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.

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.

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

(2.7)

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.

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

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

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.

3.1 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.

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.

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.

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.

4.1 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

(4.1)

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 $h\in H$ .

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 $\Omega /I^j \cong R[x]/(x^j)$ for all $j \le n+1$ by the discussion of Section 2.3. In particular, we have $I^n/I^{n+1} = R x^n$ .

The $(n,0)$ -proper defining systems relative to $\phi = \genfrac {[}{]}{0pt}{}{\chi }{n}$ and the trivial map $\theta $ to $\mathrm {U}_1(R) = \{1\}$ agree with the proper defining systems considered in [Reference SharifiSh2] for Galois groups. We give an interpretation of the resulting Massey products in terms of generalized Bockstein maps. That is, let us apply the discussion of Section 4.1 to this situation. We have $\alpha =(\chi ,\dots ,\chi ) \in Z^1(G,R)^n$ , which we denote by $\chi ^{(n)}$ . We denote $\mathfrak {U}_{\phi ,\theta }(T)$ by $\mathfrak {U}_{\genfrac {[}{]}{0pt}{}{\chi }{n}}(T)$ .

The diagram (4.1) becomes

where $p_n$ is the map attached to $(\genfrac {[}{]}{0pt}{}{\chi }{n},0)$ by Lemma 4.1.1. Explicitly, the vector $p_n(\sum _{k=0}^n a_k x^k)$ in $M_{n+1,1}(T)$ has ith entry $a_{n+1-i}$ (see the more general case proven in Lemma 4.4.1 of the next subsection).

By Lemma 3.3.3, it follows that the proper defining system $\rho _{x^n}$ relative to $\genfrac {[}{]}{0pt}{}{\chi }{n}$ that is attached to $p_n \circ f$ , where

$$ \begin{align*} f = \sum_{k=0}^{n-1} \lambda_k x^k \in Z^1(G,T \otimes_R \Omega/I^n), \end{align*} $$

satisfies $(\rho _{x^n})_{n+1-k,n+2} = \lambda _k$ for $0 \le k \le n-1$ . In particular, the element $\lambda = \lambda _0 = (\rho _{x^n})_{n+1,n+2}$ is the image of f under the map

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

induced by the augmentation $\Omega /I^n \mapsto \Omega /I = R$ .

Theorem 4.1.2 then gives us an explicit description of the values of the generalized Bockstein homomorphism on classes in $H^1(G,T \otimes _R \Omega /I^n)$ as Massey products $(\chi ^{(n)},\,\cdot \,)$ relative to $\genfrac {[}{]}{0pt}{}{\chi }{n}$ , as follows.

Theorem 4.3.1. For $f \in Z^1(G,T \otimes _R \Omega /I^n)$ , we have

$$ \begin{align*} \Psi^{(n)}([f]) = (\chi^{(n)},\lambda)_{\rho_{x^n}} \cdot x^n, \end{align*} $$

where $\rho _{x^n}$ is the proper defining system relative to $\genfrac {[}{]}{0pt}{}{\chi }{n}$ attached to f, and $\lambda $ is the image of f in $Z^1(G,T)$ .

In particular, we have the following description of the image of $\Psi ^{(n)}$ .

Theorem 2.2.4 provides the following application to the graded quotients of Iwasawa cohomology groups of $N = \ker (\chi \colon G \to H)$ .

Corollary 4.3.3. Suppose that G is p-cohomologically finite of p-cohomological dimension $2$ . Let $P_n(H)$ denote the subgroup of $H^2(G,T) \otimes _R I^n/I^{n+1}$ generated by all $(\chi ^{(n)},\lambda )_{\rho } \cdot x^n$ for proper defining systems $\rho $ relative to $\genfrac {[}{]}{0pt}{}{\chi }{n}$ and $\lambda = \rho _{n+1,n+2}$ . We have a canonical isomorphism of R-modules

$$ \begin{align*} \frac{I^nH^2_{\mathrm{Iw}}(N,T)}{I^{n+1}H^2_{\mathrm{Iw}}(N,T)} \cong \frac{H^2(G,T) \otimes_R I^n/I^{n+1}}{P_n(H)}. \end{align*} $$

4.4 Pro-bicyclic case

In this subsection, we

• • fix a surjective homomorphism $(\chi ,\psi ) \colon G \twoheadrightarrow A \times B$ , where A and B are nonzero quotients of $\mathbb {Z}_p$ ,

• • let $a,b \ge 0$ denote integers, such that $a+b=n$ and

• • suppose that R is a nonzero quotient of both A and B with $a|R| < p|A|$ if A is finite and $b|R| < p|B|$ if B is finite.

