish(5,QQ)
sage: O = H.cone().orlik_solomon_algebra(QQ)
sage: B = O.as_gca()
sage: H.cone().poincare_polynomial()
625*x^5 + 1125*x^4 + 650*x^3 + 170*x^2 + 21*x + 1
sage: [len(B.basis(i)) for i in range(6)]
[1, 21, 170, 650, 1125, 625]

tscrim · 2023-06-23T11:42:26Z

If that’s a quick (in the doctest time sense), then can we add that as an additional test? I promise, absolutely last thing. ^^;;

…mial

miguelmarco · 2023-06-26T09:56:02Z

If that’s a quick (in the doctest time sense), then can we add that as an additional test? I promise, absolutely last thing. ^^;;

Done

tscrim · 2023-06-26T10:43:53Z

Thank you.

github-actions · 2023-06-26T11:07:22Z

Documentation preview for this PR (built with commit 9a1de5e; changes) is ready! 🎉

Allow partial result in minimal_model, and translate OS-algebra to cdga

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miguelmarco self-assigned this Jun 16, 2023

miguelmarco mentioned this pull request Jun 16, 2023

Allow minimal model of cdga to return partial results #35781

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Fixed doctest

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miguelmarco and others added 3 commits June 18, 2023 16:23

Update src/sage/algebras/commutative_dga.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/algebras/orlik_solomon.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

use frozenset to look for coefficients

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Add as_gca method

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miguelmarco and others added 5 commits June 19, 2023 23:49

Update src/sage/algebras/commutative_dga.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/algebras/orlik_solomon.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/algebras/orlik_solomon.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Fix missing blankline

620a143

Update src/sage/algebras/orlik_solomon.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Allow case where generators are not indexed by 0..ny

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tscrim approved these changes Jun 22, 2023

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Add relations for all broken circuits

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miguelmarco added 2 commits June 23, 2023 10:28

Fix indices to compute monomials

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Fix doctest

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Fix sign convention

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Add test to check betti numbers match coefficients of poincare polyno…

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…mial

tscrim added s: positive review and removed s: needs review labels Jun 26, 2023

vbraun merged commit ffac6e0 into sagemath:develop Jul 1, 2023

	To: Commutative Differential Graded Algebra with generators ('a', 'b', 'c') in degrees (1, 1, 1) with relations [ab - ac + b*c] over Rational Field with differential:
	To: Commutative Differential Graded Algebra with generators ('a', 'b', 'c')
	in degrees (1, 1, 1) with relations [ab - ac + b*c] over Rational Field with differential:

		Graded Commutative Algebra with generators ('e0', 'e1', 'e2', 'e3', 'e4', 'e5', 'e6') in degrees (1, 1, 1, 1, 1, 1, 1) with relations [e1e2 - e1e3 + e2e3, e0e2 - e0e4 + e2e4, e0e1 - e0e5 + e1e5, e3e4 - e3e5 + e4e5, e0e3 - e0e6 + e3e6, e1e4 - e1e6 + e4e6, e2e5 - e2e6 + e5*e6] over Rational Field

		"""

	Commutative Differential Graded Algebra with generators ('e0', 'e1', 'e2') in degrees (1, 1, 1) with relations [e0e1 - e0e2 + e1*e2] over Rational Field with differential:
	Commutative Differential Graded Algebra with generators ('e0', 'e1', 'e2') in degrees
	(1, 1, 1) with relations [e0e1 - e0e2 + e1*e2] over Rational Field with differential:

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Allow partial result in minimal_model, and translate OS-algebra to cdga #35782

Allow partial result in minimal_model, and translate OS-algebra to cdga #35782

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miguelmarco commented Jun 16, 2023

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