Betti Numbers and number of generators - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:49:22Z http://mathoverflow.net/feeds/question/110279 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110279/betti-numbers-and-number-of-generators Betti Numbers and number of generators abd 2012-10-21T23:39:02Z 2012-10-22T14:27:48Z <p>Suppose that $R:=k[x_0,\dots,x_n]$ and $I$ is an ideal. Is there any relation between finding the minimal generators of $I$ and the graded betti numbers of the module $R/I$?</p> http://mathoverflow.net/questions/110279/betti-numbers-and-number-of-generators/110333#110333 Answer by Youngsu for Betti Numbers and number of generators Youngsu 2012-10-22T14:27:48Z 2012-10-22T14:27:48Z <p>To support J.C. Ottem's answer, let me present one example. </p> <p>Let $R = \mathbb{C}[x,y]$ and $I = (x,y^2)R$. What is the minimal graded free resolution of $R/I$, equivalently $I$? That is,</p> <p>$0 \rightarrow R(-3) \stackrel{d_1}{\rightarrow} R(-1) \oplus R(-2) \stackrel{d_0}\rightarrow R \rightarrow R/I \rightarrow 0$</p> <p>where $d_1 = (-y^2 \;\; x)$ and $d_0 = (x \;\; y^2)$.</p> <p>Now, ask what are the graded Betti numbers and minimal number of generators for $R/I$. I agree with J.C. Ottem's opinion on reviewing the definitions. I hope this helps.</p>