P-Nilpotent completion is not idempotent

Publicacions Matem`atiques, Vol 41 (1997), 481–487. P-NILPOTENT COMPLETION IS NOT IDEMPOTENT Geok Choo Tan Abstract Let Pbe an arbitrary set of primes. The P-nilpotent completion of a group Gis defined by the group homomorphism η:G→ GPwhere GP= invlim(G/ΓiG)P. Here Γ2Gis the commutator subgroup [G, G] and ΓiGthe subgroup [G, Γi−1G] when i>2. In this paper, we prove that P-nilpotent completion of an infinitely generated free group Fdoes not induce an isomorphism on the first homology group with ZPcoefficients. Hence, P-nilpotent completion is not idempotent. Another important consequence of the result in homotopy theory (as in [4]) is that any infinite wedge of circles is R-bad, where Ris any subring of rationals. 1. Introduction For a group G, we denote by Γ2Gthe commutator subgroup [G, G] and ΓiGthe subgroup [G, Γi−1G] when i>2. A group Gis nilpotent if Γi(G) is trivial for some i. The nilpotency class nil(G)ofGis the least csuch that Γc(G) is trivial. Let Pbe a set of prime numbers. There is a well-known P-localization in the category of nilpotent groups, [7]. We denote this localization on a nilpotent group Nby e:N→NP. The P-nilpotent completion or ZP-completion of a group Gis defined to be the group homomorphism η:G→G Pwhere G P= invlim(G/ΓiG)P, with irunning through all finite ordinals. For each i, the group homomorphism G→(G/ΓiG)Pdefines a localization on the category Gof groups. Its universal property gives rise to a natural map (G P/ΓiG P)P→(G/ΓiG)P. Passing to inverse limit, we obtain a natural transformation χ:(G P) P→G Pso that (( ) P,η,χ) is a monad on G. Let Fbe a free group on an infinitely countable set of generators. In [4, Proposition IV.5.4], it is proved that the abelianization of η:F→Fˆ Z= invlim(F/ΓiF) is not an isomorphism. This result is used to verify that Z-completion (which is P-nilpotent completion when Pis the set of all primes) is not idempotent in [3]. 482 G. C. Tan We study these proofs closely and obtain a similar proof of the nonidempotence of P-nilpotent completion for any set Pof primes. We use results from orthogonal pairs, idempotent monads and the P-localization on the category of nilpotent groups. Although the P-nilpotent completion is not idempotent on the category of groups, a procedure to obtain an idempotent monad from it is described in [5]. This turns out to be the minimal P-localization, which is also obtained in [2]. It is the “smallest” (in the sense that it provides the least local objects) idempotent monad which extends P-localization on the category of nilpotent groups to the category of groups. This minimal P-localization coincides with the P-nilpotent completion on groups which have finitely generated abelianization [3] and groups with stable lower central series [2]. 2. The P-nilpotent completion is not idempotent Let Cbe a category, Xbe an object of Cand f:A→Bbe a morphism of C. Then Xand fare said to be orthogonal to each other, denoted by X⊥for f⊥X,iff∗:C(B,X)∼ =C(A, X). For a class D of objects in C, the orthogonal complement of Din C, denoted by D⊥, is the class of morphisms orthogonal to every object in D. Dually, the orthogonal complement of Scan be defined for a class Sof morphisms. An orthogonal pair (S, D)inCcomprises a collection Sof morphisms in Cand a collection Dof objects in Csatisfying S=D⊥and D=S⊥. Every idempotent monad (see [8, p. 133]) (also known as localization in [6]) is associated with a unique orthogonal pair. Let a1,a 2,... be elements of a group G. We define [a1,a 2]= a−1 1a−1 2a1a2and [a1,a 2,... ,a k]=[[a1,... ,a k−1],a k] recursively for k≥3. Proposition 1. Let Fbe the free group on a1,... ,a k. For every positive integer n,[a1,... ,a k]ndoes not belong to the subgroup of Γ2F that is generated by Γk+1Fand Γ2Γ2F. Proof: Replacing Fby the quotient F/Γk+1F,Γ2Γ2F, the proposition becomes: for each n, there exists a group Gwith the following properties: (i) The commutator subgroup Γ2Gis abelian, (ii) Gis nilpotent of class k+ 1, and (iii) there exists x∈ΓkGsuch that xn=1. However, it is enough to pick a prime pthat does