ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 13 Nov 2013 11:38:20 -0600Computing basis of local artinian ring as a vector space.http://ask.sagemath.org/question/10729/computing-basis-of-local-artinian-ring-as-a-vector-space/Given a local artinian ring $A$ with residue field $\mathbb{C}$, we know two facts:
1) The ring A is isomorphic as a $\mathbb{C}$-algebra to $\mathbb{C}[x_1,\ldots,x_n]/(f_1(x_1,\ldots,x_n),\ldots,f_m(x_1,\ldots,x_n))$ where $f_i$ are some polynomials in the variables $x_1,\ldots,x_n$.
2) There exists a collection of monomials $M_t$ for $t=1,\ldots,l$ in the variables $x_1,\ldots, x_n$ such that $A$ is isomorphic to a vector space over $\mathbb{C}$ with basis {$M_1,\ldots, M_l$}.
My question is if there is a way to move from 1 to 2 in Sage. In other words, if I write down a finitely generated $\mathbb{C}$-algebra $A$, then by checking that dim A = 0, I will know that $A$ is artinian. Is there some function in Sage which will find a (ordered) basis for $A$ as a vector space when $dim A = 0$?Tue, 12 Nov 2013 07:58:13 -0600http://ask.sagemath.org/question/10729/computing-basis-of-local-artinian-ring-as-a-vector-space/Comment by Andrew Stout for <p>Given a local artinian ring $A$ with residue field $\mathbb{C}$, we know two facts:</p>
<p>1) The ring A is isomorphic as a $\mathbb{C}$-algebra to $\mathbb{C}[x_1,\ldots,x_n]/(f_1(x_1,\ldots,x_n),\ldots,f_m(x_1,\ldots,x_n))$ where $f_i$ are some polynomials in the variables $x_1,\ldots,x_n$.</p>
<p>2) There exists a collection of monomials $M_t$ for $t=1,\ldots,l$ in the variables $x_1,\ldots, x_n$ such that $A$ is isomorphic to a vector space over $\mathbb{C}$ with basis {$M_1,\ldots, M_l$}. </p>
<p>My question is if there is a way to move from 1 to 2 in Sage. In other words, if I write down a finitely generated $\mathbb{C}$-algebra $A$, then by checking that dim A = 0, I will know that $A$ is artinian. Is there some function in Sage which will find a (ordered) basis for $A$ as a vector space when $dim A = 0$?</p>
http://ask.sagemath.org/question/10729/computing-basis-of-local-artinian-ring-as-a-vector-space/?comment=16709#post-id-16709I suppose that this is an easy question. One can just compute the groebner basis for the extension of $(x_1,\ldots,x_n)$ in $A$ and then use singular to find the monomial ideal.Wed, 13 Nov 2013 11:38:20 -0600http://ask.sagemath.org/question/10729/computing-basis-of-local-artinian-ring-as-a-vector-space/?comment=16709#post-id-16709