Let H be the coimage of $(\chi ,\psi )$ so that $H \cong A \times B$ . Let $h_A,h_B \in H$ be the preimages of $(1,0),(0,1) \in A \times B$ , respectively, and let $x = [h_A]-1$ and $y=[h_B]-1$ so that $(x,y)$ is the augmentation ideal I of . We have $\Omega /I^j = R[x,y]/(x,y)^j$ for all $j \le n$ . In particular, we have

$$\begin{align*}I^n/I^{n+1} = \bigoplus_{i+j=n} R x^iy^j. \end{align*}$$

We apply the discussion of Section 4.1 to this situation. We take $\phi = \genfrac {[}{]}{0pt}{}{\chi }{a} \colon H \to \mathrm {U}_a(R)$ and $\theta = \genfrac {[}{]}{0pt}{}{\psi }{b} \colon H \to \mathrm {U}_b(R)$ . We have $\alpha =\chi ^{(a)} \in Z^1(G,R)^a$ and $\beta =\psi ^{(b)} \in Z^1(G,R)^b$ .

Set $p_{a,b} = p_{\genfrac {[}{]}{0pt}{}{\chi }{a},\genfrac {[}{]}{0pt}{}{\psi }{b}}$ for brevity. In this setting, the diagram (4.1) becomes

(4.2)

Lemma 4.4.1. The -module map $p_{a,b} \colon T \otimes _R \Omega /I^{n+1} \to \mathfrak {U}_{\genfrac {[}{]}{0pt}{}{\chi }{a},\genfrac {[}{]}{0pt}{}{\psi }{b}}(T)$ is an isomorphism satisfying

$$ \begin{align*} p_{a,b}\left(\sum_{k_1+k_2 \le n} c_{k_1,k_2} x^{k_1}y^{k_2}\right) = (c_{a+1-i,j-1})_{i,j}. \end{align*} $$

In particular, the left-hand vertical map in (4.2) is given by projection onto the factor $T \cdot x^a y^b \cong T$ .

Proof. This reduces immediately to the case that $T = R$ , since we can obtain the case of arbitrary T by R-tensor product with the identity of T. Let e be as in Lemma 4.1.1, the matrix with a single nonzero entry of $1$ in the $(a+1,1)$ -coordinate of $M_{a+1,b+1}(R)$ . The $(i,j)$ -entry of $g \star e = \genfrac {[}{]}{0pt}{}{\chi (g)}{a} e \genfrac {[}{]}{0pt}{}{\psi (g)}{b}$ is

$$\begin{align*}\binom{\chi(g)}{a+1-i}\binom{\psi(g)}{j-1}, \end{align*}$$

which agrees with the coefficient of $x^{a+1-i}y^{j-1}$ in $g \cdot 1$ by (2.9).

Corollary 4.4.2. For

$$ \begin{align*} f = \sum_{k_1+k_2 < n} \lambda_{k_1,k_2} x^{k_1}y^{k_2} \in Z^1(G,\Omega/I^n \otimes_R T) \end{align*} $$

and $\rho _{x^ay^b}$ the proper defining system relative to $(\genfrac {[}{]}{0pt}{}{\chi }{a},\genfrac {[}{]}{0pt}{}{\psi }{b})$ attached to $p_{a,b} \circ f$ by Lemma 3.3.3, we have

$$ \begin{align*} (\rho_{x^ay^b})_{a+1-k_1,a+2+k_2} = \lambda_{k_1,k_2} \end{align*} $$

for all $0 \le (k_1,k_2) < (a,b)$ . In particular, we have $(\rho _{x^ay^b})_{a+1,a+2} = \lambda _{0,0}$ , which is the image of f in $Z^1(G,T)$ under the map induced by the quotient $\Omega /I^n \to \Omega /I = R$ .

The following is then a direct consequence of Theorem 4.1.2.

Theorem 4.4.3. For $f \in Z^1(G,\Omega /I^n \otimes _R T)$ , the image of $\Psi ^{(n)}([f])$ in

$$ \begin{align*} H^2(G,T) \otimes_R I^n/I^{n+1} \cong \bigoplus_{a+b=n} H^2(G, T) \cdot x^ay^b \end{align*} $$

is

$$ \begin{align*} \sum_{a+b = n} (\chi^{(a)},\lambda,\psi^{(b)})_{\rho_{x^ay^b}} \cdot x^a y^b, \end{align*} $$

where $\rho _{x^ay^b}$ is the proper defining system relative to $(\genfrac {[}{]}{0pt}{}{\chi }{a},\genfrac {[}{]}{0pt}{}{\psi }{b})$ attached to $p_{a,b} \circ f$ and $\lambda $ is the image of f in $Z^1(G,T)$ .

Applying Theorem 2.2.4, we obtain the following description of graded quotients of Iwasawa cohomology.

