Timeline for Infinite dimensional dg-manifolds
Current License: CC BY-SA 4.0
27 events
| when toggle format | what | by | license | comment | |
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| May 11, 2023 at 21:36 | comment | added | Jon Pridham | The aim would be to take $G'=G$. As a starting point, every smooth affine atlas of the underived truncation is the truncation of an atlas for the derived stack (obstructions vanish), but you would want to lift if $G$-equivariantly. | |
| May 11, 2023 at 15:24 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 11, 2023 at 15:22 | comment | added | YkMz | If we can "lift the dg category", I am not sure whether we can replace the underlying stack $[L_1/G]$ with a finite dimensional model $[L'_1/G']$ (i.e., both L'_1 and G' are finite dimensional.). | |
| May 11, 2023 at 15:16 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 10, 2023 at 5:53 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 9, 2023 at 17:09 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 9, 2023 at 9:58 | comment | added | YkMz | What does 'lift the dg category of vector bundles' refer to ? | |
| May 7, 2023 at 14:06 | comment | added | Jon Pridham | The rough idea would be to first lift the dg category of vector bundles, but I haven't checked any details. | |
| May 7, 2023 at 8:02 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 7, 2023 at 7:57 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 7, 2023 at 7:50 | comment | added | YkMz | You mean that a replacement exists in the above example which is a dg-stack constructed by using bundles of curved dgla on $L_1$ ? Then, how can we perform the similar argument ? I am not sure whether the structure sheaf is think of as a dga like affine dg-schemes. | |
| May 5, 2023 at 17:35 | comment | added | Jon Pridham | For dg-stacks, it's rarer for such replacements to exist, but the basic condition you want is for quasi-coherent sheaves on the underlying underived stack to have vanishing positive cohomology, so a similar argument works. An example would be a quotient stack of an affine scheme by a reductive group. | |
| May 5, 2023 at 13:26 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 5, 2023 at 12:26 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 5, 2023 at 12:24 | comment | added | YkMz | The dg-stack I want to study is constructed from bundles of curved dg-lie algebras on $L^1$.In partcular, I am interested in $L$ which is related to A-infinity actions(, as written in the above edited question). | |
| May 5, 2023 at 12:17 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 4, 2023 at 9:28 | comment | added | Jon Pridham | A DGLA on its own won't give a DG stack, as you can only exponentiate $L^0$ infinitesimally. | |
| May 4, 2023 at 3:31 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 3, 2023 at 15:48 | comment | added | YkMz | It seems that the argument you provided would work well in the case of infinite affine DG manifolds. I will try to write down a proof. However, I am also curious about whether an infinite smooth DG stack obtained from the Maurer-Cartan locus of a dg-Lie algebra is quasi-isomorphic to the definition of smooth DG stacks given by Ciocan-Fontanine and Kapranov when the dg stack has a finite classical part and tangent complexes with finite cohomology at each point. | |
| May 3, 2023 at 15:38 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 2, 2023 at 12:47 | comment | added | Jon Pridham | I don't know if anyone's written it down, but the argument is just to construct a finite resolution by hand, exploiting Noetherianity and quasi-compactness. Start by forming a finite quasi-free resolution $F$ of $A$, inductively adding generators to kill the lowest homology of $cone(F->A)$, then at the final stage lift a projective $H_0A$-module by localising $F_0 $ if necessary to ensure the resolution terminates. | |
| May 2, 2023 at 9:54 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 2, 2023 at 2:52 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 1, 2023 at 15:56 | history | edited | YkMz | CC BY-SA 4.0 |
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| May 1, 2023 at 15:48 | comment | added | YkMz | You mean that infinite affine dg-manifolds with our conditions (i.e., having a finite dim classical part and tangent complexes with finite cohomology) are quasi-isomorphic to Ciocan-Fontanine and Kapranov's dg-manifolds ? If so, are their any good references ? | |
| May 1, 2023 at 13:16 | comment | added | Jon Pridham | Yes, you can relax both those conditions and you will still have a derived scheme. It's then possible for those to be quasi-isomorphic to dg-manifolds - for instance an infinite product of dg-manifolds quasi-isomorphic to a point will still be quasi-isomorphic to a point. For affine dg-manifolds, your conditions should also be sufficient. | |
| May 1, 2023 at 9:44 | history | asked | YkMz | CC BY-SA 4.0 |