On Thursday 19 March 2009, David Madore wrote:
> Dear List,
>
> I'm trying to do some algebraic geometrical / arithmetical computation
> with Sage and I find myself stuck on the following dumb problem: if I
> have an ideal J of a quotient ring R/I (where R is a polynomial ring
> and I some ideal of it), I wish to construct the ideal of R (called
> p^{-1}(J) or perhaps even simply I+J) which is obtained by pulling
> back J by the canonical map p:R->R/I. This presents no algorithmic
> difficulty, since I is given by generators (in R), and J is also given
> by generators (in R/I, but they are themselves represented by elements
> of R) and it is just a question of taking the union of these lists of
> generators (seeing those of J in R/I as arbitrary representatives in
> R) to obtain the desired idea. Unfortunately, this doesn't seem to
> work (probably because Sage doesn't know how to handle the canonical
> surjection specially?):
>
> vega david /usr/local/src/sage-3.4 $ ./sage
> ----------------------------------------------------------------------
>
> | Sage Version 3.4, Release Date: 2009-03-11 |
> | Type notebook() for the GUI, and license() for information. |
>
> ----------------------------------------------------------------------
> sage: R.<x,y> = QQ['x','y']
> sage: I = Ideal(y^2 - x^3 - x)
> sage: Rq = R.quotient(I)
> sage: p = R.hom(Rq)
> sage: J = Ideal(p(y)-1)
> sage: J
> Ideal (ybar - 1) of Quotient of Multivariate Polynomial Ring in x, y over
> Rational Field by the ideal (-x^3 + y^2 - x) sage: p.inverse_image(J)
> ---------------------------------------------------------------------------
> NotImplementedError Traceback (most recent call last)
>
> /usr/src/local/sage-3.4/<ipython console> in <module>()
>
> /usr/src/local/sage-3.4/local/lib/python2.5/site-packages/sage/rings/morphi
>sm.so in sage.rings.morphism.RingHomomorphism.inverse_image
> (sage/rings/morphism.c:3480)()
>
> NotImplementedError:
> sage: whatiwantedwasthis = I + Ideal(y-1)
> sage: whatiwantedwasthis
> Ideal (-x^3 + y^2 - x, y - 1) of Multivariate Polynomial Ring in x, y over
> Rational Field sage: p(whatiwantedwasthis)
> Ideal (0, ybar - 1) of Quotient of Multivariate Polynomial Ring in x, y
> over Rational Field by the ideal (-x^3 + y^2 - x) sage:
> p(whatiwantedwasthis) == J
> True
>
> By comparison, Macaulay2 does this:
>
> vega david ~ $ /opt/Macaulay2-1.2-r8438/bin/M2
> Macaulay 2, version 1.2
> with packages: Elimination, IntegralClosure, LLLBases,
> PrimaryDecomposition, ReesAlgebra, SchurRings, TangentCone
>
> i1 : R = QQ[x,y]
>
> o1 = R
>
> o1 : PolynomialRing
>
> i2 : I = ideal(y^2-x^3-x)
>
> 3 2
> o2 = ideal(- x + y - x)
>
> o2 : Ideal of R
>
> i3 : Rq = R/I
>
> o3 = Rq
>
> o3 : QuotientRing
>
> i4 : p = map(Rq,R,matrix{{x,y}})
>
> o4 = map(Rq,R,{x, y})
>
> o4 : RingMap Rq <--- R
>
> i5 : J = ideal(y-1)
>
> o5 = ideal(y - 1)
>
> o5 : Ideal of Rq
>
> i6 : preimage(p,J)
>
> 3
> o6 = ideal (y - 1, x + x - 1)
>
> o6 : Ideal of R
>
> i7 : use R; whatiwantedwasthis = I + ideal(y-1)
>
> 3 2
> o8 = ideal (- x + y - x, y - 1)
>
> o8 : Ideal of R
>
> i9 : o6 == whatiwantedwasthis
>
> o9 = true
>
> i10 : p(whatiwantedwasthis)
>
> o10 = ideal (0, y - 1)
>
> o10 : Ideal of Rq
>
> i11 : p(whatiwantedwasthis) == J
>
> o11 = true
>
> (Unfortunately, I can't do my computations in Macaulay2 because I need
> polynomial rings over number fields - the above example is in Q - and
> it can't handle them.)
>
> So, is there a way in Sage to pull back an ideal by a canonical map?
Quotient rings are in a particularly bad shape in Sage, here's a hack:
sage: R.<x,y> = QQ['x','y']
sage: I = Ideal(y^2 - x^3 - x)
sage: Q = R.quotient(I)
sage: p = R.hom(Q)
sage: J = Ideal(p(y)-1)
sage: J
Ideal (ybar - 1) of Quotient of Multivariate Polynomial Ring in x, y over
Rational Field by the ideal (-x^3 + y^2 - x)
sage: I2 = Ideal([f._QuotientRingElement__rep for f in J.gens()]) + I
sage: I2
Ideal (y - 1, -x^3 + y^2 - x) of Multivariate Polynomial Ring in x, y over
Rational Field
sage: I2.groebner_basis()
[x^3 + x - 1, y - 1]
that should be what you want. However, f._QuotientRingElement__rep is quite
evil because it uses the internal representation.
I'd actually give it a shot to implement the NotImplementedError above but I'm
unsure where the implementation belongs:
Not sure whether I should just patch RingHomomorphism_coercion:add a bunch of
type checks and then do something like the above if they pass and raise
NotImplementedError otherwise?
Cheers,
Martin
--
name: Martin Albrecht
_pgp: http://pgp.mit.edu:11371/pks/lookup?op=get&search=0x8EF0DC99
_otr: 47F43D1A 5D68C36F 468BAEBA 640E8856 D7951CCF
_www: http://www.informatik.uni-bremen.de/~malb
_jab: martinralbre...@jabber.ccc.de
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