Cohomology of Pro-p Demushkin Groups

• Cohomology of pro-p Demushkin groups is characterized by finite H¹, one-dimensional H², and a nondegenerate cup-product yielding a quadratic Poincaré duality (Koszul) algebra.
• The study utilizes the differential graded algebra of continuous cochains and A3-formal minimal models to reveal explicit Massey product computations and obstructions.
• Results show that A3-formality holds when q=0 or q≥5, while for p=3 and q=3, the nonvanishing canonical class indicates a definitive higher order obstruction.

A pro-p Demushkin group is a profinite group of cohomological dimension $2$ with remarkably rich structure, characterized by strong constraints on its low-degree continuous cohomology and a highly nondegenerate cup-product. The study of the cohomology of these groups, particularly in relation to formality properties in the sense of $A_\infty$-algebras, reveals deep connections between algebraic presentations, Poincaré duality, quadratic algebras, Massey products, and the intricate landscape of obstructions in Hochschild cohomology. Recent results provide precise criteria for when the differential graded algebra (DGA) of continuous cochains of a Demushkin group is $A_3$-formal, governed by the so-called $q$-invariant, with explicit calculations of the Benson–Krause–Schwede canonical class as the decisive obstruction (Pál et al., 12 Jan 2026).

1. Definition and Presentation of Pro-p Demushkin Groups

A pro-$p$ group $G$ is called a Demushkin group if it satisfies:

1. $\dim_{\mathbb{F}_p} H^1(G, \mathbb{F}_p) < \infty$,
2. $\dim_{\mathbb{F}_p} H^2(G, \mathbb{F}_p) = 1$,
3. The cup-product

$$H^1(G, \mathbb{F}_p) \times H^1(G, \mathbb{F}_p) \xrightarrow{\;\cup\;} H^2(G, \mathbb{F}_p) \cong \mathbb{F}_p$$

is a nondegenerate bilinear form.

An infinite pro-$p$ Demushkin group admits a presentation on an even number $d \geq 2$ of generators $x_1, \ldots, x_d$, subject to the sole relation

$$x_1^q [x_1, x_2] [x_3, x_4] \cdots [x_{d-1}, x_d] = 1, \quad q = p^f \ \text{or}\ q=0,$$

where $[x, y]=x^{-1}y^{-1}xy$. The integer $q$ is termed the $q$-invariant of $G$ and plays a determining role for formality properties.

2. Cohomology Algebra and Continuous Cochains

The primary object of study is the DGA of continuous cochains $(C^*(G, \mathbb{F}_p), \delta, \cup)$, where $C^n(G, \mathbb{F}_p)$ comprises all continuous functions $G^n \to \mathbb{F}_p$ with the standard inhomogeneous differential: $$(\delta\varphi)(g_1, \dots, g_{n+1}) = \varphi(g_2, \dots, g_{n+1}) + \sum_{i=1}^n (-1)^i \varphi(\dots, g_i g_{i+1}, \dots) + (-1)^{n+1} \varphi(g_1, \dots, g_n).$$ Equipped with the cup-product

$$(\varphi \cup \psi)(g_1, \dots, g_{i+j}) = \varphi(g_1, \dots, g_i)\, \psi(g_{i+1}, \dots, g_{i+j}),$$

this yields a DGA whose cohomology $H^*(G, \mathbb{F}_p)$ is a graded-commutative $\mathbb{F}_p$-algebra. Notably, $H^*(G, \mathbb{F}_p)$ is a quadratic Poincaré duality algebra of formal dimension $2$, and more precisely a Koszul algebra: $$H^*(G, \mathbb{F}_p) \cong T(H^1) / (R), \qquad \dim H^1 = d, \quad \dim H^2 = 1,$$ where $R \subset H^1 \otimes H^1$ is determined by the presentation of $G$. With a basis $\chi_1, \ldots, \chi_d$ for $H^1$, the quadratic relations in $R$ have a canonical explicit basis.

3. $A_3$-Algebras, Minimal Models, and Formality

An $A_3$-algebra over a field $F$ is a graded vector space $A = \bigoplus_{i \geq 0}A^i$ with $A^0 = F$ and structure maps

$$m_1 \!: A \rightarrow A[1], \quad m_2 \!: A \otimes A \rightarrow A, \quad m_3 \!: A^{\otimes 3} \rightarrow A[-1],$$

subject to coherence relations encoding associativity up to homotopy. Every DGA naturally inherits an $A_3$-structure (typically with $m_i = 0$ for $i \geq 3$). The cohomology $H = H^*(A)$ admits a minimal $A_3$-model $(H, m_2, m_3)$ with $m_1 = 0$ and $m_2$ the induced cup-product.

