Implement a hook to access free (graded) resolutions #34379

tscrim · 2022-08-17T00:58:02Z

In #33950, free (graded) resolutions for modules over polynomial rings were added. However, one needs to import a top-level function to do the construction. The goal of this ticket is to add a method, such as free_resolution(), to these (sub)modules (and ideals) to ease access.

Depends on #33950

CC: @kwankyu

Component: user interface

Author: Travis Scrimshaw

Branch/Commit: 6dba5e5

Reviewer: Kwankyu Lee

Issue created by migration from https://trac.sagemath.org/ticket/34379

The text was updated successfully, but these errors were encountered:

tscrim · 2022-08-17T01:16:22Z

comment:1

@kwankyu How broadly is Singular's functionality for free (graded) resolutions support to work? I can't seem to find anything definitive in the online documentation. Will it work for any commutative ring? How about any integral domain, such as

sage: R.<x,y> = QQ[]
sage: I = R.ideal([x^2 - y^2 - 1])
sage: Q = R.quo(I)
sage: Q.is_integral_domain()
True

The submodule code currently does not quite do this. (I am doing #34380 to be able to make submodules over more arbitrary integral domains.)

kwankyu · 2022-08-17T01:45:15Z

comment:2

Replying to @tscrim:

How broadly is Singular's functionality for free (graded) resolutions supposed to work?

Only quotients of free modules over multivariate polynomial rings.

I can't seem to find anything definitive in the online documentation.

This is pretty definitive:

https://www.singular.uni-kl.de/Manual/4-3-1/sing_988.htm#SEC1047

Will it work for any commutative ring?

Definitely not. Singular uses Groebner bases, which do not apply to arbitrary commutative rings. Or do I misunderstand your question?

tscrim · 2022-08-17T01:59:34Z

comment:3

Replying to @kwankyu:

Replying to @tscrim:

How broadly is Singular's functionality for free (graded) resolutions supposed to work?

Only quotients of free modules over multivariate polynomial rings.

I can't seem to find anything definitive in the online documentation.

This is pretty definitive:

https://www.singular.uni-kl.de/Manual/4-3-1/sing_988.htm#SEC1047

I don't find it so clear. What R is there? Is it a quotient of a polynomial ring like for the syzygies? The invocation of the Hilbert Syzygy Theorem does suggest that it is only for polynomial rings, but it might also be saying that it might not work for some quotients (but it might try).

Will it work for any commutative ring?

Definitely not. Singular uses Groebner bases, which do not apply to arbitrary commutative rings. Or do I misunderstand your question?

I am not an expert in (computational) commutative algebras; thank you for the explanation.

So I will likely have to raise NotImplementedError in a number of cases, but at least it will be more publicly visible.

kwankyu · 2022-08-17T02:49:58Z

comment:4

Replying to @tscrim:

I don't find it so clear. What R is there? Is it a quotient of a polynomial ring like for the syzygies? The invocation of the Hilbert Syzygy Theorem does suggest that it is only for polynomial rings, but it might also be saying that it might not work for some quotients (but it might try).

I agree that it is vague. It says that R is a quotient of K[x]_> where K[x] is a multivariate polynomial ring.

In Singular, a monomial order > can be global, local, or mixed. Depending on this, K[x]_> may define just K[x], a localization of K[x], or something more complicated.

So according to the above definition, R can be something far from the plain K[x].

I never worked in Singular with a monomial order other than global, the usual monomial orders. As far as I know, Sage also uses Singular's functionality limited to global monomial orders.

For (graded) free resolutions imported into Sage in a recent ticket, R is definitely the plain K[x].

I just looked over Singular manual, and I found no example of computing a free resolution for a module where R is other than the plain K[x], though I might be just unlucky.

