Move dim from schemes to rings

[paemurru]

Contributes to #2771 but does not close it because #2771 (comment)

[afkafkafk13]

Please handle them separately. For MPolyComplementOfKPointIdeal P is certified to be zero-dimensional so that we do not need to compute dim(P).

[paemurru]

P is certified to be zero-dimensional

It seems that this is not true.

julia> R, (x, y) = ZZ[:x, :y]
(Multivariate polynomial ring in 2 variables over ZZ, ZZMPolyRingElem[x, y])

julia> S = complement_of_point_ideal(R, [0, 0])
Complement
  of maximal ideal corresponding to rational point with coordinates (0, 0)
  in multivariate polynomial ring in 2 variables over ZZ

julia> I = ideal(R, 0)
Ideal generated by
  0

julia> L, _ = localization(R, S)
(Localization of multivariate polynomial ring in 2 variables over ZZ at complement of maximal ideal of point (0, 0), Hom: R -> localized ring)

julia> W, _ = quo(L, L(I))
(Localization of quotient of multivariate polynomial ring at complement of maximal ideal, hom: Localized ring -> Localized quotient of multivariate polynomial ring)

julia> P = prime_ideal(inverted_set(W))
Ideal generated by
  x
  y

julia> dim(P)
1

So I will leave it as it is for now.

[simonbrandhorst]

My guess is that complement_of_point_ideal silently assumes that we are working over a field? Perhaps the type and constructor should be restricted accordingly what do you think @afkafkafk13 ?

[afkafkafk13]

julia> S = complement_of_point_ideal(R, [0, 0])
Complement
of maximal ideal corresponding to rational point with coordinates (0, 0)
in multivariate polynomial ring in 2 variables over ZZ

This is indeed a problem with complement_of_point_ideal:
This ideal is NOT maximal as we do not get a field as the quotient of the ring by it. This needs to be handled in complement_of_point_ideal. My suggestion: check dimension (not number of entries!!) of the ideal before doing anything else in complement_of_point_ideal, throw an error if it is not maximal (i.e. dim(ideal) is not zero -- this should already be working in Oscar) and point the user to complement_of_prime_ideal.

[simonbrandhorst]

It is a bit fuzzy what a "point" is here. Will we get anything computationally effective for the "point" given by the maximal ideal (2,x,y) in Z[x,y]? I impression was that ComplementOfKPointIdeal just assumes a base field.

[paemurru]

I impression was that ComplementOfKPointIdeal just assumes a base field.

I think it does assume but does not check for this (although it should).

I would leave fixing complement_of_point_ideal to a separate issue.

[afkafkafk13]

ok. fine with me. But we urgently need to fix complement_of_point_ideal. Maybe we can just check

@simonbrandhorst no, we won't get anything computationally effective currently. You are right. We should just tell the user to use complement_of_prime_ideal instead, whenever we are not over a field.

Edit: This is a strategic decision and breaks some tests. I opened PR We need to discuss this. Over a ring this is not a point ideal and hence should not be used via this localization, anyway. Moreover, the localization at <x_1,\dots,x_n> in ZZ[x_1,\dots,x_n] is not what is produced here, as we are localizing w.r.t. a monomial ordering and continue working over ZZ and do not pass to QQ, even though ZZ \cap <x_1,\dots,x_n> = 0.

[afkafkafk13]

Again you might want to handle MPolyComplementOfKPointIdeal separately?

[paemurru]

I am not sure how to do this here.

[simonbrandhorst]

Thanks for looking into this.