Divided Powers and Derived De Rham Cohomology

TL;DR

This paper introduces derived divided power algebras to unify and extend classical De Rham and crystalline cohomology theories within derived algebraic geometry, providing new universal characterizations and recovering classical results.

Contribution

It develops the formalism of derived divided power algebras and characterizes derived De Rham cohomology via a universal property, unifying classical and derived theories.

Findings

Recovered classical De Rham cohomology for smooth maps

Established universal property of derived De Rham cohomology

Defined derived crystalline cohomology with classical limits

Abstract

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.