Journal of Pure and Applied Algebra

Transcription

1 Journal of Pure and Aled Algebra 23 (2009) Contents lsts avalable at ScenceDrect Journal of Pure and Aled Algebra journal homeage: wwwelsevercom/locate/jaa K 2 of fnte abelan grou algebras Yubn Gao, Guong Tang School of Mathematcal Scences, Graduate Unversty of Chnese Academy of Scences, Bejng 00049, Chna a r t c l e n f o a b s t r a c t Artcle hstory: Receved 3 March 2008 Receved n revsed form 4 Setember 2008 Avalable onlne 6 February 2009 Communcated by CA Webel In ths aer, we reresent K 2 (F[G Z ]) as the drect sum of K 2 (FG) and an elementary abelan -grou; usng ths we calculate K 2 (FG) when F s a fnte feld of odd rme characterstc and G s a fnte abelan grou of 2 -rank We also comute H (FG) DR f F s a fnte feld of odd characterstc and G s a fnte abelan -grou 2009 Publshed by Elsever BV Introducton Let F be a fnte feld of characterstc and G a fnte abelan grou A general formula for K 2 (FG) has been gven n [7], Theorem 67; t s the quotent of G AG wth A the unramfed -rng such that F A/ An uer bound for the order of K 2 (FG) was also gven n [7] However, t s not easy to determne the structure of K 2 (FG) drectly from ths quotent, even ts recse order The result of Denns and Sten n [6] (Corollary 44(a)) mles that for a cyclc grou C n, K 2 (FC n ) = For an elementary abelan -grou G, Denns, Keatng and Sten [] roved that K 2 (FG) s an elementary abelan -grou whose recse rank s also gven In Magurn [2], when F s a characterstc 2, K 2 (F[G Z 2 ]) s somorhc to the drect sum of K 2 (FG) and an elementary abelan 2-grou whose rank s determned Usng ths result, Magurn calculated K 2 (FG) when G s a fnte abelan grou wth 4-rank and F s of characterstc 2 In Secton 3 of ths aer, we extend Magurn s results to the case when s an odd rme In Secton 4, t wll be shown that to get the recse order of K 2 (FG), the only thng we need to know s the orders of kernels of Ω FG/Z K 2(FG[t]/(t k ), (t)), k mod, k >, where F s of odd characterstc, G s a fnte abelan -grou We determne the de Rham cohomology grou H DR (FG) for arbtrary abelan -grous G, and show how ths cohomology grou can be used to comute the above kernels n case G s an elementary abelan -grou 2 Prelmnares Suose k s a commutatve rng, A s a k-algebra, for an A-module M, a k-dervaton from A to M s a k-lnear ma d: A M such that d(ab) = (da)b + a(db), (a, b A) The set of all such dervatons Der k (A, M) s an A-module, whch s functoral n M A unversal k-dervaton d : A Ω A/k s defned by takng Ω A/k to be the A-module defned by generators da (a A), and relatons Ω A/K d(ab) = adb + bda, d(a + b) = da + dba, b A, as well as dc = 0, c k s called Kähler dfferentals of A over k and the unversalty s exressed by the natural somorhsm Hom A-mod (Ω A/K, M) Der k(a, M) sendng f to f d Corresondng author E-mal address: tangg@gucasaccn (G Tang) /$ see front matter 2009 Publshed by Elsever BV do:006/jjaa

2 202 Y Gao, G Tang / Journal of Pure and Aled Algebra 23 (2009) The algebra of dfferental forms over an algebra A s the exteror algebra Ω A/K = q Ω q A/K, Ω q A/K = q A Ω A/K, any element of Ω q A s called a dfferental form of degree q The morhsm d : Ω A/K Ω A/K of degree +, defned by d(a 0 da da q ) = da 0 da da q, changes (Ω, d) A/K nto a comlex (Ω A/K, d) s called the de Rham comlex of A and the cohomology algebra H DR (A) s called the de Rham cohomology of A over k Let R be a commutatve rng wth dentty Φ (R) ( 2) s defned by the followng exact sequence n [], Φ (R) K 2 (R[t]/(t )) K 2 (R[t]/(t )) The followng theorem s due to Bloch [5] Theorem 2 If R s a commutatve local F -algebra and s odd, then { Ω R/Z 0, mod, Φ (R) Ω R/Z R/Rr = m r, (, m) =, r Note that there are no formulas for Φ k (R) when k mod In fact t s very dffcult to determne them The followng result of Magurn [2] gves the structure of Ω FG/Z Theorem 22 Suose F s a fnte feld of characterstc, G s a fnte abelan grou, and ā,, ā r s an F -bass of G/G, then Ω FG/Z s a free FG-module wth bass da,, da r When R s a commutatve rng wth dentty and I s a radcal deal, K 2 (R, I) s the abelan grou whch has a resentaton wth generators the Denns Sten symbols a, b for every (a, b) R I I R and the followng relatons (D) a, b = b, a f a I; (D2) a, b + a, c = a, b + c abc f a I or b, c I; (D3) a, bc = ab, c + ac, b f a I Let R be a rng contanng R If a I and b R R, then the mage of a, b under the ma K 2 (R, I) K 2 ( R) s the Stenberg symbol { ab, b}; One can consult [3] to see more about Denns Sten symbols 3 Addng Z summands to G Suose F s a fnte feld of characterstc, G s a fnte abelan grou, and Z r = σ s a cyclc grou of order r Let A = F[G Z r ] Then there s a artal augmentaton ma ε : A F[G] sendng σ to ; the kernel of ε s I = ( σ )A Snce ε s a slt surjectve ma and K n are functors, we have a slt exact sequence whch s just a art of the long exact sequence n K -theory wth resect to the ar (A, I): K 2 (A, I) K 2 (A) K 2 (FG) So obvously K 2 (A) K 2 (FG) K 2 (A, I) From the somorhsms A FG[t]/(t r ) FG[t]/(t r ), t follows that K 2 (A, I) K 2 (FG[t]/(t r ), (t)); thus K 2 (F[G Z r ]) K 2 (FG) K 2 (FG[t]/(t r ), (t)) (3) Theorem 3 Suose F s a fnte feld of characterstc, G s a fnte abelan grou whose Sylow -subgrous s G, then K 2 (FG) s a fnte -grou annhlated by the exonent of G Proof Decomose G as the drect sum G H wth G P the Sylow -subgrou of G BY Maschke s Theorem, FH s a semsmle rng Then the Wedderburn Artn Theorem mles that FH F, where F s a fnte feld wth the same characterstc as F Now FG (FH)[G ] F G and K 2 (FG) K 2(F G ) Decomose G as a fnte drect sum of cyclc -grous G C l C l r, where l l r Let R j = F [C l C l j ], j r Usng (3) we get the followng somorhsm r K 2 (F G ) K 2 (R j [t]/(t l j+ ), (t)) K 2 (F C l ) j= The last summand vanshes; and K 2 (R j [t]/(t l j+ ), (t)) has exonent l j+ snce t s generated by a, b wth a (t), and l j+ a, b = a l j+ b lj+, b = 0, b = 0 snce a l j+ (t l j+ ) Thus K 2 (F G ) has exonent l r and accordngly K 2 (FG) s a -grou wth exonent l r

3 Y Gao, G Tang / Journal of Pure and Aled Algebra 23 (2009) When FG s not a local F -algebra, one cannot use Theorem 25 drectly to comute Φ (FG), In order to be more convenent n concrete calculaton, we artally extend Theorem 2 to the form we need Theorem 32 Suose F s a fnte feld of odd rme characterstc and G s a fnte abelan grou and R = FG Then Bloch s calculaton stll works for FG: { Ω FG/Z 0, mod, Φ (FG) Ω FG/Z R/Rr = m r, (, m) =, r Proof As n the roof of Theorem 3, FG j F jg, where G s the Sylow -subgrou of G and F j has the same characterstc as F Say G C l C lr = g g r By the defnton of Φ we have Φ (FG) Φ j (F j G ) Snce F j G s a local F -algebra, then by Theorem 2 { Ω F Φ (F j G ) j G /Z 0, mod, Ω F j G /Z F jg /(F j G ) r = m r (, m) =, r By Theorem 22, Ω FG/Z s a free FG-module wth bass dg,, dg r, Ω F j G /Z s a free F jg -module wth bass dg,, dg r, so Ω FG/Z j Ω F j G /Z as abelan grous Obvously FG/(FG) j (F jg )/(F j G ) r, the theorem now follows The followng theorem and corollary extend the results of Magurn [2] (Theorems 4 and 5) to the case when s an odd rme Theorem 33 Suose F s fnte feld wth f dmenson of the F -sace G/G Then elements, s an odd rme, G s a fnte abelan grou of order n, and r s the K 2 (F[G Z ]) K 2 (FG) Z f n ( r +( )r) Proof By (3) we have the followng somorhsm K 2 (F[G Z ]) K 2 (FG) K 2 (FG[t]/(t ), (t)) Snce a, b = a b, b, K 2 (FG[t]/(t ), (t)) s an elementary abelan -grou By the somorhsm F[G Z ] FG[Z ] FG[t]/(t ) and the exact sequence we have Φ (FG) K 2 (FG[t]/(t )) K 2 (FG[t]/(t )), 2, K 2 (FG[t]/(t ), (t)) = Φ (FG) =2 By Theorem 32, Φ (FG) Ω, FG/Z 2 <, Φ (FG) Ω FG/Z FG/(FG) Ω FG/Z s a free FG-module of rank r, so t has rank nfr as an F -vector sace The grou G has r th ower classes, so there are n( ) r elements of G that are not elements of G, hence FG/F[G ] has F -dmenson nf ( r ) Thus ) dm F (K 2 (FG[t]/(t ), (t))) = ( )nfr + nf ( r = nf ( r + ( )r ) The theorem now follows Corollary 34 Suose F s a fnte feld wth f elements, and G s a fnte abelan grou of order n wth -rank t, 2 -rank, then K 2 (FG) Z nf (t )(t )/ t Proof If = 2, the theorem s just Theorem 5 n [2]; If s an odd rme, reeated use of Theorem 33 yelds the result Snce the rocess has been shown n Theorem 5 n [2], we omt the detals Examle A drect use of Theorem 34 yelds K 2 (F 5 [Z 5 Z 5 Z 25 Z 6 ]) Z

