Schemes().hom_category()?

[Simon King]

Hi!

Ticket #10667 has the main purpose of adding fast containers for
morphisms and objects of categories". I added tests for these
containers to the category test suites.

These additional tests revealed several bugs. Therefore, the ticket
evolved into a partial overhaul of homsets and morphisms. In
sage.categories.schemes.Schemes.HomCategory, there is a comment
saying:

FIXME: what category structure is there on Homsets of
schemes?
The result above is wrong, and should be fixed during the
next
homsets overhaul.

While I am at overhauling: Is there an answer on that question? I
guess that people on this list or on sage-nt might tell whether
Schemes().hom_category() is sub-category of anything more interesting
than Sets()?

Then, in order to be able to run a full test suite on schemes: How can
one create an object of Schemes(R), R a (commutative, I guess) ring?

I tried (from the docs)
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: P.subscheme([x^2-y*z])
Closed subscheme of Projective Space of dimension 2 over Rational
Field defined by:
x^2 - y*z

However, it doesn't get its own category right:
sage: P.subscheme([x^2-y*z]).category()
Category of sets

And how can one create a morphism between two schemes?

Best regards,
Simon

[Volker Braun]

Hi Simon,

I also found that scheme morphisms currently have various issues. This is particularly important for morphisms of toric varieties, which Andrey and I are currently trying to implement. There is a big patch at  

http://trac.sagemath.org/sage_trac/ticket/11599 to clean up the scheme morphisms.

Cheers,

Volker

[Simon King]

Hi Volker,

On 5 Sep., 18:24, Volker Braun <vbraun.n...@gmail.com> wrote:
> I also found that scheme morphisms currently have various issues. This is
> particularly important for morphisms of toric varieties, which Andrey and I

> are currently trying to implement. There is a big patch at  http://trac.sagemath.org/sage_trac/ticket/11599to clean up the scheme
> morphisms.

OK. Since #10667 is rather biggish as anyway, I will not attempt to
provide testsuites for Schemes, unless #11599 soon gets a positive
review.

Best regards,
Simon