Hostname: page-component-59f8fd8595-gtxf7 Total loading time: 0 Render date: 2023-03-23T00:05:14.441Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Generalized Bockstein maps and Massey products

Published online by Cambridge University Press:  23 January 2023

Yeuk Hay Joshua Lam
Institut des Hautes Études Scientifiques, 35 Route de Chartres, Bures-sur-Yvette 91440, France; E-mail:
Yuan Liu
Department of Mathematics, University of Illinois Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801; E-mail:
Romyar Sharifi
Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095; E-mail:
Preston Wake
Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824; E-mail:
Jiuya Wang
Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, GA 30602; E-mail:


Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of $\mathbb {Q}$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology.

Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (, which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
© The Author(s), 2023. Published by Cambridge University Press

1 Introduction

At its essence, this paper revolves around the fundamental question:

How does the continuous cohomology of a profinite group G with compact coefficients compare with the cohomology of an open normal subgroup N?

As a starting point, if G has cohomological dimension d, then corestriction induces an isomorphism from the $G/N$ -coinvariants of a dth cohomology group of N to the dth cohomology group of G with the same coefficients. We view this corestriction map as the first of a sequence of generalized Bockstein maps $\Psi ^{(n)}$ for $n \ge 0$ , which we extend to closed N by considering Iwasawa cohomology. The powers of the augmentation ideal I of a completed group ring of $G/N$ yield a natural filtration on the domain of corestriction. In Section 2, we show that the nth graded piece of this augmentation filtration is isomorphic to the cokernel of $\Psi ^{(n)}$ , employing two purely homological results of Appendix A in the proof. In Section 4, we demonstrate how, in many cases, the image of $\Psi ^{(n)}$ is described by n-fold Massey products.

Massey products were first introduced by Massey [Reference MasseyMa] as a tool for proving that two topological spaces are not homotopy equivalent even when they have isomorphic cohomology rings. The best-known example involves the complement of the Borromean rings in $\mathbb {R}^3$ , three pairwise unlinked circle which are nonetheless linked, resulting in a nontrivial Massey triple product in the second cohomology. In algebra, Massey products are used to study properties of a group G that are not detected by the group cohomology ring itself. Massey products of tuples of homomorphisms on G valued in a ring R are obstructions in $H^2(G,R)$ to lifting homomorphisms to unipotent matrices from the quotient by the center.

Our initial motivation for studying this question came from Iwasawa theory. Indeed, Galois groups of number fields with restricted ramification above a prime p have p-cohomological dimension equal to $2$ , and their second cohomology groups with coefficients in p-power roots of unity are closely related to ideal class groups. In such a setting, the fundamental question above translates to comparing ideal class groups as one goes up a tower of fields, the original question of Iwasawa theory. In this vein, Mazur [Reference MazurMz] described an analogy between knot complements in $3$ -manifolds and Galois groups with restricted ramification, relating the Alexander polynomial of a knot and a characteristic ideal of an inverse limit of class groups. Morishita explored this analogy in terms of Massey products (see, for example, [Reference MorishitaMo]).

The third author studied Massey products in an Iwasawa-theoretic context, relating them to the structure of augmentation-graded pieces of limits of class groups in a nonabelian tower of Kummer extensions [Reference SharifiSh2]. This paper distills the purely algebraic results of the latter paper from their number-theoretic application. The distinct perspective using generalized Bockstein maps, that we introduce here, allows us to go beyond the procyclic setting of [Reference SharifiSh2].

Massey products of length n are defined only if certain $(n-1)$ -fold Massey products vanish. Even when defined, there is some indeterminacy in their definition, resulting from a choice of defining system, a homomorphism to the quotient of the $(n+1)$ -dimensional unipotent matrices by their center. The Massey product provides the obstruction to lifting this homomorphism to the full unipotent group. In order to view n-fold Massey products as values of $\Psi ^{(n)}$ , we define an appropriate notion of a proper defining system, reducing the aforementioned indeterminacy. The requisite definitions are given in some generality in Section 3, providing the framework for the comparison with $\Psi ^{(n)}$ in specific cases described in Section 4.

In Section 5, we demonstrate how our methods can be used to derive concrete arithmetic results by proving lower bounds on the p-ranks of class groups of finite p-ramified bicyclic and Heisenberg extensions of $\mathbb {Q}(\mu _p)$ . Though we eschew Iwasawa-theoretic applications in this paper to ground our study, a description of the augmentation filtrations of inverse limits of p-parts of class groups in $\mathbb {Z}_p$ -extensions, derived using our methods, may be found in [Reference SharifiSh4].

We also consider applications of generalized Bockstein maps to the study of absolute Galois groups of fields. Many algebraic properties of absolute Galois groups are encoded cohomologically as properties of the norm residue symbol. The celebrated norm residue isomorphism theorem of Voevodsky and Rost [Reference VoevodskyVo] (formerly the Milnor-Bloch-Kato conjecture), describes cohomology rings of absolute Galois groups with coefficients in twists of roots of unity as Milnor K-rings of the fields.

The Massey vanishing conjecture of Mináč and Tân [Reference Mináč and TânMiTa4] goes beyond the cohomological ring structure to posit that, for $n \ge 3$ , all definable n-fold Massey products with $\mathbb {F}_p$ -coefficients vanish for some choice of defining system. Earlier work of Hopkins–Wickelgren [Reference Hopkins and WickelgrenHoWi] had established this for $n = 3$ and $p = 2$ over number fields. The full $n = 3$ case of this conjecture is the triple Massey vanishing theorem of Efrat–Matzri [Reference Efrat and MatzriEfMa] and Mináč–Tân [Reference Mináč and TânMiTa3]. The introduction to Section 6 provides a more detailed, yet still incomplete, summary of the history of and rapid progress in this area. In that section, we show that certain algebraic properties of absolute Galois groups are naturally expressed in terms of generalized Bockstein maps. This perspective enables us to give a new proof for odd primes of the triple Massey vanishing theorem.

Mináč and Tân originally formulated the Massey vanishing conjecture, in part, as a way to help cohomologically characterize which profinite groups are isomorphic to absolute Galois groups of fields. We suspect that generalized Bockstein maps have an important role to play in formulating and understanding such cohomological characterizations.

We next provide a more detailed overview of our main results.

1.1 Comparing cohomology using generalized Bockstein maps

Let G be a profinite group of p-cohomological dimension $d \ge 1$ . Let H be a finitely generated pro-p quotient of G by a closed normal subgroup N. Let denote the completed $\mathbb {Z}_p$ -group ring of H, the inverse limit of the $\mathbb {Z}_p$ -group rings of the finite quotients of H. Let T be a finitely generated $\mathbb {Z}_p$ -module with a continuous action of G. This paper is concerned with the study of connecting maps in the continuous G-cohomology of the augmentation filtration of the tensor product $T \otimes _{\mathbb {Z}_p} \Omega $ . That is, if $I = \ker (\Omega \to \mathbb {Z}_p)$ denotes the augmentation ideal of $\Omega $ , then we have exact sequences

(1.1) $$ \begin{align} 0 \to T \otimes_{\mathbb{Z}_p} I^n/I^{n+1} \to T \otimes_{\mathbb{Z}_p} \Omega/I^{n+1} \to T \otimes_{\mathbb{Z}_p} \Omega/I^n \to 0 \end{align} $$

for each $n \ge 1$ , such that $\Omega /I^n$ is $\mathbb {Z}_p$ -flat. Our interest lies in the connecting homomorphisms

$$ \begin{align*} \Psi^{(n)} \colon H^{d-1}(G,T \otimes_{\mathbb{Z}_p} \Omega/I^n) \to H^d(G,T) \otimes_{\mathbb{Z}_p} I^n/I^{n+1} \end{align*} $$

attached to these sequences, which we refer to as generalized Bockstein maps, due to their similarlity to usual Bockstein maps for exact sequences of p-power order cyclic groups.

We can use the Bockstein maps to partially describe the second Iwasawa cohomology group $H^d_{\mathrm {Iw}}(N,T)$ of N with T-coefficients. This cohomology group is the inverse limit of the groups $H^d(U,T)$ under corestriction maps, where U runs over the open normal subgroups of G containing N. It is naturally endowed, through the $\mathbb {Z}_p[G/U]$ -actions on each $H^d(U,T)$ , with the structure of an $\Omega $ -module. We prove that the cokernels of the generalized Bockstein maps describe the graded quotients in the augmentation filtration of $H^d_{\mathrm {Iw}}(N,T)$ (see Theorem 2.2.4).

Theorem A. There are canonical isomorphisms

$$ \begin{align*} \frac{I^nH^d_{\mathrm{Iw}}(N,T) }{I^{n+1}H^d_{\mathrm{Iw}}(N,T)} \cong \frac{H^d(G,T) \otimes_{\mathbb{Z}_p} I^n/I^{n+1}}{\operatorname{\mathrm{im}}\Psi^{(n)}}. \end{align*} $$

The proof rests on an Iwasawa-cohomological version [Reference Lim and SharifiLiSh, Reference Fukaya and KatoFuKa] of a descent spectral sequence of Tate, applied to the terms of our exact sequences for the augmentation filtration of $\Omega $ . We verify the compatibility of these spectral sequences with generalized Bockstein maps and a connecting map in the H-homology of the $\mathbb {Z}_p$ -tensor product of $H^d_{\mathrm {Iw}}(N,T)$ with (1.1).

1.2 A brief primer on Massey products

Given a commutative ring R, a Massey product $(\chi _1, \ldots , \chi _n)$ of n homomorphisms $\chi _1, \ldots , \chi _n$ in $H^1(G,R)$ is an element of $H^2(G,R)$ that provides the obstruction to a certain problem of lifting a homomorphism formed using the tuple of characters $\chi _i$ to a homomorphism $\rho \colon G \to \mathrm {U}_{n+1}(R)$ of G to the group of $(n+1)$ -dimensional unipotent matrices in R, with $\chi _i$ providing the ith off-diagonal entry $\rho _{i,i+1}$ .

More precisely, a defining system for a Massey product $(\chi _1, \ldots , \chi _n)$ is a homomorphism $\rho \colon G \to \mathrm {U}^{\prime }_{n+1}(R)$ to the quotient of $\mathrm {U}_{n+1}(R)$ by its center, with $\rho _{i,i+1} = \chi _i$ . The Massey product $(\chi _1, \ldots , \chi _n)_{\rho }$ of $\chi _1, \ldots , \chi _n$ relative to the defining system $\rho $ is the class in $H^2(G,R)$ of the $2$ -cocycle

$$ \begin{align*} F \colon (\sigma,\tau) \mapsto \sum_{i=1}^n \rho_{1,i}(\sigma)\rho_{i,n+1}(\tau). \end{align*} $$

It vanishes if and only if $\rho $ lifts to a homomorphism $\tilde {\rho } \colon G \to \mathrm {U}_{n+1}(R)$ . In other words, the Massey product relative to $\rho $ is the obstruction to choosing the remaining upper right-hand entry $\tilde {\rho }_{1,n+1}$ to make $\tilde {\rho }$ a homomorphism, which is exactly to say that $d\tilde {\rho }_{1,n+1} = -F$ .

An n-fold Massey product $(\chi _1, \ldots , \chi _n)$ is said to be defined if a defining system for it exists. For $n=2$ , the Massey product is defined and equals the cup product $\chi _1 \cup \chi _2$ . For $n \ge 3$ , a Massey product need not be defined, and even if it is, it may have indeterminacy in its values, coming from the different choices of defining systems. A Massey product is said to contain zero or vanish if it has a defining system for which the Massey product is zero.

We shall work with profinite groups and compact coefficient rings, so our Massey products take values in continuous cohomology groups, and all cocycles and homomorphisms involved are required to be continuous. In fact, we shall allow more general Massey products valued in modules over a group ring, replacing the group of unipotent matrices with an analogous object in a generalized matrix algebra.

1.3 The images of generalized Bockstein maps

The case $d = 2$ and $H \cong \mathbb {Z}_p$ of Theorem A was first studied in [Reference SharifiSh2] from a different perspective and applied in an Iwasawa-theoretic context. Its main result has a similar form to Theorem A, but in place of the image of $\Psi ^{(n)}$ , it has a group of values of certain $(n+1)$ -fold Massey products. We relate the image of $\Psi ^{(n)}$ to Massey products for a variety of groups H.

In the situation of Section 1.1 with $H \cong \mathbb {Z}_p$ , the quotient map $G \to H$ can be thought of as an element $\chi \in H^1(G,\mathbb {Z}_p)$ . This context was considered in [Reference SharifiSh2], and a result like Theorem A is proven, but with the image of $\Psi ^{(n)}$ replaced by $(n+1)$ -fold Massey products of the form $(\chi ,\chi ,\dots ,\chi , \,\cdot \,)$ with respect to certain ‘proper’ defining systems. In Section 4.3, we show that, in the case that H is procyclic, the image of $\Psi ^{(n)}$ is generated by these same Massey products. In other words, when H is procyclic, Theorem A recovers the main result of [Reference SharifiSh2].

This raises the question of whether the relation between the values of generalized Bockstein maps and Massey products can be extended from procyclic H to more general groups. The most difficult step is to determine the appropriate notion of proper defining system. The key insight is that the proper defining systems of [Reference SharifiSh2] are those defining systems that, in a sense, partially have group-theoretic origin. That is, if H is procyclic, then for every $n>0$ , there is a group homomorphism we call the unipotent binomial matrix homomorphism

$$\begin{align*}\genfrac{[}{]}{0pt}{}{\,\cdot\,}{n} \colon H \to \mathrm{U}_{n+1}(\mathbb{Z}_p) \end{align*}$$

defined by sending a generator of H to the unipotent matrix with all $1$ ’s on the diagonal and off-diagonal and $0$ ’s elsewhere (the notation is meant to evoke binomial coefficients, the nonzero entries of $\genfrac {[}{]}{0pt}{}{x}{n}$ being binomial coefficients, see Section 4.2). A defining system $\rho \colon G \to \mathrm {U}^{\prime }_{n+2}(\mathbb {Z}_p)$ for the $(n+1)$ -fold Massey product $(\chi ,\chi ,\dots ,\chi , \cdot )$ is called proper if its restriction to the upper-left copy of $\mathrm {U}_{n+1}(\mathbb {Z}_p)$ in $\mathrm {U}^{\prime }_{n+2}(\mathbb {Z}_p)$ equals $\genfrac {[}{]}{0pt}{}{\,\cdot \,}{n} \circ \chi $ .

This suggests considering defining systems that are, at least partially, of group-theoretic origin. Let $n \ge 0$ , and let $a,b\ge 0$ be, such that $a+b=n$ . Let

$$\begin{align*}\phi \colon H \to \mathrm{U}_{a+1}(\mathbb{Z}_p), \quad \theta \colon H \to \mathrm{U}_{b+1}(\mathbb{Z}_p) \end{align*}$$

be group homomorphisms. By precomposition with $G \to H$ , these define an n-tuple of elements of $H^1(G,\mathbb {Z}_p)$ . We call that pair $(\phi , \theta )$ a partial defining system for $(n+1)$ -fold Massey products involving this n-tuple of characters. Our main general result, Theorem 3.3.4, is that a partial defining system together with a cocycle $f \in Z^1(G,T \otimes _{\mathbb {Z}_p} \Omega /I^n)$ constitutes a defining system. Moreover, a partial defining system defines a homomorphism of G-modules

$$\begin{align*}p_{\phi,\theta} \colon T \otimes_{\mathbb{Z}_p} I^n/I^{n+1} \to T, \end{align*}$$

such that $p_{\phi ,\theta }(\Psi ^{(n)}([f]))\in H^2(G,T)$ is the $(n+1)$ -fold Massey product associated to the defining system given by $(\phi ,\theta )$ and f (see Theorem 4.1.2).

We apply this general machinery to the procyclic case $H \cong \mathbb {Z}_p$ in Section 4.3, taking $(a,b) = (n,0)$ and $\phi = \genfrac {[}{]}{0pt}{}{\,\cdot \,}{n}$ . Because H is procyclic, there is an isomorphism $I^n/I^{n+1} \cong \mathbb {Z}_p$ for all n, and the map $p_{\genfrac {[}{]}{0pt}{}{\,\cdot \,}{n},1}$ is induced by this isomorphism. Hence, the values $p_{\genfrac {[}{]}{0pt}{}{\,\cdot \,}{n},1}(\Psi ^{(n)}([f]))$ completely determine the image of $\Psi ^{(n)}$ , and in this way, we show that the image of $\Psi ^{(n)}$ is given by Massey products.

For more general H, the graded quotients $I^n/I^{n+1}$ are more complicated, and we cannot hope for any $p_{\phi ,\theta }$ to be an isomorphism. However, it can happen that $I^n/I^{n+1}$ is a free module; suppose this is the case. If we can arrange that the maps $p_{\phi ,\theta }$ for varying $(\phi ,\theta )$ give a dual basis to $I^n/I^{n+1}$ , then, again, this construction gives a way to describe the image of $\Psi ^{(n)}$ in terms of Massey products. Said differently, if $I^n/I^{n+1}$ is a free module of rank d, then, by fixing a basis, we can think of $\Psi ^{(n)}([f])$ as a d-tuple of elements of $H^2(G,T)$ . If we can make d choices of pairs $(\phi ,\theta )$ , such that the maps $p_{\phi ,\theta }$ are the projectors onto these coordinates, then our results describe $\Psi ^{(n)}([f])$ as a d-tuple of Massey products.

For a general group H and general n, we do not expect that there will exist choices of $(\phi ,\theta )$ , such that the $p_{\phi ,\theta }$ constitute a dual basis to $I^n/I^{n+1}$ . However, we give some families of examples where this is the case: H is procyclic (Section 4.3), H is pro-bicyclic (Section 4.4), H is elementary abelian (Section 4.5) and H is a Heisenberg group and $n<4$ (Section 4.6). We explicate the result for $H \cong \mathbb {Z}_p^2$ in the following subsection.

1.4 The bicyclic case: an illustration

Suppose that H is isomorphic to $\mathbb {Z}_p^2$ , and let $\chi , \psi \colon H \to \mathbb {Z}_p$ denote the projections onto the two factors. For each nonnegative integer $a \le n$ , there is a partial defining system $\left (\genfrac {[}{]}{0pt}{}{\,\cdot \,}{a} \circ \chi , \genfrac {[}{]}{0pt}{}{\,\cdot \,}{n-a}\circ \psi \right )$ . Applying our general result Theorem 4.1.2 with these defining systems, we obtain the following (see Theorem 4.4.3):

Theorem B. Suppose that $\nu = (\chi ,\psi ) \colon H \to \mathbb {Z}_p^2$ is an isomorphism. Let $x, y \in I$ be, such that $x+1$ and $y+1$ are group elements mapping under $\nu $ to the standard ordered basis of $\mathbb {Z}_p^2$ . For $n \ge 2$ , the cosets of $x^ay^{n-a}$ with $0 \le a \le n$ then form a $\mathbb {Z}_p$ -basis for $I^n/I^{n+1}$ .

  1. a. To a continuous $1$ -cocycle $f \colon G \to T\otimes _{\mathbb {Z}_p} \Omega /I^n$ and $0 \le a \le n$ , we can associate a proper defining system for an $(n+1)$ -fold Massey product

    $$ \begin{align*} (\chi^{(a)},\lambda,\psi^{(n-a)}) := (\underbrace{\chi, \ldots, \chi}_{a\text{ times}}, \lambda, \underbrace{\psi, \ldots, \psi}_{n-a\text{ times}}) \in H^2(G,T), \end{align*} $$
    where $\lambda \colon G \to T$ is the composition of f with the quotient map $T \otimes _{\mathbb {Z}_p} \Omega /I^n \to T \otimes _{\mathbb {Z}_p} \Omega /I \cong T$ .
  2. b. With the notation of part a, let $[f]$ denote the class of f in $H^1(G,T \otimes _{\mathbb {Z}_p} \Omega /I^n)$ . Then

    $$ \begin{align*} \Psi^{(n)}([f]) = \sum_{a=0}^n (\chi^{(a)},\lambda,\psi^{(n-a)}) \otimes x^ay^{n-a}. \end{align*} $$

Let us illustrate Theorem B in some detail in the case that $n=2$ and $a=1$ . In this case, we have

$$ \begin{align*} \Omega/I^2 = \mathbb{Z}_p[x,y]/(x^2, xy, y^2) \end{align*} $$

in the notation of the theorem. We can therefore write the $1$ -cocycle $f \colon G \to T\otimes _{\mathbb {Z}_p} \Omega /I^2$ as

$$\begin{align*}f=\lambda+\lambda_x x+\lambda_y y, \end{align*}$$

with $\lambda _x, \lambda _y \colon G \to T$ , abbreviating the tensor product as formal multiplication. Part a of Theorem B says that f gives rise to a defining system

for the Massey triple product $(\chi , \lambda , \psi )$ . Here, the values of $\rho $ lie in the quotient of a group U of generalized upper-triangular unipotent $4$ -by- $4$ matrices by its subgroup Z of matrices with zero above-diagonal entries outside of the upper right-hand corner. The entries in the upper-right hand block are T-valued (and, in particular, $Z \cong T$ ), whereas they are $\mathbb {Z}_p$ -valued outside of it. Matrix multiplication proceeds using the $\mathbb {Z}_p$ -module structure on T. That $\rho $ is a defining system means that $\rho \colon G \to U/Z$ is a nonabelian $1$ -cocycle, where G acts on U coordinate-wise. The Massey product $(\chi ,\lambda ,\psi )_{\rho }$ relative to the defining system $\rho $ is an element of $H^2(G,T)$ providing the obstruction to lifting $\rho $ to a nonabelian $1$ -cocycle $G \to U$ .

In general, even for such a cocycle $\rho $ and therefore a Massey product $(\chi ,\lambda ,\psi )$ to exist, the cup products $\chi \cup \lambda $ and $\lambda \cup \psi $ must vanish in $H^2(G,T)$ so that cochains $\lambda _x$ and $\lambda _y$ can be chosen with $d\lambda _x = -\chi \cup \lambda $ and $d\lambda _y = -\lambda \cup \psi $ . Even then, the class $(\chi ,\lambda ,\psi )$ depends on these choices. In our description, this vanishing is encapsulated in the fact that f is a $1$ -cocycle, and the indeterminacy is removed by fixing f.

The content of part b of Theorem B is that the coefficients of $\Psi ^{(2)}([f])$ in $H^2(G,T)$ for the $\mathbb {Z}_p$ -basis $x^2$ , $xy$ and $y^2$ of $I^2/I^3$ are Massey triple products: in particular, the coefficient of $xy$ is the Massey product $(\chi ,\lambda ,\psi )_{\rho }$ for the defining system $\rho $ . More precisely, $(\chi ,\lambda ,\psi )_{\rho }$ is defined as the class of the $2$ -cocycle $F \colon G^2 \to T$ given by

$$ \begin{align*} F \colon (g,h) \mapsto \chi(g) g \lambda_x(h) + \psi(h)\lambda_y(g). \end{align*} $$

This cocycle F arises as the upper-right hand corner of $(g,h) \mapsto \tilde {\rho }(g) \cdot g \tilde {\rho }(h)$ for the naive lift of $\rho $ to a cochain $\tilde {\rho } \colon G \to U$ with zero in the upper-right hand corner. The theorem boils down to the fact that $F \cdot xy$ is also exactly the coboundary of the naive lift of f to a cochain $G \to \mathbb {Z}_p[x,y]/(x^2,y^2)$ with a zero $xy$ -coefficient.

From our perspective, the generalized Bockstein maps are more flexible than Massey products, being connecting homomorphisms more directly amenable to basic applications of homological algebra. For instance, the argument proving Theorem A for arbitrary H amounts to a diagram chase for maps of Grothendieck spectral sequences. Moreover, Theorem B allows us to study defining systems using abelian, rather than nonabelian, cocycles.

1.5 Galois groups with restricted ramification and class groups

At its core, our work is motivated by the potential arithmetic applications. One has at least something of an understanding of class groups of cyclotomic fields through Bernoulli numbers, and thereby L-functions, and most notably via the Iwasawa main conjecture (theorem of Mazur-Wiles [Reference Mazur and WilesMaWi]). However, little is known about p-adic analytic invariants describing aspects of class groups of non-CM extensions of $\mathbb {Q}$ .

One does have at least a partial understanding of the structure of p-parts of class groups of p-ramified $\mathbb {F}_p$ -extensions of $\mathbb {Q}(\mu _p)$ through known values of cup products of cyclotomic p-units, and in certain instances, one can give lower bounds on p-ranks of these groups (see [Reference SharifiSh2, Section 7]). In Section 5, we consider more complex extensions, deriving lower bounds on the p-ranks of class groups of p-ramified bicyclic and Heisenberg extensions of $\mathbb {Q}(\mu _p)$ in cases where standard genus theory does not produce any unramified extensions. The key tools in this work are Theorem A, our descriptions of the generalized Bockstein maps $\Psi ^{(n)}$ for $n \in \{1,2\}$ and computations of cup products of cyclotomic units from [Reference McCallum and SharifiMcSh].

We consider the case that the class group of $\mathbb {Q}(\mu _p)$ has p-rank $1$ . Suppose we have an $\mathbb {F}_p^2$ -extension K of $\mathbb {Q}(\mu _p)$ that is Galois over $\mathbb {Q}$ for which the cup product pairing with the Kummer cocycle of the Kummer generators of the $\mathbb {F}_p^2$ -extension vanish. Under certain assumptions on the action of $\operatorname {\mathrm {Gal}}(\mathbb {Q}(\mu _p)/\mathbb {Q})$ on these Kummer generators, we can show that the p-rank of the class group of K is at least $6$ (see Proposition 5.2.1). This $\mathbb {F}_p^2$ -extension K is then further contained in a Heisenberg extension L of $\mathbb {Q}(\mu _p)$ of degree $p^3$ that is Galois over $\mathbb {Q}$ , and the p-rank of its class group is at least $7$ (see Proposition 5.2.3). The smallest irregular prime p for which there exist $\mathbb {F}_p^2$ -extensions for which these lower bounds are shown to hold by our methods is $101$ .

In [Reference SharifiSh4], the results of this paper are applied in the setting of Iwasawa theory to study inverse limits of class groups. There, G is the Galois group of the maximal extension of a number field unramified outside a finite set of primes containing those above p, and H is the Galois group of a $\mathbb {Z}_p$ -extension. The group $H^2_{\mathrm {Iw}}(N,\mathbb {Z}_p(1))$ is closely related to, but not always isomorphic to, the inverse limit X of p-parts of class groups under norm maps in the tower of number fields defined by H. The isomorphisms of Theorem A are then used to derive exact sequences describing the graded pieces in the augmentation filtration of X.

1.6 Absolute Galois groups and Massey vanishing

Let G be a profinite group, and let p be a prime number. Let $\chi \in H^1(G,\mathbb {F}_p) = \operatorname {\mathrm {Hom}}(G,\mathbb {F}_p)$ . Consider the sequence

(1.2) $$ \begin{align} H^1(\ker \chi, \mathbb{F}_p) \xrightarrow{\mathrm{cor}} H^1(G,\mathbb{F}_p) \xrightarrow{\chi\,\cup} H^2(G,\mathbb{F}_p) \xrightarrow{\mathrm{res}} H^2(\ker \chi,\mathbb{F}_p). \end{align} $$

If $G=G_F$ is the absolute Galois group of a field F that contains a primitive pth root of unity, then this sequence is exact, as can be seen using the properties of the norm residue symbol. This exactness is an important property of absolute Galois groups: for example, it is used heavily in the proof of the norm residue isomorphism theorem (see [Reference VoevodskyVo]).

Using Theorem B, we show that:

  1. (i) The sequence (1.2) is exact at $H^1(G,\mathbb {F}_p)$ if and only if all p-fold Massey products of the form $(\chi ^{(p-1)},\lambda )$ with $\chi \cup \lambda =0$ vanish for some proper defining system.

  2. (ii) If (1.2) is exact at $H^2(G,\mathbb {F}_p)$ , then it is exact.

In light of (i), we say that a group G has the p-cyclic Massey vanishing property if the sequence (1.2) is exact at $H^1(G,\mathbb {F}_p)$ for every $\chi \in H^1(G,\mathbb {F}_p)$ . We prove the following in Theorem 6.2.1.

Theorem C. Let G be a profinite group with the p-cyclic Massey vanishing property for an odd prime p. Then every Massey triple product on $H^1(G,\mathbb {F}_p)$ which is defined contains zero.

If F is a field containing a primitive pth root of unity, then its absolute Galois group $G_F$ has the p-cyclic Massey vanishing property. Hence, Theorem C implies that every Massey triple product on $H^1(G_F,\mathbb {F}_p)$ which is defined contains zero. This is the triple Massey vanishing theorem of Efrat–Matzri [Reference Efrat and MatzriEfMa] and Mináč–Tân [Reference Mináč and TânMiTa3] for odd p (which implies the vanishing for arbitrary fields as in the latter paper). For more discussion about absolute Galois groups and the general Massey vanishing conjecture of [Reference Mináč and TânMiTa4], see the introduction to Section 6.

In our proof of Theorem C, to show that a defined Massey product $(\chi ,\lambda ,\psi )$ vanishes, we consider the coimage H of the map $(\chi ,\psi ) \colon G \to \mathbb {F}_p^2$ , let $\Omega =\mathbb {F}_p[H]$ , and let $I \subset \Omega $ be the augmentation ideal. We then apply a variant of Theorem B to this H to see that the Massey product $(\chi ,\lambda ,\psi )$ relative to a certain defining system is the obstruction to lifting $\lambda $ to a class in $H^1(G,\Omega /J)$ for a particular ideal J between $I^2$ and $I^3$ . Via an involved diagram chase, we see that the p-cyclic Massey vanishing property for the quotients of H that are the coimages of $\chi $ , $\psi $ and $\chi +\psi $ implies that this obstruction equals $\nu \cup (\chi +\psi )$ for some $\nu \in H^1(G,\mathbb {F}_p)$ . This is enough to show that the Massey product contains zero.

Theorem C raises several interesting questions that we do not attempt to address here, including whether or not the vanishing of Massey products $(\chi ^{(n)},\psi )$ for arbitrary n is sufficient to imply Massey vanishing.

2 Generalized Bockstein maps

In this section, we define generalized Bockstein maps and employ them in the study of the structure of inverse limits of cohomology groups. Throughout the paper, we use the following objects:

  • a prime number p,

  • a profinite group G,

  • a topologically finitely generated pro-p quotient H of G by a closed normal subgroup N,

  • a compact Noetherian $\mathbb {Z}_p$ -algebra R (usually taken to be a quotient of $\mathbb {Z}_p$ ),

  • the completed group ring ,

  • the augmentation ideal I of $\Omega $ , that is, the kernel of the continuous R-algebra homomorphism $\Omega \to R$ that sends every group element in H to $1$ ,

  • a positive integer n, such that $\Omega /I^n$ and $I^n/I^{n+1}$ are R-flat and

  • a compact -module T that is R-finitely generated.

Note that a compact -module is the same as a compact R-module with a continuous R-linear action of G. We will frequently take tensor products $M \otimes _R M'$ of compact -modules M and $M'$ , at least one of which is finitely generated over R. These compact R-modules (with the topology of the isomorphic completed tensor product) have the diagonal action of G.

We are concerned in this paper with the continuous cohomology groups $H^i(G,M)$ of compact -modules M for $i \ge 0$ . In particular, G-cochains are implicitly supposed to be continuous. We use square brackets to denote both classes of cocycles and group elements in completed group algebras, and we denote an element in a module and its coset in a quotient thereof by the same symbol where the context is clear.

2.1 Augmentation sequences

Since we have assumed that $\Omega /I^n$ is R-flat, the right exact sequence of compact -modules

(2.1) $$ \begin{align} 0 \to T \otimes_R I^n/I^{n+1} \to T \otimes_R \Omega/I^{n+1} \to T \otimes_R \Omega/I^n \to 0 \end{align} $$

is exact. For any $d \ge 1$ , we have the resulting connecting homomorphisms

$$\begin{align*}H^{d-1}(G,T \otimes_R \Omega/I^n) \to H^d(G,T \otimes_R I^n/I^{n+1}) \end{align*}$$

on continuous G-cohomology.

Since G acts trivially on the finitely generated R-module $I^n/I^{n+1}$ , we have a homomorphism

(2.2) $$ \begin{align} H^d(G,T) \otimes_R I^n/I^{n+1} \to H^d(G,T \otimes_R I^n/I^{n+1}) \end{align} $$

that is an isomorphism as $I^n/I^{n+1}$ is R-flat, so long as we assume either that G has finite p-cohomological dimension or that $I^n/I^{n+1}$ has a finite resolution by projective R-modules (see [Reference Lim and SharifiLiSh, Proposition 3.1.3], the proof of which does not use the assumption on R in that section). The latter condition is automatic, given that $I^n/I^{n+1}$ is flat, if R is a quotient of $\mathbb {Z}_p$ . We let

(2.3) $$ \begin{align} \Psi^{(n)} \colon H^{d-1}(G,T \otimes_R \Omega/I^n) \to H^d(G,T) \otimes_R I^n/I^{n+1} \end{align} $$

denote the resulting composite map, and we refer to it as a generalized Bockstein map.