4 204 Y Gao, G Tang / Journal of Pure and Aled Algebra 23 (2009) The order of K 2 (F q G) Suose F s a fnte feld of odd rme characterstc and G s a fnte abelan grou By the defnton of Φ (R) and the somorhsm F[G Z s] FG[t]/(t s ), we have s K 2 (F[G Z s]) = K 2 (FG) Φ (FG) =2 When mod, the order of Φ (FG) can be determned by Theorem 32 When mod, t s very dffcult to determne the recse order of Φ (FG) We have the followng commutatve dagram, let R = FG K 2 (R[t]/(t mr + ), (t)) f K 2 (R[t]/(t mr + )) K 2 (R) K 2 (R[t]/(t mr ), (t)) K 2 (R[t]/(t mr )) K 2 (R) We wll use the njectvty of f to determne whether a Denns Sten symbol s trval n K 2 (R[t]/(t mr + ), (t)) By (6) and (0) n [3], we have the followng exact sequences Ω R/Z ϕ m,r K 2 (R[t]/(t mr + ), (t)) K 2 (R[t]/(t mr ), (t)), where ϕ m,r (adb) = at mr, b So we have Φ m r +(FG) Ω FG/Z /Ker ϕ m,r If we can determne orders of all Ker ϕ m,r then we can get the order of K 2 (FG) By the facts n the roof n Theorem 3, we only need to deal wth the case when G s a fnte abelan -grou When R s a regular rng, essentally of fnte tye over a feld of ostve characterstc > 0, by Theorem 25 n [3], Ker ϕ m,r deends only on r and s the subgrou D r,r of Ω R/Z generated by {al da 0 l < r, a R} When R s not regular, for examle R = FG, where F s a fnte feld of characterstc and G s a fnte abelan -grou, by the comutatons n Lemma 0 n [3], Ker ϕ m,r stll contans D r,r but does not concde n general Here s an examle Examle 2 Suose R = F 3 [Z 3 Z 3 ], Z 3 Z 3 = σ τ Then by Theorem 22, Ω R/Z s the free R-module wth bass dσ, dτ Put m =, = 3, r =, we have Ω R/Z ϕ K 2 (R[t]/(t 4 ), (t)) f K 2 (R[t]/(t 4 )) By the defnton D,R = a l da 0 l < = da a R An easy comutaton shows that D,R s the F 3 -sace wth bass {dσ, dτ, σ dσ, τ dτ, τ dσ + σ dτ, 2σ τ dσ + σ 2 dτ, 2σ τ dτ + τ 2 dσ, τσ 2 dτ + σ τ 2 dσ } Obvously σ 2 dσ, τ 2 dτ D,R However f ϕ(σ 2 dσ ) = f ( σ 2 t 3, σ ) = { σ 3 t 3, σ } = {( t) 3, σ } = { t, σ 3 } = Smlarly, f ϕ(τ 2 dτ) = Snce f s an njectve ma, ϕ(σ 2 dσ ) = ϕ(τ 2 dτ) = Then σ 2 dσ, τ 2 dτ Ker ϕ Hence Ker ϕ D,R Theorem 4 Suose R = FG, F s a fnte feld of odd characterstc and G s a fnte abelan grou Let A = R[t]/(t mr + ) wth r, (m, ) = If a,, a s s the F -bass of G/G, A s s the free A-module wth generators da,, da s, then there s a ma K 2 (A, (t)) f 2 A (As ) Proof Frst we consder the test ma d log : K 2 (A, (t)) 2 A Ω A/Z a, b da db ab Let D : A Ω R/Z R A be defned by alyng the dervatve d : R Ω R/Z to each coeffcent, then accordng to Secton 2 of [4], there s an A-module somorhsm Ω A/Z Ω (Ω A/R R/Z R A) ( ) f df dt, Df t Then 2 Ω A A/Z 2 Ω (Ω A A/R A/R A(Ω R/Z R A)) 2(Ω A R/Z R A), and by Theorem 22, Ω R/Z s a free R-module wth generators da,, da r, so Ω R/Z R A s a free A-modules wth the same generators, now K 2 (A, (t)) d log 2 Ω π 3 A A/Z 2 (Ω A R/Z R A) ψ 2 A As The f = ψ π 3 d log s the ma we want to obtan

