  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.
