Timeline for What do we know about effective epimorphisms of derived affine schemes/manifolds?
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11 events
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| Jul 17 at 9:00 | comment | added | Jon Pridham | It's simply that the coequaliser in (derived) schemes of $\mathrm{SL}_2\times_{\mathbb{A}^2}\mathrm{SL}_2 \Rightarrow \mathrm{SL}_2$ is $\mathbb{A}^2\setminus \{0\}$, so the epimorphism $\mathrm{SL}_2 \to \mathbb{A}^2$ isn't effective. | |
| Jul 17 at 0:41 | comment | added | Brendan Murphy | I guess the idea is that an effective epimorphism of derived schemes should be an effective epi on $\pi_0$, and then should be a surjection on points (which this isn't)? And if I've calculated right then the map $\mathrm{SL}_2 \to \mathbb{A}^2 \setminus \{0\}$ is a universal effective epimorphism of schemes, because these schemes are smooth and surjective + surjective on tangent spaces. Then after taking global section this (maybe) implies that $k[(\mathbb{A}^2 \setminus \{0\})] \to k[\mathrm{SL}_2]$ is an effective monomorphism? | |
| Jul 16 at 23:07 | comment | added | Brendan Murphy | Hi @JonPridham, could you elaborate on why that map is/is not an effective epimorphism? Do you know if it's an effective epimorphism of (non-derived) affine schemes? This old MSE answer about the question was inconclusive math.stackexchange.com/questions/2403350/equalizers-in-cring/… | |
| May 4 at 5:46 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 3 at 20:33 | comment | added | Jon Pridham | @DmitriPavlov Indeed, or see Nuiten's 2017 thesis; my point was just that equivalence isn't a formal consequence of Dold-Kan. | |
| May 3 at 18:00 | comment | added | Dmitri Pavlov | @JonPridham: However, in complete analogy to the Dold–Kan correspondence, the normalized chains functor does induce a Quillen equivalence from simplicial C^∞-rings to dg-C^∞-rings, as shown in Theorem 1.1 of arxiv.org/abs/2303.12699. | |
| May 3 at 15:50 | comment | added | Jon Pridham | For an algebraic analogue, the morphism from $\mathrm{SL}_2$ to $\mathbb{A}^2$ sending a matrix to its first row is an effective epimorphism in affine schemes, but not in schemes or stacks. Derived affine $C^{\infty}$-spaces should be a bit better behaved because gluing works better, but algebraic settings would always take the colimit in sheaves. It's also worth noting that the simplicial/dg equivalence for $C^{\infty}$-rings is much more subtle than Dold-Kan (or even Eilenberg-Zilber, because the algebraic structure isn't operadic). | |
| May 3 at 13:20 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 3 at 13:14 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 3 at 12:53 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 3 at 12:47 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |