Extensions of Profinite Duality Groups

by Alexander Schmidt and Kay Wingberg

September 25, 2008

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 ∈ Mod_{p}(G) and i ≥ 0, let

D_{i}(G, A) = \underrightarrow{lim}_{U} H^{i}(U, A)^{∗},

where ^{∗} is Hom(−, ℚ_{p}/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) = \underrightarrow{lim}_{ν∈ℕ} D_{n}(G, ℤ/p^{ν}ℤ)

is called the dualizing module of G at p. Its importance lies in the functorial isomorphism

H^{n}(G, A)^{∗} ≅ Hom_{G}(A, I(G))

for all A ∈ Mod_{p}(G). This isomorphism is induced by the cup-products (V ⊆ U)

H^{n}(G, A)^{∗} ×_{p^{ν}}A^{U} ⟶ H^{n}(V, ℤ/p^{ν}ℤ)^{∗}, (ϕ, a) ⟼ (α ↦ ϕ(cor _{G}^{V}(α ∪ a)))

by passing to the limit over ν and V, and then over U. The identity-map of I(G) gives rise to the homomorphism

tr : H^{n}(G, I(G)) ⟶ ℚ_{p}/ℤ_{p} ,

called the trace map.

The profinite group G is called a duality group at p of dimension n if for all i ∈ ℤ and all finite G-modules A ∈ Mod_{p}(G), the cup-product and the trace map

H^{i}(G, Hom(A, I(G)) × H^{n−i}(G, A) ⟶^{∪} H^{n}(G, I(G)) ⟶^{tr} ℚ_{p}/ℤ_{p}

yield an isomorphism

H^{i}(G, Hom(A, I(G))) ≅ H^{n−i}(G, A)^{∗}.

Remark: In [Ve], J.-L. Verdier used the name strict Cohen-Macaulay at p for what we call a profinite duality group at p here. In [Pl], 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:

F C(G, p): H^{i}(G, A) is finite for all finite A ∈ Mod_{p}(G) and for all i ≥ 0.

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

F C(G, p): H^{i}(U, ℤ/pℤ) is finite for all open subgroups U of G and all i ≥ 0.

By a duality theorem due to J. Tate, see [Ta] Thm. 3 or [Ve] 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, ℤ/pℤ) = 0 for 0 ≤ 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 < ∞, 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 → H → G → G/H → 1 if the kernel satisfies F C(H, p). Weaker forms of Theorem 1 were first proved by A. Pletch (for D_{p}^{n}-groups, see [Pl]^{1}) and by the second author (for Poincaré groups, see [Wi]).

1The proof given by Pletch in [Pl] is only correct for pro-p-groups as the author assumes that finitely generated projective modules over the complete group ring ℤ_{p}[[G]] are free.

Theorem 1. Let

1 ⟶ H ⟶ G ⟶ G/H ⟶ 1

be an exact sequence of profinite groups such that condition F C(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)^{∨} ≅ I(H)^{∨} ˆ⊗_{ℤ_{p}} I(G/H)^{∨},

where ^{∨} is the Pontryagin dual and ˆ⊗_{ℤ_{p}} is the tensor-product in the category of compact ℤ_{p}-modules.

Remark: The assumption F C(H, p) is necessary, as the following examples show:

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

2. Let D be a duality group at p of dimension 2, F a duality group at p of dimension 1 and G = F ∗ D their free product. The kernel of the projection G ↠ 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

Proposition 2. Let

1 ⟶ H ⟶ G ⟶ G/H ⟶ 1

be an exact sequence of profinite groups. Assume that FC(H, p) holds. Then there is a spectral sequence of homological type

E_{ij}^{2} = D_{i}(G/H, ℤ/pℤ) ⊗ D_{j}(H, ℤ/pℤ) ⟹ D_{i+j}(G, ℤ/pℤ).

