wrong category for parent of composition of number field endomorphisms

[fchapoton]

namely

sage: K.<a, b> = NumberField([x^2 - 2, x^2 - 3])
sage: e, u, v, w = End(K)
sage: e.abs_hom().parent().category()
Category of homsets of number fields
sage: u.abs_hom().parent().category()
Category of homsets of number fields
sage: v.abs_hom().parent().category()
Category of homsets of number fields
sage: (v*v).abs_hom().parent().category()
Category of homsets of unital magmas and additive unital additive magmas

The last one should be the same as the other ones..

CC: @tscrim

Component: number fields

Author: Frédéric Chapoton

Branch/Commit: 768891b

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/23647

[fchapoton]

comment:1

Travis, can you help, please ? This prevents from using richcmp for number fields morphisms.

[tscrim]

comment:2

So the problem comes down to this:

sage: v.parent() is End(K)
True
sage: v.parent() is Hom(K, K, category=NumberFields())
True
sage: (v*v).parent() == End(K)
True
sage: Hom(K, K, category=Rings()) is End(K)  # The issue
False
sage: Hom(K, K, category=Rings()) is (v*v).parent()
True

The source of which, when you go down _composition_, comes from this:

sage: v.category_for()  # Which calls
Category of rings
sage: v.parent().homset_category()
Category of rings

The latter should be NumberFields(), not Rings() because the subclass

sage.rings.number_field.morphism.NumberFieldHomset

does not pass NumberFields() as its category. However, I tried passing the category of the domain, but this causes failures because ComplexLazyField is not a number field and issues with CyclotomicField that I don't understand.

[fchapoton]

comment:3

Do you have branch for what you tried ?

[tscrim]

comment:4

No, I don't as it didn't work. However, I basically just added this to number_field/morphism.py:

class NumberFieldHomset(RingHomset_generic):  # This was already there
    def __init__(self, R, S, category=None):
        if category is None:
            from sage.categories.number_fields import NumberFields
            category = NumberFields()
        RingHomset_generic(self, R, S, category)

and the other variant I tried was

    def __init__(self, R, S, category=None):
        if category is None:
            category = R.category()
        RingHomset_generic(self, R, S, category)

[fchapoton]

comment:5

I have made another try, that causes interesting doctest failures:

Failed example:
    G = End(K); G
Expected:
    Automorphism group of Cyclotomic Field of order 12 and degree 4
Got:
    Set of Homomorphisms from Cyclotomic Field of order 12 and degree 4 to Cyclotomic Field of order 12 and degree 4

EDIT: ok, ok. This is indeed a group.

[fchapoton]

comment:6

Here is my branch.

---

New commits:

ec77695

tentative

[fchapoton]

Commit: ec77695

[fchapoton]

Branch: public/23647

[fchapoton]

comment:7

in vanilla sage:

sage: End(QQ)
Set of Homomorphisms from Rational Field to Rational Field
sage: End(QuadraticField(7))
Automorphism group of Number Field in a with defining polynomial x^2 - 7
sage: End(CyclotomicField(7))
Automorphism group of Cyclotomic Field of order 7 and degree 6
sage: End(UniversalCyclotomicField())
Set of Homomorphisms from Universal Cyclotomic Field to Universal Cyclotomic Field
sage: End(GF(5))
Automorphism group of Finite Field of size 5

and

[tscrim]

comment:8

This does work, but it is a very big hack IMO. I think it might be worth trying to do a more comprehensive fix. At least as a first step, we can put QQ in NumberFields: #23761.

[tscrim]

comment:9

I believe UniversalCyclotomicField() is not a number field as I believe is an infinite-dimensional field extension of Q.

[fchapoton]

comment:10

let us see what the patchbots say

---

New commits:

56c31af

trac 23647

[fchapoton]

Changed branch from public/23647 to public/23647-v2

[fchapoton]

Changed commit from ec77695 to 56c31af

[sagetrac-git]

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

49ddc5d

trac 23647 fix doctest

[sagetrac-git]

Changed commit from 56c31af to 49ddc5d

[fchapoton]

comment:12

This seems to work (somehow .__init__ was forgotten in the previous tentatives)

But

• it does not fix the problem in py3 richcmp for number fields morphism #23875
• it may behave uncorrectly for morphisms from number fields to rings that are not fields

[sagetrac-git]

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

3676535

trac 23647 better category handling in Number fields morphisms

[sagetrac-git]

Changed commit from 49ddc5d to 3676535

[fchapoton]

comment:14

Here is another tentative, let us see what patchbot says.

---

New commits:

3676535

trac 23647 better category handling in Number fields morphisms

[sagetrac-git]

Changed commit from 3676535 to 768891b

[sagetrac-git]

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

768891b

trac 23647 add doctest

[fchapoton]

comment:17

Green patchbot. Travis, do you think that this is good enough ?

We could also try to find the best category given the target category..

[tscrim]

Author: Frédéric Chapoton

[tscrim]

comment:18

I think it improves things and is fairly close to a proper (IMO) fix of determining the category directly from the input. We can worry about doing more if this becomes an issue again somewhere else.

[tscrim]

Reviewer: Travis Scrimshaw

[vbraun]

Changed branch from public/23647-v2 to 768891b