Timeline for Is a group scheme determined by its category of representations?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 29, 2017 at 7:27 | comment | added | Artur Jackson | @Anton Yes, thank you. This picture is quite nice. I handn't attempted to think about that aspect 2-categorically. | |
| Jan 28, 2017 at 6:29 | comment | added | მამუკა ჯიბლაძე | @Anton Thank you, it is much more solid than what I would ever imagine! And you are right, I will try to formulate a separate question. | |
| Jan 28, 2017 at 0:27 | comment | added | Anton Fetisov | (cont...) A fibre functor should map this 2-category of representations to the 2-category of $\mathbb C$-abelian categories. Invertible representations defined above will map to invertible categories, which for $R = \mathbb C$ are all isomorphic to $Vect_{\mathbb C}$, an isomorphism defined by a line over $\mathbb C$. Thus a fiber functor gives a point on the double dual curve, i.e. on $E$. An automorphism of fiber functor also corresponds to a point on double dual, thus we reconstruct $E$ tannakially. The details are complicated, you should post a question if you are interested. | |
| Jan 28, 2017 at 0:27 | comment | added | Anton Fetisov | @მამუკაჯიბლაძე Conceptually, you should consider the 2-category of representations of $E$ on (sufficiently good) abelian categories over $R$ (here $R$ is the base ring which I will assume $=\mathbb C$ for fun). For example, a representation of $E$ on $Vect_{\mathbb C}$ is the same as a representation of cogroup $QCoh(E)$ on $Vect_{\mathbb C}$ ($QCoh(E)$ has obvious multiplication and inherits comultiplication from $E$), which is the same as a line bundle on $E$. Thus linear 2-representations of $E$ correspond to points on the dual elliptic curve. | |
| Jan 27, 2017 at 16:31 | history | edited | Artur Jackson | CC BY-SA 3.0 |
I misread the OP.
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| Jan 27, 2017 at 16:25 | comment | added | Artur Jackson | Haha, I somehow did not notice the question asked for `affine'! Yes of course the affine case is the celebrated theorem. | |
| Jan 27, 2017 at 16:24 | comment | added | მამუკა ჯიბლაძე | @BenWieland Sorry this is way above my level, so what follows may be completely senseless, but I rather had in mind some kind of "meromorphic representations" of $E$ (as a group). Something like a system of line bundles $(L_i)_{1\leqslant i\leqslant n}$ on $E$ and sections $s_{ij}$ of $L_i^*\otimes L_j$, with $s_{ij}(xy)=\sum_ks_{ik}(x)s_{kj}(y)$, where the latter sum is understood as a convolution of sections of $(L_i^*\otimes L_k)\boxtimes(L_k^*\otimes L_j)$... | |
| Jan 27, 2017 at 15:13 | comment | added | Ben Wieland | @მამუკაჯიბლაძე Here is a counterexample, suggesting that it is hopeless to reconstruct $BE$ from the category of sheaves: $\mathbb P^1$ and $E$ are complete, so that GAGA applies to their categories of coherent sheaves. If you could reconstruct $BE$ from the category of sheaves on $E$, that would suggest that you could describe maps $\mathbb P^1\to BE$ in terms of sheaves. But there are more analytic maps (Hopf surfaces) than algebraic. The total spaces aren't projective, but $\mathbb P^1$ and $E$ are. Lurie isolates this as uniquely bad. | |
| Jan 27, 2017 at 6:40 | comment | added | მამუკა ჯიბლაძე | In fact there is something serious and unknown to me here. Let us look at it this way: when your "standard cogenerator" does not work on some particular object you use instead of maps to this cogenerator sections of bundles with fibre the cogenerator. E. g. there are not enough functions on projective varieties, so you use sections of line bundles on them instead. Now switch on the group structure; you have a single object groupoid in schemes with not enough functors to $\text{Vect}$. Following the above principle one should use some kind of $\text{Vect}$-bundles over the groupoid? | |
| Jan 27, 2017 at 6:33 | comment | added | მამუკა ჯიბლაძე | $E$ is not affine :P | |
| Jan 27, 2017 at 6:08 | review | Late answers | |||
| Jan 27, 2017 at 6:33 | |||||
| Jan 27, 2017 at 5:54 | history | edited | Artur Jackson | CC BY-SA 3.0 |
Added some extra info that I thought the OP would enjoy.
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| Jan 27, 2017 at 5:49 | history | answered | Artur Jackson | CC BY-SA 3.0 |