Group-Scheme Cohomology

• Group-scheme cohomology is a generalization of classical group cohomology to affine group schemes in Tannakian contexts, extracting invariants from representations.
• It employs Ext groups in tensor categories to relate de Rham cohomology with differential, algebraic, and topological invariants in diverse geometric settings.
• The theory underpins applications in differential Galois theory, moduli of connections, and deformation problems using spectral sequences, duality, and homotopy invariance.

Group-scheme cohomology is the generalization of classical group cohomology to the setting where the group is replaced by an affine group scheme, typically over a field of characteristic zero. This theory, central to modern Tannakian formalism, underpins the study of differential Galois groups, the fundamental groups of algebraic and noncommutative spaces, and their representations. It enables the extraction of cohomological invariants from actions of these group schemes and provides the crucial link between geometric, representation-theoretic, and differential-topological phenomena.

1. Affine Group Schemes and Representations

Let $G$ be an affine group scheme over a field $k$ of characteristic zero. The category $\operatorname{Rep}(G)$ of finite-dimensional (or ind-) linear representations of $G$ over $k$ is a rigid symmetric tensor category. Group-scheme cohomology for a $G$-representation $V$ is defined as

$$H^i(G, V) := \operatorname{Ext}^i_{\operatorname{Rep}(G)}(k, V),$$

where the functor $V \mapsto V^G = \operatorname{Hom}_G(k, V)$ is left exact, and $H^i(G, V)$ is its right-derived functor. This framework generalizes the notion of group cohomology for abstract groups to the Tannakian and algebraic context, enabling the use of injective resolutions in $\operatorname{Rep}(G)$ and compatibility with spectral sequences and duality theories (Bao et al., 25 Mar 2025).

2. Tannakian Formalism and Differential Fundamental Groups

Let $X$ be a smooth, projective, geometrically connected curve over a field $k$ of characteristic zero. The abelian tensor category $\operatorname{MIC}^c(X)$ consists of finite-rank algebraic vector bundles on $X$ endowed with $k$-linear integrable connections. For a fixed $k$-point $x \in X(k)$, the fiber functor $\omega_x$ assigns to each object its fiber at $x$: $\omega_x \colon (E, \nabla) \mapsto E|_x.$ By Tannakian duality, $\operatorname{MIC}^c(X)$ is equivalent as a tensor category to $\operatorname{Rep}_k(G)$, where $G = \pi_1^{\mathrm{diff}}(X, x)$ is the affine group scheme

$G = \operatorname{Aut}^{\otimes}(\omega_x).$

This group scheme, known as the differential (or Tannakian) fundamental group, encodes the monodromy of connections on $X$ and governs the category of their representations (Bao et al., 25 Mar 2025).

3. Group-Scheme Cohomology and Comparison Theorems

Given a vector bundle with integrable connection $(E, \nabla)$ corresponding to $V = E|_x$ in $\operatorname{Rep}(G)$, there is a canonical comparison map between group-scheme cohomology and the de Rham cohomology of $E$: $$\delta^i \colon H^i(G, V) \;\longrightarrow\; H^i_{\mathrm{dR}}(X, E).$$ For curves of genus $g \geq 1$, this map is always an isomorphism for all $i \geq 0$, reflecting the fact that the Tannakian fundamental group is a "de Rham $k(\pi,1)$-space". Explicitly, this yields:

• $H^0(G, V) \cong H_{\mathrm{dR}}^0(X, E)$, the space of horizontal sections,
• $H^1(G, V) \cong H_{\mathrm{dR}}^1(X, E)$, the de Rham cohomology in degree one,
• $H^2(G, V) \cong H_{\mathrm{dR}}^2(X, E)$, which is one-dimensional for the trivial connection, and
• $H^i(G, V) = 0$ for $i > 2$ (Bao et al., 25 Mar 2025).

This correspondence parallels classical topology — for $X = C$ a Riemann surface, $\pi_1^{\mathrm{diff}}(C)$ and its cohomology coincide, via this equivalence, with the traditional topological invariants.

4. Group-Scheme Cohomology in Differential Galois Theory

In differential algebra, the notion of the differential Galois group $G_{\mathrm{PV}}$ of a linear differential equation over a differential field $K$ is that of an affine group scheme over the field of constants $C$. The category of finite-dimensional differential modules is Tannakian, with the fiber functor given by the space of solutions. The correspondence is now: $$H^i(G_{\mathrm{PV}}, V) = \operatorname{Ext}^i_{\operatorname{DiffMod}(K)}(K, E),$$ for the differential module $E$ corresponding to $V$. Moreover, the "general" Galois group introduced by Umemura and its functorial cohomology coincide with the Picard–Vessiot Galois group and its cohomology (Saito, 2012). Thus, group-scheme cohomology provides the natural setting for extension problems, deformation theory, and computation of differential invariants in this algebraic context.

5. Group-Scheme Cohomology for Noncommutative Spaces

Van Suijlekom and Winkel extend the concept of fundamental group and its cohomology to noncommutative ("NC") spaces described by differential graded algebras (dgas). The category of finitely generated projective bimodules with flat bimodule connection over a dga $(A, d)$ forms a neutral Tannakian category under suitable conditions. Its Tannakian dual is an affine group scheme $\pi_1(A, d)$, and group-scheme cohomology $H^i(\pi_1(A, d), V)$ classifies extensions and higher invariants of flat bimodules in this setting (Suijlekom et al., 2019). For commutative dgas such as the de Rham complex of a manifold, this recovers standard representation-theoretic cohomology; for noncommutative structures (e.g., the noncommutative torus), one obtains the pro-algebraic "envelope" of the topological group underlying the NC space.

6. Fundamental Properties and Vanishing Theorems

Key structural results for group-scheme cohomology include:

• Cohomological dimension: For $\pi_1^{\mathrm{diff}}(X)$ of a smooth projective curve, the cohomological dimension is $2$, i.e., $H^i(G, V) = 0$ for $i > 2$ (cf. classical $\pi_1$ of a complex curve).
• Functoriality: Group-scheme cohomology is functorial under group scheme morphisms induced by morphisms of spaces or dgas.
• Morita and Homotopy Invariance: The cohomology theory is preserved under derived Morita equivalence of dgas and homotopy equivalence of spaces, reflecting invariance under "noncommutative" topological transformations (Suijlekom et al., 2019).

7. Computational Tools and Applications

Computation of group-scheme cohomology leverages spectral sequences (e.g., the Hodge–de Rham spectral sequence), duality (Poincaré pairing), and exact sequences arising from filtrations or Galois correspondences. Applications include:

• Analysis of obstructions to deformation of analytic structures,
• Classification of differential equations up to gauge equivalence,
• Study of moduli of flat connections,
• Investigation of the role of exotic spheres and smoothing invariants in diffeomorphism groups (Wang, 2023, Wang, 7 Apr 2025),
• Computation of invariants in the noncommutative geometry of dgas and quantum spaces (Suijlekom et al., 2019).

Group-scheme cohomology thus serves as the organizing principle for the algebraic and topological study of fundamental groups, both classical and generalized, across a spectrum of geometric, algebraic, and analytic contexts.