Proof. Let g run through the open normal subgroups of G. Then gH/H ≅ g/g ∩ H runs through the open normal subgroups of G/H. For a G-module A ∈ Mod_{p}(G), we consider the Hochschild-Serre spectral sequence

E(g, g∩H, A) : E_{2}^{ij}(g, g∩H, A) = H^{i}(g/g∩H, H^{j}(g∩H, A)) ⟹ H^{i+j}(g, A).

If g′ ⊆ g is another open normal subgroup of G, then the corestriction yields a morphism

cor : E(g′, g′ ∩ H, A) ⟶ E(g, g ∩ H, A)

of spectral sequences. The map

E_{2}^{ij}(g′, g′ ∩ H, A) ⟶ E_{2}^{ij}(g, g ∩ H, A)

is the composite of the maps

H^{i}(g′/g′ ∩ H, H^{j}(g′ ∩ H, A)) ⟶^{cor _{g∩H}^{g′∩H}} H^{i}(g′/g′ ∩ H, H^{j}(g ∩ H, A))

⟶^{cor _{g/g∩H}^{g′/g′∩H}} H^{i}(g/g ∩ H, H^{j}(g ∩ H, A))

and the map between the limit terms is the corestriction

cor _{g}^{g′} : H^{i+j}(g′, A) ⟶ H^{i+j}(g, A).

For 2 ≤ r ≤ ∞ we set

E_{ij}^{2} = D_{ij}^{r}(G, H, A) := \underrightarrow{lim}_{g} E_{r}^{ij}(g, g ∩ H, A)^{∗}.

As taking duals and direct limits are exact operations, the terms D_{ij}^{r}(G, H, A), 2 ≤ r ≤ ∞, 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 ∩ H ⊆ h, then these groups are g/g ∩ H-modules. We see that

D_{ij}^{2}(G, H, A) = \underrightarrow{lim}_{\begin{matrix}h⊆H \\ h⊴G \end{matrix}} \underrightarrow{lim}_{\begin{matrix}g⊆G \\ g∩H⊆h \end{matrix}} H^{i}(g/g ∩ 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_{ij}^{2}(G, H, ℤ/pℤ) ≅ D_{i}(G/H, ℤ/pℤ) ⊗ D_{j}(H, ℤ/pℤ)

for all i and j. To this end note that all occurring abelian groups are 𝔽_{p}- vector spaces, so that ^{∗} is Hom(−, 𝔽_{p}). Further note that for vector spaces V, W over a field k the homomorphism

V^{∗} ⊗ W^{∗} ⟶(V ⊗ W)^{∗}, ϕ ⊗ ψ ⟼ (v ⊗ w ↦ ϕ(v)ψ(w))

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′ ⊆ g be open subgroups of G such that g acts trivially on the finite group H^{j}(h, ℤ/pℤ). Then, by [NSW] (1.5.3)(iv), the diagram

H^{i}(g′/g′ ∩ H, ℤ/pℤ) ⊗ H^{j}(h, ℤ/pℤ) _{∼} ⟶^{∪} H^{i}(g′/g′ ∩ H, H^{j}(h, ℤ/pℤ))
↓^{cor ⊗id}  ↓^{cor}
H^{i}(g/g ∩ H, ℤ/pℤ) ⊗ H^{j}(h, ℤ/pℤ) _{∼} ⟶^{∪} H^{i}(g/g ∩ H, H^{j}(h, ℤ/pℤ))

commutes. For fixed h, we therefore obtain isomorphisms

D_{i}(G/H, ℤ/pℤ) ⊗ H^{j}(h, ℤ/pℤ)^{∗}

≅ ( \underrightarrow{lim}_{g} H^{i}(g/g ∩ H, ℤ/pℤ)^{∗}) ⊗ H^{j}(h, ℤ/pℤ)^{∗}
≅ \underrightarrow{lim}_{g} H^{i}(g/g ∩ H, ℤ/pℤ)^{∗} ⊗ H^{j}(h, ℤ/pℤ)^{∗}
≅ \underrightarrow{lim}_{g} (H^{i}(g/g ∩ H, ℤ/pℤ) ⊗ H^{j}(h, ℤ/pℤ))^{∗}
≅ \underrightarrow{lim}_{g} H^{i}(g/g ∩ H, H^{j}(h, ℤ/pℤ))^{∗}.

