How to compute syzygy module of an ideal in a quotient ring?

asked 2017-04-18 04:34:27 -0500

KittyL gravatar image

updated 2017-04-18 04:35:09 -0500

I am trying to compute the syzygy module of an ideal generated by two polynomials <p,q> modulo I, where I is another ideal. This means to compute a generating set [(p1,q1),...,(ps,qs)] of the module {(g,h): gp+hq is in I}. I know that in Sage, we can use singular command to compute syzygy module:

R.<x,y> = PolynomialRing(QQ, order='lex')
M = I.syzygy_module();M
[       -2        -1         1]
[       -y 2*x^2 + y         0]

But this does not work with modulo I:

R.<x,y> = PolynomialRing(QQ, order='lex')
M = I.syzygy_module();M
Ideal (-b^2, b^2) of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
Error in lines 4-4
Traceback (most recent call last):

Is there a way to do that?

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If your generators for i are i1,...,is, you could just compute the syzygy module of <f,g,i1,...,is> and project on the first two coordinates. On the level of grobner basis computations, that's what you'd probably end up doing anyway.

nbruin gravatar imagenbruin ( 2017-04-19 00:43:36 -0500 )edit

That is a very smart solution. Thank you very much!

KittyL gravatar imageKittyL ( 2017-04-20 11:57:56 -0500 )edit