Math. Z. (2012) 270:1057–1065DOI 10.1007/s00209-010-0840-0 Mathematische Zeitschrift

On Galois cohomology of semisimple groups over localand global fields of positive characteristic, II

Nguyêñ Quôc Thang

Received: 3 July 2008 / Accepted: 23 December 2010 / Published online: 13 January 2011© Springer-Verlag 2011

Abstract We show that the recent results of Prasad and Rapinchuk (Adv. Math. 207(2),646–660, 2006) on the existence and uniqueness of certain global forms of semisimple alge-braic groups with given local behaviour in the case of number fields still hold in the case ofglobal function fields.

Keywords Forms of linear algebraic groups · Galois cohomology · Flat cohomology

Mathematics Subject Classification (2000) Primary 11E72; Secondary 18G50 · 20G10

1 Introduction

In this paper, which is a continuation of [20], we give some applications of results proved inloc.cit. Let k be a field, V the set of all non-equivalent valuations of k, and let for each v ∈ Vthere be given connected linear algebraic groups Gv defined over the completion kv of k at v.Let the groups Gv have splitting fields lv which are finite over kv , i.e., each Gv has solvableradical R(Gv) defined and split over lv , and the semisimple lv-group Gv/R(Gv) is lv-split.Now assume that there exists a connected linear algebraic k-group G0 which is k-split in thesense above such that G0 ×kv lv � Gv ×kv lv , for all v ∈ Vk , which means that Gv are formsof G0. One may ask naturally

(a) Does there exist a connected linear algebraic k-group G such that G × kv � Gv over kvfor all v ?

(b) If so, how many are there ?

Partially supported by NAFOSTED and Esnault-Viehweg’s Leibniz program at University of Essen,Germany.

N. Q. Thang (B)Institute of Mathematics, 18-Hoang Quoc Viet, Hanoi, Vietname-mail: nqthang@math.ac.vn

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Such groups G will be called global forms in the sequel. These questions remind us, forexample, reciprocity law for central simple algebras (Hasse–Brauer–Noether theorem), andthey are of interest in their ownright. It is not our task to formulate a more general form ofquestion, which will eventually lead, say, to the theory of “companions” as it is formulatedin [10]. We are interested in a more special class of fields k and of groups G (in order to geta reasonable answer), namely global fields k and absolutely almost simple k-groups G withtheir local forms Gv , where each Gv is an inner twist of G. In [1], the authors have provedthe surjectivity of localization map for certain algebraic groups and apply it to give a proof ofthe existence of cocompact subgroups in the group of rational points of connected reductivegroups defined over local fields of characteristic 0. In fact, some of the ideas given there canbe extended to give a stronger assertion. Namely, in a recent paper [15], the authors haveproved the following result, which extends a similar result in [1] and is of its own interest(see also footnote on page 124 of [11]).

Fix an absolutely almost simple quasi-split group G0 defined over a field k and consideran inner k-form G of G0. Let � = Gal(ks/k) be the absolute Galois group. For the notionof Tits index and other related notions used below, we refer the readers to [22]. Denote by�(G, k) the Tits index of G over k, and �(G, k)d the set of all circled (i.e., distinguished)vertices of �(G, k). Then there is so called ∗-action of � on �(G, k), and we denote all∗-orbits on �(G0, k) by �i , i = 1, . . . , r . We have the following theorem.

Theorem 1 With above notation, assume that k is a global field, and G0 is simply connected.Fix a non-archimedean valuation v0 of k, and assume that there are given kv-forms, whichare inner twists Gv of G0 for all v ∈ V \{v0}, such that for almost all v,Gv is quasi-splitover kv .

(a) There exists a k-form, which is an inner twist G of G0 and is kv-isomorphic to Gv forall v ∈ V \{v0}.

(b) If an isotropic k-form G satisfying a) as above exists then there exists an index i, 1 ≤i ≤ r , such that �i ⊂ �(Gv, kv)d for all v ∈ V \{v0} and the k-rank of G is less orequal to the number of orbits satisfying the above inclusion.

