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Prime ideals and "Point on Spectrum"

asked 2011-03-15 16:13:32 +0100

Weaam gravatar image

updated 2011-03-16 13:55:09 +0100

niles gravatar image

The spectrum of the ring of integers $\mathbb{Z}$ consists of the prime ideals, i.e. $Spec(\mathbb{Z}) = \cup_{p \space prime}p\mathbb{Z} \cup (0)$.

i1: S = Spec(ZZ)
i2: nZ = ZZ.ideal(6)
i3: S(nZ)
o3: Point on Spectrum of Integer Ring defined by the Principal ideal (6) of Integer Ring

i4: nZ.is_prime()
o4: False

Obviously, nZ is not a prime ideal, as 6 is composite. Hence by definition, it is not in $Spec(\mathbb{Z})$. So what does "Point on Spectrum" means exactly in Sage?


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answered 2011-03-15 16:42:53 +0100

niles gravatar image

Spec means the same thing in Sage that it does everywhere else. It's just that Spec just doesn't check to see whether its input is a prime ideal: it relies on SchemeTopologicalPoint_prime_ideal: You can tell this because the __call__ method of sage.schemes.generic.spec.Spec is simply

    return point.SchemeTopologicalPoint_prime_ideal(self, x)

However SchemeTopologicalPoint_prime_ideal doesn't check to see whether the input ideal is prime either! It does allow an optional argument check which will perform the check, but this is disabled by default. Here is the code from sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal.__init__:

    R = S.coordinate_ring()
    from sage.rings.ideal import Ideal
    P = Ideal(R, P)
    # ideally we would have check=True by default, but
    # unfortunately is_prime() is only implemented in a small
    # number of cases
    if check and not P.is_prime():
        raise ValueError, "The argument %s must be a prime ideal of %s"%(P, R)
    SchemeTopologicalPoint.__init__(self, S)
    self.__P = P

So if you were calling SchemeTopologicalPoint_prime_ideal directly, you could pass check=True to have it check for you:

sage: S = Spec(ZZ)
sage: nZ = ZZ.ideal(6)
sage: from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal as primept
sage: primept(S,nZ)
Point on Spectrum of Integer Ring defined by the Principal ideal (6) of Integer Ring
sage: primept(S,nZ,check=True)
ValueError                                Traceback (most recent call last)
ValueError: The argument Principal ideal (6) of Integer Ring must be a prime ideal of Integer Ring

Unfortunately, the __call__ method of Spec doesn't take a check argument, and it doesn't pass its additional keyword arguments along using **kwds, so there isn't a way to have Spec check for you directly.

To me, this all seems confusing, shoddy, and disappointing; you should file a ticket on Trac for this (if there isn't one already). If you're in a situation where you need this functionality, I would suggest adding a line of code to check whether the ideal is prime before you pass it to Spec.

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If someone is filing a trac ticket about this, then I have two comments: first, note this problem: `Spec(IntegerModRing(9)).an_element()` has output `Point on Spectrum of Ring of integers modulo 9 defined by the Principal ideal (2) of Ring of integers modulo 9`. Second, is somewhat related.

John Palmieri gravatar imageJohn Palmieri ( 2011-03-15 17:08:03 +0100 )edit

@niles It makes perfect sense now. Thank you.

Weaam gravatar imageWeaam ( 2011-03-15 23:02:16 +0100 )edit

@John Palmieri. I shouldn't fill a trac ticket then? Thanks for your efforts.

Weaam gravatar imageWeaam ( 2011-03-15 23:15:56 +0100 )edit

@Weaam -- I think you should file the ticket. Whoever does, just report the link here when you do so we know about it.

niles gravatar imageniles ( 2011-03-16 09:35:07 +0100 )edit

@niles -- Thanks again for your efforts. I filed the ticked (#10949).

Weaam gravatar imageWeaam ( 2011-03-16 12:58:28 +0100 )edit

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Asked: 2011-03-15 16:13:32 +0100

Seen: 1,024 times

Last updated: Mar 15 '11