Corollary 4.4.4. Suppose that G is p-cohomologically finite of p-cohomological dimension $2$ . Let $P_n(H)$ denote the subgroup of $H^2(G,T) \otimes _R I^n/I^{n+1}$ consisting of all sums $\sum _{a+b = n} (\chi ^{(a)},\lambda ,\psi ^{(b)})_{\rho _{x^ay^b}} \cdot x^a y^b$ , where the $\rho _{x^ay^b}$ and $\lambda $ are associated to a cocycle in $Z^1(G,\Omega /I^n \otimes _R T)$ as in Proposition 4.4.3. We then have a canonical isomorphism of R-modules

$$ \begin{align*} \frac{I^nH^2_{\mathrm{Iw}}(N,T)}{I^{n+1}H^2_{\mathrm{Iw}}(N,T)} \cong \frac{H^2(G,T) \otimes_R I^n/I^{n+1}}{P_n(H)}. \end{align*} $$

4.5 Elementary abelian p-groups

The pattern seen in the cyclic and bicyclic cases does not continue for all finitely generated abelian pro-p groups. To see why, consider the case that $H \cong \mathbb {F}_p^3$ with basis $(\gamma _1,\gamma _2,\gamma _3)$ and dual basis $(\chi _1,\chi _2,\chi _3)$ . For $x_i = [\gamma _i]-1$ , we have a basis of $I^n/I^{n+1}$ consisting of monomials $x_1^{i_1}x_2^{i_2}x_3^{i_3}$ with $i_1+i_2+i_3=n$ . Following the pattern of the cyclic and bicyclic cases, one might guess that the coefficient of $x_1^{i_1}x_2^{i_2}x_3^{i_3}$ in $\Psi ^{(n)}([f])$ is an $(n+1)$ -fold Massey product involving $i_j$ copies of each $\chi _j$ and another cocycle $\lambda $ determined by f. However, this pattern fails already for $n=3$ and the coefficient of $x_1x_2x_3$ : any 4-fold Massey product involving the $\chi _i$ must have two of these characters beside each other, and thus, to be defined, the cup product of those two characters must vanish. Since these cup products will not vanish in general, we cannot hope for such a general statement to hold.

Nevertheless, at least in some cases, one can still describe the generalized Bockstein maps $\Psi ^{(n)}$ in terms of Massey products, at the expense of taking a nonstandard basis for $I^n/I^{n+1}$ . In this subsection, we assume that $H \cong \mathbb {F}_p^r$ for some $r \ge 1$ . Correspondingly, we take $n < p$ and $R = \mathbb {F}_p$ .

We let $V^\vee =\operatorname {\mathrm {Hom}}(V,\mathbb {F}_p)$ for an abelian group V. For any element $\chi \in H^\vee $ , we have a homomorphism

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

Precomposing with $G \to H$ , we may view $\chi $ as a character of G. This gives an $(n,0)$ -partial defining system, and we set $p_{\chi ,n} = p_{\genfrac {[}{]}{0pt}{}{\chi }{n},0}$ for brevity. By (4.1), the map $p_{\chi ,n}$ induces a map $p_{\chi ,n} \colon I^n/I^{n+1} \to \mathbb {F}_p$ , so $p_{\chi ,n} \in (I^n/I^{n+1})^\vee $ . This defines a function $p_{-,n} \colon H^\vee \to (I^n/I^{n+1})^\vee $ .

Let us fix an isomorphism $H \xrightarrow {\sim } \mathbb {F}_p^r$ , which, in turn, fixes an ordered dual basis $(\gamma _i)_{i=1}^r$ of H. Setting $x_i = [\gamma _i] - 1 \in \Omega $ , this provides an identification

(4.3) $$ \begin{align} \Omega/I^{n+1} = \mathbb{F}_p[x_1,\dots,x_n]/(x_1, \ldots, x_r)^{n+1}. \end{align} $$

Then $I^n/I^{n+1}$ has a basis given by $x_1^{d_1}\cdots x_r^{d_r}$ with $(d_1,\ldots ,d_r)$ ranging over r-tuples of nonnegative integers with $d_1+\dots +d_r=n$ . We compute $p_{\chi ,n}$ on this basis.

Lemma 4.5.1. Let $\chi \in H^{\vee }$ . For any nonnegative integers $d_1, \ldots , d_r$ with sum n, we have

$$\begin{align*}p_{\chi,n}(x_1^{d_1}\cdots x_r^{d_r})=\prod_{i=1}^r \chi(\gamma_i)^{d_i}. \end{align*}$$

Proof. We have

$$ \begin{align*} p_{\chi,n}(x_i) = \left(\genfrac{[}{]}{0pt}{}{\chi(\gamma_i)}{n}-1 \right)e=((1+u_n)^{\chi(\gamma_i)}-1)e \in \mathfrak{U}_{\genfrac{[}{]}{0pt}{}{\chi}{n}}, \end{align*} $$

where $u_n$ is as in Section 4.2 and e is as in Lemma 4.1.1. Note that $u_n^n$ has a $1$ in its $(1,n+1)$ entry and all other entries $0$ , and $u_n^{n+1} = 0$ . For $d_1+\dots +d_r=n$ , the value $p_{\chi ,n}(x_1^{d_1}\cdots x_r^{d_r})$ is the $(1,n+1)$ -entry of the matrix

$$ \begin{align*} \prod_{i=1}^r ((1+u_n)^{\chi(\gamma_i)}-1)^{d_i} = \prod_{i=1}^r (\chi(\gamma_i)u_n)^{d_i} = \left(\prod_{i=1}^r \chi(\gamma_i)^{d_i} \right) u_n^n, \end{align*} $$

proving the lemma.

