Timeline for The notion of multiplicity in algebraic geometry
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Feb 28 at 6:28 | comment | added | imtrying46 | Do you have a reference for the formula $e(\mathfrak{m}/(f),R/(f))=\operatorname{ord}(f)e(\mathfrak{m},R)$? | |
| S Sep 28, 2013 at 0:46 | history | suggested | Michael Albanese | CC BY-SA 3.0 |
Replaced \\, | \\, with \mid so math would render properly.
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| Sep 27, 2013 at 23:59 | review | Suggested edits | |||
| S Sep 28, 2013 at 0:46 | |||||
| Aug 8, 2013 at 2:44 | review | Suggested edits | |||
| Aug 8, 2013 at 3:16 | |||||
| Apr 24, 2013 at 13:00 | comment | added | Jesko Hüttenhain | Thanks a bunch. I accepted this answer because it helps me most for what I am doing, but thanks to Filippo Edoardo and Will Savin for the very helpful explanations. | |
| Apr 24, 2013 at 12:58 | vote | accept | Jesko Hüttenhain | ||
| Apr 23, 2013 at 21:15 | history | edited | Youngsu | CC BY-SA 3.0 |
corrected typo
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| Apr 23, 2013 at 21:15 | comment | added | Youngsu | Sorry about that. You are absolutely correct. Let me edit that part. One example is when $R = \mathbb{C}[[x]]$. For $f \in R$ One can not talk about a the highest degree in general, but there is well defined lowest degree. This is the order of $f$. Or where the Taylor series of $f$ at $0$ starts. | |
| Apr 23, 2013 at 18:42 | comment | added | Jesko Hüttenhain | Your definition of $\mathrm{ord}(f)$ confuses me because for one thing, it does not depend on $f$ at all and second, $I\in m^i$ looks a lot like you ment to write $f\in m^i$. | |
| Apr 23, 2013 at 18:28 | comment | added | Youngsu | The definition of ord is what I used. It is in the second last sentence of the first paragraph. One good fact that might help you is that for a local ring $(R,m)$, $e(m,R) = e(gr_m(R)_+, gr_m(R))$ where $gr_m(R)_+ = \oplus_{i \ge 1} [gr_m(R)]_i$. Maybe it is worth looking at the definition of the Hilbert-Samuel multiplicity in commutative algebra, the degree in algebraic geometry. Also, the associativity formula in commutative algebra can give a connection between geometric multiplicity in algebraic geometry. | |
| Apr 23, 2013 at 7:22 | comment | added | Jesko Hüttenhain |
I suppose $\mathrm{ord}(f)=\sup\{ i\mid f\in m^i \}$, yes? So basically, I can use my definition in the case where $X$ is a regular scheme, which might just be good enough for me. +1 and thanks!
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| Apr 23, 2013 at 5:31 | history | answered | Youngsu | CC BY-SA 3.0 |