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Filtrations and cohomology II: the Gauss-Manin connection

Benjamin Antieau

TL;DR

This work develops a universal Gauss--Manin connection for filtered derived invariants, including Hochschild filtration, infinitesimal cohomology, and derived de Rham cohomology. It constructs complete multiplicative Gauss--Manin filtrations with explicit graded pieces ${gr}^i_{GM} ext{HH}_{fil}(S/k)\\simeq ext{HH}_{fil}(S/R)\otimes_R \text{ins}^i\Lambda^i L_{R/k}[i]$ and analogous expressions for ${\widehat{\mathbbl{Π}}}_{S/k}$ and ${\widehat{dR}}_{S/k}$, together with Griffiths transversality. The paper develops a derived bifiltered-algebra framework, descent theory, and several key applications: nilinvariance in infinitesimal cohomology, the Quillen spectral sequence, a refined HKR filtration, and a conceptual viewpoint on characteristic-zero prismatic cohomology via the Gauss--Manin connection. In particular, it recovers classical Katz--Oda theory in smooth cases, explains t-de Rham comparisons, and provides tools for mixed-characteristic phenomena. The methods unify homological and algebro-geometric structures through a coherent, universal, and computable formalism.

Abstract

We use derived methods to study the Gauss-Manin connection in Hochschild homology, infinitesimal cohomology, and derived de Rham cohomology. As applications, we give new approaches to nilinvariance, the Quillen spectral sequence, and the HKR filtration. We extend the results of Bhatt's work on de Rham cohomology in characteristic zero to infinitesimal cohomology in mixed characteristic and show that the comparison to Hartshorne's algebraic de Rham complex "is" the Gauss-Manin connection. Finally, we explain the main features of prismatic cohomology in characteristic zero via the Gauss-Manin connection.

Filtrations and cohomology II: the Gauss-Manin connection

TL;DR

This work develops a universal Gauss--Manin connection for filtered derived invariants, including Hochschild filtration, infinitesimal cohomology, and derived de Rham cohomology. It constructs complete multiplicative Gauss--Manin filtrations with explicit graded pieces and analogous expressions for and , together with Griffiths transversality. The paper develops a derived bifiltered-algebra framework, descent theory, and several key applications: nilinvariance in infinitesimal cohomology, the Quillen spectral sequence, a refined HKR filtration, and a conceptual viewpoint on characteristic-zero prismatic cohomology via the Gauss--Manin connection. In particular, it recovers classical Katz--Oda theory in smooth cases, explains t-de Rham comparisons, and provides tools for mixed-characteristic phenomena. The methods unify homological and algebro-geometric structures through a coherent, universal, and computable formalism.

Abstract

We use derived methods to study the Gauss-Manin connection in Hochschild homology, infinitesimal cohomology, and derived de Rham cohomology. As applications, we give new approaches to nilinvariance, the Quillen spectral sequence, and the HKR filtration. We extend the results of Bhatt's work on de Rham cohomology in characteristic zero to infinitesimal cohomology in mixed characteristic and show that the comparison to Hartshorne's algebraic de Rham complex "is" the Gauss-Manin connection. Finally, we explain the main features of prismatic cohomology in characteristic zero via the Gauss-Manin connection.

Paper Structure

This paper contains 13 sections, 32 theorems, 88 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Notation.
  4. Acknowledgments.
  5. Symmetric monoidality of (co)homology
  6. The horizontal connection
  7. The filtered horizontal connection
  8. Derived algebraic structures
  9. Descent
  10. Completions and derived infinitesimal cohomology
  11. The Quillen spectral sequence
  12. Another look at the HKR filtration
  13. Characteristic zero prismatic cohomology

Key Result

Theorem 1.1

Let $k\rightarrow R\rightarrow S$ be maps of derived commutative rings.

Figures (2)

  • Figure 1: The bifiltered picture for $\Delta_!(\mathrm{F}^\star M)$ in terms of the filtered pieces of $\mathrm{F}^\star M$. Here, we place $\mathrm{F}^{i,j}(\Delta_!(\mathrm{F}^\star M))$ in the $(-i,j)$-coordinate of the usual $xy$-plane.
  • Figure 4: The double complex picture for the filtered flat filtration $\mathrm{F}^{\star,\star}_\mathrm{h}\mathrm{dR}_{S/k}$.

Theorems & Definitions (87)

  • Theorem 1.1: Proof \ref{['proof:main']}
  • Theorem 1.2: Theorem \ref{['thm:nilinvariance']}
  • Theorem 1.3: Theorem \ref{['thm:charp']}
  • Theorem 1.4: Theorem \ref{['thm:quillen']}
  • Theorem 1.5: Theorem \ref{['thm:hkr']}
  • Theorem 1.6: Theorem \ref{['thm:our_prism']}
  • Proposition 2.1
  • proof
  • Remark 2.3
  • Corollary 2.4
  • ...and 77 more