Revisiting derived crystalline cohomology

We prove that the $\infty$-category of surjections of animated rings is projectively generated, introduce and study the notion of animated PD-pairs - surjections of animated rings with a "derived" PD-structure. This allows us to generalize classical results to non-flat and non-finitely-generated situations. Using animated PD-pairs, we develop several approaches to derived crystalline cohomology and establish comparison theorems. As an application, we generalize the comparison between derived and classical crystalline cohomology from syntomic (affine) schemes (due to Bhatt) to quasisyntomic schemes. We also develop a non-completed animated analogue of prisms and prismatic envelopes. We prove a variant of the Hodge-Tate comparison for animated prismatic envelopes from which we deduce a result about flat cover of the final object for quasisyntomic schemes, which generalizes several known results under smoothness and finiteness conditions.