Deflation and Residuation for Class Formation

Transcription

1 Journal of Algebra 245, ) doi: /jabr , available online at on Deflation and Residuation for Class Formation Kuniaki Horie Department of Mathematics, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa , Japan and Mitsuko Horie Department of Mathematics, Ochanomizu University, Otsuka, Bunkyo-ku, Tokyo , Japan Communicated by Walter Feit Received September 14, 2000 in memory of mrs. kenkichi iwasawa. In the present paper, recalling briefly a deflation map and a residuation map in the cohomology theory of finite groups, we shall see that a certain pair of isomorphisms of Tate s main theorem for class formations establishes a correspondence of the associated residuation map to the associated deflation map Theorem 1). This result, supplementary to the results of Artin and Tate, will further provide us with a commutative diagram for Galois cohomology groups, which just generalizes a fundamental commutative diagram for number knots Theorem 2) Academic Press 1 Let G be any finite group. Let denote the ring of rational) integers. By a G-module, we mean as usual a leftmodule over G, the group ring of G over. LetU and V be G-modules. Let U G V denote the quotient group of U V modulo the subgroup of U V generated by σu σv u v σ G u U v V. For each u v U V,we denote by u G v the class of u v in U G V. For each integer m 0, /01 $35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved.

2 608 horie and horie let G m denote the direct product of m copies of G and let P G m denote the free G-module with basis G m. Of course, we understand P G 0 to be the left regular module of G. Identifying U G P G 0 with the additive group of U we let n = n G U denote, for each integer n>0, the homomorphism U G P G n U G P G n 1 such that n u G σ 1 σ n ) = σ 1 1 u G σ 2 σ n 1 i u G σ 1 σ i 1 σ i σ i+1 σ i+2 σ n n 1 + i=1 + 1 n u G σ 1 σ n 1 with u U σ 1 G σ n G. Itfollows that Im n+1 G U Ker n G U for every integer n>0. Let G denote the sum in G of all elements of G and let U G denote the G-submodule of U consisting of the elements of U fixed by every elementof G, sothat G U is a G-submodule of U G.Let GU denote the G-submodule of U consisting of all u U with G u = 0 and let I G U denote the G-submodule of U generated by 1 σ U for all σ G I G U G U. Now, for any r, let H r G U denote the rth Tate cohomology group of G with coefficients in U. Then, in particular, H 0 G U =U G / G U H 1 G U = G U/I G U and, in the case r 2, H r G U = Ker r 1 G U ) / Im r G U ) The additive group of, which we also denote by, will be considered a trivial G-module. Accordingly, for any integer n>0, we identify G P G n with the additive group of G n. Next, G and U being as above, let N be any normal subgroup of G so that U N becomes a G/N-module. Take any integer m 0. Then the deflation map H m G U H m G/N U N in the sense of Kawada [5] and Weiss [9] is induced, in the case m = 0, by the identity map U G U N G/N and, in the case m 1, by the homomorphism U G P G m 1 U N G/N P G/N m 1 which maps u G σ 1 σ m 1 to N u G/N σ 1 N σ m 1 N

3 deflation and residuation 609 for all u σ 1 σ m 1 U G m 1. On the other hand, let U N = U/I N U so that U N becomes a G/N-module. Then the residuation map) H m 2 G U H m 2 G/N U N in the sense of Nakayama [6] is induced by the homomorphism U G P G m+1 U N G/N P G/N m+1 which maps u G σ 1 σ m+1 to a + IN U ) G/N σ 1 N σ m+1 N for all u σ 1 σ m+1 U G m+1. We shall often denote the deflation map and the residuation map above by Dfl and Rsd, respectively. Remark. If U is a trivial G-module such as, then two homomorphisms Dfl H n G U H n G/N U Rsd H n G U H n G/N U are obtained for every integer n 2, and it follows cf. [9, Sect. 3]) that Dfl ζ = N Rsd ζ for every ζ H n G U 2 Let be a class formation in the sense of Artin and Tate [1], the Galois group of, A the formation module of, and the nonempty) setof all fields of. Then, by the definition of a formation, there exists an indexed family E E of subgroups of with finite indices such that = E E A ) Although A is a -module, we shall write multiplicatively the law of composition in A cf. Section 3). The image of each ρ a A under the action of on A will be denoted by a ρ, whence b στ = b τ σ for each σ τ b 2 A Now, take a field F of F. For any E in with E F, let E/F = F / E and let A E denote the E-level of : A E = A E = a A a σ = a for all σ E Note that A E naturally becomes a E/F -module. Let r be any integer. Let ξ E/F denote the fundamental class in H 2 E/F A E and E/F the homomorphism H r E/F H r+2 E/F A E such that E/F ζ =ξ E/F ζ for each ζ H r E/F