Remark 2.1.1. We may replace the assumption that $\Omega /I^n$ is R-flat with the assumption that T is R-flat in order that (2.1) still holds. We may also replace the assumption that $I^n/I^{n+1}$ is R-flat with the assumption that G has p-cohomological dimension d and still have an isomorphism as in (2.2) (to see this, choose a presentation of $I^n/I^{n+1}$ by finitely generated free R-modules and use the right exactness of the dth cohomology functor and the tensor product, noting that $H^d(G,T^r) \cong H^d(G,T) \otimes _R R^r$ for any r). With either replacement, $\Psi ^{(n)}$ is still a map as in (2.3).

2.2 Graded quotients of Iwasawa cohomology groups

Recall that N denotes the kernel of the surjection $G \to H$ . Our interest in this section is in the Iwasawa cohomology groups

$$ \begin{align*} H^i_{\mathrm{Iw}}(N,T) = \varprojlim_{N \leq U \unlhd^o G} H^i(U,T) \end{align*} $$

for $i \ge 1$ , where the inverse limit is taken with respect to corestriction maps over open normal subgroups U of G containing N. Note that the Iwasawa cohomology groups are relative to the larger group G, though this is omitted from our notation. Since each $H^i(U,T)$ is a $R[G/U]$ -module and the actions are compatible with corestriction, the group $H^i_{\mathrm {Iw}}(N,T)$ is endowed with the structure of an $\Omega $ -module.

Remark 2.2.1. If H is finite, then $H^i_{\mathrm {Iw}}(N,T) = H^i(N,T)$ .

Let us define two notions that we need. First, a profinite group $\mathcal {G}$ is p-cohomologically finite if $\mathcal {G}$ has finite p-cohomological dimension and $H^i(\mathcal {G},M)$ is finite for every finite $\mathbb {Z}_p[\mathcal {G}]$ -module M and $i \ge 0$ . Second, a compact p-adic Lie group is a profinite group that has an open pro-p subgroup, any closed subgroup of which can be topologically generated by r elements for some fixed r. Equivalently, a compact p-adic Lie group is any profinite group continuously isomorphic to a closed subgroup of $\mathrm {GL}_n(\mathbb {Z}_p)$ for some $n \ge 1$ .

We make the following assumptions for the rest of this section:

  • G is p-cohomologically finite of p-cohomological dimension d,

  • R is a complete commutative local Noetherian $\mathbb {Z}_p$ -algebra with finite residue field and

  • either

    1. (i) H is a compact p-adic Lie group or

    2. (ii) T has a finite resolution by a complex of -modules free of finite rank over R.

Recall that the zeroth H-homology group of a compact $\Omega $ -module M is its coinvariant module $M_H \cong M/IM$ . In our setting, corestriction provides an isomorphism on coinvariants in degree d (see [Reference Neukirch, Schmidt and WingbergNSW, Proposition 3.3.11]), which is to say that we have a natural isomorphism

(2.4) $$ \begin{align} \frac{H^d_{\mathrm{Iw}}(N,T)}{IH^d_{\mathrm{Iw}}(N,T)} \cong H^d(G,T). \end{align} $$

This gives rise to a Grothendieck spectral sequence for the implicit composition of right exact functors, which is a version of Tate’s descent spectral sequence for Iwasawa cohomology.

Proposition 2.2.2 (Fukaya-Kato, Lim-Sharifi).

The $\Omega $ -modules $H^i_{\mathrm {Iw}}(N,T)$ are finitely generated for all $i \ge 0$ . Moreover, we have a first quadrant homological spectral sequence of R-modules

$$ \begin{align*} E^2_{i,j}(T) = H_i(H,H^{d-j}_{\mathrm{Iw}}(N,T)) \Rightarrow E_{i+j}(T) = H^{d-i-j}(G,T), \end{align*} $$

where d is the p-cohomological dimension of G.

This result is proven in [Reference SerreSe2, Theorem 1, Reference TateTa] if H is finite, and it follows from [Reference Fukaya and KatoFuKa, Proposition 1.6.5] if (ii) holds and from [Reference Lim and SharifiLiSh, Propositions 3.1.3 and 3.2.4] if (i) holds.

The isomorphism (2.4) and the other edge maps on coinvariant groups in this spectral sequence are given by the inverse limits of corestriction maps. This isomorphism forces the nth graded quotient $I^nA/I^{n+1}A$ in the augmentation filtration of $A = H^d_{\mathrm {Iw}}(N,T)$ to be a quotient of $H^d(G,T) \otimes _R I^n/I^{n+1}$ using the surjective map

$$ \begin{align*} A/IA \otimes_R I^n/I^{n+1} \to I^nA/I^{n+1}A \end{align*} $$

induced by the map $A \times I^n \to I^nA$ given by the multiplication $(a,x) \mapsto xa$ . As we shall see, this quotient is in fact $\operatorname {\mathrm {coker}} \Psi ^{(n)}$ .

Recall that we have assumed that $\Omega /I^n$ is R-flat. Moreover, the fact that H is topologically finitely generated implies that $\Omega /I^n$ is finitely generated over R.

Lemma 2.2.3. Let A be an $\Omega $ -module, and consider the exact sequence

(2.5) $$ \begin{align} 0 \to A \otimes_R I^n/I^{n+1} \to A \otimes_R \Omega/I^{n+1} \to A \otimes_R \Omega/I^n \to 0. \end{align} $$

The connecting homomorphism

$$ \begin{align*} \partial_n \colon H_1(H,A \otimes_R \Omega/I^n) \to A_H \otimes_R I^n/I^{n+1} \end{align*} $$

in the H-homology of (2.5) has cokernel isomorphic to $I^n A/I^{n+1} A$ .

Proof. We have compatible, natural isomorphisms of R-modules

$$ \begin{align*} (A \otimes_R \Omega/I^m)_H \cong \Omega/I^m \otimes_{\Omega} A \cong A/I^mA \end{align*} $$

for $m \ge 1$ given on $a \in A$ and $\omega \in \Omega $ (or its quotient by $I^m$ ) by

$$ \begin{align*} a \otimes \omega \mapsto \iota(\omega) \otimes a \mapsto \iota(\omega)a, \end{align*} $$

where $\iota \colon \Omega \to \Omega $ is the unique continuous R-linear map given by inversion of group elements on H. Note that the switch of terms in the tensor product in the first isomorphism is necessitated by the fact that A is a left $\Omega $ -module (in fact, these become isomorphisms of $\Omega $ -modules since $a \otimes \omega h^{-1} \mapsto h \cdot \iota (\omega ) \otimes a$ for $h \in H$ under the first map).

By the long exact sequence in H-homology and the above isomorphisms, the cokernel of interest is identified with the kernel of the quotient map $A/I^{n+1}A \to A/I^nA$ , hence, the result.

We now come to our theorem.

Theorem 2.2.4. For each $n \ge 1$ , there is a canonical isomorphism

$$ \begin{align*} \frac{I^n H^d_{\mathrm{Iw}}(N,T)}{I^{n+1} H^d_{\mathrm{Iw}}(N,T)} \cong \frac{H^d(G,T) \otimes_R I^n/I^{n+1}}{\operatorname{\mathrm{im}} \Psi^{(n)}} \end{align*} $$

of R-modules, where d is the p-cohomological dimension of G.

Proof. There are isomorphisms

$$ \begin{align*} H^d_{\mathrm{Iw}}(N,T \otimes_R M) \cong H^d_{\mathrm{Iw}}(N,T) \otimes_R M \end{align*} $$

for any compact -module M finitely generated over R, since G has p-cohomological dimension d. In particular, the following sequence is exact:

$$ \begin{align*} 0 \to H^d_{\mathrm{Iw}}(N,T \otimes_R I^n/I^{n+1}) \to H^d_{\mathrm{Iw}}(N,T \otimes_R \Omega/I^{n+1}) \to H^d_{\mathrm{Iw}}(N,T \otimes_R \Omega/I^n) \to 0. \end{align*} $$

We consider the connecting homomorphism in H-homology:

(2.6) $$ \begin{align} \partial^{(n)} \colon H_1(H,H^d_{\mathrm{Iw}}(N, T) \otimes_R \Omega/I^n) \to H^d_{\mathrm{Iw}}(N,T)_H \otimes_R I^n/I^{n+1}. \end{align} $$

We next apply Lemma A.0.1 of the appendix, which says that edge maps to total terms in homological Grothendieck spectral sequences are compatible with connecting maps. Here, the spectral sequence is that of Proposition 2.2.2, which is associated to the composition of functors $F = H_0(H, \,\cdot \,)$ and $F' = H^d_{\mathrm {Iw}}(N, \,\cdot \,)$ , noting that $F \circ F' \cong H^d(G,\,\cdot \,)$ via corestriction. The connecting homomorphisms are from degrees $1$ to $0$ and are associated to the short exact sequence of (2.1).

In this setting, the lemma provides a commutative square related to the diagram


but with $L_1(F \circ F')(T \otimes _R \Omega /I^n)$ in place of $H^{d-1}(G,T \otimes _R \Omega /I^n)$ . By Lemma A.0.2, which is a simple consequence of the universality of left-derived functors, we have a surjection from the latter object to the former, compatible with their connecting homomorphisms to $H^d(G,T) \otimes _R I^n/I^{n+1}$ . This allows us to make the replacement while maintaining the surjectivity of the left vertical map, so we indeed have the commutative square (2.7).

By Lemma 2.2.3, the isomorphism in the statement of the theorem is the map on cokernels of the horizontal maps in (2.7).

Although not used in this paper, for the purposes of Iwasawa-theoretic applications, it is useful to have a slightly stronger version of Theorem 2.2.4. So, we remark that it has the following generalization, with virtually no additional complications (given that the results of [Reference Lim and SharifiLiSh, Reference Fukaya and KatoFuKa] hold in this generality).

Remark 2.2.5. Let $\mathcal {G}$ be a profinite group, and let $\Gamma $ be a quotient of $\mathcal {G}$ by a closed normal subgroup G. Let $\mathcal {H}$ be a quotient of $\mathcal {G}$ by a closed normal subgroup N that is contained in G, and let $H = G/N$ as before. We then have $\Gamma \cong \mathcal {H}/H$ . That is, we have a commutative diagram of exact sequences

Take T to be a compact -module finitely generated over R, and replace the assumptions on G and H from the beginning of this subsection with the identical assumptions on $\mathcal {G}$ and $\mathcal {H}$ , respectively. We have Iwasawa cohomology groups $H^i_{\mathrm {Iw}}(N,T)$ and $H^i_{\mathrm {Iw}}(G,T)$ , which are now taken relative to the larger group $\mathcal {G}$ . These are finitely generated as modules over and , respectively, and we have, as before, a spectral sequence

$$ \begin{align*} E^2_{i,j}(T) = H_i(H,H^{d-j}_{\mathrm{Iw}}(N,T)) \Rightarrow E_{i+j}(T) = H^{d-i-j}_{\mathrm{Iw}}(G,T) \end{align*} $$

but now of $\Lambda $ -modules. In exactly the same manner as before, this gives rise to isomorphisms

$$ \begin{align*} \frac{I^n H^d_{\mathrm{Iw}}(N,T)}{I^{n+1} H^d_{\mathrm{Iw}}(N,T)} \cong \frac{H^d_{\mathrm{Iw}}(G,T) \otimes_R I^n/I^{n+1}}{\operatorname{\mathrm{im}} \Psi^{(n)}}, \end{align*} $$

again, of $\Lambda $ -modules.

2.3 The abelian case

We turn to the direct computation of generalized Bockstein maps on $1$ -cocycles for abelian H. That is, let us now take H to be a finitely generated, abelian pro-p group, and let us take R to be a quotient of $\mathbb {Z}_p$ . We give an explicit formula for $\Psi ^{(n)}$ under a hypothesis on the size of R that ensures our flatness hypothesis is satisfied. If H has no nonzero p-torsion, no hypothesis is needed.

We begin with the following simple lemma.

Lemma 2.3.1. Let s and t be positive integers with $n < p^{t-s+1}$ . Then $(1+x)^{p^t}-1$ is in the ideal $(x^{n+1},p^s)$ of $\mathbb {Z}[x]$ .

Proof. Recall that $p^s$ divides $\binom {p^t}{i}$ for $0 < i < p^{t-s+1}$ . Therefore

$$ \begin{align*} (1+x)^{p^t} = \sum_{i=0}^{p^t} \binom{p^t}{i}x^i \equiv 1 \bmod (x^{n+1},p^s), \end{align*} $$

so long as $n < p^{t-s+1}$ .

Let $h_1, \dots , h_r$ be a minimal set of generators for H, labeled such that $h_1, \dots , h_c$ have finite orders $p^{t_1}\le \dots \le p^{t_c}$ and $h_{c+1},\dots , h_r$ have infinite order, for some $0 \le c \le r$ . Define $x_i = [h_i] - 1 \in \Omega $ for $1 \le i \le r$ , where $[h_i]$ denotes the group element of $h_i$ , so that $I = (x_1, \dots , x_r)$ . We then have

We have $c>0$ if and only if H is not $\mathbb {Z}_p$ -free, in which case, we suppose that $R=\mathbb {Z}/p^s\mathbb {Z}$ with $n < p^{t_1-s+1}$ . By Lemma 2.3.1, we have

$$\begin{align*}\Omega/I^j \cong \frac{R[x_1,\ldots,x_r]}{(x_1,\ldots,x_r)^j} \end{align*}$$

for $j \le n+1$ . Moreover, $I^n/I^{n+1}$ is a free R-module with a basis consisting of the monomials in the variables $x_i$ of degree n. In particular, the generalized Bockstein map $\Psi ^{(n)}$ is defined. We may view any element $q \in T \otimes _R \Omega /I^n$ as having the form

$$ \begin{align*} q = \sum_{k_1+ \cdots +k_r < n} \alpha_{k_1,\ldots,k_r} x_1^{k_1} \cdots x_r^{k_r}, \end{align*} $$

where the sum is taken over r-tuples $(k_1, \ldots , k_r)$ of nonnegative integers with sum less than n and with $\alpha _{k_1,\ldots ,k_r} \in T$ , omitting the notation for the tensor product in such an expression. Setting $\|k\| = k_1 + \cdots + k_r$ for an r-tuple $(k_1, \ldots , k_r)$ , let’s simplify this notation as

(2.8) $$ \begin{align} q = \sum_{\|k\| < n} \alpha_k x^k, \end{align} $$

where $x^k = x_1^{k_1} \cdots x_r^{k_r}$ .

Let $\pi \colon G \to H$ denote the quotient map. For each i, let

$$ \begin{align*} A_i = \begin{cases} \mathbb{Z}/p^{t_i}\mathbb{Z} & \text{if } 1 \le i \le c, \\ \mathbb{Z}_p & \text{if } c < i \le r. \end{cases} \end{align*} $$

For $1 \le i \le r$ , let $\chi _i \colon G \to A_i$ be the homomorphisms determined by

$$ \begin{align*} \pi(g) = \prod_{i=1}^r h_i^{\chi_i(g)} \end{align*} $$

for $g \in G$ . The action of $g \in G$ on q as in (2.8) is given by multiplication by $\prod _{i=1}^r (1+x_i)^{\chi _i(g)}$ . That is, we have the formula

(2.9) $$ \begin{align} g \cdot q = \sum_{\|k\| < n} \left(\sum_{0 \le k' \le k} \binom{\chi(g)}{k'} g\alpha_{k-k'} \right) x^k, \end{align} $$

where the second sum is over r-tuples $k'$ of nonnegative integers with $k^{\prime }_i \le k_i$ for each i and where we have set $\binom {\chi (g)}{k'} = \binom {\chi _1(g)}{k^{\prime }_1} \cdots \binom {\chi _r(g)}{k^{\prime }_r}$ .

Note that our assumption on the cardinality of R can be rephrased as saying that either $c=0$ or R is a quotient of $A_1$ such that $|R| < \frac {p}{n}|A_1|$ . With our notation and this assumption established, we can give an explicit formula for $\Psi ^{(n)}$ .

Proposition 2.3.2. Let $f \colon G \to T \otimes _R \Omega /I^n$ be a $1$ -cocycle, and write

$$ \begin{align*} f = \sum_{\|k\| < n} \lambda_k x^k \end{align*} $$

with $\lambda _k \colon G \to T$ . Then $\Psi ^{(n)}$ takes the class of f to the class of the $2$ -cocycle

$$ \begin{align*} (g,h) \mapsto \sum_{\|k\| = n} \left( \sum_{0 < k' \le k} \binom{\chi(g)}{k'} g \lambda_{k-k'}(h) \right) x^k, \end{align*} $$

where the first sum is taken over r-tuples $k = (k_1, \ldots , k_r)$ of nonnegative integers summing to n and the second sum is taken over nonzero r-tuples $k'$ of nonnegative integers with $k^{\prime }_i \le k_i$ for all i.

Proof. Consider the set-theoretic section

(2.10) $$ \begin{align} s_n \colon T \otimes_R \Omega/I^n \to T \otimes_R \Omega/I^{n+1} \end{align} $$

that takes a sum as in (2.8) to the same expression in the larger module. Let $\tilde {f} = s_n \circ f$ . By definition, $\Psi ^{(n)}([f])$ is the class of $d\tilde {f}$ , where

$$ \begin{align*} d\tilde{f}(g,h) = \tilde{f}(g) + g\tilde{f}(h) - \tilde{f}(gh) \end{align*} $$

for $g, h \in G$ . Since f is a cocycle, the right-hand side of this expression is equal to the degree n part of $g\tilde {f}(h)$ , which by (2.9) is exactly as in the statement of the proposition.

For general H, pro-p but not necessarily abelian, we can use this computation to see that $\Psi ^{(1)}$ is given by cup products. We consider the case that H is a quotient of G, such that the abelianization $H^{\mathrm {ab}}$ of H is finitely generated and pro-p. As before, but now for $H^{\mathrm {ab}}$ in place of H, there are nonnegative integers $r \ge c$ and positive integers $t_1 \le \cdots \le t_c$ , such that

(2.11) $$ \begin{align} H^{\mathrm{ab}} \cong \bigoplus_{i=1}^r A_i, \end{align} $$

where $A_i = \mathbb {Z}/p^{t_i}\mathbb {Z}$ for $i=1,\dots ,c$ and $A_i=\mathbb {Z}_p$ for $i=c+1,\dots , r$ . For $i=1,\dots , r$ , we let $\chi _i \colon G \to A_i$ denote the quotient map $G \to A_i$ . We take $n= 1$ , and our condition on the cardinality of R becomes $s \le t_1$ when $c \ge 1$ .

Fix generators $h_1, \ldots , h_r$ of H, such that each $h_i$ maps to $1 \in A_i$ under the composition of the quotient map and the isomorphism in (2.11). There is an isomorphism $I/I^2 \cong H^{\mathrm {ab}} \otimes _{\mathbb {Z}_p} R$ taking the image of $x_i = [h_i]-1$ to $h_i \otimes 1$ .

Proposition 2.3.3. Let H be a finitely generated pro-p group with $H^{\mathrm {ab}}$ as in (2.11), let I be the augmentation ideal in and let $\chi _i$ and $x_i$ for $1 \le i \le r$ be as in the previous paragraph. For any $1$ -cocycle $f \colon G \to T$ , we have

$$\begin{align*}\Psi^{(1)}([f]) = \sum_{i=1}^r (\chi_i \cup f) x_i \in H^2(G,T) \otimes_R I/I^2. \end{align*}$$

Proof. Let with augmentation ideal $I' \subset \Omega '$ . Both $\Omega /I$ and $\Omega '/I'$ are identified with R via the augmentation maps, and there are also compatible isomorphisms between the graded quotients $I/I^2$ and $I'/(I')^2$ and the R-module $H^{\mathrm {ab}} \otimes _{\mathbb {Z}_p} R$ . It follows that the canonical map $\Omega \to \Omega '$ induces an isomorphism $\Omega /I^2 \cong \Omega '/(I')^2$ . Thus, $\Psi ^{(1)}$ equals the first generalized Bockstein map for $H^{\mathrm {ab}}$ , and the proposition follows from the case $n=1$ of Proposition 2.3.2.

This result was previously studied by the third author in the context of Iwasawa theory, where these maps are referred to as reciprocity maps with restricted ramification (see, for instance, [Reference SharifiSh3, Lemma 4.1] for its introduction). In the following section, we study analogous results for $\Psi ^{(n)}$ with $n> 1$ in terms of higher Massey products.

3 Massey products

In this section, we review the definitions of Massey products and defining systems, with some modifications from the standard definitions in order to allow for nontrivial coefficient modules. We also introduce the notions of partial and proper defining systems.

3.1 Upper-triangular generalized matrix algebras

The notion of Massey products that we will use is conveniently stated using the theory of generalized matrix algebras, as found in [Reference Bellaïche and ChenevierBeCh, Section 1.3, pp. 19–21]. We require only a simple upper-triangular version of these algebras. Let n be a positive integer, and let R be a commutative ring.

