---
author:
- Alexander Schmidt and Kay Wingberg
date: 'September 25, 2008'
title: '**Extensions of profinite duality groups**'
---

Let $G$ be a profinite group and let $p$ be a prime number. By $\Mod_p(G)$ we denote the category of discrete $p$-primary $G$-modules. For $A\in \Mod_p(G)$ and $i\geq 0$, let $$D_i(G,A)=\varinjlim_U\, H^i(U,A)^*,$$ where $\null^*$ is $\Hom(-,{{\mathbb Q}}_p/{{\mathbb Z}}_p)$, the direct limit is taken over all open subgroups $U$ of $G$ and the transition maps are the duals of the corestriction maps. $D_i(G,A)$ is a discrete $G$-module in a natural way. Assume that $n=\cd_p\,G$ is finite. Then the $G$-module $$I(G)=\varinjlim_{\nu\in {{\mathbb N}}}\,D_n(G,{{\mathbb Z}}/p^\nu{{\mathbb Z}})$$ is called the [**dualizing module**]{} of $G$ at $p$. Its importance lies in the functorial isomorphism $$H^n(G,A)^*\cong\Hom_G(A,I(G))$$ for all $A\in\Mod_p(G)$. This isomorphism is induced by the cup-products ($V\subseteq U$) $$H^n(G,A)^* \times \null_{p^\nu} A^U \longrightarrow H^n(V,{{\mathbb Z}}/p^\nu{{\mathbb Z}})^*,\ (\phi,a)\longmapsto \big(\alpha \mapsto \phi(\cor_G^V(\alpha \cup a))\big)$$ by passing to the limit over $\nu$ and $V$, and then over $U$. The identity-map of $I(G)$ gives rise to the homomorphism $$\tr :H^n(G,I(G))\lpfeil {{\mathbb Q}}_p/{{\mathbb Z}}_p\,,$$ called the [**trace map**]{}.

The profinite group $G$ is called a [**duality group at $p$ of dimension $n$**]{} if for all $i\in{{\mathbb Z}}$ and all finite $G$-modules $A\in\Mod_p(G)$, the cup-product and the trace map $$H^i(G,\Hom(A,I(G)) \times H^{n-i}(G,A)\lpfeil^\cup
H^n(G,I(G))\lpfeil^{\tr}{{\mathbb Q}}_p/{{\mathbb Z}}_p$$

yield an isomorphism

$$H^i(G,\Hom(A,I(G)))\cong H^{n-i}(G,A)^*.$$

[**Remark:**]{} In [@V], J.-L. Verdier used the name [**strict Cohen-Macaulay at $p$**]{} for what we call a profinite duality group at $p$ here. In [@48], A. Pletch defined $D_p^n$-groups (and called them duality groups at $p$ of dimension $n$). The $D_p^n$-groups of Pletch are exactly the duality groups at $p$ (in our sense) which, in addition, satisfy the following finiteness condition:

$FC(G,p)$: [*$H^i(G,A)$ is finite for all finite $A\in \Mod_p(G)$ and for all $i\geq 0$*]{}.

Since any finite, discrete $G$-module is trivialized by an open subgroup $U$ of $G$, condition $FC(G,p)$ can also be rephrased in the form:

$FC(G,p)$: [*$H^i(U,{{\mathbb Z}}/p{{\mathbb Z}})$ is finite for all open subgroups $U$ of $G$ and all $i\geq 0$*]{}.

By a duality theorem due to J. Tate, see [@T] Thm.3 or [@V] Prop.4.3 or [@NSW] (3.4.6), a profinite group $G$ of cohomological $p$-dimension $n$ is a duality group at $p$ if and only if $$D_i(G,{{\mathbb Z}}/p{{\mathbb Z}})=0 \quad \hbox{ for $0\leq i<n$}.$$ As a consequence we see that every open subgroup of a duality group at $p$ is a duality group at $p$ as well (of the same cohomological dimension), and if an open subgroup of $G$ is a duality group at $p$ and $\cd_p\,G<\infty$, then $G$ is a duality group at $p$ of the same cohomological dimension (use [@NSW] (3.3.5)(ii)). Furthermore, any profinite group of cohomological $p$-dimension $1$ is a duality group at $p$.