4 610 horie and horie Here, for each pair ξ ζ in H 2 E/F A E H r E/F ξ ζ denotes the image of ξ ζ under the cup product H 2 E/F A E H r E/F H r+2 E/F A E As is well known, E/F isomorphism cf. [8]). is an isomorphism, and we shall call it the Tate Theorem 1. Let m be a nonnegative integer. Given any K and L in with L K F L F put = K / L and identify K/F with L/F / as a group acting on A K ) in the obvious manner. Then the diagram H m 2 L/F Rsd H m 2 K/F L/F H m L/F A F Dfl K/F H m K/F A K is commutative. Proof. Let f be any element of the fundamental class ξ L/F in H 2 L/F A L. For each s K/F, we fix an element s of L/F with s = s. Letf K denote the map 2 K/F A L such that f K s t = f η s t f η st 1 f η s for each s t 2 K/F. Since f is a 2-cocycle 2 L/F A L and a normal subgroup of L/F, we then have, for any ρ, ) ρ f η s t = f ρη s t f ρ η s t 1 f ρ η s = f η s t f ρ ηst 1 f ρ η s f η st 1 f ρ ηst f ρ η 1 f η st 1)ρ = ) ρ f η s = f η s f ρ η s 1 f ρ η

5 deflation and residuation 611 These imply f K s t ρ = f K s t, whence f K s t belongs to A K. Furthermore, for any σ τ 2 L/F, f σ τ f τ η 1 f η τ ) σ = f σ τ f στ η 1 f σ τη f σ τ 1 f ση τ f σ ητ 1 f σ η = f στ η 1 f ση τ f σ η = f K σ τ f στ η 1 f η στ f σ η f η σ 1 so that f K σ τ f σ τ L K = f τ η 1 f η τ σ f στ η 1 f η στ 1 f σ η 1 f η σ Hence, we know notonly thatf K is a 2-cocycle 2 K/F A K, butalso that the class of f K in H 2 L K K/F A K is mapped to ξl/f by the inflation map H 2 K/F A K H 2 L/F A L, which is injective. Thus, f K belongs to ξ K/F. Now, we assume m 2. Let z = σ 1 σ m+1 vary through m+1 L/F and, for each positive integer i m + 1, put s i = σ i in K/F = L/F /. Taking any class ζ in H m 2 L/F, let a z z z be any elementof ζ, with each a z in. Since the class ξ L/F ζ in H m L/F A L contains z a z f σ1 σ 2 σ 1σ 2 1 ) itfollows from L/F that z ) σ1 σ 2 1 a z f σ 1 σ 2 η Dfl ξ L/F ζ 1) On the other hand, we have ) a z f K s 1 s 2 σ 1σ 2 1 ξ K/F Rsd ζ 2) z )

6 612 horie and horie by f K ξ K/F and z a z s 1 s m+1 Rsd ζ. Given any σ L/F, put g σ = f η σ f η σ 1 Then, for each τ L/F, g σ τ = f τη σ f τ ησ 1 f τ η f τη σ 1 f τ ησ f τ η 1 = f τη σ f τη σ 1 in particular, g σ is fixed by every elementof, namely, belongs to A K. It further follows that f K τ σ 1 f τ σ η = f ητ σ 1 f η τσ f η τ 1 f ητ σ f η τσ 1 f η τ =g σ τ g τσ 1 g τ so that f σ 1 σ 2 η σ 1σ 2 ) 1 K/F +f K s 1 s 2 σ 1σ 2 1 = g σ 2 σ 1 2 g σ 1 σ 2 σ 1σ 2 1 g σ 1 σ 1σ 2 1) = g σ 2 σ 1 2 +g σ 1 σ 2 σ 1σ 2 1 m + 1 i ) g σ 1 σ 1 1 s 2 s i 1 s i s i+1 s i+2 s m+1 i=2 + 1 m+1 ) g σ 1 σ 1 1 s 2 s m g σ 1 σ 1 1 ) σ 1 2 m + 1 i 1 ) g σ 1 σ 1 1 s 2 s i 1 s i s i+1 s i+2 s m+1 i=2 + 1 m ) g σ 1 σ 1 1 s 2 s m

