9:15 AM
I have a silly question. In writing up a current project, which concerns derived schemes which I'd like to model as locally ringed spaces with sheaves of animated rings on them, I have a particular construction on derived affine schemes of finite presentation (which on animated algebras is a left Kan extension). I'd like to globalize this in the most efficient way possible to qcqs schemes. Classically, I'd use a descent datum type of approach, taking qcqs X with explicit atlas U_i
then show compatibility along intersections. To do this for derived schemes however, I don't see anywhere in standard literature (Toen, Gaitsgory-Rozenblyum, Lurie, etc) in any model (DG-algebras, spectra, etc) this approach used to build derived schemes explicitly, so I assume there is a reason that naively using descent datum along affine covers with various weak equivalences is the issue
as the relevant cocycle diagram is only commutative up to homotopy
I guess my question is the following (given the above), if I have a collection of derived affine schemes U_i for which \coprod U_i forms a Zariski atlas for a qcqs scheme X (so there are finitely many U_i, (derived) intersections of U_i's are affine, and also admit finite atlas), then replacing U_i with V_i the image of some functor, how can I check the associated prestack (which I guess I'm not sure how best to describe) is itself a derived scheme?
It feels like something like commuting with principal localization (or perhaps commuting with derived fiber products of affines) should be "enough" but I want to explain the details more precisely and am having trouble explicating this in the standard resources (e.g., I don't want to invoke spectra, nor Lurie's geometries etc)