Will it work for any commutative ring?

free_resolution() could be attached to at least

quotients of polynomial rings
ideals of polynomial rings
quotients of free modules over polynomial rings
submodules of free modules over polynomial rings

tscrim · 2022-08-17T03:23:55Z

comment:5

It seems that #33950 doesn't handle the univariate case either:

sage: R.<x> = QQ[]
sage: I = R.ideal([x^4 + 3*x^2 + 2])
sage: FreeResolution(I)
<repr(<sage.homology.free_resolution.FreeResolution at 0x7fb357e0f9d0>) failed: TypeError: unknown argument type '<class 'sage.rings.polynomial.ideal.Ideal_1poly_field'>'>

sage: M = R^3
sage: v = M([x^4 + 3*x^2 + 2, x^3 + x - 2, x + 1])
sage: w = M([x^3 + 2*x + 4, x^2 + 3*x + 2, 2*x + 3])
sage: S = M.submodule([v, w])
sage: S
Free module of degree 3 and rank 2 over Univariate Polynomial Ring in x over Rational Field
Echelon basis matrix:
[                                                 1 -3/17*x^4 - 11/68*x^3 - 9/34*x^2 + 31/68*x + 11/34  -2/17*x^4 - 5/34*x^3 - 7/34*x^2 + 11/68*x + 49/68]
[                                                 0           3*x^5 + 2*x^4 + 7*x^3 + 6*x^2 + 6*x + 12            2*x^5 + 2*x^4 + 5*x^3 + 7*x^2 - 2*x + 2]
sage: FreeResolution(S)
<repr(<sage.homology.free_resolution.FreeResolution at 0x7fb357ef4700>) failed: TypeError: unknown argument type '<class 'sage.matrix.matrix_polynomial_dense.Matrix_polynomial_dense'>'>

Is this expected?

From the Singular documentation, I am guessing it also requires the polynomial ring to be over a field (mainly Z is not allowed either)?

tscrim · 2022-08-17T03:23:55Z

Dependencies: #32324

tscrim · 2022-08-17T03:28:24Z

comment:6

Of course, I can trick Sage into making the univariate polynomial ring believe it is a multivariate one implemented in Singular. Although I would hope there would be an easier mechanism for doing this.

tscrim · 2022-08-17T03:28:58Z

comment:7

For example

sage: R.<x> = PolynomialRing(QQ, implementation="singular")
sage: type(R)
<class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>
sage: I = R.ideal([x^4 + 3*x^2 + 2])
sage: FreeResolution(I)
S^1 <-- S^1 <-- 0
sage: M = R^3
sage: v = M([x^4 + 3*x^2 + 2, x^3 + x - 2, x + 1])
sage: w = M([x^3 + 2*x + 4, x^2 + 3*x + 2, 2*x + 3])
sage: S = M.submodule([v, w])
sage: FreeResolution(S)
S^3 <-- S^2 <-- 0

tscrim · 2022-08-17T03:46:03Z

comment:8

Another way of asking the above is there an easy way to make the univariate case work without converting everything to the Singular implementation?

tscrim · 2022-08-17T03:53:04Z

comment:9

Univariate polynomial rings over fields are PIDs and submodules of free modules over PIDs are free, right? So perhaps we should just handle this case within Sage?

tscrim · 2022-08-17T04:05:44Z

comment:10

A very minor typographical bug:

sage: GradedFreeResolution(I, degrees=[2,1], shifts=[-3])
S(--3) <-- S(--2)⊕S(-1) <-- S(-2) <-- 0

kwankyu · 2022-08-17T04:16:56Z

comment:11

Replying to @tscrim:

You meant #33950.

Univariate polynomial rings over fields are PIDs and submodules of free modules over PIDs are free, right?

Right.

So perhaps we should just handle this case within Sage?

Perhaps Singular requires K of K[x] to be a field. With R univariate polynomial ring, there is nothing interesting with free resolutions, as all submodules are free as you said.

tscrim · 2022-08-17T04:22:35Z

comment:12

Replying to @kwankyu:

Replying to @tscrim:

You meant #33950.

Yes; I fixed this above.

So perhaps we should just handle this case within Sage?

Perhaps Singular requires K of K[x] to be a field. With R univariate polynomial ring, there is nothing interesting with free resolutions, as all submodules are free as you said.