7 Therefore, z deflation and residuation 613 ) σ1 σ 2 1 a z f σ 1 σ 2 η a z fk s 1 s 2 σ 1σ 2 1 ) + z = m a z + 1 i a zi + 1 m+1 a z z i=1 m K/F A K z ) ) g σ2 σ 1 2 ) a z g σ1 σ 1 1 s 2 s m+1 )) where z = σ 2 σ m+1 σ 1 and, for each positive integer i m, z i = σ 2 σ i σ 1 σ1 1 σ i+1 σ i+2 σ m+1 However, we obtain from z a z ζ that ) m a z + 1 i a zi + 1 m+1 a z = 0 σ 1 i=1 with σ 2 σ m+1 fixed. Hence, by 1) and 2), the theorem is proved. An argument quite similar to the above gives us a proof of the theorem for m = 1, as follows. Let ζ be any class in H 3 L/F and let b σ τ σ τ be an elementof ζ with each b σ τ in. Then σ L/F b σ τ b σ σ 1 τ + b τ σ =0 for every τ L/F ) bσ τ στ 1 f σ τ η f K σ τ b σ τ στ 1 Dfl ξ L/F ζ ξ K/F Rsd ζ Hence, ξ K/F Rsd ζ Dfl ξ L/F ζ 1 f K σ τ 1 ) bσ τ στ 1 f σ τ η

8 614 horie and horie = g τ τ 1 g στ στ 1 g σ στ 1) b σ τ = g τ bσ τ b σ σ 1τ+bτ σ τ 1 g σ b σ τσ 1 g σ b σ τ στ 1 = We therefore have g σ στ 1 σ 1) b σ τ ξ K/F Rsd ζ =Dfl ξ L/F ζ which is to be proved. Letus finally consider the case m=0. LetI denote the group generated by ρσ ρ σ ρ σ L/F in the additive group of L/F. Then H 2 L/F is nothing but the quotient of the additive group of L/F modulo I. Take any τ L/F. Itfollows thatthe class ξ L/F τ + I in H 0 L/F A L contains σ L/F f σ τ 1. Hence, f σ τ 1 Dfl ξ L/F τ + I σ L/F On the other hand, we obtain However, we may assume that s K/F f K s τ 1 ξ K/F Rsd τ + I τ s s K/F i.e., τ = τ so that f K s τ = f η s τ 1 f η s τ f η s 1 s K/F s K/F = f σ τ 1 σ L/F Therefore, Dfl ξ L/F τ + I = ξ K/F Rsd τ + I This obviously concludes the proof of the theorem for the case m = 0.

9 deflation and residuation Let k be a global field, namely, either a finite algebraic number field or an algebraic function field in one variable over a finite constant field. Let be a separable algebraic closure of k. Both k and will be fixed throughout the following. We shall also assume all global fields to be contained in. Given any global field F, let F denote the multiplicative group of F, J F the idele group of F, and C F the idele class group of F. By means of the canonical injection F J F, we consider F to be a subgroup of J F so that C F = J F /F.LetP be the set of all primes of k. For each P, we fix a prime of lying above and denote by the completion of at and by F the completion of F in. Now, for later convenience, let us give typical examples of a class formation, as follows. Let 0 be the set of global fields containing k. LetC denote the inductive limit of C E for all E 0 taken with respect to the canonical injections C E C E for all extensions E /E of global fields in 0. Then we obtain a class formation Gal /k Gal /E E 0 C We understand here that C Gal /E = C E for each E 0 regarding C E as a subgroup of C via the canonical injection C E C.Of course, for any Galois extension E /E of global fields in 0, Gal E /E is identifed with Gal /E /Gal /E, and the Tate isomorphism H r Gal E /E H r+2 Gal E /E C E is defined for every r. Next, let be any prime of k; P. Itfollows that is a Galois extension over k, the completion of k at. Let be the set of all finite extensions over k in. Then a class formation is obtained, where Gal /k Gal / denotes the multiplicative group of. 4 Let r be any integer and let F be any global field which is a Galois extension over k. LetG F denote the Galois group of F over k and, for any G F -module A, let H r F A denote the rth Tate cohomology group of G F with coefficients in A: G F = Gal F/k H r F A =Hr G F A

