Timeline for Representability of the tensor functor with a vector space
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 6, 2017 at 14:32 | comment | added | Anton Fonarev | This has been discussed, see mathoverflow.net/questions/6764/… | |
| Apr 6, 2017 at 12:46 | comment | added | Laurent Moret-Bailly | Another proof: let $F$ be the functor in question. Then the natural map $F(k[[t]])\to\varprojlim_n F\left(k[[t]]\,/\,(t^{n+1})\right)$ is not surjective. It would be bijective if $F$ were representable. | |
| Apr 6, 2017 at 12:32 | comment | added | Jason Starr | Here is a proof when the dimension of $V$ is countably infinite. For $R=k[\epsilon]/\langle \epsilon^2\rangle$, consider the subspace $V\otimes_k \epsilon R $ of $V\otimes_k R$. If the functor is representable by a $k$-scheme $X$, for the stalk of the structure sheaf $(\mathcal{O},\mathfrak{m})$ at the $k$-point of $X$ representing the zero vector in $V$, the vector space above equals the vector space $\text{Hom}_k(\mathfrak{m}/\mathfrak{m}^2,k)$. This vector space is uncountably generated if $\mathfrak{m}/\mathfrak{m}^2$ has infinite dimension. | |
| Apr 6, 2017 at 12:31 | comment | added | Jason Starr | This argument is also the proof of Claim 3.1 in my paper at the following URL: arxiv.org/pdf/math/0602646.pdf | |
| Apr 6, 2017 at 11:41 | review | First posts | |||
| Apr 6, 2017 at 11:43 | |||||
| Apr 6, 2017 at 11:41 | history | asked | infi | CC BY-SA 3.0 |