not divide nand find a p-group Gsuch that Γ2Gis abelian and Gis nilpotent of class k+ 1. For any positive integer m, consider the Z/p vector space Von a P-nilpotent completion is not idempotent 483 basis {v1,v 2,... ,v pm}and let σ∈GL(V) where σ(vi)=vi+vi+1 if i≤pm−1 vpmif i=pm. For each positive integer j≤pm, σj(vi)=         j l=0 j lvi+lif i≤pm−j r l=0 j lvi+lif i>p m−j, where r=pm−i so that the order of σis pm. The semi-direct product group of Vand σ is a p-group whose commutator subgroup is abelian and has nilpotency class pm(see [1] and [9]). By choosing m≥k+ 1 and factoring this semi-direct product group by the k+ 1-th lower central term we obtain a group Gwith the required properties. Let Pbe a fixed set of prime numbers. We use the notation n∈Pto mean all prime divisors of nare in Pand Pto denote the complement of Pin the set of all primes. A group Gis said to be P-local if the map g→ gnis a bijection for all n∈P. A group homomorphism f:G→K is said to be (i) P-injective if for any two elements g1,g 2∈Gsuch that f(g1)=f(g2), there exists an integer n∈Psuch that gn 1=gn 2; (ii) P-surjective if for every k∈K, there exists an integer n∈P such that kn∈Imf; and (iii) P-bijective if fis both P-injective and P-surjective. On the category of nilpotent groups, there is a well-known P-localization [7], which is denoted by e:N→NPfor each nilpotent group N, where NPis P-local nilpotent and eis a P-bijection. The P-nilpotent completion or ZP-completion of a group Gis defined to be the group homomorphism η:G→G Pinduced by the group homomorphisms GG/ΓiGe →(G/ΓiG)P, where G P= invlim(G/ΓiG)P, with irunning through all finite ordinals. For each i, the above group homomorphism G→(G/ΓiG)Pdefines an idempotent monad on Gwhose universal property enables us to complete the following diagram G P−−−−→(G P/ΓiG P)P    (G/ΓiG)P 484 G. C. Tan by a unique map (G P/ΓiG P)P→(G/ΓiG)P. Passing to inverse limits, we obtain a natural transformation χ:(G P) P→G Pso that (( ) P,η,χ) is a monad on G. Let Gbe the category of groups and Gbe the full subcategory of groups Gsuch that the natural homomorphism G P→(G P) Pis an isomorphism. Then ( ) Prestricts to an idempotent monad on G. Let (S,D) be the associated orthogonal pair. Since every abelian group A satisfies A P∼ =AP, all abelian groups are objects of G; moreover, all P-local abelian groups are in D. For any group Gin G, the completion homomorphism η:G→G Pis in Sand hence it is orthogonal to all P-local abelian groups. From this fact it follows that, for all groups Gin G, the natural map (G/Γ2G)P→(G P/Γ2G P)P induced by ηis an isomorphism. Thus, if Gis in G, then H1(G;ZP)∼ = H1(G P;ZP). For any group G, we denote by γithe projection of Gonto G/ΓiG,by θithe natural epimorphism from G Ponto (G/ΓiG)P,by¯ηthe abelianization of η:G→G P, and by ethe P-localization homomorphism. Since (G/Γ2G)Pis abelian, there is a unique homomorphism ¯ θ2:G P/Γ2G P→ (G/Γ2G)Psuch that ¯ θ2γ2=θ2. Now we have ¯ θ2¯ηγ2=¯ θ2γ2η=θ2η=eγ2. Since γ2is surjective, we infer that ¯ θ2¯η=e. Under the assumption that the group Gis in the subcategory G, both ¯ηand eare P-bijections. It follows that ¯ θ2is a P-bijection as well. Hence, we have proved the following result. Proposition 2. For a group G, if the natural homomorphism G P→(G P) Pis an isomorphism, then the homomorphism H1(G;ZP)→ H1(G P;ZP)induced by the P-completion map G→G Pand the homomorphism H1(G P;ZP)→H1(G;ZP)induced by the projection G P→ (G/Γ2G)Pare isomorphisms, and they are inverse to each other. We next prove that if Fis a free group on an infinite set of generators, then ¯ θ2is not P-injective. This implies that Fis not in G, as desired. Thus, we shall assume that ¯ θ2is P-injective and arrive at a contradiction. Pick a countable subset of free generators of Fand label them as {aij}, where 1 ≤j≤i. Denote by Fmthe free group generated by am1,... ,a mm. Let πmbe the projection of Fonto Fmsending all other P-nilpotent completion is not idempotent 485 generators to 1. Likewise, we denote by ˆπm, the induced homomorphism F P→(Fm) Pand by ηm, the completion map Fm→(Fm) P. Consider the element b=(b2,b 3,...)