We show that the class of profinite duality groups is closed under group
extensions provided that the kernel satisfies some finiteness condition. This
extends earlier results of Pletch and of Wingberg

Schmidt, Alexander

Wingberg, Kay

English

arXiv

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Universita¨t RegensburgMathematikExtensions of profinite duality groupsAlexander Schmidt and Kay WingbergPreprint Nr. 25/2008Extensions of profinite duality groupsAlexander Schmidt and Kay WingbergSeptember 25, 2008Let G be a profinite group and let p be a prime number. By Modp(G)we denote the category of discrete p-primary G-modules. For A ∈ Modp(G)and i ≥ 0, letDi(G,A) = lim−→UH i(U,A)∗,where ∗ is Hom(−,Qp/Zp), the direct limit is taken over all open subgroupsU of G and the transition maps are the duals of the corestriction maps.Di(G,A) is a discrete G-module in a natural way. Assume that n = cdpG isfinite. Then the G-moduleI(G) = lim−→ν∈NDn(G,Z/pνZ)is called the dualizing module of G at p. Its importance lies in the func-torial isomorphismHn(G,A)∗ ∼= HomG(A, I(G))for all A ∈ Modp(G). This isomorphism is induced by the cup-products(V ⊆ U)Hn(G,A)∗ × pνAU −→ Hn(V,Z/pνZ)∗, (φ, a) 7−→(α 7→ φ(cor VG(α ∪ a)))by passing to the limit over ν and V , and then over U . The identity-map ofI(G) gives rise to the homomorphismtr : Hn(G, I(G))−→Qp/Zp ,called the trace map.The profinite group G is called a duality group at p of dimension nif for all i ∈ Z and all finite G-modules A ∈ Modp(G), the cup-product andthe trace map1H i(G,Hom(A, I(G))×Hn−i(G,A) ∪−→Hn(G, I(G)) tr−→Qp/Zpyield an isomorphismH i(G,Hom(A, I(G))) ∼= Hn−i(G,A)∗.Remark: In [Ve], J.-L. Verdier used the name strict Cohen-Macaulayat p for what we call a profinite duality group at p here. In [Pl], A. Pletchdefined Dnp -groups (and called them duality groups at p of dimension n). TheDnp -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 ∈ Modp(G) and for all i ≥ 0.Since any finite, discrete G-module is trivialized by an open subgroup U ofG, condition FC(G, p) can also be rephrased in the form:FC(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. 4.3or [NSW] (3.4.6), a profinite group G of cohomological p-dimension n is aduality group at p if and only ifDi(G,Z/pZ) = 0 for 0 ≤ i < n.As a consequence we see that every open subgroup of a duality group atp 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 cdpG < ∞, thenG 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 1is a duality group at p.We call a profinite group G virtually a duality group at p of (virtual)dimension vcdpG = n if an open subgroup U of G is a duality group at pof dimension n.The objective of this paper is to give a proof of Theorem 1 below, whichstates that the class of duality groups is closed under group extensions 1→H → G → G/H → 1 if the kernel satisfies FC(H, p). Weaker forms ofTheorem 1 were first proved by A. Pletch (for Dnp -groups, see [Pl]1) and bythe second author (for Poincaré groups, see [Wi]).1The proof given by Pletch in [Pl] is only correct for pro-p-groups as the author assumesthat finitely generated projective modules over the complete group ring Zp[[G]] are free.2Theorem 1. Let1−→H −→G−→G/H −→ 1be an exact sequence of profinite groups such that condition FC(H, p) is sat-isfied. 