Derived Infinitesimal Cohomology

• Derived infinitesimal cohomology is a universal filtered derived commutative cohomology theory that characterizes functions on infinitesimal thickenings via a canonical Hodge filtration.
• It leverages spectral sequences and derived symmetric powers to connect classical de Rham, Hochschild, and prismatic cohomologies in a unified framework.
• Its framework elucidates deformation theory and obstructions in algebraic and complex geometry, providing concrete tools for studying geometric structures.

Derived infinitesimal cohomology is the universal filtered derived commutative cohomology theory that encodes formal thickenings and higher-order infinitesimal neighborhoods in algebraic and complex geometry. It characterizes the derived functorial behavior of functions on infinitesimal thickenings, equipped with a canonical Hodge filtration whose associated graded is given by symmetric powers of the cotangent complex. Derived infinitesimal cohomology forms the connective tissue between classical infinitesimal methods, derived enhancements of de Rham cohomology, and prismatic cohomology, supplying a foundational framework for advances in derived algebraic and complex geometry, arithmetic geometry, and deformation theory.

1. Conceptual Foundations and Formalism

The derived infinitesimal cohomology of a map of commutative associative rings $k \to R$ is canonically constructed in two equivalent ways (Antieau, 18 Nov 2025, Antieau, 3 Nov 2025):

• As the totalization of the cochain complex $R\Gamma((R/k)_{\inf},\mathcal{O})$, where $(R/k)_{\inf}$ is the derived infinitesimal site—objects are $k$-algebras over $R$ with locally nilpotent kernel, and covers are faithfully flat maps.
• As the initial object $F_H^\star(R/k)$ in the $\infty$-category of complete filtered derived commutative $k$-algebras with a graded functor $gr^0$ whose left adjoint is $F_H^\star(-/k)$, the functor assigning Hodge-filtered derived infinitesimal cohomology.

This object possesses a canonical, exhaustive, and complete decreasing Hodge filtration on its underlying module. Formally, for a derived commutative $k$-algebra $R$, one has

$$F_H^\star(R/k) \simeq \bigoplus_{n\geq 0} \mathrm{LSym}_R^n( L_{R/k}[-1])$$

where $L_{R/k}$ is the cotangent complex and $\mathrm{LSym}$ is the derived symmetric power. The $i$th filtered piece is $\bigoplus_{n\geq i}\mathrm{LSym}_R^n( L_{R/k}[-1])$ (Antieau, 3 Nov 2025).

The totalization of $F_H^\star(R/k)$ recovers the complex of global functions on all infinitesimal thickenings of $\Spec R$, and the spectral sequence associated to the filtration has

$$E_1^{p,q} = H^{q}\bigl( \mathrm{LSym}_R^p(L_{R/k}[-1]) \bigr) \implies \pi_{q-p}(F_H^\star(R/k))$$

2. Comparison with Other Cohomology Theories

Derived infinitesimal cohomology interpolates between several classical and modern notions:

• De Rham Cohomology: The process of “crystallization” (i.e., imposing crystalline divided-power conditions on the infinitesimal filtration) produces the derived de Rham complex [(Antieau, 3 Nov 2025), Thm 7.1]:

$$\divideontimes F_H^\star(R/k) \simeq F_H^\star dR_{R/k}$$

For smooth $R$ over $k$ of characteristic zero, the filtration splits, and one recovers the classical Hodge–de Rham spectral sequence, with

$$F_H^i(R/k) \simeq \bigoplus_{n\geq i} \mathrm{Sym}_R^n(\Omega^1_{R/k}[-1])$$

and $F_H^i dR_{R/k} \simeq \Omega^i_{R/k}[-i]$.

• Hochschild Homology: The HKR filtration on Hochschild homology is the Hodge filtration on infinitesimal cohomology relative to $HH(R/k)$ [(Antieau, 18 Nov 2025), Thm 9.1]:

$$HH_{\mathrm{fil}}(R/k) \simeq F_H^\star{}_{R/HH(R/k)}$$

• Prismatic Cohomology: In characteristic zero, prismatic cohomology is equivalent to Hodge-complete derived de Rham with its Gauss–Manin connection, i.e., $R\Gamma((R/A)_{\prism},\mathcal{O}) \simeq F_H^\star{}_{R/A}$ [(Antieau, 18 Nov 2025), Thm 10.2]. In $p$-adic geometry, relative prismatic cohomology is constructed as an envelope over a free $\delta$-algebra on the Hodge-filtered infinitesimal complex (Holeman, 2023).