We call a profinite group $G$ [**virtually a duality group at $p$ of (virtual) dimension $\vcd_p \,G=n$**]{} if an open subgroup $U$ of $G$ is a duality group at $p$ of dimension $n$.

The objective of this paper is to give a proof of Theorem \[1\] below, which states that the class of duality groups is closed under group extensions $1\to H\to G\to G/H \to 1$ if the kernel satisfies $FC(H,p)$. Weaker forms of Theorem \[1\] were first proved by A. Pletch (for $D_p^n$-groups, see [@48][^1]) and by the second author (for Poincaré groups, see [@71]).

\[1\] Let $$1 \lpfeil H\lpfeil G\lpfeil G/H\lpfeil 1$$

be an exact sequence of profinite groups such that condition $FC(H,p)$ is satisfied. Then the following assertions hold:

[(i)]{} If $G$ is a duality group at $p$, then $H$ is a duality group at $p$ and $G/H$ is virtually a duality group at $p$.

[(ii)]{} If $H$ and $G/H$ are duality groups at $p$, then $G$ is a duality group at $p$.

Moreover, in both cases we have: $$\cd_p\,G=\cd_p\,H+\vcd_p\,G/H,$$

and there is a canonical $G$-isomorphism $$I(G)^\vee\cong I(H)^\vee \; {{\mathop{\hat\otimes}}}_{{{\mathbb Z}}_p}\; I(G/H)^\vee,$$ where $\null^\vee$ is the Pontryagin dual and ${{\mathop{\hat\otimes}}}_{{{\mathbb Z}}_p}$ is the tensor-product in the category of compact ${{\mathbb Z}}_p$-modules.

[**Remark:**]{} The assumption $FC(H,p)$ is necessary, as the following examples show:

-   Let $G$ be the free pro-$p$-group on two generators $x,y$ and let $H\subset G$ be the normal subgroup generated by $x$. Then $H$ is free of infinite rank, $G/H$ is free of rank one and $1 \pfeil H\pfeil G\pfeil G/H\pfeil 1$ is an exact sequence in which all three groups are duality groups of dimension one.

-   Let $D$ be a duality group at $p$ of dimension $2$, $F$ a duality group at $p$ of dimension $1$ and $G=F\ast D$ their free product. The kernel of the projection $G\twoheadrightarrow D$ has cohomological $p$-dimension $1$, hence is a duality group a $p$ of dimension $1$. The group $G$ has cohomological $p$-dimension $2$ but is is not a duality group at $p$.

In the proof of Theorem \[1\], we make use of the following

\[spectral\] Let $$1 \lpfeil H\lpfeil G\lpfeil G/H\lpfeil 1$$ be an exact sequence of profinite groups. Assume that FC($H,p$) holds. Then there is a spectral sequence of homological type $$E^2_{ij}= D_i(G/H,{{\mathbb Z}}/p{{\mathbb Z}}) \otimes D_j(H,{{\mathbb Z}}/p{{\mathbb Z}}) \Longrightarrow D_{i+j}(G,{{\mathbb Z}}/p{{\mathbb Z}}).$$

Let $g$ run through the open normal subgroups of $G$. Then $gH/H \cong g/g\cap H$ runs through the open normal subgroups of $G/H$. For a $G$-module $A\in\Mod_p(G)$, we consider the Hochschild-Serre spectral sequence $$E(g,g\cap H,A):\ E_2^{ij}(g,g\cap H,A)=H^i(g/g\cap H, H^j(g\cap H,A))\Longrightarrow H^{i+j}(g,A).$$

