Timeline for How many commutative local algebras are there?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2018 at 8:34 | comment | added | Zach Teitler | It is not exactly a Hilbert scheme. Hilbert schemes (of zero-dimensional subschemes of affine space) parametrize algebras $A=k[x_1,\dotsc,x_n]/I$ where $\dim_k(A)$ is a fixed integer. But the power $l$ so that $J^l=0$ can vary as $A$ varies across the Hilbert scheme. That power—more precisely, the value $l-1$, the maximum so that $J^{l-1} \neq 0$—is called the *socle degree*, and $n$ is called the embedding dimension of the algebra. You're asking for the algebras of given embedding dimension and socle degree. | |
| S Jul 29, 2017 at 15:03 | history | suggested | LSpice | CC BY-SA 3.0 |
Minor proofreading (chiefly <> -> \langle\rangle)
|
| Jul 29, 2017 at 14:32 | review | Suggested edits | |||
| S Jul 29, 2017 at 15:03 | |||||
| Jul 29, 2017 at 12:18 | answer | added | Benjamin Steinberg | timeline score: 5 | |
| Jul 29, 2017 at 8:48 | comment | added | Mare | Maybe we can restrict at first to $n \leq 3$ and $l \leq 4$ at first with $q=2$ or $q=3$. That sounds doable for a computer. | |
| Jul 29, 2017 at 2:36 | comment | added | Mohan | I doubt whether these can be written down for large $\dim_K K[x_1,\ldots, x_n]/I$. | |
| Jul 29, 2017 at 0:58 | comment | added | Mare | Yes but maybe choosing q to be 2 or 3 and l at most 4 one might hope that a computer can list all algebras to do some experiments with them. | |
| Jul 29, 2017 at 0:58 | history | edited | Mare | CC BY-SA 3.0 |
added 224 characters in body
|
| Jul 29, 2017 at 0:46 | comment | added | Mohan | These are parametrized by the Hilbert scheme and they can be very complicated. I doubt whether there are any reasonable way you can write them all down. | |
| Jul 29, 2017 at 0:20 | history | asked | Mare | CC BY-SA 3.0 |