(c) Let L be the minimal splitting field of G0. Assume that v0 is not split in L if [L : k] = 2.Then there exists an isotropic k-form G as in a) if there is some orbit �i satisfying b),and there exists a k-form G whose k-rank is equal to the total number of such orbits.

In [15], the authors prove the theorem in the case of number fields (cf. [[15], Theorem1]). In next section we prove this theorem in the case of global function fields. In Sect. 3,under some conditions, we prove the finiteness of the number of global inner twists whichare k-forms and discuss the uniqueness. It should be mentioned that many arguments of theproof given in the case of number field carry over to the case of function field, but not all. Itis our purpose to fill in the details needed in the case of function fields. As in [15], one of themain tool is Lemma 3 below, which is an analog of Lemma 1 of [15], which in turn extendsProposition 1.6 of [1]. We present two approaches to the proof of this lemma.

2 Theorem 1: the case of global function fields

Our first main observation is the following.

Proposition 2 Theorem 1 holds if k is assumed to be a global function field.

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Proof As in [1] and [15], one of the main ingredients is to prove that for a semisimple k-groupG, the localization map

ϕS : H1(k,G) → ⊕v∈SH1(kv,G)

is surjective, where in [1], S is finite, and in [15], S = V \{v0}. (Here, as in [15], by con-vention, we understand the sum ⊕ just as coproduct.) The other arguments can be repeatedalmost verbatim, so we are focusing only on the surjectivity of ϕS .

Let G be the universal k-covering of G, F the central subgroup of G, such that G � G/F .Then due to the vanishing of H1 of simply connected semisimple groups over local and globalfunction fields (due to Bruhat - Tits [2] and Harder [8], respectively), and the surjectivity ofthe coboundary maps �K : H1(K ,G) → H2

f l(K , F) (see [4,20]), where K = k or kv andthe subscript f l means the flat cohomology, one can check that the surjectivity of ϕS , whereS = V \{v0}, is equivalent to the surjectivity of the corresponding localization map

ψS : H2f l(k, F) → ⊕v∈SH2

f l(kv, F).

To prove the surjectivitiy of ψS , one may use the methods given in [21], pp. 4303–4304,while we prove the same statement for finite sets S.

Thus, Proposition 2 follows from the following lemma. In fact, we need this lemma onlyin the case of multiplicative groups F , but we take a chance to present it in a bit more generalform. Lemma 3 (Cf. [1,15]) Let k be a global function field of characteristic p > 0, S = V \{v0}as above. For any finite commutative k-group scheme F, the localization map

ψS : H2f l(k, F) → ⊕v∈SH2

f l(kv, F)

is surjective.

Proof of Lemma 3 Essentially, there are two approaches to the proof of Lemma 3. The firstimitates the number field case, by extending the argument involving classical result of Poi-tou–Tate on the duality of finite Galois modules to the function field case. Here one needs aresult of Frey [5], extending Poitou–Tate theorem on global function fields.

First proof. The first imitates the number field case, by extending the argument involvingclassical result of Poitou–Tate on the duality of finite Galois modules to the function fieldcase. Here one needs a result of Frey ([5], Satz D2), extending Poitou–Tate theorem on globalfunction fields and then just repeats the same proof suggested by Tate [1].

Second proof. This second proof avoids using Poitou–Tate duality in function field case in itsfull generality, and makes use only of the fact that this theorem holds for finite Galois moduleof order prime to the characteristic, the proof of which is known before the appearance of [5],and basically the classical proof in number field case can be repeated (see [5], Introduction,and [14]; see also [12]).

We know that (see e.g. [18]) there exists a sequence

{0} ⊆ Am ⊆ A0 ⊆ F,

where F/A0 is étale, A0 is connected, A0/Am is of additive type and Am is of multiplica-tive type, which are all finite commutative k-group schemes. Consider the following exactsequence

0 → A0 → F → F/A0 → 0

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and related exact sequence of flat cohomology

H2f l(k, A0) → H2

f l(k, F) → H2f l(k, F/A0) → H3

f l(k, A0) = 0

where the last term above is 0, since the cohomological dimension of k is 2 (see [11]). Assum-ing the assertion for A0 and F/A0, the corresponding assertion for F holds by chasing thediagram. Since the same argument holds when we consider the exact sequence

0 → Am → A0 → A0/Am → 0,

we are reduced to proving the assertion in the case F is of étale (resp. additive, resp. multi-plicative) type.

