Projective Spaces are not not cached

[Gareth Ma]

In the documentation of `ProjectiveSpace` constructor, it says

> Projective spaces are not cached, i.e., there can be several with
> the same base ring and dimension (to facilitate gluing
> constructions).

However, that doesn't seem to be true, since (1) it's a `UniqueRepresentation`,
and (2) `ProjectiveSpace(3, ZZ) is ProjectiveSpace(3, ZZ)` returns `True`. Is
this a bug? And I guess more importantly, is this a problem in practice?

@Kwankyu since I believe you are familiar with algebraic geometry in Sage

Best regards,
Gareth

[Kwankyu Lee]

On Wednesday, June 12, 2024 at 2:17:43 AM UTC+9 Gareth Ma wrote:

In the documentation of `ProjectiveSpace` constructor, it says

> Projective spaces are not cached, i.e., there can be several with
> the same base ring and dimension (to facilitate gluing
> constructions). 

However, that doesn't seem to be true, since (1) it's a `UniqueRepresentation`,

Right.

 

and (2) `ProjectiveSpace(3, ZZ) is ProjectiveSpace(3, ZZ)` returns `True`.

Indeed.

 

Is this a bug?

According to git blame, it seems that the comment is from 2009 but "class ProjectiveSpace_ring(UniqueRepresentation" is from 2018. I guess that the comment was not simply deleted when "UniqueRepresentation" was added (there is no relevant doctest after the comment).

 

And I guess more importantly, is this a problem in practice?

I have no more experience than you regarding gluing construction with projective spaces.  

According to the above comment, gluing construction may not work because projective space is now "UniqueRepresentation". Clearly this was not a problem  in practice up to now, perhaps because there is not much gluing construction in sage. But if you are working with gluing constructions, you may need to refactor ProjectiveSpace_ring. 

In my opinion, in gluing several projective spaces, these spaces should at least differ in coordinate variable names even if with the same base ring and dimension, and hence caching should not interfere with gluing construction.

Good luck.

Kwankyu