Timeline for Reconstruct a variety from the category of locally free sheaves
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 15, 2022 at 9:59 | history | edited | user494739 | CC BY-SA 4.0 |
deleted 36 characters in body
|
| Nov 15, 2022 at 1:14 | comment | added | Martin Brandenburg | @DanielLoughran A category is more than just its objects | |
| Nov 15, 2022 at 0:51 | comment | added | LSpice | Your post ends mid-sentence: "In particular, we are interested in". | |
| Nov 15, 2022 at 0:50 | history | edited | LSpice | CC BY-SA 4.0 |
Typo
|
| Nov 14, 2022 at 23:39 | comment | added | SashaP | @DanielLoughran The set of isomorphism classes of objects does not remember much indeed, but the data of the category (which includes all morphisms between modules) does recover $R$: in the PID example you can recover $R$ as the endomorphisms of the only non-zero (i.e. non-initial) object that is not a coproduct of two non-zero objects. | |
| Nov 14, 2022 at 23:36 | answer | added | SashaP | timeline score: 2 | |
| Nov 14, 2022 at 20:50 | comment | added | Daniel Loughran | Can someone please explain why the following isn't a counter-example? Take $R$ a PID. Then any finitely generated locally free $R$ module is free. So it seems that the category of locally free $R$ modules contains very little information in this case. | |
| Nov 14, 2022 at 17:14 | comment | added | Martin Brandenburg | Some unfinished thoughts: Let $X$ be a scheme with $\mathbf{Vect}(X) \simeq \mathbf{Vect}(\mathbb{P}^n)$ as $k$-linear tensor categories. There is an invertible object $\mathcal{L} \in \mathbf{Vect}(X)$ and morphisms $s_0,\dotsc,s_n : \mathcal{O}_X \to \mathcal{L}$ which correspond to $\mathcal{O}(1)$ and $t_0,\dotsc,t_n$ under the equivalence. Clearly $(s_0,\dotsc,s_n) : \mathcal{O}_X^n \to \mathcal{L}$ is an epi in $\mathbf{Vect}(X)$, but in order to construct $f : X \to \mathbb{P}^n$ with the universal property of $\mathbb{P}^n$, we need that it is an epi in $\mathbf{Qcoh}(X)$. Unclear! | |
| Nov 14, 2022 at 14:31 | history | edited | user494739 | CC BY-SA 4.0 |
added 179 characters in body
|
| Nov 14, 2022 at 13:01 | comment | added | Martin Brandenburg | The affine case is easy since a commutative ring R is not just (this is well-known) isomorphic to the center of the category of R-modules, but - with the same proof - also of the category of finitely generated projective R-modules. (ncatlab.org/nlab/show/center) | |
| Nov 14, 2022 at 4:45 | comment | added | Martin Brandenburg | Of course you want to consider schemes with the resolution property, which means there are enough locally free sheaves. Projective schemes have this property, though. But the main problem here is that exact sequences in $\mathbf{Vect}(X)$ can be "weird", by which I mean that they don't have to be exact in $\mathbf{Qcoh}(X)$. For example, $2 : \mathbb{Z} \to \mathbb{Z}$ is an epimorphism in the category of all locally free $\mathbb{Z}$-modules, in fact all torsionfree $\mathbb{Z}$-modules, but of course not in the category of all $\mathbb{Z}$-modules. This already nukes many "obvious proofs". | |
| Nov 14, 2022 at 4:43 | comment | added | Martin Brandenburg | First of all, Rosenberg's reconstruction theorem holds for all quasi-separated schemes (see arxiv.org/abs/1310.5978). | |
| Nov 13, 2022 at 19:52 | comment | added | user494739 | Thank you for your advice, just edited. | |
| Nov 13, 2022 at 19:51 | history | edited | user494739 | CC BY-SA 4.0 |
added 114 characters in body
|
| Nov 13, 2022 at 19:41 | comment | added | Neil Strickland | It will make a lot of difference whether you regard $\text{Vect}_a(X)$ as just a category, or a $\mathbb{C}$-linear category, or a $\mathbb{C}$-linear tensor category, so you should specify that. | |
| S Nov 13, 2022 at 19:23 | review | First questions | |||
| Nov 13, 2022 at 20:07 | |||||
| S Nov 13, 2022 at 19:23 | history | asked | user494739 | CC BY-SA 4.0 |