If $g'\subseteq g$ is another open normal subgroup of $G$, then the corestriction yields a morphism $$\cor:E(g',g'\cap H,A)\lpfeil E(g,g\cap H,A)$$ of spectral sequences. The map $$E_2^{ij}(g',g'\cap H,A) \lpfeil E_2^{ij}(g,g\cap H,A)$$ is the composite of the maps $$H^i\big(g'/g'\cap H, H^j(g'\cap H,A)\big)
\lpfeil^{\cor_{g\cap H}^{g'\cap H}} H^i\big(g'/g'\cap H,H^j(g\cap H,A)\big)$$ $$\lpfeil^{\cor_{g/g\cap H}^{g'/g'\cap H}} H^i\big(g/g\cap H,H^j(g\cap H,A)\big)$$ and the map between the limit terms is the corestriction $$\cor^{g'}_g:H^{i+j}(g',A)\lpfeil H^{i+j}(g,A).$$ For $2\leq r\leq\infty$ we set $$E^2_{ij}=D^r_{ij}(G,H,A):=\varinjlim_g\,E^{ij}_r(g,g\cap H,A)^*.$$ As taking duals and direct limits are exact operations, the terms $D^r_{ij}(G,H,A)$, $2\leq r \leq \infty$, establish a homological spectral sequence which converges to $D_n(G,A)$. If $h$ runs through the open subgroups of $H$ which are normal in $G$, then the cohomology groups $H^j(h,A)$ are $G$-modules in a natural way. If $g$ is open in $G$ with $g\cap H\subseteq h$, then these groups are $g/g\cap H$-modules. We see that

$$D^2_{ij}(G,H,A)=\varinjlim_{\substack{h\subseteq H\\h\trianglelefteq G}}\
\varinjlim_{\substack{g\subseteq G\\ g\cap H\subseteq h}}\,H^i(g/g\cap H,H^j(h,A))^*,$$

where for both limits the transition maps are (induced by) $\cor^*$. In order to conclude the proof of the proposition, it remains to construct isomorphisms $$D^2_{ij}(G, H,{{\mathbb Z}}/p{{\mathbb Z}})\cong D_{i} (G/H,{{\mathbb Z}}/p{{\mathbb Z}}) \otimes D_{j}(H,{{\mathbb Z}}/p{{\mathbb Z}})$$ for all $i$ and $j$. To this end note that all occurring abelian groups are ${{\mathbb F}}_p$-vector spaces, so that $\null^*$ is $\Hom(-,{{\mathbb F}}_p)$. Further note that for vector spaces $V,W$ over a field $k$ the homomorphism $$V^*\otimes W^* \lpfeil (V\otimes W)^*,\ \phi\otimes \psi \longmapsto \big(v\otimes w\mapsto \phi(v)\psi(w)\big)$$ is an isomorphism provided that $V$ or $W$ is finite-dimensional. Let $h$ be an open subgroup of $H$ which is normal in $G$ and let $g'\subseteq g$ be open subgroups of $G$ such that $g$ acts trivially on the finite group $H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}})$. Then, by [@NSW] (1.5.3)(iv), the diagram $$\renewcommand{\arraystretch}{1.6}
\begin{array}{ccc}
 H^{i}(g'/g'\cap H,{{\mathbb Z}}/p{{\mathbb Z}})\otimes H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}}) & \lriso{\cup} & H^{i}\big(g'/g'\cap H, H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}})\big)\\
 \phantom{xxxmN}\mapd{\cor \otimes \mathit{id}}&&\mapd{\cor}\\
 H^{i}(g/g\cap H,{{\mathbb Z}}/p{{\mathbb Z}})\otimes H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}}) &  \lriso{\cup} & H^{i}\big(g/g\cap H, H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}})\big)
\end{array}
\renewcommand{\arraystretch}{1}$$ commutes. For fixed $h$, we therefore obtain isomorphisms

