Homotopy Theory

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12:36 AM

Hi all, (hopefully) quick question! Sorry if it's basic but I am a curious grad student.

Do étale morphisms of derived schemes enjoy the same lifting property that (formally) étale morphisms of schemes do? The lifting property in particular is here: https://stacks.math.columbia.edu/tag/02HG and my definition of étale morphisms between derived schemes is due to Toën in DAG: namely, locally of finite presentation and vanishing relative cotangent complex.

So if I have a diagram like the stacks project link, but of derived schemes (maybe only commuting up to homotopy...?), is it true that I g…

The only thing tripping me up is that I'm not sure what "first order thickening" means for a derived scheme. If that's nonsense, maybe just replace it with something like Spec A -> Spec A[t]/t^k for some simplicial ring A.

Oh, and I suppose the whole idea of diagrams only commuting up to homotopy still trips me up as well. :)

7 hours later…

Maxime Ramzi

7:59 AM

@BryanShih did you mean henselian ? I'm not sure what limit you're talking about, but either way, thanks to remark 3.8 (2), the claim holds (let g have coefficients in I and x be the solution to the equation in B. Then x is the unique solution in B/I as well, but there, 0 is a solution, so that x = 0 in B/I - in other words, x is in I)

7 hours later…