The key lemma is then the following.

Proof. Using our identification (4.3), any nonzero $F \in I^n/I^{n+1}$ has a unique representative also denoted F in $\mathbb {F}_p[x_1,\dots ,x_r]$ that is homogeneous of degree n and Lemma 4.5.1 implies that

$$\begin{align*}p_{\chi,n}(F) = F(\chi(\gamma_1),\dots,\chi(\gamma_r)). \end{align*}$$

Writing $\Gamma \colon H^\vee \to \mathbb {F}_p^r$ for the isomorphism given by $\Gamma (\chi ) = (\chi (\gamma _1),\dots ,\chi (\gamma _r))$ , this can be succinctly written as $p_{\chi ,n}(F)=F(\Gamma (\chi ))$ .

For a finite set S and an $s \in S$ , we denote by $\mathbb {F}_p^S$ the vector space of functions $S \to \mathbb {F}_p$ and by $1_s \in \mathbb {F}_p^S$ the indicator function of s. The lemma may then be rephrased as the statement that the linearization $\tilde {p}_{-,n}$ of $p_{-,n}$ , given by

$$\begin{align*}\tilde{p}_{-,n} \colon \mathbb{F}_p^{H^\vee} \to (I^n/I^{n+1})^\vee, \quad 1_\chi \mapsto p_{\chi,n} \end{align*}$$

is surjective, or, equivalently, that the dual map

$$\begin{align*}\tilde{p}_{-,n}^\vee \colon I^n/I^{n+1} \to (\mathbb{F}_p^{H^\vee})^\vee \end{align*}$$

is injective. For any nonzero $F \in I^n/I^{n+1}$ , since $n<p$ , the finite field Nullstellensatz provides the existence of $v \in \mathbb {F}_p^r$ for which $F(v) \ne 0$ . Then

$$ \begin{align*} \tilde{p}_{-,n}^\vee(F)(1_{\Gamma^{-1}(v)}) = p_{\Gamma^{-1}(v),n}(F)= F(v) \ne 0, \end{align*} $$

so $\tilde {p}_{-,n}^\vee (F) \ne 0$ .

We now come to our result expressing values of the generalized Bockstein maps as sums of ‘cyclic’ Massey products. If $\chi \in H^{\vee }$ and $f \in Z^1(G, T \otimes _R \Omega /I^n)$ , then we say that a proper defining system relative to $\genfrac {[}{]}{0pt}{}{\chi }{n}$ is attached to f if it is attached to the image of f in $Z^1(G, T \otimes _R \Omega _{\chi }/I_{\chi }^n)$ , where $\Omega _{\chi } = R[H/\ker \chi ]$ and $I_{\chi }$ is its augmentation ideal.

Theorem 4.5.4. There exist $N \ge 1$ and $\chi _1, \ldots , \chi _N \in H^\vee $ , such that $(p_{\chi _i,n})_{i=1}^N$ is an ordered $\mathbb {F}_p$ -basis of $(I^n/I^{n+1})^\vee $ . For any such $(\chi _i)_{i=1}^N$ , let $(y_i)_{i=1}^N$ be the basis of $I^n/I^{n+1}$ dual to $(p_{\chi _i,n})_{i=1}^N$ . Then for any $f \in Z^1(G, T \otimes _R \Omega /I^{n})$ , we have

$$\begin{align*}\Psi^{(n)}([f]) = \sum_{i=1}^N (\chi_i^{(n)},\lambda)_{\rho_i} \cdot y_i, \end{align*}$$

where $\rho _i$ is the proper defining system relative to $\genfrac {[}{]}{0pt}{}{\chi _i}{n}$ attached to f and $\lambda $ is the image of f in $Z^1(G,T)$ .

Proof. The first statement is clear from Lemma 4.5.2. For the second statement, let

$$\begin{align*}\Psi^{(n)}([f]) = \sum_{i=1}^N c_i \cdot y_i, \end{align*}$$

for some $c_i \in H^2(G,T)$ . Since $p_{\chi _1,n},\dots ,p_{\chi _N,n}$ is the dual basis to $y_1,\dots ,y_N$ , we have $c_i = p_{\chi _i,n}(\Psi ^{(n)}([f]))$ for $1 \le i \le N$ . But by Theorem 4.1.2, we have

$$ \begin{align*} p_{\chi_i,n}(\Psi^{(n)}([f])) = (\chi_i^{(n)},\lambda)_{\rho_i}.\\[-36pt] \end{align*} $$

4.6 Heisenberg case

In this section, assume that $H=\mathrm {U}_3(A)$ for a nonzero quotient A of $\mathbb {Z}_p$ , and that R is a quotient of A, such that either $R=\mathbb {Z}_p$ or $n|R| < p|A|$ . We study the generalized Bockstein maps $\Psi ^{(n)}$ in the cases $n = 2$ and $n = 3$ .