Definition 3.1.1. An n-dimensional upper-triangular generalized matrix algebra $\mathcal {A}$ over R (or, R-UGMA) is an R-algebra formed out of the data of

  • finitely generated R-modules $A_{i,j}$ for $1 \le i \le j \le n$ with $A_{i,j} = R$ if $i=j$ and

  • R-module homomorphisms $\varphi _{i,j,k} \colon A_{i,j} \otimes _R A_{j,k} \to A_{i,k}$ for all $1 \le i \le j \le k \le n$ which are induced by the given R-actions if $i = j$ or $j = k$ ,

such that the two resulting maps

$$ \begin{align*} A_{i,j} \otimes_R A_{j,k} \otimes_R A_{k,l} \to A_{i,l} \end{align*} $$

coincide for all $1 \le i < j < k < l \le n$ . The tuple $(A_{i,j},\varphi _{i,j,k})$ defines an R-algebra $\mathcal {A}$ with underlying R-module

$$ \begin{align*} \mathcal{A}=\bigoplus_{1 \le i \le j \le n} A_{i,j} \end{align*} $$

and multiplication given by matrix multiplication: that is, for $a = (a_{i,j})$ and $b = (b_{i,j})$ in $\mathcal {A}$ , the $(i,j)$ -entry $(ab)_{i,j}$ of $ab$ is

$$\begin{align*}(ab)_{i,j} = \sum_{k=i}^j \varphi_{i,k,j}(a_{i,k} \otimes b_{k,j}). \end{align*}$$

Our interest is in the multiplicative group $\mathcal {U} = \mathcal {U}(\mathcal {A})$ of unipotent matrices in a UGMA $\mathcal {A}$ , that is, those $a = (a_{i,j})$ with $a_{i,i} = 1$ for all i. We shall often take the quotient $\mathcal {U}' = \mathcal {U}'(\mathcal {A})$ of this $\mathcal {U}$ by its central subgroup $\mathcal {Z} = \mathcal {Z}(\mathcal {A})$ of unipotent central elements, that is, those $a \in \mathcal {U}$ with $a_{i,j} = 0$ for all $(i,j) \neq (1,n)$ .

The following is the key example for our purposes.

Example 3.1.2. Let M be a finitely generated R-module, and let m be a positive integer less than n. We define an n-dimensional R-UGMA $\mathcal {A}_n(M,m)$ as follows. Set

$$\begin{align*}A_{i,j} = \begin{cases} M &\text{if }i \le m < j, \\ R &\text{ otherwise,}\end{cases} \end{align*}$$

and take the maps $\varphi _{i,j,k}$ to be the R-module structure maps. This makes sense since, given $i \le j \le k$ , at least one of $A_{i,j}$ and $A_{j,k}$ must be R, as m cannot satisfy both $m < j$ and $j \le m$ .

Let us write $\mathcal {U}_n(M,m)$ for $\mathcal {U}(\mathcal {A}_n(M,m))$ and $\mathcal {U}^{\prime }_n(M,m)$ for $\mathcal {U}'(\mathcal {A}_n(M,m))$ . To make this easier to visualize, note that we can write $\mathcal {U}_n(M,m)$ in ‘block matrix’ form as

$$ \begin{align*} \mathcal{U}_n(M,m) = \begin{pmatrix} \mathrm{U}_m(R) & \mathrm{M}_{m,n-m}(M) \\ 0 & \mathrm{U}_{n-m}(R) \end{pmatrix}, \end{align*} $$

where $\mathrm {U}_k(R) \leqslant \mathrm {GL}_k(R)$ denotes the group of upper-triangular unipotent matrices and $\mathrm {M}_{k,l}(M)$ denotes the additive group of k-by-l matrices with entries in M for positive integers k and l. The latter group is endowed with a left $\mathrm {U}_k(R)$ -action and a commuting right $\mathrm {U}_l(R)$ -action. Put differently, $\mathcal {A}_n(M,m)$ itself is a sort of $2$ -by- $2$ generalized matrix algebra, allowing noncommutative rings on the diagonal and bimodules in the nondiagonal entries.

We actually need to use profinite UGMAs defined just as in Definition 3.1.1 using profinite rings R and compact R-modules $A_{i,j}$ but now assuming that the induced multiplication maps $A_{i,j} \times A_{j,k} \to A_{i,k}$ are continuous. Alternatively, the maps $\varphi _{i,j,k}$ can be replaced by maps of completed tensor products over R in the definition.

Though unnecessary, to keep things simple, let us suppose that the compact R-modules $A_{i,j}$ in a profinite R-UGMA are R-finitely generated. This forces them to have the adic topology for any directed system of ideals that are open neighborhoods of zero. Moreover, their tensor products and completed tensor products are then abstractly isomorphic, and so we may, in particular, view the tensor products $A_{i,j} \otimes _R A_{j,k}$ themselves as compact R-modules (for a slightly longer discussion of this, see [Reference Lim and SharifiLiSh, Section 2.3]).

Note that any profinite R-UGMA $\mathcal {A}$ has a topology as a finite direct product of the compact R-modules $A_{i,j}$ , and $\mathcal {U}$ inherits the subspace topology.

We also want to make a second modification, allowing a continuous action of G.

Definition 3.1.3. For a profinite ring R and a profinite group G, a profinite $(R,G)$ -UGMA is the data of a profinite R-UGMA $\mathcal {A}$ together with a continuous G-action on each $A_{i,j}$ , such that

  • the action on $A_{i,i} = R$ is trivial for all i and

  • the maps $\varphi _{i,j,k}$ are maps of -modules, where $A_{i,j} \otimes _R A_{j,k}$ is given the diagonal action of G.

We remark that, aside from issues of finite generation, the difference between a profinite -UGMA and a profinite $(R,G)$ -UGMA is that in the former, each , whereas in the latter, each $A_{i,i}$ is R with the trivial G-action. We are interested in the latter structure.

Example 3.1.4. If R is a profinite ring and T is a compact -module (that is R-finitely generated), then the R-UGMA $\mathcal {A}_n(T,m)$ of Example 3.1.2 has a natural structure of a profinite $(R,G)$ -UGMA by letting G act on $A_{i,j}$ via its action on T if $i \le m < j$ and trivially otherwise.

3.2 Defining systems and Massey products

Let R be a profinite ring, let G be a profinite group and let $n \ge 2$ . Let $T_1, \dots , T_n$ be compact -modules that are R-finitely generated for simplicity, and let $\chi _i \colon G \to T_i$ be continuous 1-cocycles for $1 \le i \le n$ . In this section, we define Massey products of these cocycles, which will be 2-cocycles that depend on a number of choices constituting a defining system.

Definition 3.2.1. A defining system for the Massey product of $\chi _1,\dots ,\chi _n$ is the data of

  • an $(n+1)$ -dimensional profinite $(R,G)$ -UGMA $\mathcal {A}$ and

  • a (nonabelian) continuous 1-cocycle $\rho \colon G \to \mathcal {U}'$ ,

such that $A_{i,i+1}=T_i$ for $1 \le i \le n$ and the composition of $\rho $ with projection to $A_{i,i+1}$ is $\chi _i$ .

Given a defining system $\rho \colon G \to \mathcal {U}'$ , there is a unique function $\tilde {\rho } \colon G \to \mathcal {U}$ lifting $\rho $ and having zero as the $(1,n+1)$ -entry of $\tilde {\rho }(g)$ for all $g \in G$ . We let $\rho _{i,j} \colon G \to A_{i,j}$ be the map given by taking the $(i,j)$ -entry of $\tilde {\rho }$ .

Definition 3.2.2. Given a defining system $\rho $ , the n-fold Massey product $(\chi _1,\dots ,\chi _n)_\rho \in H^2(G,A_{1,n+1})$ is the class of the 2-cocycle

$$\begin{align*}(g,h) \mapsto \sum_{i=2}^n \varphi_{1,i,n+1}(\rho_{1,i}(g) \otimes g \rho_{i,n+1}(h)) \end{align*}$$

that sends $(g,h)$ to the $(1,n+1)$ -entry of $\tilde {\rho }(g)\cdot g \tilde {\rho }(h)$ .

In the remainder of the paper, we will restrict our attention to the setting of the $(n+1)$ -dimensional profinite $(R,G)$ -UGMAs of the form $\mathcal {A}_{n+1}(T,m)$ defined in Examples 3.1.2 and 3.1.4. This means, in particular, that we only consider n-fold Massey products for which there is an m with $1 \le m \le n$ , such that $T_m = T$ and $T_i = R$ for $i \neq m$ . In particular, we will always have $(\chi _1,\dots ,\chi _n)_\rho \in H^2(G,T)$ .

In [Reference SharifiSh2], the third author considered the case in which $m = n$ and $\chi _1 = \cdots = \chi _{n-1}$ in a Galois-cohomological setting. In that case, the key idea for relating Massey products to graded pieces of Iwasawa cohomology groups was to consider only a restricted set of defining systems referred to as proper defining systems. We will consider a more general notion of proper defining system that depends on extra data we call a partial defining system. In [Reference SharifiSh2], the partial defining system comes from unipotent binomial matrices, which we review in Section 4.2 below.

3.3 Massey products relative to proper defining systems

Fix an integer $n \ge 2$ and two integers $a,b \ge 0$ with $a+b=n$ . Let $Z^i(G,M)$ for a profinite -module M denote the group of continuous i-cocycles on G valued in M. Choose tuples

$$ \begin{align*} \alpha=(\alpha_1,\dots,\alpha_a) \in Z^1(G,R)^a \ \mathrm{and}\ \beta = (\beta_1,\dots,\beta_b) \in Z^1(G,R)^b \end{align*} $$

and a compact -module T that is finitely generated as an R-module.

We next consider a pair of homomorphisms that constitute a part of the defining systems for $(n+1)$ -fold Massey products $(\alpha _1, \ldots , \alpha _a, \lambda , \beta _1, \ldots , \beta _b)$ , where $\lambda \in Z^1(G,T)$ is allowed to vary. We write the collection of such Massey products as $(\alpha ,\,\cdot \,,\beta )$ for short.

Definition 3.3.1. A partial defining system for $(n+1)$ -fold Massey products $(\alpha , \,\cdot \,, \beta )$ is a pair of homomorphisms

$$ \begin{align*} \phi \colon G \to \mathrm{U}_{a+1}(R)\ \mathrm{and}\ \theta \colon G \to \mathrm{U}_{b+1}(R), \end{align*} $$

such that $\alpha $ is the off-diagonal of $\phi $ and $\beta $ is the off-diagonal of $\theta $ , that is, $\phi _{i,i+1} = \alpha _i$ for $1 \le i \le a$ and $\theta _{i,i+1} = \beta _i$ for $1 \le i \le b$ .

More specifically, an $(a,b)$ -partial defining system is a partial defining system restricting to some pair $(\alpha ,\beta ) \in Z^1(G,R)^a \times Z^1(G,R)^b$ .

Recall that

$$ \begin{align*} \mathcal{U}_{n+2}(T,a+1) = \begin{pmatrix} \mathrm{U}_{a+1}(R) & M_{a+1,b+1}(T) \\ & \mathrm{U}_{b+1}(R) \end{pmatrix}. \end{align*} $$

We may then write the quotient by the unipotent central matrices as

$$ \begin{align*} \mathcal{U}^{\prime}_{n+2}(T,a+1) = \begin{pmatrix} \mathrm{U}_{a+1}(R) & M^{\prime}_{a+1,b+1}(T) \\ & \mathrm{U}_{b+1}(R) \end{pmatrix} \end{align*} $$

for $M^{\prime }_{a+1,b+1}(T) = M_{a+1,b+1}(T)/T$ , where T is identified with the matrices that are zero outside the $(1,b+1)$ -entry.