Multiplicative case. First we assume that F is k-split. In this case, it is just a direct product overk of the k-group schemes of type μm,m are natural numbers. We may invoke the fundamen-tal exact sequences of global class field theory (for Brauer groups) (Hasse–Brauer–Noethertheorem)

0 → Br(k) → ⊕vBr(kv) → Q/Z → 0,

0 → m Br(k) → ⊕v m Br(kv) → Z/mZ → 0,

where m is any natural number, and mC denotes the subgroup of m-torsion of C .We note as in [15], that the first sequence for Brauer groups implies that we have an

isomorphism

Br(k) � ⊕v �=v0 Br(kv),

hence also an isomorphism

m Br(k) � ⊕v �=v0 m Br(kv).

In particular, the localization map

H2f l(k, μm) → ⊕v �=v0 H2

f l(kv, μm)

is surjective as required.In the general case, since F is a commutative group scheme, the theory of trace due to

Deligne tells us (see [6], pp. 201–202 for more details) that for any finite field extension L/k,there exists a corestriction (norm) homomorphism

NiL/k : Hi

f l(L , F) → Hif l(k, F)

for all i ≥ 0. In fact, we have the following exact sequence of k-group schemes

1 → F ′ → RL/k(F)N→ F → 1,

where RL/k denotes the functor of restriction of scalars, and N denotes the norm. Then forall i ≥ 0 we have canonical isomorphism (see loc. cit)

Hif l(k, RL/k(F)) � Hi

f l(L , FL)

where FL = F ×k L is obtained by base change k → L and N induces the homomorphismN i

L/k above. Since the cohomological dimension of k is 2, it follows that N 2L/k is surjective.

Hence it suffices to prove the assertion over L for H2f l(L , FL). It is well-known that (see e.g.

[[17], Expose VIII]) the finite multiplicative k-group scheme F is split (diagonalisable) overa finite separable extension K/k. Let L be a finite Galois (over k) extension of K , which is a

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splitting field of F, k ⊂ K ⊂ L . Then we are reduced to proving the assertion for the groupFL , which are split over L . In this case, it amounts to proving the surjectivity of the map

m Br(L) → ⊕w∈VL (v0) m Br(Lw),

where VL(v0) denotes the set of all non-equivalent valuationsw of L , which have restrictionsw|k non-equivalent to v0, and this is obvious.

Étale case. As in the multiplicative case, by considering finite field extensions L/k, andthe corestriction (norm) homomorphisms

NiL/k : Hi

f l(L , F) → Hif l(k, F)

for all i ≥ 0, coming from exact sequence of k-group schemes

1 → F ′ → RL/k(F)N→ F → 1,

it suffices to prove the assertion over L . If F is of étale type, then by taking L a sufficientlylarge field, which splits off F we may assume that from the beginning that F is a constantfinite group scheme over k (or the same, F is a trivial Gal(ks/L)-module. Further, it is known(see e.g. [18]), that we have a decomposition F = F1 × F2, into direct product over k, whereF1 (resp. F2) is product of constant étale finite group schemes of order of p-power (resp. oforders prime to p). We are thus reduced further to considering the case F = Z/nZ, whereeither n = pr , or (n, p) = 1. If n = pr , then the Cartier dual F D of F is isomorphic to μpr ,and thus by Shatz’s duality for finite commutative group schemes ([18,19]) over local fields,H2

f l(kv, F) is trivial for all v, so the assertion is trivial in this case. If (n, p) = 1, then F isa finite Galois module and it is known that the Poitou–Tate duality theorem holds for F . Inparticular, for the convenience of the readers, we recall that the following partial result holds

Theorem (see [5,12,14]). Let k be a global field of characteristic p, F a finite Galois module oforder prime to p. Let�0 := I m(H0(k, F) → P0 := ∏

v H0(kv, F)), �2 := I m(H2(k, F) →P2 := ∐

v H2(kv, F)). Then there is a perfect duality between P0 and P2 in the sense of Pon-trjagin, and �0 is the orthogonal complement of �2 with respect to this duality.