A DGA $A$ is $A_3$-formal if, in its minimal $A_3$-model, $m_3$ may be chosen to vanish. Equivalently, $A$ is an $A_3$-algebra whose $m_3$-obstruction class—the Benson–Krause–Schwede canonical class $\gamma_A$ in Hochschild cohomology,

$\gamma_A \in HH^{3,-1}(H^*(A)),$

satisfies $\gamma_A = 0$.

4. Obstructions, Massey Products, and Hochschild Cohomology

Triple Massey products in $A$ provide explicit manifestations of the $m_3$-structure. The canonical class $\gamma_A$ in Hochschild cohomology accounts for the nontriviality of these higher operations. For pro-$p$ Demushkin groups, one can construct explicit maps: $$f_1: H^1 \to Z^1(G, \mathbb{F}_p), \qquad f_2: R \to C^1(G, \mathbb{F}_p),$$ and define the cocycle

$$\Psi_3(\chi_a, \chi_b, \chi_c) = -\chi_a \cup f_2(\chi_b\chi_c) - f_2(\chi_a \chi_b)\cup \chi_c,$$

representing $\gamma_G \in HH^{3,-1}(H^*)$. The obstruction vanishes if and only if a lift $h: R \to H^1$ exists such that

$$\partial h(\chi_a \chi_b \chi_c) = \Psi_3(\chi_a, \chi_b, \chi_c).$$

The explicit calculation of $\Psi_3$ for pro-$p$ Demushkin groups is achieved by constructing cochain homotopies using continuous homomorphisms $G \to U_3(\mathbb{F}_p)$ and explicit matrix computations in $U_4(\mathbb{F}_p)$.

5. Main Results: $A_3$-Formality Criteria for Demushkin Groups

Let $p$ be an odd prime and $G$ a pro-$p$ Demushkin group with $q$-invariant $q \in \{0, p^f\}$:

• If $q=0$ or $q \geq 5$, then the canonical class $\gamma_{(G, \mathbb{F}_p)}$ vanishes and $(G, \mathbb{F}_p)$ is $A_3$-formal.
• If $p = 3$ and $q = 3$, then $\gamma_{(G, \mathbb{F}_3)} \neq 0$ and $(G, \mathbb{F}_3)$ is not $A_3$-formal.

This dichotomy is established by explicit computation of the relevant cocycle. For $q \neq 3$, all triple-Massey-type cocycles are boundaries; hence the canonical class vanishes. For $p = q = 3$, the obstruction persists, exemplified by the unsolvability of a matrix equation in $U_4(\mathbb{F}_3)$, following Dwyer’s criterion (Pál et al., 12 Jan 2026).

6. Structure and Koszulity of Cohomology Rings

$H^*(G, \mathbb{F}_p)$ is a quadratic, graded-commutative, Poincaré duality algebra of formal dimension $2$. The algebraic presentation can be made explicit:

• Generators: $\chi_1, \ldots, \chi_d$ in $H^1$.
• Relations: quadratic, given by

$$\left\{ \chi_{2k-1} \otimes \chi_{2k} + \chi_{2k} \otimes \chi_{2k-1} \mid k = 1, \dots, d/2 \right\} \cup \{\chi_i \otimes \chi_j \mid |i-j| \neq 1\}.$$

• Cup-product:

$$\chi_i \cup \chi_j = \begin{cases} 0 & |i-j| \neq 1, \ \delta_{i+1, j}\,\omega & j = i+1, \end{cases}$$

where $\omega$ is the Poincaré duality generator in $H^2$.

By a direct argument on Hilbert series, $H^*(G, \mathbb{F}_p)$ is shown to be Koszul, and $HH^{3,-1}(H^*)$ is thus computable on the Koszul complex.

7. Broader Context and Implications

The explicit $A_3$-formality dichotomy for pro-$p$ Demushkin groups at odd primes, governed by the $q$-invariant, delineates the occurrence of higher Massey product obstructions precisely. This result refines understanding of the structure of Galois representations, the role of quadratic algebras in group cohomology, and the realization of $A_\infty$-formality in arithmetic topological contexts. Advanced explicit computations, particularly in unipotent matrix groups over finite fields and their connection to cohomological operations, exemplify the synergy of homological algebra, profinite group theory, and algebraic geometry in modern mathematical investigations (Pál et al., 12 Jan 2026).