Definition 3.3.2. Given a $1$ -cocycle $\lambda \colon G \to T$ , a proper defining system for an $(n+1)$ -fold Massey product $(\alpha ,\lambda ,\beta )$ relative to a partial defining system $(\phi ,\theta )$ is a continuous $1$ -cocycle

$$\begin{align*}\rho \colon G \to \mathcal{U}^{\prime}_{n+2}(T,a+1) \end{align*}$$

of the form

$$\begin{align*}\rho = \begin{pmatrix} \phi & \kappa \\ 0 & \theta \end{pmatrix} \end{align*}$$

for some $\kappa \colon G \to M^{\prime }_{a+1,b+1}(T)$ with $\kappa _{a+1,1} = \lambda $ .

The advantage of proper defining systems is that they are parameterized by abelian, rather than nonabelian, cocycles. To show this, we introduce a compact -module $\mathfrak {U}_{\phi ,\theta }(T)$ , such that proper defining systems in T relative to $(\phi ,\theta )$ correspond to 1-cocycles with values in $\mathfrak {U}_{\phi ,\theta }(T)$ .

Consider the compact R-module $\mathfrak {U}_{n+2}(R)$ that is the R-module of strictly upper-triangular $(n+2)$ -dimensional square matrices. The group $\mathrm {U}_{n+2}(R)$ acts continuously on $\mathfrak {U}_{n+2}(R)$ by conjugation. We consider a -submodule $\mathfrak {U}_{a,b}(R)$ of $\mathfrak {U}_{n+2}(R)$ given by

$$\begin{align*}\{ x=(x_{ij}) \in M_{n+2}(R) \ | \ x_{ij} =0 \text{ if } j \le a+1 \text{ or } i \ge a+2 \}. \end{align*}$$

In other words, breaking $M_{n+2}(R)$ into blocks using the partition $n+2=(a+1)+(b+1)$ and using block-matrix notation, we have

$$\begin{align*}\mathfrak{U}_{a,b}(R) = \begin{pmatrix} 0 & M_{a+1,b+1}(R) \\ 0 & 0 \end{pmatrix}. \end{align*}$$

Given a partial defining system $(\phi , \theta )$ , we consider $\mathfrak {U}_{a,b}(R)$ as a G-module via the continuous homomorphism

$$\begin{align*}G \to \mathrm{U}_{n+2}(R), \quad g \mapsto \begin{pmatrix} \phi(g) & 0 \\ 0 & \theta(g) \end{pmatrix}. \end{align*}$$

We define an -module $\mathfrak {U}_{\phi ,\theta }(T)$ as $\mathfrak {U}_{a,b}(R) \otimes _R T$ with the diagonal G-action. We also have the following equivalent definition, which has the benefit of being more explicit:

  • $\mathfrak {U}_{\phi ,\theta }(T) = M_{a+1,b+1}(T)$ as an R-module,

  • the action map $G \to \operatorname {\mathrm {End}}(M_{a+1,b+1}(T))$ is given, for $g \in G$ and $x \in M_{a+1,b+1}(T)$ , by

    $$\begin{align*}g \star x = \phi(g) \cdot gx \cdot \theta(g)^{-1}, \end{align*}$$
    where $gx$ means apply the g action on T to each matrix entry, and the multiplication denoted by ‘ $\cdot $ ’ is of matrices.

Going forward, we use the latter description of $\mathfrak {U}_{\phi ,\theta }(T)$ , so consider it as consisting of $(a+1)$ -by- $(b+1)$ matrices, rather than as a subgroup of $M_{n+2}(R)$ . Note that $\mathfrak {U}_{\phi ,\theta }(T)$ contains a copy of T as an -submodule by inclusion in the $(1,b+1)$ -entry. Let

$$ \begin{align*} \mathfrak{U}^{\prime}_{\phi,\theta}(T) = \mathfrak{U}_{\phi,\theta}(T)/T. \end{align*} $$

Let $x \mapsto \tilde {x}$ denote the R-module section $\mathfrak {U}_{\phi ,\theta }'(T) \to \mathfrak {U}_{\phi ,\theta }(T)$ given by filling in the $(1,b+1)$ -entry as $0$ .

Lemma 3.3.3. Let $(\phi ,\theta )$ be a partial defining system for Massey products $(\alpha ,\,\cdot \,,\beta )$ . Then the map that takes a continuous $1$ -cocycle $\kappa ' \colon G \to \mathfrak {U}^{\prime }_{\phi ,\theta }(T)$ to a map $\rho \colon G \to \mathcal {U}^{\prime }_{n+2}(T,a)$ given by

$$\begin{align*}\rho=\begin{pmatrix} \phi & \kappa'\theta \\ 0 & \theta \end{pmatrix} \end{align*}$$

is a bijection between $Z^1(G,\mathfrak {U}^{\prime }_{\phi ,\theta }(T))$ and the set of proper defining systems in T relative to $(\phi ,\theta )$ .

Proof. Given a cochain $\kappa ' \colon G \to \mathfrak {U}^{\prime }_{\phi ,\theta }(T)$ , set $\kappa = \kappa '\theta \colon G \to \mathfrak {U}^{\prime }_{\phi ,\theta }(T)$ . We have to check that

$$ \begin{align*} \rho = \begin{pmatrix} \phi & \kappa \\ 0 & \theta \end{pmatrix} \end{align*} $$

is a cocycle if and only if $\kappa '$ is a cocycle. Matrix multiplication tells us that $\rho $ is a cocycle if and only if

(3.1) $$ \begin{align} \kappa(gh)= \phi(g) g\kappa(h) + \kappa(g)\theta(h). \end{align} $$

The cochain $\kappa '$ is a cocycle if and only if the second equality holds in the following string of equalities

$$ \begin{align*} \kappa(gh)& =\kappa'(gh)\theta(gh) \\ & =( g\star \kappa'(h)+\kappa'(g))\theta(gh) \\ & =( \phi(g) g\kappa'(h) \theta(g)^{-1} + \kappa'(g)) \theta(g) \theta(h)\\ &= \phi(g) g\kappa'(h) \theta(h) + \kappa'(g)\theta(g) \theta(h) \\ &= \phi(g) g\kappa(h) + \kappa(g)\theta(h), \end{align*} $$

hence, the result.

The value of the Massey product associated to a proper defining system is also a value of a connecting homomorphism for an exact sequence attached to the underlying partial defining system.

Theorem 3.3.4. Let $(\phi ,\theta )$ be a partial defining system for $(\alpha ,\,\cdot \,,\beta )$ . Let $\kappa ' \in Z^1(G,\mathfrak {U}^{\prime }_{\phi ,\theta }(T))$ , and let $\rho = \left (\begin {smallmatrix} \phi & \kappa ' \theta \\ & \theta \end {smallmatrix}\right )$ be the associated proper defining system as in Lemma 3.3.3. Consider the short exact sequence

$$\begin{align*}0 \to T \to \mathfrak{U}_{\phi,\theta}(T) \to \mathfrak{U}^{\prime}_{\phi,\theta}(T) \to 0. \end{align*}$$

Then the image of the class of $\kappa '$ under the connecting map

$$\begin{align*}\partial \colon H^1(G,\mathfrak{U}^{\prime}_{\phi,\theta}(T)) \to H^2(G,T) \end{align*}$$

is the $(n+1)$ -fold Massey product $(\alpha _1,\dots ,\alpha _a, \kappa ^{\prime }_{a+1,1}, \beta _1,\dots ,\beta _b)_{\rho }$ .

Proof. Let $\kappa = \kappa ' \theta \colon G \to \mathfrak {U}^{\prime }_{\phi ,\theta }(T)$ , and let $\tilde {\kappa }$ be its unique lift to $\mathfrak {U}_{\phi ,\theta }(T)$ with $\tilde {\kappa }(g)$ having zero $(1,b+1)$ -entry for all $g \in G$ . The map $\tilde {\kappa }' = \tilde {\kappa }\theta ^{-1} \colon G \to \mathfrak {U}_{\phi ,\theta }(T)$ is then a lift of $\kappa '$ . By definition, the image of $\kappa $ is represented by the $2$ -cocycle that is given by taking the $(1,b+1)$ -entry of $d\tilde {\kappa }'$ . We have

$$ \begin{align*} d\tilde{\kappa}'(g,h) &= \tilde{\kappa}'(g) + g \star \tilde{\kappa}'(h) - \tilde{\kappa}'(gh) \\ &= \tilde{\kappa}(g)\theta(g)^{-1} + \phi(g) g \tilde{\kappa}(h) \theta(h)^{-1} \theta(g)^{-1} - \tilde{\kappa}(gh)\theta(gh)^{-1}\\ &= (\tilde{\kappa}(g)\theta(h) + \phi(g)g\tilde{\kappa}(h) - \tilde{\kappa}(gh))\theta(gh)^{-1}. \end{align*} $$

Since $\kappa $ satisfies (3.1), we have $\tilde {\kappa }(g)\theta (h) + \phi (g)g\tilde {\kappa }(h) - \tilde {\kappa }(gh) \in T$ , and T is fixed under the action of right multiplication by an element of $\mathrm {U}_{b+1}(R)$ . Since $\tilde {\kappa }(gh)$ has zero $(1,b+1)$ -entry, the $(1,b+1)$ -entries of $d\tilde {\kappa }'(g,h)$ and $\tilde {\kappa }(g)\theta (h) + \phi (g)g\tilde {\kappa }(h)$ are equal.

The Massey product $(\alpha _1,\dots ,\alpha _a, \kappa _{a+1,1}, \beta _1,\dots ,\beta _b)_{\rho }$ (and note that $\kappa _{a+1,1} = \kappa ^{\prime }_{a+1,1}$ ) is the $(1,n+2)$ -entry of $\tilde {\rho }(g) \cdot g\tilde {\rho }(h)$ , where

$$ \begin{align*} \tilde{\rho} = \begin{pmatrix} \phi & \tilde{\kappa} \\ 0 & \theta \end{pmatrix}. \end{align*} $$

The result then follows from the fact that

$$ \begin{align*} \tilde{\rho}(g) \cdot g\tilde{\rho}(h) = \begin{pmatrix} \phi(g) & \tilde{\kappa}(g) \\ 0 & \theta(g) \end{pmatrix}\begin{pmatrix} \phi(h) & g\tilde{\kappa}(h) \\ 0 & \theta(h) \end{pmatrix} = \begin{pmatrix} \phi(gh) & \phi(g)g\tilde{\kappa}(h) + \tilde{\kappa}(g)\theta(h) \\ 0 & \theta(gh) \end{pmatrix}.\\[-44pt] \end{align*} $$

In fact, the proof of Theorem 3.3.4 gives an explicit map $Z^1(G,\mathfrak {U}^{\prime }_{\phi ,\theta }(T)) \to Z^2(G,T)$ , taking a $1$ -cocycle $\kappa '$ to the $(1,b+1)$ -entry of $d\tilde {\kappa }'$ , for the specific lift $\tilde {\kappa }'$ of $\kappa '$ defined therein.

4 Massey products as values of Bockstein maps

We return to the setting and notation of Section 2. We first discuss a general result that gives partial information about the generalized Bockstein map $\Psi ^{(n)}$ in terms of Massey products. Then we discuss specific examples where this information completely determines $\Psi ^{(n)}$ .