The assertion of the lemma now follows from the same proof suggested by Tate [1].Additive case. If F is of additive type, then the assertion is trivial, since H2

f l(L , FL) = 0in this case (see [18], Proposition 12).

3 Uniqueness and finiteness of number of global forms

In [15], (Section 4, Theorem 3) there was discussed the uniqueness of global forms G underthe following restrictions: the local kv-groups Gv are inner twists of a given quasi-split k-group G0. It was indicated in loc.cit., Remark 6, that in the case of number fields the globalform may not be unique. Then we may ask about the finiteness of the number of inner globalforms, and if the same is true in the case of function field. We record Theorem 3 of loc.citbelow in a statement which includes also the case of function fields. We give the proof onlyfor the last statement, since the proof of other parts of the theorem in the case of functionfield remains the same, by making use of Theorem A of [20], and also analogous facts fromlocal and global class field theory for function fields.

Theorem 4 (Cf. [15], Theorem 3) Let G0 be an absolutely almost simple simply connectedgroup defined and quasi-split over a global field k, G0 the adjoint k-group correspondingto G0, F0 the center of G0, and v0 a non-archimedean valuation of k. Assume that for all

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v �= v0, there are given local kv-groups Gv which are inner twists of G0, and consider thek-form G of G0, which is locally kv-isomorphic to Gv for all v �= v0.

1. The k-form G of G0 is unique if and only if the localization map

α : H1(k, G0) → ⊕v �=v0 H1(kv, G0)

is injective.2. α is injective if and only if the following localization map

β : H2f l(k, F0) → ⊕v �=v0 H2

f l(kv, F0)

is injective.3. Let L be the minimal splitting field of G0, P=L (resp. P is a cubic extension of k contained

in L) if [L : k] �= 6 (resp. [L : k] = 6, i.e., G0 is of trialitarian type 6 D4). Then β isinjective if and only if v0 is not split in P.

4. In general, the uniqueness may not hold and there are only finitely many k-isomorphismclasses of above indicated such k-forms G.

Proof of Theorem 4, 1–3 It is the same as in [15], by using Harder’s Theorem andKneser’s Theorem in function field case (the bijectivity of H1(l, G0) → H2(l, F0), wherel = k or kv).

We need to check the finiteness of the number of global forms, for which the followinglemma is useful.

Lemma 5 Let k be a global field, S ⊂ V a cofinite subset of valuations of k (i.e., V \S isfinite), which omits at least one non-archimedean valuation of k. Let

1 → F → Gπ→ G → 1

be the exact sequence corresponding to central k-isogeny π : G → G of semisimple groups,where G is simply connected. Consider the localization maps

αS : H1(k,G) → ⊕v∈SH1(kv,G),

βS : H2f l(k, F) → ⊕v∈SH2

f l(kv, F).

Then

1. K er αS is finite if K er βS is finite. If S contains all archimedean valuations, or k is aglobal function field, the converse also holds.

2. If G satisfies cohomological Hasse principle, or k is a global function field, then K er αS

is trivial (resp. αS is injective) if and only if K erβS is trivial (resp. βS is injective).

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Proof We have the following commutative diagram

0 → H1(k, G)u→ ⊕v �∈SH1(kv, G) → 0

↓ ↓ r ↓ t

0 → K er αSs→ H1(k,G)

αS→ ⊕v �∈SH1(kv,G) → 0

α′ ↓ ↓ β ′ ↓ γ ′

0 → K er βSs′→ H2

f l(k, F)βS→ ⊕v �∈SH2

f l(kv, F) → 0

↓ ↓

0 0

In this diagram, s, s′ are just inclusions (and thus are injective), and r, t, β ′, γ ′ are connect-ing maps. Here the coboundary map β ′ is surjective (resp. bijective) by Kneser’s Theorem [9],if k is a number (resp. p-adic) field, and is in fact bijective by [20], Theorem A, if k is a localor global function field, (cf. also [4], where it was proved that (unpublished) δ′ is bijective(resp. surjective) if k is a local (resp. global) function field). Thus the middle and the rightvertical rows are exact. The top map u is a bijection due to well-known cohomological Hasseprinciple for G. The middle and bottom horizontal rows are exact according to Proposition1 of [15], if k is a number field, and according to the similar statement in case of functionfields (see the proof of Proposition 2).

