Timeline for Two (other) rings...are they isomorphic?
Current License: CC BY-SA 3.0
14 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Nov 15, 2014 at 22:37 | vote | accept | Nicholas Proudfoot | ||
| Nov 15, 2014 at 22:37 | answer | added | Nicholas Proudfoot | timeline score: 11 | |
| Nov 13, 2014 at 16:07 | comment | added | joro | @NicholasProudfoot I misunderstood, sorry. Will delete the nonsense comment. | |
| Nov 13, 2014 at 16:01 | comment | added | Nicholas Proudfoot | @joro: I'm a little confused; does it make sense to talk about the zeta series when you are dealing with a local ring? I thought that the zeta function had to do with counting points, and these guys each have only one point. (I definitely agree that, if they were quotients of polynomial rings, these two rings would not be isomorphic!) | |
| Nov 12, 2014 at 22:20 | answer | added | David E Speyer | timeline score: 6 | |
| Nov 12, 2014 at 20:03 | answer | added | Vladimir Dotsenko | timeline score: 5 | |
| Nov 12, 2014 at 15:07 | comment | added | Nicholas Proudfoot | That's correct. I don't think that there will be any general statement that gets me the isomorphism for free. If the rings are isomorphic (which I hope that they are), I think it will involve some trick that is specific to this example. (Of course, I hope that it will generalize to other similar examples that arise in the problem that I'm considering). | |
| Nov 12, 2014 at 15:02 | comment | added | Francesco Polizzi | Anyway, already for isolated singularities, it is not true in general that the analytic (or formal) type of the singularity is the same of the type of the tangent cone. For instance, take $$\mathbb{C}[[x, y, z, w]]/(x^3+y^3+z^3+w^3), \quad \mathbb{C}[[x, y, z, w]]/(x^3+y^3+z^3+w^3+xyzw).$$ Then these two germs are not isomorphic: in fact they have the same Milnor number ($16$) but different Tjurina number ($16$ and $15$, respectively). | |
| Nov 12, 2014 at 14:54 | comment | added | Francesco Polizzi | Just a remark: in this case, the singularities of your hypersurfaces at the origin are not isolated. | |
| Nov 12, 2014 at 14:49 | comment | added | Nicholas Proudfoot | Ack, sorry! I introduced some typos while translating between the previous version and this one. Both are supposed to be local rings at the origin on singular hypersurfaces in $\mathbb{C}^4$. The first hypersurface is homogeneous of degree 3, while the second has an extra term of degree 4. Hence the first hypersurface is isomorphic to the tangent cone (at the origin) of the second hypersurface. | |
| Nov 12, 2014 at 14:46 | history | edited | Nicholas Proudfoot | CC BY-SA 3.0 |
added 17 characters in body
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| Nov 12, 2014 at 14:22 | comment | added | Steven Landsburg | Your rings have different dimensions and hence are not isomorphic. Is there perhaps a typo in the question? | |
| Nov 12, 2014 at 14:12 | history | asked | Nicholas Proudfoot | CC BY-SA 3.0 |