Finite Group Schemes: Structures & Invariants

• Finite group schemes are algebraic objects defined on finite-type, affine schemes with a group law, incorporating non-reduced and infinitesimal structures.
• They extend classical finite groups by enabling rich cohomological methods such as CFG and support variety theory, linking geometry with representation theory.
• Advanced studies involve deformation theory, essential dimension, and invariant theory, all crucial for understanding modular representations and arithmetic applications.

A finite group scheme is a group scheme over a base ring or field (typically Noetherian or algebraically closed in positive characteristic) whose underlying scheme is affine and of finite type, with coordinate algebra finite as a module over the base. Finite group schemes generalize finite abstract groups by allowing non-reduced structures, such as infinitesimal group schemes and group schemes of positive characteristic. Their paper is foundational in areas such as modular representation theory, algebraic geometry, arithmetic geometry, and the theory of algebraic stacks. This article surveys the core structural, cohomological, and representation-theoretic aspects of finite group schemes, emphasizing recent developments and their implications.

1. Structure and Classification

Finite group schemes can be classified into various types according to their geometric, arithmetic, and structural properties:

• Reduced (étale) vs. Infinitesimal Types: Reduced finite group schemes correspond to finite étale algebras, i.e., schemes representing abstract finite groups. Infinitesimal finite group schemes, characterized by nilpotent behavior of the structure sheaf, arise predominantly in positive characteristic and are closely tied to restricted Lie algebras and Frobenius kernels.
• Trigonalisable and Simple Group Schemes: Over an algebraically closed field of characteristic $p>0$, a group scheme is trigonalisable if it has a normal unipotent subgroup with diagonalizable quotient. Simple finite group schemes associated to generalized Witt algebras (such as $I(m;n)$) exhibit nontrivial connections to Cartan type Lie algebras and can be systematically analyzed via their automorphism group schemes and triangulations (Otabe, 13 Sep 2024).
• Semicommutative Group Schemes: Semicommutative finite group schemes are constructed from commutative ones by iterated semidirect products with commutative kernels and taking quotients by normal subgroups. Constant semicommutative group schemes correspond to those assembled by successively solving split embedding problems with abelian kernels. Invariants such as the $p$-rank, essential dimension, and notion of retract rationality further refine the landscape (Darda et al., 2022, Otabe, 13 Sep 2024, Chang et al., 2016, Fakhruddin, 2019).
• Group Schemes of Square-Free Order: When the order (rank) is square-free, every finite flat group scheme over a base scheme can be decomposed (after a finite étale base change) into a semidirect product of a commutative direct sum of prime order subgroup schemes and a finite étale group scheme (Murty et al., 2016).
• Deformation and Non-commutative Cases: The deformation theory and classification of non-commutative finite flat group schemes (e.g., $G_\lambda$ as a non-abelian semi-direct product of $\alpha_p$ and $\mu_{p^m}$) have direct bearing on questions such as whether group schemes of a given type are killed by their order (affirmed in (Torti, 18 Nov 2024)).

2. Cohomological Methods and Finite Generation

The cohomology of finite group schemes, realized as $H^*(G, A)$ for a $G$-algebra $A$, is a central tool:

• Cohomological Finite Generation (CFG): Building on Friedlander–Suslin’s finite generation of the cohomology of finite group schemes over fields, van der Kallen’s theorem extends CFG to arbitrary Noetherian bases: for any finite group scheme $G$ and finitely generated $G$-algebra $A$, $H^*(G,A)$ is finitely generated as a $k$-algebra (Goméz et al., 3 Oct 2025, Pevtsova, 2014). This result leverages reduction to cohomology for $GL_n$, techniques of power reductivity, and Grosshans filtrations.
• Support Varieties and $\pi$-Points: The spectrum of the cohomology ring, or its projectivization, underlies the support variety theory for the modular representation category of $G$. The concept of $\pi$-points and $\Pi$-support connects the representation theory of $G$-modules with the geometry of $H^*(G, k)$, refining and extending the detection of projectivity and facilitating the classification of thick subcategories (Pevtsova, 2014).
• Generalized Cohomological Invariants: The theory now encompasses refined structures such as non-maximal rank varieties, locally closed subvarieties stratified by Jordan type, and the zero loci of extension classes—all of which enable more intricate stratifications of moduli and representation categories (Friedlander et al., 2011).

3. Representation Theory and Geometric Classification

The modular representation theory of finite group schemes incorporates deep geometric, cohomological, and homological structures:

• Stable Module Categories and Stratification: The stable category of representations admits tensor triangulated structure and is stratified as a module category over the cohomology ring $H^*(G, k)$, with thick tensor ideals classified via the Balmer spectrum, itself homeomorphic to $\mathrm{Spec}\, H^*(G,k)$ for compact objects (Barthel et al., 2023, Pevtsova, 2014). Costratification and fiberwise detection principles enable reduction to the paper of modules over the residue fields.
• Support, Cosupport, and Tensor Category Theory: Explicit formulas describe the supports and cosupports of modules and their tensor products. For example, for $M, N$ in the derived category,

$$\mathrm{supp}_G(M \otimes_R N) = \mathrm{supp}_G M \cap \mathrm{supp}_G N,\quad \mathrm{cosupp}_G\ \mathrm{Hom}(M,N) = \mathrm{supp}_G M \cap \mathrm{cosupp}_G N.$$

This theoretical framework governs the classification of thick and localizing subcategories.

