#20512: 0 by 0 minor of a matrix should belong to the base ring
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Reporter: kedlaya | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-7.2
Component: linear algebra | Resolution:
Keywords: schemes, Jacobian, matrix, | Merged in:
minors | Reviewers:
Authors: | Work issues:
Report Upstream: N/A | Commit:
Branch: | Stopgaps:
Dependencies: |
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Changes (by kedlaya):
* keywords: schemes, Jacobian => schemes, Jacobian, matrix, minors
* cc: nbruin (removed)
* component: algebraic geometry => linear algebra
Old description:
> This shouldn't return an error:
> {{{
> sage: P.<x,y,z> = ProjectiveSpace(2, QQ)
> sage: X = P.subscheme([])
> sage: X.Jacobian_matrix() # This works
> []
> sage: X.Jacobian() #This doesn't
> ...
> AttributeError: 'sage.rings.integer.Integer' object has no attribute
> 'reduce'
> }}}
> I think the mathematically correct answer is that X.Jacobian() should
> equal the ideal (1). This is consistent with the definition in the
> docstring:
> {{{
> * the `d\times d` minors of the Jacobian matrix, where `d` is
> the :meth:`codimension` of the algebraic scheme, and
>
> * the defining polynomials of the algebraic scheme. Note that
> some authors do not include these in the definition of the
> Jacobian ideal. An example of a reference that does include
> the defining equations is [LazarsfeldJacobian].
> }}}
> In this case d=0, and the unique 0 by 0 minor of any matrix (empty or
> not) is equal to 1. And anyway, the Jacobian ideal of the full ambient
> space should cut out the empty subscheme.
New description:
This shouldn't return an error:
{{{
sage: P.<x,y,z> = ProjectiveSpace(2, QQ)
sage: X = P.subscheme([])
sage: X.Jacobian_matrix() # This works
[]
sage: X.Jacobian() #This doesn't
...
AttributeError: 'sage.rings.integer.Integer' object has no attribute
'reduce'
}}}
I think the mathematically correct answer is that X.Jacobian() should
equal the ideal (1). This is consistent with the definition in the
docstring:
{{{
* the `d\times d` minors of the Jacobian matrix, where `d` is
the :meth:`codimension` of the algebraic scheme, and
* the defining polynomials of the algebraic scheme. Note that
some authors do not include these in the definition of the
Jacobian ideal. An example of a reference that does include
the defining equations is [LazarsfeldJacobian].
}}}
In this case d=0, and the unique 0 by 0 minor of any matrix (empty or not)
is equal to 1. And anyway, the Jacobian ideal of the full ambient space
should cut out the empty subscheme.
--
Comment:
In fact, the problem seems to be that the 0 by 0 minor of a matrix is
returned as 1 in the ring of integers, not the base ring of the matrix.
Retitled and reclassified accordingly.
--
Ticket URL: <http://trac.sagemath.org/ticket/20512#comment:2>
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