It actually doesn't seem to be too hard to have a check for this and tweak the code to handle these cases.

kwankyu · 2022-08-17T06:48:20Z

comment:13

Replying to @tscrim:

A very minor typographical bug:

sage: GradedFreeResolution(I, degrees=[2,1], shifts=[-3])
S(--3) <-- S(--2)⊕S(-1) <-- S(-2) <-- 0

How did you get this? What is I?

tscrim · 2022-08-17T06:57:02Z

comment:14

I am using I from comment:7. The point is that positive shifts are represented by double negatives.

kwankyu · 2022-08-17T07:10:54Z

comment:15

Replying to @tscrim:

I am using I from comment:7.

I is not homogeneous, and the size of degrees is not the number of variables. So it is a wrong input.

The point is that positive shifts are represented by double negatives.

I see. Please fix this here.

tscrim · 2022-08-17T07:25:06Z

comment:16

Replying to @kwankyu:

Replying to @tscrim:

I am using I from comment:7.

I is not homogeneous, and the size of degrees is not the number of variables. So it is a wrong input.

Actually, there is no way it is the one from comment:7...I just forgot which one I used in my haste. ^^;;

The point is that positive shifts are represented by double negatives.

I see. Please fix this here.

Will do.

tscrim · 2022-08-17T08:26:06Z

Author: Travis Scrimshaw

tscrim · 2022-08-17T08:26:06Z

Commit: dfb3384

tscrim · 2022-08-17T08:26:06Z

Branch: public/rings/free_gr_res_hook-34379

tscrim · 2022-08-17T08:26:06Z

comment:17

Okay, it is finally all ready. It took me a bit of time to figure out that when the matrix over-determines the system over a PID, then we should compute the echelon form.

One point of controversy might be the repr for the multigraded cases. My change effects the prints (0,...,0) and (-a, -b) instead of 0 and -(a, b), respectively. I think this is better to indicate the multigrading by being verbose here. I can also change it so that M((0,0)) gets printed as M(0, 0) instead, which might also be better.

New commits:

`0232e9e`	`Add free resolutions`
`bdbb479`	`Merge branch 'develop' into t/33950/free-resolution`
`736d448`	`Moving the main computation to be done on demand.`
`5285e67`	`Moving Singular conversion code to libs/singular; converting resoltuion files to Python files.`
`5917129`	`Changing the doc for clarity and to add some more descriptions.`
`3f1d1c3`	`Merge branch 'develop' into t/33950/public/rings/free_multigraded_resolutions-33950`
`ab725f2`	`Fix spaces`
`89fc91c`	`Adding hooks for (graded) free resolutions and handling cases when given a free module.`
`dfb3384`	`Rewriting the _repr_module() of graded resolutions to negate the grading.`

sagetrac-git · 2022-08-22T07:32:47Z

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

`f7eff26`	`Fixing last details.`

sagetrac-git · 2022-08-22T07:32:47Z

Changed commit from dfb3384 to f7eff26

tscrim · 2022-08-22T07:33:19Z

comment:19

This should now pass the patchbot.

tscrim · 2022-08-22T07:33:58Z

Changed dependencies from #32324 to #33950

sagetrac-git · 2022-09-07T11:41:12Z

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

`250c5cb`	`Some edits`

kwankyu · 2022-09-07T11:57:33Z

comment:60

In FreeModule, why do you check if the module is a free module only when module is a matrix? You wanted to provide free resolutions for free modules.

In sage.rings.ideal, you provided free_resolution() hook. In this context, an ideal is of a general commutative ring, but your code seems to assume that the ring is a univariate polynomial ring over a field. You need to check this first. If the ring is other than that, raise a NotImplemented error.

tscrim · 2022-09-07T12:32:51Z

comment:61

Replying to Kwankyu Lee:

In FreeModule, why do you check if the module is a free module only when module is a matrix? You wanted to provide free resolutions for free modules.

This is what we agreed on: special check for matrices because you wanted the code to handle it but not add a method to the matrix classes. Thus, the main entry point needs to process it. Everything else is delegated to free_resolution().