• Classification by Representation Type: The $p$-rank of a finite group scheme influences its representation type: when $p$-rank is at most one, group schemes (modulo their largest linearly reductive normal subgroup) have either finite or domestic representation type (Chang et al., 2016, Farnsteiner, 2012). For principal blocks, classification as domestic or of polynomial growth is equivalent to being Morita equivalent to a trivial extension of a radical square-zero tame hereditary algebra, closely related to binary polyhedral group schemes (Farnsteiner, 2012).
• Generalized F-Signature: For actions on symmetric algebras $S = \operatorname{Sym}(V)$ with "small" group schemes $G$, the $F$-signature of $S^G$ distinguishes linearly reductive and non-linearly reductive cases:

$s(S^G) = \begin{cases} 1/\dim_k k[G] & \text{if $G$ is linearly reductive} \ 0 & \text{otherwise} \end{cases}$

(Hashimoto et al., 2023). Generalizations account for reflexive modules and link to the asymptotic behavior of Frobenius push-forwards.

4. Invariants, Degree Bounds, and Extensions

The invariant theory and extension theory of finite group schemes mirror but also diverge from their abstract group analogues:

• Noether’s Degree Bound and Molien’s Formula: For linearly reductive group schemes, Noether’s bound $\beta(G) \le |G|$ holds for the degrees of generators of the invariant ring. This fails for infinitesimal group schemes, for which no uniform degree bound exists ($\beta(G) = \infty$). The Hilbert–Poincaré series of invariants can be computed using a generalized Molien formula, via Witt vector lifts and compatibility with reductions to characteristic zero (Kemper et al., 30 May 2025).
• Central Extensions and Torsors: Central extensions of finite commutative group schemes by a sheaf $F$ are described via explicit "extension data," where objects are $F$-torsors on $G$ equipped with a trivialization satisfying a cocycle condition. Cohomological exact sequences relate $\operatorname{Ext}_S(G,F)$ to low-degree Hochschild cohomology and Cartier dual torsors, giving concrete computational tools for classification and connections to Selmer groups of abelian varieties (Bruin, 2022).
• Level Structures and Moduli Obstructions: Level structures on group schemes, essential in arithmetic geometry, can be defined canonically for specific group schemes (e.g., Oort–Tate group schemes of rank $p$), but cannot be globally defined on the entire stack of finite flat commutative group schemes due to the failure of functoriality (Guan, 2020).

5. Torsors, Galois Theory, and Arithmetic Applications

Torsors under finite group schemes connect closely with arithmetic statistics, Galois theory, and conjectures in arithmetic geometry:

• Counting Torsors and the Stacky Malle Conjecture: For a finite étale tame $F$-group scheme $G$ over a global field, heights generalizing the discriminant can be defined on $G$-torsors. The Stacky Batyrev–Manin conjecture, and its analogue of the Malle conjecture, predict asymptotics for the number of $G$-torsors of bounded height:

$$\#\{x \in BG(F) : H(x) \le B\} \sim C B^{a(c)} (\log B)^{b(c) - 1}$$

with explicit description of $a(c)$, $b(c)$, and the leading constant involving arithmetic invariants such as Tate–Shafarevich groups (Darda et al., 2022).

• Quantitative Inverse Galois Theory: For semicommutative finite group schemes, there exist positive-density families of connected torsors (i.e., Galois extensions) of bounded height, answering a refined version of the inverse Galois problem. Explicit lower bounds on the counting function are established, e.g., for the alternating group $\mathfrak{A}_4$, the number of connected torsors grows at least as $CB^{1/2}$ (Darda et al., 2022).
• Equidistribution and Adelic Properties: In the commutative case, $G$-torsors are shown to equidistribute in the space of local torsors (an adelic space) with respect to a canonical measure, as the height bound increases. Equidistribution reflects the deep interplay between local and global arithmetic, and underlies the quantitative analysis of rational points on stacks (Darda et al., 2022).

6. Rationality and Essential Dimension

The rationality properties of classifying spaces and the essential dimension of group schemes provide a bridge to stable birational geometry:

• Retract Rationality of $BG$: For finite connected group schemes $G$ of certain types (trigonalizable, or associated with generalized Witt algebras in specific cases), the classifying stack $BG$ is retract rational: its function field is stably birational to that of affine space. This is established via structural results on automorphism group schemes and by reducing rationality to lifting problems for torsors (Otabe, 13 Sep 2024).
• Essential Dimension One: The complete classification of finite group schemes of essential dimension one shows that such schemes can be embedded in $\operatorname{PGL}_2$, have one-dimensional Lie algebra, and lift to $\operatorname{GL}_2$ (Fakhruddin, 2019).

7. Deformation Theory and Order-Killedness

A remarkable structural property, generalizing classical Lagrange's theorem, is established in families:

• Order-Killedness for Deformations: Any finite flat group scheme over a local Artin ring whose special fiber is in a certain explicit family of non-commutative $k$-group schemes is annihilated by its order. The classification of deformations is given in terms of infinitesimal parameters and polynomial deformations of the group law (Torti, 18 Nov 2024).

---

Finite group schemes, through their algebraic, geometric, and cohomological features, serve as a natural interface between algebraic geometry, arithmetic, and the modular representation theory. Current research integrates sophisticated cohomological techniques, explicit geometric models, and arithmetic statistics, deepening the understanding of their structure, classification, and invariants, while forging new connections to homotopical and algebraic stack-theoretic approaches.