Let

(4.4) $$ \begin{align} x= \left[\left(\begin{smallmatrix}1 & 1 & 0 \\ 0& 1 & 0 \\ 0 & 0 & 1\end{smallmatrix}\right)\right]-1, \quad y=\left[\left(\begin{smallmatrix}1 & 0 & 0 \\ 0& 1 & 1 \\ 0 & 0 & 1\end{smallmatrix}\right)\right]-1, \quad z= \left[\left(\begin{smallmatrix}1 & 0 & 1 \\ 0& 1 & 0 \\ 0 & 0 & 1\end{smallmatrix}\right)\right]-1 \in \Omega. \end{align} $$

Then I is the two-sided ideal generated by x and y, and $I/I^2 \cong Rx \oplus Ry$ . Let $\chi , \psi \colon G \to A$ be the unique characters factoring through H such that

$$\begin{align*}\chi\left(\left(\begin{smallmatrix}1 & 1 & 0 \\ 0& 1 & 0 \\ 0 & 0 & 1\end{smallmatrix}\right)\right) = 1, \quad \chi\left(\left(\begin{smallmatrix}1 & 0 & 0 \\ 0& 1 & 1 \\ 0 & 0 & 1\end{smallmatrix}\right)\right) = 0, \quad \psi\left(\left(\begin{smallmatrix}1 & 1 & 0 \\ 0& 1 & 0 \\ 0 & 0 & 1\end{smallmatrix}\right)\right) = 0, \quad \mathrm{and} \quad \psi\left(\left(\begin{smallmatrix}1 & 0 & 0 \\ 0& 1 & 1 \\ 0 & 0 & 1\end{smallmatrix}\right)\right) = 1. \end{align*}$$

Then $(\chi , \psi ) \colon G \to A \times A$ defines a homomorphism.

Lemma 4.6.1. The R-module $I^2/I^3$ is freely generated by the image of the set

$$ \begin{align*} S_2 = \{ x^2, y^2, yx, z \}, \end{align*} $$

and $I^3/I^4$ is R-freely generated by the image of

$$ \begin{align*} S_3=\{x^3,xz,yx^2,y^2x,y^3,yz\}. \end{align*} $$

Proof. For any n, Lemma 2.3.1 and the condition that $n|R| < p|A|$ in the case that A is finite are enough to guarantee that the quotient $\Omega /I^{n+1}$ is isomorphic to the analogous quotient with A replaced by $\mathbb {Z}_p$ , so we may suppose in this proof that $H = \mathrm {U}_3(\mathbb {Z}_p)$ .

Let $\Sigma $ be the noncommutative -power series ring $\Sigma $ in variables x and y. It follows from the standard presentation of $\mathrm {U}_3(\mathbb {Z}_p)$ as a finitely generated pro-p group that is the quotient of $\Sigma $ by the ideal generated by

(4.5) $$ \begin{align} w = (1+y)(1+x)z - (xy-yx). \end{align} $$

The augmentation ideal I of $\Omega $ is $(x,y)$ , so $I^n$ is generated by the monomials in x and y of degree at least n. Using (4.5), we can reduce this to

$$ \begin{align*} I^n = (y^j x^i z^k \mid i+j+2k \ge n). \end{align*} $$

It is therefore enough to check that the image of the set $S_n$ is R-linearly independent in $I^n/I^{n+1}$ for $n\in \{2,3\}$ .

Consider $\Sigma $ as a graded R-algebra with x, y and z in degrees $1$ , $1$ and $2$ , respectively. Let $J_n$ denote the ideal of elements of $\Sigma $ of degree at least n. Suppose that $f\in \Sigma $ lies in the intersection of the R-span of the elements of $S_n$ with $(w)+J_{n+1}$ . When $n=2$ , one can easily see that $f=0$ . When $n=3$ , there are $a, b, c, d \in R$ , such that

$$\begin{align*}f+J_4 = (ax+by)w + w(cx+dy)+J_4. \end{align*}$$

By the hypothesis on f, the degree $3$ terms above are in the R-span of $S_3$ , which forces $a=b=c=d=0$ , and, hence, $f=0$ .

Let us first consider the case $n = 2$ . By Lemma 4.6.1, we see that $I^2/I^3$ is a free R-module on the set $S_2 = \{x^2,y^2,yx,z\}$ . We consider the three partial defining systems

$$\begin{align*}\phi_{x^2} = \genfrac{[}{]}{0pt}{}{\chi}{2}, \quad \phi_{y^2} = \genfrac{[}{]}{0pt}{}{\psi}{2}, \quad \phi_z \colon H \to \mathrm{U}_3(R) \times \mathrm{U}_1(R) = \mathrm{U}_3(R), \end{align*}$$

with $a=2$ and $b=0$ , where $\phi _z$ is the quotient map on coefficients and the partial defining system

$$\begin{align*}\phi_{yx} = (\chi,\psi) \colon H \to \mathrm{U}_2(R) \times \mathrm{U}_2(R) = R \times R \end{align*}$$

for $a=b=1$ . By Theorem 3.3.4, the partial defining systems $\phi _{x^2}$ , $\phi _{y^2}$ , $\phi _{z}$ and $\phi _{yx}$ correspond to Massey products $(\chi ,\chi , \,\cdot \,)$ , $(\psi ,\psi ,\,\cdot \,)$ , $(\chi ,\psi ,\,\cdot \,)$ and $(\chi , \,\cdot \,, \psi )$ , respectively.