In sage.rings.ideal, you provided free_resolution() hook. In this context, an ideal is of a general commutative ring, but your code seems to assume that the ring is a univariate polynomial ring over a field. You need to check this first. If the ring is other than that, raise a NotImplemented error.

I require the ideal is principal, which is automatically free because it has a single generator. This case holds for all commutative rings, right? All other cases will raise an NotImplementedError either via the is_principal() call or from the result being False.

kwankyu · 2022-09-07T13:55:16Z

comment:62

Replying to Travis Scrimshaw:

In sage.rings.ideal, you provided free_resolution() hook. In this context, an ideal is of a general commutative ring, but your code seems to assume that the ring is a univariate polynomial ring over a field. You need to check this first. If the ring is other than that, raise a NotImplemented error.

I require the ideal is principal, which is automatically free because it has a single generator. This case holds for all commutative rings, right?

Yes.

All other cases will raise an NotImplementedError either via the is_principal() call or from the result being False.

I see. Okay.

tscrim · 2022-09-07T14:16:34Z

comment:63

If you’re good with my changes and the patchbot comes back green (or you are confident with the appropriate tests being run), then positive review?

kwankyu · 2022-09-07T14:29:13Z

comment:64

Replying to Travis Scrimshaw:

Replying to Kwankyu Lee:

In FreeModule, why do you check if the module is a free module only when module is a matrix? You wanted to provide free resolutions for free modules.

This is what we agreed on: special check for matrices because you wanted the code to handle it

No. I wanted that the matrix input is kept in (__init__() of) free resolution subclasses.
But I even agreed to remove it if you want because of maintenance burden.

For FreeResolution dispatcher, I think that not accepting matrix input is the right way. I think I agreed upon this...

kwankyu · 2022-09-07T14:41:10Z

comment:65

You removed

        if isinstance(module, Ideal_generic):
            S = module.ring()
            if len(module.gens()) == 1 and S in IntegralDomains():
                is_free_module = True
        elif isinstance(module, Module_free_ambient):
            S = module.base_ring()
            if (S in PrincipalIdealDomains()
                or isinstance(module, FreeModule_generic)):
                is_free_module = True

from FreeResolution dispatcher, are these cases handled in free_resolution()? Or you decided not to handle those cases?

tscrim · 2022-09-07T14:58:59Z

comment:66

Replying to Kwankyu Lee:

Replying to Travis Scrimshaw:

Replying to Kwankyu Lee:

In FreeModule, why do you check if the module is a free module only when module is a matrix? You wanted to provide free resolutions for free modules.

This is what we agreed on: special check for matrices because you wanted the code to handle it

No. I wanted that the matrix input is kept in (__init__() of) free resolution subclasses.
But I even agreed to remove it if you want because of maintenance burden.

This isn't really that much of a burden, especially since we don't fundamentally want to work with the module but the underlying matrix. We have a good justification for the special case of matrices.

For FreeResolution dispatcher, I think that not accepting matrix input is the right way. I think I agreed upon this...

If the class accepts a matrix, then the dispatcher should be able to handle the matrix. I think it would be really strange to have this discrepancy. Plus the constructor does some things to sanitize the input; in particular, it makes sure the matrix is immutable as this can lead to some very subtle bugs that are very hard to track down. (For example, you pass a mutable matrix into the constructor, compute the resolution, mutate the matrix; now the resolution does not necessarily match the input matrix.)

Actually, this makes me think of a case it should also handle: a free resolution input (where it would just return the input up to maybe stripping/adding the grading).

tscrim · 2022-09-07T14:59:43Z

comment:67

Replying to Kwankyu Lee:

You removed

        if isinstance(module, Ideal_generic):
            S = module.ring()
            if len(module.gens()) == 1 and S in IntegralDomains():
                is_free_module = True
        elif isinstance(module, Module_free_ambient):
            S = module.base_ring()
            if (S in PrincipalIdealDomains()
                or isinstance(module, FreeModule_generic)):
                is_free_module = True

from FreeResolution dispatcher, are these cases handled in free_resolution()? Or you decided not to handle those cases?

