Meta Ticket: Utilize coercion framework for scheme points and morphisms

[bhutz]

While there is some coercion-like behavior for schemes objects, it is done in an ad-hoc way that does not utilize the coercion framework and is overdue to be updated.

The following three scenarios should work using the coercion model.

• Coercion between ambient spaces whose base rings are compatible.

For example, \mathbb{P}^n(ZZ) should be coercible to \mathbb{P}^n(QQ) since ZZ is coercible to QQ. This may involve a finding a larger common parent.

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: T.<u,v> = ProjectiveSpace(ZZ,1)
sage: Q = P(1,1);Q2 = T(1,1)
sage: Q == Q2
False

This would be a major a paradigm shift for schemes, since currently the first comparison for points is equality of the ambient spaces. The current behavior is in direct contradiction to other areas of Sage, for example, matrix spaces.

sage: m1=matrix(QQ,2,2,[1,1,0,1])
sage: m2=matrix(ZZ,2,2,[1,1,0,1])
sage: m1==m2
True

This would also allow functions to act on points when the domain of the function and ambient space of the point are compatible.

• Coercion of other objects to schemes points

  These other objects could be any object which makes sense to coerce such as: base ring elements or list/tuples of base ring elements.

  While it is sometimes possible to initialize a point in some of these scenarios, since it is not done through the coercion framework, comparisons fail.

sage: P.<u,v>=ProjectiveSpace(ZZ,1)
sage: P(0) == 0
False

• left and right actions for base rings and matrices on scheme points and morphisms

   There is a natural `PGL` action on projective space and `GL` action on affine space. It would be natural to use the aspect of the coercion model.
    
   There is also a natural action of the base ring on projective points and morphisms. This is currently done with the scale_by() function, which modifies the object in place. However, there are good arguments for why this should be changed (see ticket #22268).
  

CC: @novoselt @pfili @sagetrac-atowsley

Component: algebraic geometry

Keywords: sd104, SI2019

Issue created by migration from https://trac.sagemath.org/ticket/23047

[bhutz]

Changed author from paulfili, atowsley to none

[bhutz]

comment:3

The three tickets for this functionality are the following:

• adding SchemeMorphism_points to coercion framework #23805 - coercion for points

• Fix is_preperiodic function domain check #23814 - matrix action on maps

• implement left action of matrices on scheme points #23816 - matrix action on points

[sagetrac-atowsley]

comment:4

The three tickets for this functionality are the following: (change 23814 to 23813)

#23805 - coercion for points

#23813 - matrix action on maps

#23816 - matrix action on points

[bhutz]

comment:5

all tickets finished as of sage-days 104

[bhutz]

comment:6

not sure how to go about closing a meta ticket like this. I'm marking it positive as all aspects are now completed.

[bhutz]

Changed keywords from none to sd104, SI2019

[fchapoton]

comment:8

Not sure how to close that kind of ticket either, so used "works for me"