Timeline for Ideals with the same Hilbert series
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 15, 2018 at 16:14 | comment | added | Igor Makhlin | @AviSteiner Here all I meant was the codimension of the vector subspace $I$ inside the vector space $R$, i.e. the dimension of the quotient ring. Anyway, thank you for your advice, I'll look into it. | |
| Feb 15, 2018 at 14:38 | comment | added | Avi Steiner | @imakhlin There are two things that are commonly called the codimension of an ideal $I$ in a (commutative unital) ring $R$. In Encyclopedia of Mathematics, these are called the height and coheight. When $R$ has finite Krull dimension $n$, both of these quantities are at most $n$. So, I don’t know what you mean by codimension. | |
| Feb 15, 2018 at 14:26 | comment | added | Igor Makhlin | @AviSteiner Lol, sorry, I'm referring to the codimension of the ideal in the ring =D The quotient is infinite-dimensional, in particular, its Hilbert function is a Hilbert series rather than a Hilbert polynomial. | |
| Feb 15, 2018 at 2:18 | comment | added | Avi Steiner | @imakhlin They’re ideals in a polynomial ring of dimension n. So, their codimemsion (however you’re defining it) is at most n. Certainly not infinite. | |
| Feb 14, 2018 at 23:51 | comment | added | Igor Makhlin | @AaronDall Thank you! Could those be the same as the lex-segment ideals from Miller's and Sturmfels's book? I read the corresponding section quite a while ago but, if I'm not mistaken, they show that these ideals are, in a sense, extremal among those with the same Hilbert function and not much apart from that. Should I look elsewhere? | |
| Feb 14, 2018 at 23:40 | comment | added | Igor Makhlin | @AviSteiner Thank you! My knowledge of Hilbert schemes is very basic so I will certainly take a closer look. I must, however, say that in the cases I'm interested in $I$ and $J$ are of infinite codimension. Would Hilbert schemes still be of help here? | |
| Feb 14, 2018 at 10:01 | comment | added | Aaron Dall | Check out lexicographic ideals as well. | |
| Feb 14, 2018 at 1:31 | comment | added | Avi Steiner | Look into Hilbert schemes | |
| Feb 13, 2018 at 23:55 | history | asked | Igor Makhlin | CC BY-SA 3.0 |