Timeline for Why is there no stack of $\ell$-adic sheaves on a curve?
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| Jun 20, 2013 at 4:20 | comment | added | Jacob Lurie | I suppose my original answer is misleading: the "Betti" moduli space is defined over the rational numbers (even over the integers), so you could base change it to any field you like. And when you base change it to the $\ell$-adics, you get something closely related to your $\Loc_{n}^{\ell}(X)$. However, the Betti moduli space depends only on the topology of $X$, not on its complex structure. Geometric Langlands is about the deRham moduli spaces, where there is a connection between the algebraic structure on $X$ and on the moduli space. | |
| Jun 20, 2013 at 4:14 | comment | added | Jacob Lurie | You could make that definition. However, the business about $\infty$-topoi and higher stacks is a red herring, because $BGL_n$ is an ordinary stack. You also don't get very much structure this way: for example, any $\ell$-adic sheaf $F$ with Zariski-dense monodromy will have an open neighborhood in your $Loc^{\ell}_{n}(X)$ which is isomorphic to BG_m. In particular, $Loc^{\ell}_{n}(X)$ doesn't allow you to "move continuously" between nonisomorphic sheaves with dense monodromy. | |
| Jun 20, 2013 at 0:01 | comment | added | anon | where $(X^{et}\otimes \mathbb{Q}_{\ell})^{sch}$ Toen's $\ell$-adic schematic homotopy type of $X$? | |
| Jun 19, 2013 at 23:53 | comment | added | anon | If $X$ is a curve over $\mathbb{C}$ can't you build the moduli of rank $n$ de Rham local systems on $X$ in a straightforward way using $(X,x)^{DR}$, the Toen de Rham homotopy type of $X$. Namely $Loc^{DR}_n(X) = Map( (X,x)^{DR}, BGL_n )$, where we're in the $\infty$-topos of (higher) stacks over $\mathbb{C}$. If $X$ is a curve over a finite field (of characteristic not equal to $\ell$), couldn't we just define $Loc^{\ell}_n(X) = Map ( (X^{et}\otimes \mathbb{Q}_{\ell})^{sch} , BGL_n)$, now in the topos of (higher) stacks over $\mathbb{Q}_{\ell}$. | |
| Jun 19, 2013 at 21:48 | comment | added | anon | Does this mean that the correct replacement in the $\ell$-adic setting should be a rigid analytic stack? Perhaps something akin to Galois deformation spaces in the sense of Mazur? | |
| Jun 19, 2013 at 19:34 | history | answered | Jacob Lurie | CC BY-SA 3.0 |