4.1 Partial defining systems and Bockstein maps

Fix integers $a,b \ge 0$ , such that $a+b=n$ and group homomorphisms

$$ \begin{align*} \phi \colon H \to \mathrm{U}_{a+1}(R)\ \mathrm{and}\ \theta \colon H \to \mathrm{U}_{b+1}(R), \end{align*} $$

so viewing $\phi $ and $\theta $ as maps from G via precomposition with the quotient map, the pair $(\phi ,\theta )$ is an $(a,b)$ -partial defining system. We let $\alpha = (\phi _{i,i+1})_i$ and $\beta = (\theta _{i,i+1})_i$ , so this partial defining system is of Massey products $(\alpha , \,\cdot \,, \beta )$ . If $b = 0$ , we often refer to the pair $(\phi ,\theta )$ simply as $\phi $ .

Lemma 4.1.1. Let $e \in \mathfrak {U}_{a,b}(R)$ be the matrix with $(a+1,1)$ -entry equal to $1$ and all other entries $0$ . There is a continuous -module homomorphism $p_{\phi ,\theta } \colon \Omega /I^{n+1} \to \mathfrak {U}_{a,b}(R)$ given on the cosets of images of group elements by

$$ \begin{align*} p_{\phi,\theta}([h]) = \phi(h) \cdot e \cdot \theta(h)^{-1}. \end{align*} $$

The image of $I^n$ is contained in the submodule of matrices that are zero outside of their $(1,b+1)$ -entries.

Proof. The map $\tilde {p}_{\phi ,\theta } \colon \Omega \to \mathfrak {U}_{a,b}(R)$ inducing $p_{\phi ,\theta }$ is given by the action of H on $e \in \mathfrak {U}_{a,b}(R)$ via the composite homomorphism

$$\begin{align*}H \xrightarrow{\rho_{\phi,\theta}} \mathrm{U}_{n+2}(R) \xrightarrow{\mathrm{ad}} \operatorname{\mathrm{Aut}}(\mathfrak{U}_{a,b}(R)), \end{align*}$$

where $\rho _{\phi ,\theta } \colon H \to \mathrm {U}_{n+2}(R)$ is given by

$$ \begin{align*} \rho_{\phi,\theta}(h) = \begin{pmatrix} \phi(h) & 0 \\ 0 & \theta(h) \end{pmatrix} \end{align*} $$

and $\mathrm {ad}$ denotes the conjugation action. The action of G on $\Omega $ is given by the homomorphism $G \to H$ , and the action of G on $\mathfrak {U}_{a,b}(R)$ is given by the composite of this map with $H \to \operatorname {\mathrm {Aut}}(\mathfrak {U}_{a,b}(R))$ , so $\tilde {p}_{\phi ,\theta }$ is G-equivariant. We must show it factors through $\Omega /I^{n+1}$ .

Let be the augmentation ideal. Since the H-action factors through $\mathrm {U}_{n+2}(R)$ , we have $I^k\mathfrak {U}_{a,b}(R) \subseteq J^k\mathfrak {U}_{a,b}(R)$ for all k. It is easy to see inductively that

$$\begin{align*}J^k\mathfrak{U}_{n+2}(R) = \{ (a_{ij}) \in M_{n+2}(R) \ | \ a_{ij}=0 \text{ if } j-i \le k\}. \end{align*}$$

In particular, $J^{n+1}\mathfrak {U}_{n+2}(R)=0$ and

$$\begin{align*}J^n\mathfrak{U}_{n+2}(R)= \{ (a_{ij}) \in M_{n+2}(R) \ | \ a_{ij}=0 \text{ if } (i,j) \ne (1,n+2)\}. \end{align*}$$

Still viewing $\mathfrak {U}_{a,b}(R)$ as a subgroup of $\mathfrak {U}_{n+2}(R)$ , the containments

$$\begin{align*}I^k\mathfrak{U}_{a,b}(R) \subseteq J^k\mathfrak{U}_{a,b}(R) \subseteq J^k\mathfrak{U}_{n+2}(R) \end{align*}$$

imply the result.

Lemma 4.1.1 implies that there is a map of short exact sequences of -modules


where $p_{\phi ,\theta }$ is the tensor product with T of the map in Lemma 4.1.1 coming from $\rho _{\phi ,\theta }$ . As a direct consequence of this commutativity and Theorem 3.3.4, we have the following.

Theorem 4.1.2. Let $\phi \colon G \to \mathrm {U}_{a+1}(R)$ and $\theta \colon G \to \mathrm {U}_{b+1}(R)$ restrict to $\alpha \in Z^1(G,R)^a$ and $\beta \in Z^1(G,R)^b$ as above. Let $f \in Z^1(G,T \otimes _R \Omega /I^n)$ , and let $\rho $ denote the proper defining system relative to $(\phi ,\theta )$ associated to $p_{\phi ,\theta } \circ f$ by Lemma 3.3.3. Then we have

$$\begin{align*}p_{\phi,\theta} (\Psi^{(n)}([f])) = (\alpha,(p_{\phi,\theta} \circ f)_{a+1,1},\beta)_{\rho} \end{align*}$$

in $H^2(G,T)$ . Here, the maps $p_{\phi ,\theta }$ on the left and right are those induced on cohomology by the left and right vertical maps in (4.1).

We will give examples of groups H and integers n, such that there is a set X of choices of $(\phi ,\theta )$ for which the map

$$\begin{align*}H^2(G,T) \otimes_R I^n/I^{n+1} \xrightarrow{\prod_{(\phi,\theta) \in X} p_{\phi,\theta}} \prod_{(\phi, \theta) \in X} H^2(G,T) \end{align*}$$

is injective. In such cases, Theorem 4.1.2 shows that the generalized Bockstein map $\Psi ^{(n)}$ is determined by Massey products. In the rest of this section, we consider some specific examples in detail.

4.2 Unipotent binomial matrices

We introduce the unipotent binomial matrices, which are a source of many partial defining systems. Let n denote a positive integer, and let p be a prime number.

Let $u_n$ denote the $(n+1)$ -dimensional nilpotent upper triangular matrix

$$\begin{align*}u_n = \begin{pmatrix} 0 &\hspace{6pt} 1 &\hspace{6pt} 0 &\hspace{6pt} \cdots &\hspace{6pt} 0 \\ &\hspace{6pt} 0 &\hspace{6pt} 1 &\hspace{6pt} \ddots &\hspace{6pt} \vdots \\ &\hspace{6pt}&\hspace{6pt} 0 &\hspace{6pt} \ddots &\hspace{6pt} 0 \\ &\hspace{6pt}&\hspace{6pt}&\hspace{6pt} \ddots &\hspace{6pt} 1 \\ &\hspace{6pt}&\hspace{6pt}&\hspace{6pt}&\hspace{6pt} 0 \end{pmatrix}. \end{align*}$$

For any $k \ge 1$ , the matrix $u_n^k$ has $(i,j)$ -entry $1$ if $j-i=k$ and $0$ otherwise. In particular, we have $u_n^{n+1} = 0$ .

Let $\genfrac {[}{]}{0pt}{}{\,\cdot \,}{n} \colon \mathbb {Z}_p \to \mathrm {U}_{n+1}(\mathbb {Z}_p)$ denote the unique continuous homomorphism to $(n+1)$ -dimensional unipotent matrices with $\mathbb {Z}_p$ -entries such that $\genfrac {[}{]}{0pt}{}{1}{n}=1+u_n$ . By the binomial theorem, for $a\in \mathbb {Z}$ , we have

$$\begin{align*}\genfrac{[}{]}{0pt}{}{a}{n} = (1+u_n)^a = \sum_{k=0}^n \binom{a}{k} u_n^k = \begin{pmatrix} 1 &\hspace{6pt} a &\hspace{6pt} \binom{a}{2} &\hspace{6pt} \cdots &\hspace{6pt} \binom{a}{n} \\ &\hspace{6pt} 1 &\hspace{6pt} a &\hspace{6pt} \ddots &\hspace{6pt} \vdots \\ &\hspace{6pt}&\hspace{6pt} 1 &\hspace{6pt} \ddots &\hspace{6pt} \binom{a}{2} \\ &\hspace{6pt}&\hspace{6pt}&\hspace{6pt} \ddots &\hspace{6pt} a \\ &\hspace{6pt}&\hspace{6pt}&\hspace{6pt}&\hspace{6pt} 1 \end{pmatrix}. \end{align*}$$

If $t \ge s$ and $n < p^{t-s+1}$ , then the composite map

$$\begin{align*}\mathbb{Z} \xrightarrow{\genfrac{[}{]}{0pt}{}{\,\cdot\,}{n}} \mathrm{U}_{n+1}(\mathbb{Z}_p) \to \mathrm{U}_{n+1}(\mathbb{Z}/p^s\mathbb{Z}) \end{align*}$$

that sends a to $(1+u_n)^a$ modulo $p^s$ factors through $\mathbb {Z}/p^t\mathbb {Z}$ by Lemma 2.3.1 applied with $x = u_n$ . By abuse of notation, we again denote the resulting map $\mathbb {Z}/p^t\mathbb {Z} \to \mathrm {U}_{n+1}(\mathbb {Z}/p^s\mathbb {Z})$ by $\genfrac {[}{]}{0pt}{}{\,\cdot \,}{n}$ . In particular, the map $\binom {\,\cdot \,}{n} \colon \mathbb {Z} \to \mathbb {Z}/p^s\mathbb {Z}$ given by $a \mapsto \binom {a}{n} \pmod {p^s}$ factors through $\mathbb {Z}/p^t\mathbb {Z}$ , and we abuse notation to also denote the resulting map $\mathbb {Z}/p^t\mathbb {Z} \to \mathbb {Z}/p^s\mathbb {Z}$ by $\binom {\,\cdot \,}{n}$ .

The following lemma, phrased conveniently for our purposes, summarizes the above discussion.

Lemma 4.2.1. Let A be a quotient of the ring $\mathbb {Z}_p$ and R be a quotient of A. Let H be a profinite group and $\chi \colon H \to A$ be a continuous homomorphism. Suppose that either $A = \mathbb {Z}_p$ or $|R| < \frac {p}{n} |A|$ . Then there is a homomorphism

$$ \begin{align*} \genfrac{[}{]}{0pt}{}{\chi}{n} \colon H \to \mathrm{U}_{n+1}(R), \end{align*} $$

defined by $\genfrac {[}{]}{0pt}{}{\chi }{n}(h) = \genfrac {[}{]}{0pt}{}{\chi (h)}{n}$ for all $h\in H$ .

Proof. If $|R| = p^s$ and $|A| = p^t$ , then $|R| < \frac {p}{n}|A|$ if and only if $n < p^{t-s+1}$ .

4.3 Procyclic case

In this subsection, we fix a surjective homomorphism $\chi \colon G \to A$ , where A is a nonzero quotient of $\mathbb {Z}_p$ . We suppose that our ring R is a nonzero quotient of A with $A = \mathbb {Z}_p$ or $n|R| < p|A|$ . We define H to be the coimage of $\chi $ , so $H \cong A$ . We fix $h \in H$ to be the preimage of $1 \in A$ and let $x=[h]-1 \in \Omega $ , which is a generator of the augmentation ideal I. Our assumption on the size of R implies that