As for $n = 3$ , the graded quotient $I^3/I^4$ is a free R-module on $S_3$ of Lemma 4.6.1. For each $s \in S_3$ , we define a partial defining system $\phi _s$ (viewed as a pair of homomorphisms) as follows:

$$ \begin{align*} \phi_{x^3}\colon H \xrightarrow{\genfrac{[}{]}{0pt}{}{\chi}{3},0} \mathrm{U}_4(R) \times \mathrm{U}_1(R), \\ \phi_{xz}\colon H \xrightarrow{\mathrm{id},\chi} \mathrm{U}_3(R) \times \mathrm{U}_2(R), \\ \phi_{yx^2}\colon H \xrightarrow{\psi,\genfrac{[}{]}{0pt}{}{\chi}{2}} \mathrm{U}_2(R) \times \mathrm{U}_3(R), \\ \phi_{y^2x}\colon H \xrightarrow{\genfrac{[}{]}{0pt}{}{\psi}{2},\chi} \mathrm{U}_3(R) \times \mathrm{U}_2(R), \\ \phi_{y^3}\colon H \xrightarrow{\genfrac{[}{]}{0pt}{}{\psi}{3},0} \mathrm{U}_4(R) \times \mathrm{U}_1(R), \\ \phi_{yz}\colon H \xrightarrow{\psi,\mathrm{id}} \mathrm{U}_2(R) \times \mathrm{U}_3(R). \end{align*} $$

By Theorem 3.3.4, each partial defining system corresponds to a collection of Massey products as follows:

$$ \begin{align*} \phi_{x^3} \longleftrightarrow (\chi,\chi,\chi,\,\cdot\,), \\ \phi_{xz} \longleftrightarrow (\chi,\psi,\,\cdot,\chi), \\ \phi_{yx^2} \longleftrightarrow (\psi,\,\cdot,\chi,\chi), \\ \phi_{y^2x} \longleftrightarrow (\psi,\psi,\,\cdot,\chi), \\ \phi_{y^3} \longleftrightarrow (\psi,\psi,\psi,\,\cdot\,), \\ \phi_{yz}\longleftrightarrow (\psi,\,\cdot,\chi,\psi). \end{align*} $$

For each $s \in S_n$ with $n \in \{2,3\}$ , the diagram (4.1) becomes

where the maps $p_s$ are induced by the map $\phi _s$ , and we have used the shorthand $\mathfrak {U}_s(T)$ for $\mathfrak {U}_{\phi _s}(T)$ (and similarly for the quotients). Note that $p_s \colon T \otimes _R I^n/I^{n+1} \to T$ is just the R-tensor product of the likewise-defined $p_s \colon I^n/I^{n+1} \to R$ with the identity on T. The maps $p_s \colon I^n/I^{n+1} \to R$ for $s \in S_n$ form the dual basis to the R-basis $S_n$ of $I^n/I^{n+1}$ . This can be seen by an omitted direct computation, proceeding as in the following example.

Example 4.6.2. Suppose that $n = 2$ , and take $s = z \in S_2$ . Recall that $\phi _z \colon H \to \mathrm {U}_3(R)$ is given by the canonical surjection $A \to R$ on coefficients. By definition of $p_z \colon \Omega /I^3 \to \mathfrak {U}_{\phi _z}(R) = M_{3,1}(R)$ in Lemma 4.1.1, we have

$$ \begin{align*} p_z([h]) = \phi_z(h) \left( \begin{smallmatrix} 0 \\0\\1 \end{smallmatrix} \right) \in M_{3,1}(R) \end{align*} $$

for all $h \in H$ . Recalling that $x+1$ , $y+1$ and $z+1$ are the group elements of matrices as in (4.4), we compute

(4.6) $$ \begin{align} \begin{aligned} &p_{z}(x^2) = \left( \begin{smallmatrix} 0 \\0\\1 \end{smallmatrix} \right) - 2 \left( \begin{smallmatrix} 0 \\0\\1 \end{smallmatrix} \right) + \left( \begin{smallmatrix} 0 \\0\\1 \end{smallmatrix} \right) = 0,\\ & p_{z}(y^2)= \left( \begin{smallmatrix} 0 \\2\\1 \end{smallmatrix} \right) - 2 \left( \begin{smallmatrix} 0 \\1\\1 \end{smallmatrix} \right) + \left( \begin{smallmatrix} 0 \\0\\1 \end{smallmatrix} \right) = 0,\\ & p_{z}(yx) = \left( \begin{smallmatrix} 0 \\1\\1 \end{smallmatrix} \right) - \left( \begin{smallmatrix} 0 \\1\\1 \end{smallmatrix} \right) + \left( \begin{smallmatrix} 0 \\0\\1 \end{smallmatrix} \right) - \left( \begin{smallmatrix} 0 \\0\\1 \end{smallmatrix} \right) = 0, \\ & p_{z}(z) = \left( \begin{smallmatrix} 1 \\0\\1 \end{smallmatrix} \right) - \left( \begin{smallmatrix} 0 \\0\\1 \end{smallmatrix} \right) = \left( \begin{smallmatrix} 1\\0\\0\\ \end{smallmatrix}\right), \end{aligned} \end{align} $$