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 isvirtually 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:cdpG = cdpH + vcdpG/H,and there is a canonical G-isomorphismI(G)∨ ∼= I(H)∨ ⊗ˆZp I(G/H)∨,where ∨ is the Pontryagin dual and ⊗ˆZp is the tensor-product in the categoryof compact Zp-modules.Remark: The assumption FC(H, p) is necessary, as the following examplesshow:1. Let G be the free pro-p-group on two generators x, y and let H ⊂ Gbe the normal subgroup generated by x. Then H is free of infiniterank, G/H is free of rank one and 1→H→G→G/H→ 1 is an exactsequence 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 atp of dimension 1 and G = F ∗ D their free product. The kernel ofthe projection G  D has cohomological p-dimension 1, hence is aduality group a p of dimension 1. The group G has cohomologicalp-dimension 2 but is is not a duality group at p.In the proof of Theorem 1, we make use of the followingProposition 2. Let1−→H −→G−→G/H −→ 1be an exact sequence of profinite groups. Assume that FC(H, p) holds. Thenthere is a spectral sequence of homological typeE2ij = Di(G/H,Z/pZ)⊗Dj(H,Z/pZ) =⇒ Di+j(G,Z/pZ).3Proof. 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-moduleA ∈ Modp(G), we consider the Hochschild-Serre spectral sequenceE(g, g∩H,A) : Eij2 (g, g∩H,A) = H i(g/g∩H,Hj(g∩H,A)) =⇒ H i+j(g, A).If g′ ⊆ g is another open normal subgroup of G, then the corestriction yieldsa morphismcor : E(g′, g′ ∩H,A)−→E(g, g ∩H,A)of spectral sequences. The mapEij2 (g′, g′ ∩H,A)−→Eij2 (g, g ∩H,A)is the composite of the mapsH i(g′/g′ ∩H,Hj(g′ ∩H,A)) cor g′∩Hg∩H−→ H i(g′/g′ ∩H,Hj(g ∩H,A))cor g′/g′∩Hg/g∩H−→ H i(g/g ∩H,Hj(g ∩H,A))and the map between the limit terms is the corestrictioncor g′g : H i+j(g′, A)−→H i+j(g, A).For 2 ≤ r ≤ ∞ we setE2ij = Drij(G,H,A) := lim−→gEijr (g, g ∩H,A)∗.As taking duals and direct limits are exact operations, the termsDrij(G,H,A),2 ≤ r ≤ ∞, establish a homological spectral sequence which converges toDn(G,A). If h runs through the open subgroups of H which are normal inG, then the cohomology groups Hj(h,A) are G-modules in a natural way. Ifg is open in G with g ∩H ⊆ h, then these groups are g/g ∩H-modules. Wesee thatD2ij(G,H,A) = lim−→h⊆HhEGlim−→g⊆Gg∩H⊆hH i(g/g ∩H,Hj(h,A))∗,where for both limits the transition maps are (induced by) cor ∗. In order toconclude the proof of the proposition, it remains to construct isomorphismsD2ij(G,H,Z/pZ) ∼= Di(G/H,Z/pZ)⊗Dj(H,Z/pZ)4for all i and j. To this end note that all occurring abelian groups are Fp-vector spaces, so that ∗ is Hom(−,Fp). Further note that for vector spacesV,W over a field k the homomorphismV ∗ ⊗W ∗−→(V ⊗W )∗, φ⊗ ψ 7−→(v ⊗ w 7→ φ(v)ψ(w))is an isomorphism provided that V or W is finite-dimensional. Let h be anopen subgroup of H which is normal in G and let g′ ⊆ g be open subgroupsof G such that g acts trivially on the finite group Hj(h,Z/pZ). Then, by[NSW] (1.5.3)(iv), the diagramH i(g′/g′ ∩H,Z/pZ)⊗Hj(h,Z/pZ) ∪∼−→ H i(g′/g′ ∩H,Hj(h,Z/pZ))ycor ⊗id ycorH i(g/g ∩H,Z/pZ)⊗Hj(h,Z/pZ) ∪∼−→ H i(g/g ∩H,Hj(h,Z/pZ))commutes. For