Timeline for The notion of multiplicity in algebraic geometry
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 23, 2013 at 18:13 | comment | added | Youngsu | The ring $k[x,y] /(x^2, x^3-y^2)$ is isomorphic to $k[x,y]/(x^2,y^2)$. The monomials of $k[x,y]$ that are not zero in this rings are $1,x,y,xy$. As you mentioned $xy <-> t^5$ is not zero. | |
| Apr 23, 2013 at 10:51 | comment | added | Martin Brandenburg | For the DVR case, induct on the order of $f$ and use the additivity of the length on short exact sequences. | |
| Apr 23, 2013 at 7:14 | comment | added | Jesko Hüttenhain | Not that it really matters, but I thought $3$ sounded reasonable: We have the ring $k[t^2,t^3]/(t^4)=k[x,y]/(x^2,x^3-y^2)$ where $x=t^2$, $y=t^3$ and $xy=t^5$. One more question, though: You say that the Definitions agree when the local ring is a DVR: Do you have a reference? | |
| Apr 23, 2013 at 5:31 | comment | added | Youngsu | I meant to say Theorem 14.10 which is essentially a corollary of Theorem 14.9. | |
| Apr 23, 2013 at 4:50 | comment | added | Youngsu | I think the length is $4$ since $t^5 \notin (t^4)$. As mentioned in Filippo Alberto Edoardo's answer the failure might be coming from the point that the associated graded ring (tangent cone) is not a domain. In this example the associated graded ring is isomorphic to $k[X,Y]/(X^2)$ which is not a domain. Theorem 14.8 in Matsumura's book may give you some algebraic hint on the question and example. | |
| Apr 22, 2013 at 22:13 | comment | added | Mariano Suárez-Álvarez | a true two. ${}{}$ | |
| Apr 22, 2013 at 21:07 | history | answered | Will Sawin | CC BY-SA 3.0 |