and note that $\left ( \begin {smallmatrix} 1\\0\\0\\ \end {smallmatrix}\right )$ gives the identity of $R \subset \mathfrak {U}_{\phi _z}(R)$ .

By Theorem 4.1.2, we then have the following.

Theorem 4.6.3. For $n \in \{2,3\}$ and $f \in Z^1(G,T \otimes _R \Omega /I^n)$ , the element $\Psi ^{(n)}([f])$ of

$$ \begin{align*} H^2(G,T) \otimes_R I^n/I^{n+1} \cong \bigoplus_{s \in S_n} H^2(G,T) s \end{align*} $$

is the sum

(4.7) $$ \begin{align} (\chi,\chi,\lambda)_{\rho_{x^2}} x^2 + (\chi,\lambda,\psi)_{\rho_{yx}} yx + (\psi,\psi,\lambda)_{\rho_{y^2}}y^2 + (\chi,\psi,\lambda)_{\rho_{z} }z \end{align} $$

for $n = 2$ and the sum

(4.8) $$ \begin{align} \begin{split} (\chi,\chi,&\chi,\lambda)_{\rho_{x^3}} x^3 + (\chi,\psi,\lambda,\chi)_{\rho_{xz}} xz + (\psi,\lambda,\chi,\chi)_{\rho_{yx^2}}yx^2 \\ &+ (\psi,\psi,\lambda,\chi)_{\rho_{y^2x} }y^2x+ (\psi,\psi,\psi,\lambda)_{\rho_{y^3} }y^3+ (\psi,\lambda,\chi,\psi)_{\rho_{yz} }yz \end{split} \end{align} $$

for $n = 3$ , where each $\rho _s$ for $s \in S_n$ is the proper defining system relative to $\phi _s$ attached to $p_s \circ f$ by Lemma 3.3.3, and $\lambda $ is the image of f in $Z^1(G,T)$ .

As before, Theorem 2.2.4 then provides the following isomorphisms.

Corollary 4.6.4. Suppose that G is p-cohomologically finite of p-cohomological dimension $2$ . For $n \in \{2,3\}$ , let $P_n(H)$ denote the subgroup of $H^2(G,T) \otimes _R I^n/I^{n+1}$ consisting of all sums in (4.7) for $n = 2$ and in (4.8) for $n = 3$ . We then have a canonical isomorphism of R-modules

$$ \begin{align*} \frac{I^nH^2_{\mathrm{Iw}}(N,T)}{I^{n+1} H^2_{\mathrm{Iw}}(N,T)} \cong \frac{H^2(G,T) \otimes_R I^n/I^{n+1}}{P_n(H)}. \end{align*} $$

5.1 Notation and preliminaries

In this subsection, we recall some standard facts regarding the mod p unramified outside p cohomology of the pth cyclotomic field, for an odd prime p. Most of these may be found, for instance, in [Reference McCallum and SharifiMcSh]. Let $\mathrm {Cl}_K$ denote the ideal class group of a number field K. Let S denote the set of primes over p in any number field. Let $\mathrm {Cl}_{K,S}$ denote the S-class group of K, which is to say the class group of the ring $\mathcal {O}_{K,S}$ of S-integers of K. Let $G_{K,S}$ denote the Galois group of the maximal unramified outside S, or p-ramified, extension of K.

For any number field K and prime p, Kummer theory provides an exact sequence

$$\begin{align*}0 \to \mathcal{O}_{K,S}^\times \otimes_{\mathbb{Z}} \mathbb{F}_p \to H^1(G_{K,S},\mu_p) \to \mathrm{Cl}_{K,S}[p] \to 0 \end{align*}$$

and a canonical injection

of $\mathbb {F}_p[\Delta ]$ -modules. The latter injection is an isomorphism if K is a p-ramified, purely imaginary extension of $\mathbb {Q}$ with a unique prime over p. We shall write Massey products of elements of $H^1(G_{F,S},\mu _p)$ as products of elements of $F^{\times }/F^{\times p}$ whose Kummer cocycles (in this case, characters) give classes in $H^1(G_{F,S},\mu _p)$ , as opposed to the cocycles themselves.

