Divided Powers and Derived De Rham Cohomology

We develop the formalism of derived divided power algebras, and revisit the theory of derived De Rham and derived crystalline cohomology in this framework. We characterize derived De Rham cohomology of a derived commutative algebra $A$ over a base $R$, together with the Hodge filtration on it, in terms of the universal property as the largest filtered divided power thickening of $A$. We show that our approach recovers the classical De Rham cohomology in the case of a smooth map $R\rightarrow A$, and therefore in general, recovers the derived De Rham cohomology in the sense of Illusie. Along the way, we develop some generalities on square-zero extensions and derivations in derived algebraic geometry and apply them to give the universal property of the first Hodge truncation of the derived De Rham cohomology. Finally, we define derived crystalline cohomology relative to a general divided power base, show that it satisfies the main properties of the crystalline cohomology and coincides with the classical crystalline cohomology in the smooth case.