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How can I construct modules as quotient algebras?

asked 2011-05-05 16:24:53 +0200

Mike gravatar image

I would like to do the following: given a field k and an ideal I of A = k[x1,...,x_n], I would like to construct the k-module A/I and, when A/I is finite as a k-module, compute its rank. Is this possible in Sage?

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answered 2011-05-06 13:24:34 +0200

benjaminfjones gravatar image

updated 2011-05-06 13:25:59 +0200

It is easy to compute what you want in Sage if you have a specific field k and specific number of generators. The method you want is the vector_space_dimension method of the ideal I. Here is an example from the documentation for that method that involves the field QQ (rational numbers) and two generators u, v:

sage: R.<u,v> = PolynomialRing(QQ)
sage: g = u^4 + v^4 + u^3 + v^3
sage: I = ideal(g) + ideal(g.gradient())
sage: I
Ideal (u^4 + v^4 + u^3 + v^3, 4*u^3 + 3*u^2, 4*v^3 + 3*v^2) of Multivariate Polynomial Ring in u, v over Rational Field
sage: I.dimension()
sage: I.vector_space_dimension()
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Asked: 2011-05-05 16:24:53 +0200

Seen: 399 times

Last updated: May 06 '11