$
\ds D_{i}(G/H,{{\mathbb Z}}/p{{\mathbb Z}}) \otimes H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}})^*
$ $$\begin{array}{lcl}
\phantom{uuuuuuuuuuuuuuuuuuuu}&\cong&
\ds\big(\varinjlim_g H^{i}(g/g\cap H,{{\mathbb Z}}/p{{\mathbb Z}})^*\big) \otimes H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}})^* \\
&\cong&\ds\varinjlim_g \; H^{i}(g/g\cap H,{{\mathbb Z}}/p{{\mathbb Z}})^* \otimes H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}})^* \\
&\cong&\ds\varinjlim_g \big(H^{i}(g/g\cap H,{{\mathbb Z}}/p{{\mathbb Z}}) \otimes H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}})\big)^* \\
&\cong&\ds\varinjlim_g H^{i}\big(g/g\cap H, H^{j}(h,{{\mathbb Z}}/p{{\mathbb Z}})\big)^*.
\end{array}$$ Passing to the limit over $h$, we obtain the required isomorphism $$D_{i} (G/H,{{\mathbb Z}}/p{{\mathbb Z}}) \otimes D_{j}(H,{{\mathbb Z}}/p{{\mathbb Z}}) \cong D^2_{ij}(G,H,{{\mathbb Z}}/p{{\mathbb Z}}).$$

\[lemma\] Under the assumptions of Proposition [\[spectral\]]{}, let $i_0$ and $ j_0$ be the smallest integers such that $D_{i_0}(G/H,{{\mathbb Z}}/p{{\mathbb Z}})\neq 0$ and $D_{j_0}(H,{{\mathbb Z}}/p{{\mathbb Z}})\neq 0$, respectively. Then $D_{i_0+j_0}(G,{{\mathbb Z}}/p{{\mathbb Z}})\neq 0$.

The spectral sequence constructed in Proposition \[spectral\] induces an isomorphism $$D_{i_0+j_0}(G,{{\mathbb Z}}/p{{\mathbb Z}}) \cong D_{i_0}(G/H,{{\mathbb Z}}/p{{\mathbb Z}}) \otimes D_{j_0}(H,{{\mathbb Z}}/p{{\mathbb Z}})\neq 0.$$

Assume that $G$ is a duality group at $p$ of dimension $d$. Let $\cd_p\, H=m$ and $n=d-m$. Then there exists an open subgroup $H_1$ of $H$ such that $H^m(H_1,{{\mathbb Z}}/p{{\mathbb Z}})\neq 0$. Let $G_1$ be an open subgroup of $G$ such that $H_1=G_1\cap H$. Then $G_1$ is a duality group at $p$ of dimension $d$, $\cd_p\, H_1=m$ and $G_1/H_1$ is an open subgroup of $G/H$. We consider the exact sequence $$1\lpfeil H_1\lpfeil G_1 \lpfeil G_1/H_1 \lpfeil 1.$$ As $H^m(H_1,{{\mathbb Z}}/p{{\mathbb Z}})$ is finite and nonzero, we have $\vcd_p\,G_1/H_1=n$, see [@NSW] (3.3.9). Furthermore, $D_i(G_1,{{\mathbb Z}}/p{{\mathbb Z}})=0$, $i<n+m$. Using Corollary \[lemma\], we see that $D_i(G_1/H_1,{{\mathbb Z}}/p{{\mathbb Z}})=0$ for all $i<n$ and $D_j(H_1,{{\mathbb Z}}/p{{\mathbb Z}})=0$ for all $j<m$. Thus $G_1/H_1$, hence $G/H$, is virtually a duality group at $p$ of dimension $n$, and $H_1$, and so $H$, is a duality group at $p$ of dimension $m$. This shows (i).

