Timeline for Betti Numbers and number of generators
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Oct 24, 2012 at 14:42 | comment | added | Youngsu | Hi. You always need to put the condition "graded minimal free resolution". Check this condition with your definition of (graded) Betti numbers. The entries of $d_0$ are a generating set of $I$. Here $(x, y^2) = I$. This can be understood how you build a resolution. I am a bit confused by your example. $R/I=<1,y>$. Did you mean that $I = (1,y)$? If so then you wouldn't get anything since $I = R$. | |
| Oct 23, 2012 at 17:28 | vote | accept | abd | ||
| Oct 23, 2012 at 17:28 | comment | added | abd | Thanks Youngsu, your example helped me to understand the definition more. please correct me if I understood that wrong, now from the free resolution we can find the betti numbers easily and now the first module in the sequence $R(-1)\oplus R(-2)$ will be generated by two polynomials of degree 1 and 2 respectively. Is that means the ideal $I$ will be generated by polynomials of degree 1 and 2? Also how the differential map relate to the generators of $I$? and what we can get from the betti number about $R/I=<1,y>$? Is there any thing we can extract about the quotient? | |
| Oct 22, 2012 at 14:27 | history | answered | Youngsu | CC BY-SA 3.0 |