fixed h, we therefore obtain isomorphismsDi(G/H,Z/pZ)⊗Hj(h,Z/pZ)∗∼=(lim−→gH i(g/g ∩H,Z/pZ)∗)⊗Hj(h,Z/pZ)∗∼= lim−→gH i(g/g ∩H,Z/pZ)∗ ⊗Hj(h,Z/pZ)∗∼= lim−→g(H i(g/g ∩H,Z/pZ)⊗Hj(h,Z/pZ))∗∼= lim−→gH i(g/g ∩H,Hj(h,Z/pZ))∗.Passing to the limit over h, we obtain the required isomorphismDi(G/H,Z/pZ)⊗Dj(H,Z/pZ) ∼= D2ij(G,H,Z/pZ).Corollary 3. Under the assumptions of Proposition 2, let i0 and j0 be thesmallest integers such that Di0(G/H,Z/pZ) 6= 0 and Dj0(H,Z/pZ) 6= 0,respectively. Then Di0+j0(G,Z/pZ) 6= 0.Proof. The spectral sequence constructed in Proposition 2 induces an iso-morphismDi0+j0(G,Z/pZ) ∼= Di0(G/H,Z/pZ)⊗Dj0(H,Z/pZ) 6= 0.5Proof of Theorem 1. Assume that G is a duality group at p of dimension d.Let cdpH = m and n = d−m. Then there exists an open subgroup H1 of Hsuch that Hm(H1,Z/pZ) 6= 0. Let G1 be an open subgroup of G such thatH1 = G1 ∩H. Then G1 is a duality group at p of dimension d, cdpH1 = mand G1/H1 is an open subgroup of G/H. We consider the exact sequence1−→H1−→G1−→G1/H1−→ 1.As Hm(H1,Z/pZ) is finite and nonzero, we have vcdpG1/H1 = n, see [NSW](3.3.9). Furthermore, Di(G1,Z/pZ) = 0, i < n + m. Using Corollary 3,we see that Di(G1/H1,Z/pZ) = 0 for all i < n and Dj(H1,Z/pZ) = 0 forall j < m. Thus G1/H1, hence G/H, is virtually a duality group at p ofdimension n, and H1, and so H, is a duality group at p of dimension m. Thisshows (i).Assume now that H and G/H are duality groups at p of dimension mand n. Then, cdpG = n+m by [NSW] (3.3.8), and in the spectral sequenceof Proposition 2 we have E2ij = 0 for (i, j) 6= (n,m). Hence Dr(G,Z/pZ) = 0for r 6= 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 runthrough all open subgroups of H which are normal in G and g runs throughthe open subgroups of G. Since m = cdpH, the Hochschild-Serre spectralsequence induces isomorphismsHm+n(g,Z/pνZ) ∼= Hn(g/g ∩H,Hm(g ∩H,Z/pνZ)),and we obtainI(G) ∼= lim−→νlim−→gHm+n(g,Z/pνZ)∗∼= lim−→νlim−→hlim−→gHn(g/g ∩H,Hm(h,Z/pνZ))∗∼= lim−→νlim−→hlim−→g,resH0(g/g ∩H,Hom (Hm(h,Z/pνZ), I(G/H)))∼= lim−→νlim−→hHom(Hm(h,Z/pνZ), I(G/H))∼= Homcts(lim←−νlim←−hHm(h,Z/pνZ), I(G/H))∼= Homcts((lim−→νlim−→hHm(h,Z/pνZ)∗)∨, I(G/H))∼= Homcts (I(H)∨, I(G/H)) ∼=(I(H)∨ ⊗ˆZp I(G/H)∨)∨(see [NSW] (5.2.9) for the last isomorphism). This completes the proof ofthe theorem.6References[NSW] Neukirch, J., Schmidt, A., Wingberg, K. Cohomology of NumberFields. 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. Coho-mologie Galoisienne. Lecture Notes in Mathematics 5, Springer 1964(Cinquième édition 1994)[Ve] Verdier, J.-L. Dualité dans la cohomologie des groupes profinis. Annexe2 to Chap.I in Serre, J.-P. Cohomologie Galoisienne. Lecture Notes inMathematics 5, Springer 1964 (Cinquième édition 1994)[Wi] Wingberg, K. On Poincaré groups. J. London Math. Soc. 33 (1986)271–278Alexander Schmidt, NWF I - Mathematik, Universität Regensburg, D-93040 Regens-burg, Deutschland. email: alexander.schmidt@mathematik.uni-regensburg.deKay Wingberg, Mathematisches Institut, Universität Heidelberg, Im NeuenheimerFeld 288, 69120 Heidelberg, Deutschland. email: wingberg@mathi.uni-heidelberg.de7

Extensions of profinite duality groups

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