10 616 horie and horie We then write r F or, simply, F for the Tate isomorphism HF r H r+2 F C F. Now, given any P, let G F denote the decomposition group for F/k of the restriction /F, F the multiplicative group of F, and j the injective homomorphism from F to J F such that, for each α F and each prime of F, the -componentof j α is α or 1 according as lies above or not. The Galois group of F over k will be identified with G F in the obvious manner: Gal F /k =G F.Let H r G F Hr+2 F J F be the composite of the Tate isomorphism H r G F Hr+2 G F F, the corestriction map H r+2 G F F Hr+2 F F, and the homomorphism H r+2 F F Hr+2 F J F induced by j. On the other hand, let Cor denote the corestriction map H r G F Hr F. We then obtain homomorphisms φ r F = φ F H r G F Hr+2 F J F P γ r F = γ F H r G F Hr F P such that φ r F c = c γ r F c = Cor c P P for every element c = c P of P Hr G F. Note that φ F is known to be an isomorphism. The following result is also well known cf. [2, Chap. 7]): Lemma. The diagram H r G F γ F H r F P φ F F H r+2 F J F H r+2 F C F is commutative, where the lower horizontal map is the homomorphism from H r+2 F J F to H r+2 F C F induced by the natural surjection J F C F. Let r still be an integer and F a finite Galois extension over k in. Let j r F denote the homomorphism HF r F HF r J F induced by the canonical injection F J F. Then the natural exact sequence 1 F J F C F 1

11 gives rise to an exact sequence deflation and residuation 617 HF r 1 J F HF r 1 C F HF r F j r F HF r J F Hence, by the above lemma, we have an isomorphism F Coker γ r 3 F Ker j r F mapping the class in Coker γ r 3 F of each c HF r 3 to the image of r 3 F c under the connecting homomorphism HF r 1 C F HF r F. Now, let m be any nonnegative integer and let L/K be an extension of global fields which are Galois extensions over k. Then the natural exactsequences 1 L J L C L 1 1 K J K C K 1 yield the following commutative diagram with exact rows cf. [9, Sect. 4]): HL m 1 J L HL m 1 C L HL m L HL m J L HL m C L Dfl Dfl Dfl Dfl Dfl H m 1 K J K H m 1 C K H m K H m J K H m C K K In particular, the deflation map HL m L HK m K defines a homomorphism D L/K Ker j m L Ker j m K Next, for each P, let R denote the residuation map H m 3 G L H m 3 G K, G K being identified with the quotient of G L modulo the kernel of the restriction map G L G K ; Gp K = G L /Gal L /K Itthen follows that the diagram H m 3 G Cor L H m 3 G L K K K R Rsd Cor H m 3 G K H m 3 G K is commutative since the upper and lower corestriction maps are induced by the inclusions G L m+2 G m+2 L and G K m+1 G m+1 K, respectively. Hence, we get a commutative diagram P H m 3 G L H m 3 γ L L Rsd P H m 3 G K H m 3 γ K K

12 618 horie and horie Here the vertical map on the left is the group homomorphism mapping each c P in P H m 3 G L to R c P in P H m 3 G K. The residuation map HL m 3 HK m 3 therefore defines a homomorphism R L/K Coker γ m 3 L Coker γ m 3 K Furthermore, Theorem 1 gives us a commutative diagram H m 3 L L HL m 1 C L Rsd Dfl H m 3 K HK m 1 C K We thus obtain the following result: Theorem 2. The diagram Coker γ m 3 L L Ker j m L K R L/K D L/K K Coker γ m 3 K Ker j m K is commutative. Finally, let m = 0 in Theorem 2. For any finite Galois extension F over k in, let N F/k denote the norm map J F J k and ν F/k the number knotof F/k; ν F/k = k N F/k J F /N F/k F We then easily see that Ker j 0 L = ν L/k Ker j 0 K = ν K/k D L/K αn L/k L = αn K/k K for every α k N L/k J L Hence, Theorem 2 yields a commutative diagram Coker γ 3 L L ν L/k R L/K 3) Coker γ 3 K K ν K/k where the vertical map on the right is the homomorphism from ν L/k to ν K/k induced by the canonical injection k N L/k J L k N K/k J K.

13 deflation and residuation 619 Remark. The commutative diagram 3) plays an important role in the study of number knots cf. [3, 4, 7]). This paper was motivated at first by giving an explicit proof of 3), which the authors could not find in the literature. REFERENCES 1. E. Artin and J. Tate, Class Field Theory, Benjamin, Elmsford, NY, J. W. S. Cassels and A. Fröhlich, Eds., Algebraic Number Theory, Academic Press, New York, K. Horie and M. Horie, Relations among certain number knots, preprint. 4. W. Jehne, On knots in algebraic number theory, J. Reine Angew. Math. 311/ ), Y. Kawada, Algebraic Number Theory in Japanese), Kyoritsu, Tokyo, T. Nakayama, A remark on fundamental exact sequences in cohomology of finite groups, Proc. Japan Acad. Ser. A Math. Sci ), T. Tannaka, H. Kuniyoshi, F. Terada, and S. Takahashi, Number-theoretical properties of cohomology groups in Japanese), Sūgaku ), J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math ), E. Weiss, A deflation map, J. Math. Mech ),