1. Assume that K er βS is finite. First we notice that since s is just an inclusion,

Card(K er α′) ≤ Card(K er β ′).

Since K er β ′ is exactly the image of H1(k, G) in H1(k,G), so

Card(K er β ′) ≤ Card(H1(k, G)).

The well-known cohomological Hasse principle for simply connected semisimple groupsover global fields (cf. [8,9,13]) tells us that we have a bijection

H1(k, G) �∏

v∈∞H1(kv, G),

if k is a number field, and H1(k, G) = 0 if k is a function field. In both cases the 1-Galoiscohomology set H1(k, G) is finite (see [16]), hence so is K er α′. By standard twistingmethod (see [7,9,16]), one checks that all fibers of α′ are finite, hence so is K er αS ,since K er βS is finite.If k is a number field and S contains all archimedean valuations (resp. if k is a globalfunction field), then α′ is surjective, since β ′ is surjective and γ ′ is bijective acording toKneser’s Theorems (resp. according to Theorem A of [20]), thus K er βS if K er αS isfinite.

2. If G satisfies cohomological Hasse principle, then the same proof of [15], Theorem 3,1) applies. Now we assume that k is a global function field. From the proof of 1) aboveand that β ′ and γ ′ are all bijections (Theorem A of [20]) we see that in this case, α′ has

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trivial kernel (in fact, bijective). The twisting argument then shows that αS is a bijectionand the assertion follows.

Proof of Theorem 4, 4 From 3 it follows that in general the uniqueness may not hold (bychoosing, say [L : k] = 2, i.e., G0 is quasi-split and non-split k-group, and v0 such thatv0 splits off in L). To check the finiteness of the global forms considered, by Lemma 5, weneed only check the finiteness of K er βS . If k is a number field, the finiteness of K er βS iswell-known (cf. [11], Chap. 1), where S is just cofinite and needs not be V \{v0}, and we justtake G = G0, F = F0 in notation of the theorem. If k is a function field, then let L/k be asplitting Galois extension for F0

(F0 is of multiplicative type). Denote R := RL/k(F0), and consider the exact sequenceof k-group schemes

1 → F ′ i→ RN→ F0 → 1,

and the following commutative diagram with exact rows

0 → K er δSr→ H2

f l(k, F ′) δS→ ⊕v∈SH2f l(kv, F ′) → 0

α ↓ ↓ β ↓ γ

0 → K er γSs→ H2

f l(k, R)γS→ ⊕v∈SH2

f l(kv, R) → 0

α′ ↓ ↓ β ′ ↓ γ ′

0 → K er βSt→ H2

f l(k, F0)βS→ ⊕v∈SH2

f l(kv, F0) → 0

↓ ↓ ↓

0 0 0

where α, β, γ (resp. α′, β ′, γ ′) are induced from i (resp. N ). Since β ′, γ ′, δS are surjective bythe proof of Proposition 2, it follows that the same is true for α′. Hence to prove the finitenessof K er βS , it suffices to prove the same assertion for K er γS . By using the isomorphisms

H2f l(k, R) � H2

f l(L , FL),H2f l(kv, R) �

∏

wi |vH2

f l(Lwi , FLwi),

where {wi } denotes the finite set of all extensions of v to L , and the fact that F is split overL , we are reduced to proving the finiteness of the kernel of the following homomorphism

H2f l(L , μm) → ⊕w∈T H2

f l(Lw,μm),

where T denotes the set of all valuations of L , which are extensions of valuations of Sto L . Hence T is cofinite and the assertion now is obvious, as it follows from the Hasse–Brauer–Noether theorem. (In the case that S omits only one non-archimedean valuation v0, Tis the set of all valuations of L , which are extensions of valuations of k to L except those,which are the extensions of v0. Hence if v0 does not split in L , then, as it follows from thecase of the groups μm, K er γS is trivial, hence so is K er βS , i.e., the global k-form with the

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properties indicated in Theorem 4 exists uniquely, giving another argument in the proof ofPart 3) of the theorem.) Acknowledgments This paper was written while I was visiting Universität Essen at the invitation of Pro-fessors H. Esnault and E. Viehweg in the program of the Leibniz prize. I would like to thank them heartilyfor their warm hospitality and excellent working conditions, and especially Hélène Esnault and the referee forcritical remarks and valuable advices which led to better presentation of the paper.