Assume now that $H$ and $G/H$ are duality groups at $p$ of dimension $m$ and $n$. Then, $\cd_p G=n+m$ by [@NSW] (3.3.8), and in the spectral sequence of Proposition \[spectral\] we have $E^2_{ij}=0$ for $(i,j)\neq (n,m)$. Hence $D_r(G,{{\mathbb Z}}/p{{\mathbb Z}})=0$ for $r\neq n+m$ showing that $G$ is a duality group at $p$ of dimension $n+m$.

In order to prove the assertion about the dualizing modules, let $h$ run through all open subgroups of $H$ which are normal in $G$ and $g$ runs through the open subgroups of $G$. Since $m=\cd_p\,H$, the Hochschild-Serre spectral sequence induces isomorphisms $$H^{m+n}(g,{{\mathbb Z}}/p^\nu{{\mathbb Z}})\cong H^n(g/g\cap H,H^m(g\cap H,{{\mathbb Z}}/p^\nu{{\mathbb Z}})),$$ and we obtain

$$\begin{array}{rcl}
I(G) & \cong& \ds\varinjlim_\nu\,\ds\varinjlim_g\,H^{m+n}(g,{{\mathbb Z}}/p\null^\nu{{\mathbb Z}})^*  \\
     & \cong &\ds\varinjlim_\nu\,\ds\varinjlim_{h}\,
         \ds\varinjlim_{g}\,H^n\big(g/g\cap H,H^m(h,{{\mathbb Z}}/p\null^\nu{{\mathbb Z}})\big)^*\\
     & \cong &\ds\varinjlim_\nu\,\ds\varinjlim_{h}\,
         \ds\varinjlim_{g,\res}\,H^0\big(g/g\cap H,\Hom\,(H^m(h,{{\mathbb Z}}/p\null^\nu{{\mathbb Z}}),I(G/H))\big) \\
     & \cong & \ds\varinjlim_\nu\,\ds\varinjlim_{h}\,\Hom(H^m(h,{{\mathbb Z}}/p\null^\nu{{\mathbb Z}}), I(G/H))\\
     &\cong &\Hom_{cts}(\ds\varprojlim_\nu\,\ds\varprojlim_{h}\,H^m(h,{{\mathbb Z}}/p\null^\nu{{\mathbb Z}}), I(G/H))\\
     &\cong &\Hom_{cts}\big((\ds\varinjlim_\nu\,\ds\varinjlim_{h} H^m(h,{{\mathbb Z}}/p^\nu{{\mathbb Z}})^*)^\vee ,I(G/H)\big)\\
     & \cong & \Hom_{cts}\,(I(H)^\vee,I(G/H))\cong\big(I(H)^\vee{\mathop{\hat\otimes}}_{{{\mathbb Z}}_p} I(G/H)^\vee\big)^\vee
\end{array}$$ (see [@NSW] (5.2.9) for the last isomorphism). This completes the proof of the theorem.

[999]{}

Neukirch, J., Schmidt, A., Wingberg, K. [*Cohomology of Number Fields.*]{} sec.ed. Springer 2008

Pletch, A. [*Profinite duality groups I.*]{} J. Pure Applied Algebra [**16**]{} (1980) 55–74 and 285–297

Tate, J. [*Letter to Serre.*]{} Annexe 1 to Chap.I in Serre, J.-P. [*Cohomologie Galoisienne.*]{} Lecture Notes in Mathematics [**5**]{}, Springer 1964 (Cinquième édition 1994)

Verdier, J.-L. [*Dualité dans la cohomologie des groupes profinis.*]{} Annexe 2 to Chap.I in Serre, J.-P. [*Cohomologie Galoisienne.*]{} Lecture Notes in Mathematics [**5**]{}, Springer 1964 (Cinquième édition 1994)

Wingberg, K. [*On Poincaré groups.*]{} J. London Math. Soc. [**33**]{} (1986) 271–278

[^1]: The proof given by Pletch in [@48] is only correct for pro-$p$-groups as the author assumes that finitely generated projective modules over the complete group ring ${{\mathbb Z}}_p{[\![}G{]\!]}$ are free.