Passing to the limit over h, we obtain the required isomorphism

D_{i}(G/H, ℤ/pℤ) ⊗ D_{j}(H, ℤ/pℤ) ≅ D_{ij}^{2}(G, H, ℤ/pℤ).

Corollary 3. Under the assumptions of Proposition 2, let i_{0} and j_{0} be the smallest integers such that D_{i_{0}}(G/H, ℤ/pℤ) ≠ 0 and D_{j_{0}}(H, ℤ/pℤ) ≠ 0, respectively. Then D_{i_{0}+j_{0}}(G, ℤ/pℤ) ≠ 0.

Proof. The spectral sequence constructed in Proposition 2 induces an isomorphism

D_{i_{0}+j_{0}}(G, ℤ/pℤ) ≅ D_{i_{0}}(G/H, ℤ/pℤ) ⊗ D_{j_{0}}(H, ℤ/pℤ) ≠ 0.

Proof of Theorem 1. 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}, ℤ/pℤ) ≠ 0. Let G_{1} be an open subgroup of G such that H_{1} = G_{1} ∩ 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 ⟶ H_{1} ⟶ G_{1} ⟶ G_{1}/H_{1} ⟶ 1.

As H^{m}(H_{1}, ℤ/pℤ) is finite and nonzero, we have vcd_{p} G_{1}/H_{1} = n, see [NSW] (3.3.9). Furthermore, D_{i}(G_{1}, ℤ/pℤ) = 0, i < n + m. Using Corollary 3, we see that D_{i}(G_{1}/H_{1}, ℤ/pℤ) = 0 for all i < n and D_{j}(H_{1}, ℤ/pℤ) = 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 2 we have E_{ij}^{2} = 0 for (i, j) ≠ (n, m). Hence D_{r}(G, ℤ/pℤ) = 0 for r ≠ 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, ℤ/p^{ν}ℤ) ≅ H^{n}(g/g ∩ H, H^{m}(g ∩ H, ℤ/p^{ν}ℤ)),

and we obtain

I(G) ≅ \underrightarrow{lim}_{ν} \underrightarrow{lim}_{g} H^{m+n}(g, ℤ/p^{ν}ℤ)^{∗}
≅ \underrightarrow{lim}_{ν} \underrightarrow{lim}_{h} \underrightarrow{lim}_{g} H^{n}(g/g ∩ H, H^{m}(h, ℤ/p^{ν}ℤ))^{∗}
≅ \underrightarrow{lim}_{ν} \underrightarrow{lim}_{h} \underrightarrow{lim}_{g,res} H^{0}(g/g ∩ H, Hom (H^{m}(h, ℤ/p^{ν}ℤ), I(G/H)))
≅ \underrightarrow{lim}_{ν} \underrightarrow{lim}_{h} Hom(H^{m}(h, ℤ/p^{ν}ℤ), I(G/H))
≅ Hom_{cts}(\underleftarrow{lim}_{ν} \underleftarrow{lim}_{h} H^{m}(h, ℤ/p^{ν}ℤ), I(G/H))
≅ Hom_{cts}((\underrightarrow{lim}_{ν} \underrightarrow{lim}_{h} H^{m}(h, ℤ/p^{ν}ℤ)^{∗})^{∨}, I(G/H))
≅ Hom_{cts} (I(H)^{∨}, I(G/H)) ≅ (I(H)^{∨} ˆ⊗_{ℤ_{p}} I(G/H)^{∨})^{∨}

(see [NSW] (5.2.9) for the last isomorphism). This completes the proof of the theorem.

References

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

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

[Ta] 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)

[Ve] 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)

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

Alexander Schmidt, NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Deutschland. email: alexander.schmidt@mathematik.uni-regensburg.de

Kay Wingberg, Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Deutschland. email: wingberg@mathi.uni-heidelberg.de