Timeline for Homogeneous ideal and its system of generators
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 9, 2012 at 9:02 | comment | added | Wilberd van der Kallen | That means using the map $R\to R/J$. | |
| Oct 8, 2012 at 16:52 | comment | added | Knot | I am so sorry because I am not an expert in commutative algebra. What do you mean by factoring $R$ by the ideal $J$ generated by all elements of degrees at least $N$ ? | |
| Oct 8, 2012 at 12:42 | comment | added | Wilberd van der Kallen | Well, I was in the setting of Karl, not Proposition 1.5.15 of Bruns and Herzog. Actually B & H tell on the same page that if one wants the degrees unique one should restrict to cases like Karl considers. (They call the degrees $\beta_{0i}$.) But their argument is a bit sophisticated. I was simply observing that if one factors $R$ by the ideal $J$ generated by all elements of degree at least $N$, then the minimal system $S$ maps to a minimal system plus zeroes. Now just watch how many generators become zero as $N$ varies. That tells you how many generators there were in each degree. | |
| Oct 8, 2012 at 8:13 | comment | added | Knot | @Wilberd van der Kallen : Sorry, Could you please make it more precisely ? | |
| Oct 8, 2012 at 8:12 | comment | added | Knot | Oh, it was in the definition of a homogeneous ideal : A homogeneous ideal $I$ is homogeneous if it is generated by homogeneneous element. | |
| Oct 8, 2012 at 7:46 | comment | added | Wilberd van der Kallen | If one applies the positive answer to your original question to truncated versions of the ring, one should get the degrees also. Here by a truncated version I mean that you factor out all elements of degree greater than some fixed number. | |
| Oct 8, 2012 at 7:39 | comment | added | Wilberd van der Kallen | Presumably your generators are homogeneous? | |
| Oct 8, 2012 at 4:50 | comment | added | Knot | Thank you very much, Youngsu. So, what about the degree of the elements in the minimal set of generators ? | |
| Oct 8, 2012 at 2:06 | history | answered | Youngsu | CC BY-SA 3.0 |