These are handled by the respective free_resolution() methods in, e.g., ideal.

kwankyu · 2022-09-07T15:16:53Z

comment:69

Replying to Travis Scrimshaw:

Actually, this makes me think of a case it should also handle: a free resolution input (where it would just return the input up to maybe stripping/adding the grading).

Why? This is strange. PolynomialRing(PolynomialRing(QQ)) does not work in that way.

kwankyu · 2022-09-07T15:18:17Z

comment:70

trac is buggy...

tscrim · 2022-09-07T15:23:20Z

comment:71

Replying to Kwankyu Lee:

Replying to Travis Scrimshaw:

Actually, this makes me think of a case it should also handle: a free resolution input (where it would just return the input up to maybe stripping/adding the grading).

Why? This is strange. PolynomialRing(PolynomialRing(QQ)) does not work in that way.

Right, that was dumb of me. I was thinking it was more Element-like (e.g., Partition), but it is Parent-like (or function-like).

Do you think it would be useful to implement a method in graded free resolutions like as_ungraded() that forgets the grading?

kwankyu · 2022-09-07T15:47:09Z

comment:72

Replying to Travis Scrimshaw:

Do you think it would be useful to implement a method in graded free resolutions like as_ungraded() that forgets the grading?

No as far as I know. Let's leave that as a future work for who needs it.

kwankyu · 2022-09-07T15:48:07Z

comment:73

Now let this go.

sagetrac-git · 2022-09-07T16:20:22Z

Changed commit from 250c5cb to 6dba5e5

sagetrac-git · 2022-09-07T16:20:22Z

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. New commits:

`6dba5e5`	`Hot fixes`

tscrim · 2022-09-07T23:34:02Z

comment:76

Thank you. This improved the code from my original proposal.

It took me a minute to realize that this is equivalent to the way I am used to seeing the definition of a resolution with 0 <- M <- F_1 <- .... However, this is definitely the better way to present this given the string representation.

kwankyu · 2022-09-08T00:31:04Z

comment:77

Replying to Travis Scrimshaw:

Thank you. This improved the code from my original proposal.

Thank you for the work.

If we want to be more mathematical, matrix input may be entirely removed. But we may regard matrix input as an added conveniency feature. I am happy in either way.

It took me a minute to realize that this is equivalent to the way I am used to seeing the definition of a resolution with 0 <- M <- F_1 <- .... However, this is definitely the better way to present this given the string representation.

A resolution of M is also said to be a resolution of F_1/M. See pages 22-23 of

https://faculty.math.illinois.edu/Macaulay2/Book/ComputationsBook/book/book.pdf

It seems that "resolution of M" is more conventional, but I think "resolution of F_1/M" is more reasonable. If you want to augment F_1 <- ... <- 0 at the left to get an exact sequence, that should be

0 <- F_1/M <- F_1 <- ... <- 0

where the canonical map F_1/M <- F_1 is our 0-th differential map and F_1/M is our target().

vbraun · 2022-09-22T22:35:33Z

Changed branch from public/rings/free_gr_res_hook-34379 to 6dba5e5

tscrim added this to the sage-9.7 milestone Aug 17, 2022

tscrim added c: user interface labels Aug 17, 2022

tscrim added the s: needs review label Aug 17, 2022

kwankyu added s: positive review and removed s: needs review labels Sep 7, 2022

sagetrac-git mannequin added s: needs review and removed s: positive review labels Sep 7, 2022

kwankyu added s: positive review and removed s: needs review labels Sep 7, 2022

mkoeppe modified the milestones: sage-9.7, sage-9.8 Sep 19, 2022

vbraun removed the s: positive review label Sep 22, 2022

vbraun closed this as completed in 3063144 Sep 22, 2022

tscrim mentioned this issue Jan 29, 2023

Syncronize implementations of graded free resolutions of graded modules #34386

Open

Implement a hook to access free (graded) resolutions #34379

Implement a hook to access free (graded) resolutions #34379

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