id summary reporter owner description type status priority milestone component resolution keywords cc merged author reviewer upstream work_issues branch commit dependencies stopgaps
20512 0 by 0 minor of a matrix should belong to the base ring kedlaya "This shouldn't return an error:
{{{
sage: P. = 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.
" defect closed major sage-7.2 linear algebra fixed schemes, Jacobian, matrix, minors Kiran Kedlaya Frédéric Chapoton N/A 06a432a988286bfae4a86e140e53022a9cbc4fc1 06a432a988286bfae4a86e140e53022a9cbc4fc1