Filtrations and cohomology I: crystallization
Benjamin Antieau
TL;DR
This paper develops a unified framework for filtrations on derived commutative rings and analyzes three key filtered cohomology theories: HKR-filtered Hochschild homology $\mathrm{HH}_{\mathrm{fil}}(R/k)$, Hodge-filtered derived de Rham cohomology $\mathrm{F}^\star_\mathrm{H}\mathrm{dR}_{R/k}$, and Hodge-filtered infinitesimal cohomology $\mathrm{F}^\star_\mathrm{H}{\boldsymbol{\Pi}}_{R/k}$, along with their completions. It establishes graded identifications $\mathrm{gr}^i\mathrm{HH}(R/k)\simeq\Lambda^i\mathrm{L}_{R/k}[i]$, $\mathrm{gr}^i\mathrm{dR}_{R/k}\simeq\Lambda^i\mathrm{L}_{R/k}[-i]$, and $\mathrm{gr}^i{\boldsymbol{\Pi}}_{R/k}\simeq\mathrm{LSym}^i_R(\mathrm{L}_{R/k}[-1])$, all expressed via derived functors $\mathrm{L}_{R/k}$ and $\mathrm{LSym}$. The central contribution is the crystallization adjunction, producing a natural map $\mathrm{F}^\star_\mathrm{H}{\boldsymbol{\Pi}}_{R/k}\to\mathrm{F}^\star_\mathrm{H}\mathrm{dR}_{R/k}$ for nonnegatively filtered objects, and, upon completion, identifying $\mathrm{gr}^*\mathrm{HH}_{\mathrm{fil}}(R/k)[-2\*,}$ with $\mathrm{F}^\star_\mathrm{H}\widehat{\mathrm{dR}}_{R/k}$. The paper also develops the machinery of filtered/graded derived rings (via filtered/graded monads and polynomial calculus), analyzes characteristic-$p$ phenomena, and outlines a program for future work on Gauss–Manin connections and derived cohomology theories (F&C.II, etc.). This crystallization perspective connects infinitesimal and crystalline viewpoints and sets the stage for a deeper understanding of filtrations in prismatic and related 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.