Now let $F = \mathbb {Q}(\zeta _p)$ for an odd prime p and a primitive pth root of unity $\zeta _p$ . Note that $\mathcal {O}_{F,S} = \mathbb {Z}[\zeta _p,\frac {1}{p}]$ and $\mathrm {Cl}_{F,S} = \mathrm {Cl}_F$ , since the prime $(1-\zeta _p)$ over p is principal. Let $\Delta = \operatorname {\mathrm {Gal}}(F/\mathbb {Q})$ , and let $\omega \colon \Delta \to \mathbb {Z}_p^\times $ be unique lift of the mod p cyclotomic character. For $j \in \mathbb {Z}$ , the $\omega ^j$ -isotypical component, or eigenspace, of a $\mathbb {Z}_p[\Delta ]$ -module M is

$$ \begin{align*} M^{(j)} = \{ m \in M \mid \delta m = \omega(\delta)^j m \text{ for all } \delta \in \Delta \}. \end{align*} $$

We say that a positive even integer $k < p$ is an irregular index for p if $\mathrm {Cl}_F[p]^{(1-k)} \ne 0$ , or equivalently, p divides the numerator of the kth Bernoulli number $B_k$ . As p divides the denominator of $B_{p-1}$ , every irregular index k for p satisfies $k \le p-3$ .

We suppose that p satisfies Vandiver’s conjecture that $\mathrm {Cl}_{\mathbb {Q}(\zeta _p+\zeta _p^{-1})}[p] = 0$ . By Leopoldt’s reflection principle, this implies that for each irregular index k, the eigenspace $\mathrm {Cl}_F[p]^{(1-k)}$ is cyclic, so we fix a generator and let $\alpha _k \in H^1(G_{F,S},\mu _p)^{(1-k)}$ be its unique lift. This also allows us to identify $H^2(G_{F,S},\mu _p)^{(1-k)}$ with $\mathbb {F}_p$ via the isomorphisms

$$\begin{align*}\mathbb{F}_p \xrightarrow{1 \mapsto \alpha_k} H^1(G_{F,S},\mu_p)^{(1-k)} \xrightarrow{\sim} \mathrm{Cl}_F[p]^{(1-k)} \xrightarrow{\sim} (\mathrm{Cl}_F \otimes_{\mathbb{Z}} \mathbb{F}_p)^{(1-k)} \xrightarrow{\sim} H^2(G_{F,S},\mu_p)^{(1-k)}, \end{align*}$$

where the isomorphism $(\mathrm {Cl}_F \otimes _{\mathbb {Z}} \mathbb {F}_p)^{(1-k)} \xrightarrow {\sim } \mathrm {Cl}_F[p]^{(1-k)}$ is multiplication by a power of p.

For an odd integer i, we define

$$\begin{align*}\eta_i \in (\mathcal{O}_{F,S}^\times \otimes_{\mathbb{Z}} \mathbb{F}_p)^{(1-i)} \end{align*}$$

to be the projection of $1-\zeta _p$ into that eigenspace. We often refer to the index i as taking values in $\mathbb {Z}/(p-1)\mathbb {Z}$ . Via Kummer theory, we identify $\eta _i$ with an element of

$$\begin{align*}H^1(G_{F,S},\mu_p)^{(1-i)} \cong (H^1(G_{F,S},\mu_p) \otimes_{\mathbb{Z}} \mu_p^{\otimes (i-1)})^\Delta \cong H^1(G_{F,S},\mu_p^{\otimes i})^\Delta \end{align*}$$

and $\zeta _p$ with an element of $H^1(G_{F,S},\mu _p)^{(1)}$ . Vandiver’s conjecture for p is equivalent to the statement that every $\eta _i$ is nontrivial. We codify all this and a bit more in the following remark.

Given a p-extension $L/F$ that is unramified outside p and for which $L/\mathbb {Q}$ is Galois, we can consider its Galois group $H = \operatorname {\mathrm {Gal}}(L/F)$ , which is of course normal inside $\operatorname {\mathrm {Gal}}(L/\mathbb {Q})$ . Set $\Omega = \mathbb {F}_p[H]$ as before. We have an action of $G_{\mathbb {Q},S}$ on $\Omega $ , such that $g \in G_{\mathbb {Q},S}$ sends the group element $[h]$ of $h \in H$ to $[\bar {g}h\bar {g}^{-1}]$ , where $\bar {g}$ is the image of g in G.

Since $G_{F,S}$ is normal in $G_{\mathbb {Q},S}$ , this $G_{\mathbb {Q},S}$ -action (together with right conjugation on $G_{F,S}$ in the usual fashion) induces a $\operatorname {\mathrm {Gal}}(L/\mathbb {Q})$ -action on $H^*(G_{F,S},\mu _p \otimes _{\mathbb {F}_p} I^n/I^m)$ for every $0 \le n < m$ . For $m = n+1$ , this action factors through $\Delta $ . We fix a lift of $\Delta $ to a subgroup of $\operatorname {\mathrm {Gal}}(L/\mathbb {Q})$ so that we may speak of the $\Delta $ -action on these cohomology groups for all $n < m$ , though, in general, this action depends upon the choice of lift. The generalized Bockstein maps $\Psi ^{(n)}$ are then $\Delta $ -equivariant.