5 Y Gao, G Tang / Journal of Pure and Aled Algebra 23 (2009) Let R be as above, R Ω R/Z the algebra of dfferental forms over R, ( R Ω R/Z, d) the de Rham comlex of R, and H DR (R) the de Rham cohomology of R Corollary 42 Let A, R and f be as above, ϕ mr : Ω R/Z K 2(A, (t)), then the cocycle Z of the de Rham comlex ( Ω R R/Z, d) s equal to the Ker(f ϕ m,r ) Proof By the defnton, f ϕ m,r s the comoste of the followng mas Ω R/Z ϕ m,r K 2 (A, (t)) d log 2 Ω π 3 A A/Z 2 (Ω A R/Z R A) ψ 2 A (As ) where f ϕ m,r (adb) = ψ π 3 d log( at mr, b ) = ψ π 3 ( d(atmr ) db ) = ψ(t mr (da ) (db )) = t mr da db By abt mr Theorem 22, Ω R/Z s the free R-module wth bass da,, da s, hence Ω 2 R/Z s the free R-module wth bass da da j, < j s, and 2 A As s a free A-module wth the same generators Defne an R-homomorhsm g : Ω 2 R/Z 2 A As by g(da db) = t mr da db, obvously g s njectve The g d s a homomorhsm Ω R/Z 2 A As such that g d (adb) = t mr da db = f ϕ m,r (adb) Hence g d = f ϕ m,r Snce g s an njectve ma, we have Ker(f ϕ m,r ) = Ker(g d ) = Ker d = Z Theorem 43 Suose F s a fnte feld of odd rme characterstc, G s a fnte abelan -grou wth cyclc decomoston G = x x n, ord(x ) = l, n Let G = j x j Then H DR (FG) s an F-sace wth bass S = {gxj dx n, 0 j < l, j mod, g G } Proof If T = f α,,α n X α X α n n F [X,, X n ], the olynomal rng n X,, X n over F, the formal artal dervatve T s defned by X T = α f α,,α n X α X α X X α n n If x FG, then x = H(x,, x n ) for some olynomal H(X,, X n ) n F[X,, X n ], thus dx = n H = (x X,, x n )dx Ω FG/Z By Theorem 22, Ω FG/Z s a free FG-module wth bass dx,, dx n If v Ω FG/Z, then v = n H j (x,, x n )dx j, j= where H j (X,, X n ) F[X,, X n ], j =,, n Then dv = ( Hj H ) (x,, x n )dx dx j X <j X j Snce Ω 2 FG/Z s the free FG-module wth bass dx dx j, < j n, then dv = 0 f and only f H j X = H X j, < j n (4) We can wrte H (X,, X n ) n the followng H (X,, X n ) = l =0 Q (X 2,, X n )X, wth Q F[X 2,, X n ] Let w = l 2 mod ( + ) Q (x 2,, x n )x + Then v dw = H (x,, x n )dx + n H j (x,, x n )dx j, j=2 where H = Q Snce d(v dw) = dv ddw = 0, then v dw Z, by (4) we have H X j = H j, 2 j n X (42)

6 206 Y Gao, G Tang / Journal of Pure and Aled Algebra 23 (2009) Suose H j (X,, X n ) = l = Q j (X 2,, X n )X, j 2 then by (42) mod Q (X 2,, X n ) X j l X = Q j (X 2,, X n )X = By comarng the degree of X of the two olynomals above, we conclude that both sdes are equal to 0, so we have Q (X 2,, X n ) = P (X 2,, X n ), mod H j (X,, X n ) = P j (X, X 2,, X n ), j = 2,, n Now we have found v Z wth v = v n H DR (FG), and ( ) n v = P (x,, 2 x n )x dx + P j (x, x 2,, x n )dx j j=2 Now usng nducton and reeatng the above rocess we can eventually fnd v n Z such that v n = v n H DR and ( ) n v n = T j (x,, x,, x n )xj dx = j Obvously v n can be generated by S, S Z, S B = {0}, and S s an F-ndeendent set snce Ω FG s an F-sace wth bass {gdx g G, n} So S s an F-bass of H DR (FG) Corollary 44 Let F be as above, G an elementary abelan -grou wth ndeendent generators x,, x n Then H DR (FG) s an n-dmensonal F-vector sace wth bass {x dx n} Proof Snce G s an elementary abelan -grou, G Theorem 43 Prooston 45 Let F and G be as n Theorem 43 Then the coboundary B of F-vector sace =, =,, n Now the concluson follows mmedately from (Ω FG/Z, d ) has bass S = {dg