Filtrations and cohomology I: crystallization

TL;DR

This paper compares various filtered cohomology theories and demonstrates that de Rham cohomology can be viewed as the crystallization of infinitesimal cohomology, unifying different perspectives in algebraic geometry.

Contribution

It introduces a comparison framework for multiple filtered derived cohomology notions and establishes a new conceptual link between de Rham and infinitesimal cohomology.

Findings

De Rham cohomology is the crystallization of infinitesimal cohomology.

Comparison of HKR-filtered Hochschild homology and Hodge-filtered de Rham cohomology.

Clarifies relationships among different filtered cohomology theories.

Abstract

We compare several different notions of filtered derived commutative ring, discussing HKR-filtered Hochschild homology, Hodge-filtered de Rham cohomology, and the lesser-known Hodge-filtered infinitesimal cohomology. Our main result is that de Rham cohomology is the crystallization of infinitesimal cohomology.