3. Filtration, Spectral Sequences, and Algebraic Properties

The filtrations of derived infinitesimal cohomology are characterized by:

• Hodge Filtration: Decreasing, complete, and exhaustive by construction. The associated graded in the smooth case is

$$\operatorname{gr}^i F_H^\star(R/k) = \mathrm{LSym}_R^i(L_{R/k}[-1])$$

• Spectral Sequences: The filtered structure produces spectral sequences of the form

$$E_1^{p,q} = \pi_q(\mathrm{LSym}_R^p( L_{R/k}[-1])) \implies \pi_{q-p}( F_H^\star(R/k))$$

and, after crystallization, the classical de Rham spectral sequence

$$E_1^{p,q} = H^{q-p}(R,\Omega^p_{R/k}) \implies H^{q-p}(\widehat{dR}_{R/k})$$

• Koszul-Type Resolutions: For $R=A/I$, $F_H^i(R/k) \simeq I^i$ is the classical Rees algebra filtration. Crystallization produces the divided-power envelope, agreeing with filtered derived de Rham (Antieau, 3 Nov 2025).

The Poincaré lemma holds in this context: for every $R$, $F_H^0(R/k) \simeq k$ (Antieau, 3 Nov 2025).

4. Geometry, Deformations, and Gauss–Manin Connection

Derived infinitesimal cohomology is foundational in deformation theory and obstruction calculus. In the Dolbeault analytic setting, the infinitesimal Dolbeault groups $H^{r,q}_{\inf}(X,n)$ control infinitesimal deformations, higher-order Kodaira–Spencer classes, and analytic obstructions on $n$th-order thickenings of complex varieties (Lin et al., 2023).

The Gauss–Manin connection emerges from the bifiltered structure associated to morphisms $k \to R \to S$ of derived commutative rings:

• The totalization along one filtration direction yields a new filtered complex whose associated graded is

$$\operatorname{gr}^n_{\mathrm{GM}} F_H^\star{}_{S/k} \simeq F_H^\star{}_{S/R} \widehat{\otimes}_R \Lambda^n L_{R/k}[-n]$$

• This induces the canonical Gauss–Manin differential $\nabla$ satisfying Griffiths transversality [(Antieau, 18 Nov 2025), Thm 4.12].

Nilinvariance (Kashiwara’s Lemma), compatibility with completion, and the behavior under regular morphisms (as in the comparison with Hartshorne’s algebraic de Rham complex) reinforce its role as the universal formal thickening invariant.

5. Prismatic and $p$-adic Infinitesimal Geometry

The machinery of derived infinitesimal cohomology underlies the construction of relative prismatic cohomology (Holeman, 2023):

• One applies the free $\delta$-algebra functor to the Hodge-filtered complete infinitesimal cohomology, and then an $I$-adic envelope, yielding the initial $\delta$-algebra with prescribed reduction, thereby matching the universal property of Bhatt and Scholze’s prismatic cohomology.
• This approach bypasses site-theoretic methods, enabling a purely algebraic construction of prismatic cohomology from the bar–Čech–Alexander presentation of infinitesimal cohomology.

In characteristic $p$, for perfect (quasisyntomic) $k$-algebras $R$, the infinitesimal cohomology coincides with the Rees algebra $I^{\star} R^\flat$ (perfectoidization), matching known structures in prismatic theory (Antieau, 3 Nov 2025).

6. Applications and Interactions with Representation Theory

Infinitesimal and derived cohomologies govern the structure and support varieties of representations of finite group schemes, Drinfeld doubles, and related Hopf algebras. For Drinfeld doubles of Frobenius kernels, the finite generation of cohomology, the structure of spectral sequences, and the explicit algebra map involving support varieties in terms of Lie algebra data are controlled by deformation classes that are infinitesimal in nature (Friedlander et al., 2017).

In the field of tensor categories, derived infinitesimal techniques inform the deformation theory of monoidal functors and tensorators, explicitly appearing in calculations of higher Davydov–Yetter cohomology and the tangent spaces to varieties of $R$-matrices for Hopf algebras (Faitg et al., 28 Nov 2024).

7. Further Developments and Research Directions

Research on derived infinitesimal cohomology is rapidly advancing along several axes:

• Further refinement of the adjoint functor presentations and monad structures on filtered derived commutative rings.
• Systematic studies of comparability between prismatic, infinitesimal, and other integral $p$-adic cohomologies across mixed and equal characteristic settings.
• Generalization of the Gauss–Manin framework to relative and logarithmic contexts, with applications to noncommutative, higher, and derived geometry.
• Extension of analytic and Dolbeault–type infinitesimal theories to the paper of higher order obstructions and moduli of complex structures (Lin et al., 2023).

The theory of derived infinitesimal cohomology provides both concrete computational tools—such as spectral sequences, support variety maps, and explicit obstruction formulas—and a universal categorical language for understanding the passage from local to global in the infinitesimal topology of derived algebraic geometry. Its unifying role among deformation theory, arithmetic geometry, and derived complex geometry positions it as a central object in modern mathematics (Antieau, 18 Nov 2025, Antieau, 3 Nov 2025, Holeman, 2023, Friedlander et al., 2017, Lin et al., 2023, Faitg et al., 28 Nov 2024).