∈Fˆ Z, where b2= 1 and, for m≥2, bm+1 is the class of [a21,a 22][a31,a 32,a 33]···[am1,... ,a mm] in F/Γm+1F. Since the natural map Fˆ Z→F Pis injective, we may view bas an element of F Pas well. In fact we have ˆπm(b)=ηm([am1,... ,a mm]). Since 1 = θ2(b)=¯ θ2(γ2(b)) and ¯ θ2is assumed to be P-injective, it follows that γ2(b)n= 1 for some n∈P. Hence, bn∈Γ2F P. Therefore we may write bn=[u1,u 2]···[u2k−1,u 2k], with ui∈F Pfor all i. Now, for each i, we have θ2(ui)ti=e(γ2(zi)), for some ti∈Pand zi∈Fbecause eis P-surjective and γ2is surjective. Since only a finite number of generators of Fare involved in zi,wehave πm(zi) = 1 for all iand all mexcept for a finite number of indices m1,... ,m r. Choose any m=m1,... ,m r, which will remain fixed in the rest of the argument. Let ψmbe the unique homomorphism that renders the following diagram commutative: F P θ2 (F/Γ2F)P e ←− (F/Γ2F)γ2 ←− F   ˆπm  ψm  πm   (Fm) P θ2 (Fm/Γ2Fm)P e ←− (Fm/Γ2Fm)γ2 ←− Fm For each i,wehave ψm(θ2(ui))ti=ψm(e(γ2(zi))) = e(γ2(πm(zi))) = 1. Since the target of ψmis a P-local group, we infer that ψm(θ2(ui)) = 1, and hence θ2(ˆπm(ui)) = 1. Therefore, θm+1(ˆπm(ui)) belongs to the kernel of the reduction map (Fm/Γm+1Fm)P→(Fm/Γ2Fm)P, that is, θm+1(ˆπm(ui)) ∈(Γ2Fm/Γm+1Fm)P 486 G. C. Tan for all i. Now observe that θm+1(ηm([am1,... ,a mm]n)) = θm+1(ˆπm(bn)) =[θm+1(ˆπm(u1)),θ m+1(ˆπm(u2))]···[θm+1(ˆπm(u2k−1)),θ m+1(ˆπm(u2k))], which is an element of (Γ2Γ2Fm/Γm+1Fm)P. Hence, there is an integer q∈Pand an element x∈Γ2Γ2Fm/Γm+1Fmsuch that θm+1(ηm([am1,... ,a mm]n))q=e(x). Since we have the commutative diagram Fm ηm −−−−→(Fm) P γm+1    θm+1 Fm/Γm+1Fm e −−−−→(Fm/Γm+1Fm)P where Fm/Γm+1Fmis torsion-free and hence the localization map eis injective, we infer that γm+1([am1,... ,a mm]nq)=x. It follows that [am1,... ,a mm]nq belongs to the subgroup of Fmgenerated by Γ2Γ2Fmand Γm+1Fm. This contradicts Proposition 1. We have thus shown Theorem 3. Let Fbe a free group on an infinite set of generators. Then, for any set of primes P, the natural homomorphisms H1(F;ZP)→ H1(F P;ZP)and η:F P→(F P) Pboth fail to be isomorphisms. We thus conclude that P-nilpotent completion is not idempotent on the category of groups. As in [4, Proposition IV.5.4], it follows from our theorem and [4, Proposition IV.5.3] that Corollary 4. Any infinite wedge of circles is R-bad, where Ris any subring of the rationals. Acknowledgements. The author wishes to thank Jon Berrick, Carles Casacuberta and Garth Warner for many helpful comments and the referee for his/her suggestions on the presentation of this paper. P-nilpotent completion is not idempotent 487 References 1. R. Baer, Nilpotent groups and their generalizations, Trans. Amer. Math. Soc. 47 (1940), 393–434. 2. A. J. Berrick and G. C. Tan, The minimal extension of P-localization on groups, Math. Proc. Cambridge Philos. Soc. 118 (1996), 243–255. 3. A. K. Bousfield, Homological localization towers for groups and π-modules, Mem. Amer. Math. Soc. 10(186) (1977), ?. 4. A. K. Bousfield and D. M. Kan,“Homotopy Limits, Completions and Localizations,” Lecture Notes in Math. 304, SpringerVerlag, Berlin, 1972. 5. C. Casacuberta, A. Frei and G. C. Tan, Extending localization functors, J. Pure Appl. Algebra 103 (1995), 149–165. 6. C. Casacuberta, G. Peschke and M. Pfenniger, Orthogonal pairs in categories and localization, in “Proc. Adams Memorial Symposium,” London Math. Soc. Lecture Note Ser. 175, Camb. Univ. Press, Cambridge, 1992, pp. 211–223. 7. P. Hilton, G. Mislin and J. Roitberg,“Localization of Nilpotent Groups and Spaces,” North-Holland Math. Studies 15, NorthHolland, Amsterdam, 1975. 8. S. MacLane,“Categories for the Working Mathematician,” Graduate Texts in Math. 5, Springer-Verlag, Berlin, 1972. 9. M. Suzuki,“Group Theory II,” Grundlehren der mathematischen Wissenchaften 248, A series of Comprehensive Studies in Mathematics, Springer Verlag, 1986. Department of Mathematics Faculty of Science National University of Singapore 10 Kent Ridge Crescent SINGAPORE 119260 Primera versi´o rebuda el 3 de Maig de 1996, darrera versi´o rebuda el 20 de Juny de 1997