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

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')
f=2*x^2+y
g=y
h=2*f+g
I=ideal(f,g,h)
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')
S.<a,b>=R.quo(x^2+y^2)
I=ideal(a^2,b^2);I
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?

edit retag close merge delete

Sort by ยป oldest newest most voted

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.

more

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

( 2017-04-20 18:57:56 +0200 )edit