References

1. Borel, A., Harder, G.: Existence of cocompact subgroups of reductive groups over local fields. J. ReineUnd Angew. Math. Bd. 298, 53–64 (1978)

2. Bruhat, F., Tits, J.: Groupes algébriques sur un corps local. Chapitre III. Compléments et applications àla cohomologie galoisienne. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, 671–698 (1987)

3. Demazure, M., Gabriel, P.: Groupes algébriques. Tom 1, Paris (1970)4. Douai, J.-C.: 2-Cohomologie galoisienne des groupes semi-simples, Thèse, Université des Sciences et

Tech. de Lille 1 (1976)5. Frey, G.: Dualitätssatz für endlichen kommutative Gruppenschemata über Kongruenzfunktionenkörp-

ern. J. Reine Und Angew. Math. Bd. 301, 46–58 (1978)6. Gille, P.: La R-équivalence sur les groupes réductifs définis sur un corps de nombres. Pub. Math. I. H. E.

S., t. 86, 199–235 (1997)7. Giraud, J.: Cohomologie non-abélienne, Grund. des Math. Wiss. Bd, vol. 179. Springer-Verlag, Ber-

lin (1971)8. Harder, G.: Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III. J. Reine Angew. Math.

Bd. 274/275, 125–138 (1975)9. Kneser, M.: Lecture on Galois cohomology of classical groups, Tata Inst. Fund. Res. (1969)

10. Mazur, B.: On the passage from local to global in number theory. Bull. Amer. Math. Soc. (N.S.) 29(1),14–50 (1993)

11. Milne, J.: Arithmetic duality theorems, 2nd ed. BookSurge, Charleston (2006)12. Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields. 2nd edn. Grund. der Math. Wiss.,

Bd, vol. 323. Springer-Verlag, Berlin (2008)13. Platonov, V., Rapinchuk, A.: Algebraic groups and Number theory. Academic Press, Dublin (1993)14. Poitou, G. (ed.): Cohomologie galoisienne des modules finis. Séminaire de l’Institut de Mathématiques

de Lille, sous la direction de G. Poitou. Travaux et Recherches Mathématiques, No. 13 Dunod, Paris(1967)

15. Prasad, G., Rapinchuk, A.: On the existence of isotropic forms of semi-simple algebraic groups overnumber fields with prescribed local behavior. Adv. Math. 207(2), 646–660 (2006)

16. Serre, J.-P.: Cohomologie galoisienne, Lecture Notes in Math. vol. 5, 5th éd. Springer-Verlag, Berlin(1995)

17. Demazure, M., Grothendieck, A. (éd.): Schémas en groupes, Lecture Notes in Math, vol. 151–153.Springer-Verlag, Berlin (1970)

18. Shatz, S.: Cohomology of Artinian group schemes over local fields. Annals of Mathematics 79,411–449 (1964)

19. Shatz, S.: Profinite groups: arithmetic and geometry, Annals of Math. Studies, vol. 67. Princeton Univer-sity Press, New Jersy (1972)

20. Thang, N.Q.: On Galois cohomology of semisimple groups over local and global fields of positive char-acteristic. Math. Z. Bd. 259(2), 457–470 (2008)

21. Thang, N.Q., Tân, N.D.: On the Galois and flat cohomology of unipotent algebraic groups over local andglobal function fields. I. J. Algebra 319, 4288–4324 (2008)

22. Tits, J.: Classification of algebraic semisimple groups, in “Algebraic Groups and Discontinuous Sub-groups” (Proc. Sympos. Pure Math., vol. 9, Boulder, Colo., 1965) pp. 33–62 Amer. Math. Soc., Provi-dence, R.I., (1966)

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