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 $R/I$?



Betti Numbers and number of generators
To support J.C. Ottem's answer, let me present one example. 


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,


$0 \rightarrow R(-3) \stackrel{d\_1}{\rightarrow} R(-1) \oplus R(-2) \stackrel{d\_0}\rightarrow R \rightarrow R/I \rightarrow 0 $


where $d\_1 = (-y^2 \;\; x)$ and $d\_0 = (x \;\; y^2)$.


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.