g G G } as an Proof Suose x FG, then x = H(x,, x n ), where H(X,, X n ) s a olynomal n F[X,, X n ] Thus dx = n H = (x X,, x n )dx Hence dx = 0 f and only f H X = 0, n Ths mles H(X,, X n ) = H (X,, X n ) for some olynomal H (X,, X n ) If g,, g m G G, m = f dg = 0, f ( ) m F, that s d = f g = 0, so m f g = H (x,, x), n H F[X,, X n ] = We conclude that f = 0, m, S s an F-ndeendent set Obvously B s generated by S, now the rooston s roved Theorem 46 Suose F s a fnte feld of odd rme characterstc, G s an elementary abelan -grou wth ndeendent generators g,, g n Set ϕ m,r : Ω ( FG/Z K 2 FG[t]/(t m r + ), (t) ), (m, ) =, r Then Ker ϕ m,r has bass S = {dg, g dg g G {}, n} as an F-vector sace Proof By Theorem 43, Ker ϕ m,r S, where S s the F-vector sace generated by S By Lemma (0) n [3], {dg g G {}} Ker ϕ m,r Snce f : K 2 ( FG[t]/(t m r + ), (t) ) K 2 ( FG[t]/(t m r + ) ) s njectve and f ϕ(g dg ) = f ( g t mr, g ) = { t mr, g } = { t mr, } =, t mles that S Ker ϕ m,r Thus Ker ϕ m,r = S The ndeendence of S follows from Prooston 45 Theorem 47 Let F be a fnte feld of odd rme characterstc, G s an arbtrary fnte abelan -grou Let S = {a l da, g r d g 0 l < r, g G, g r = }, then S Ker ϕm,r

7 Y Gao, G Tang / Journal of Pure and Aled Algebra 23 (2009) Proof By Theorem 25 n [3], D r,fg = a l da a FG, 0 l < r Ker ϕ m,r Snce f : K 2 ( FG[t]/(t m r + ), (t) ) K 2 (FG[t]/(t mr + )) s njectve, and f ϕ m,r (g r dg) = f ( g r t mr, g ) = { g r t mr, g} = {( gt m ) r, g} = { gt m, g r } = { gt m, } = Thus g r dg Ker ϕ m,r, so S Ker ϕm,r Remark For an arbtrary fnte abelan -grou G, we guess that Ker ϕ m,r s generated by {a l da, g r dg a FG, 0 l < r, g G, g r = } By Theorem 46, ths s true when G s an elementary abelan -grou Acknowledgements Ths work was suorted by the Natonal Natural Scence Foundaton of Chna (No ) and Hundred Talent rogram of Chnese Academy of Scences References [] RK Denns, ME Keatng, MR Sten, Lower bounds for the order of Wh 2 (G), Math Ann 223 (976) [2] B Magurn, Exlct K 2 of some fnte grou rngs, J Pure Al Algebra 209 (2007) 80 9 [3] J Stenstra, On K 2 and K 3 of truncated olynomal rngs, n: Algebrac K -Theory (Evanston, 980), n: Lecture Notes n Math, vol 854, Srnger, Berln, 97 [4] L Roberts, S Geller, K 2 of some truncated olynomal rngs, n: Rng Theory Waterloo, n: Lecture Notes n Math, vol 734, Srnger Verlag, 978 [5] S Bloch, Algebrac K -theory and crystallne cohomology, Publ Math IHES 47 (977) [6] RK Denns, MR Sten, K 2 of dscrete valuaton rngs, Adv Math 8 (2) (975) [7] R Olver, Lower bounds for K to 2 Ẑ π and K 2 (Zπ), J Algebra 94 (2) (985) Further readng [] CS Seshadr, L oératon de Carter Alcatons Exosé 6 Sémnare C Chevalley, 4 ( ) [2] D Husemoller, Cyclc Homology, Srnger-Verlag, 99 [3] W Van der Kallen, J Stenstra, The relatve K 2 of truncated olynomal rngs, J Pure Al Algebra 34 (2 2) (984) [4] AL Borel, Lnear Algebrac Grous, WA Benjamn, New York, 969