VỀ MỞ RỘNG PHÂN BẬC CỦA NHÓM PHẠM TRÙ BỆN

Header Page of 258 Bộ giáo dục đào tạo TRường đại học vinh - CH TH KIM PHNG V M RNG PHN BC CA NHểM PHM TR BN Luận án tiến sĩ toán học NGH AN - 2014 Footer Page of 258 Header Page of 258 Bộ giáo dục đào tạo TRường đại học vinh - CH TH KIM PHNG V M RNG PHN BC CA NHểM PHM TR BN Chuyờn ngnh: i s v Lý thuyt s Mó s: 62 46 01 04 Luận án tiến sĩ toán học Người hướng dẫn khoa học: PGS TS NGUYN TIN QUANG PGS TS NGễ S TNG NGH AN - 2014 Footer Page of 258 Header Page of 258 i LI CAM OAN Lun ỏn ny c hon thnh ti Trng i hc Vinh, di s hng dn ca PGS TS Nguyn Tin Quang v PGS TS Ngụ S Tựng Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca tụi v cỏc ng tỏc gi Cỏc kt qu lun ỏn l trung thc, c cỏc ng tỏc gi cho phộp s dng v cha tng c cụng b trc ú Tỏc gi Ch Th Kim Phng Footer Page of 258 Header Page of 258 ii LI CM N Lun ỏn ny c hon thnh di s hng dn ca PGS TS Nguyn Tin Quang v PGS TS Ngụ S Tựng Tỏc gi xin c by t lũng bit n sõu sc ti Thy Nguyn Tin Quang v Thy Ngụ S Tựng Tỏc gi xin cm n NCS Phm Th Cỳc v s cng tỏc vit bi bỏo chung v tho lun nhng bi toỏn cú liờn quan Trong quỏ trỡnh hon thnh lun ỏn, tỏc gi ó nhn c s quan tõm v gúp ý ca PGS TS Nguyn Thnh Quang, PGS TS Lờ Quc Hỏn, TS Nguyn Th Hng Loan, cỏc thnh viờn B mụn i s, Khoa S phm Toỏn hc, Trng i hc Vinh cựng cỏc nh khoa hc v bn bố ng nghip Tỏc gi xin chõn thnh cm n v nhng s giỳp quý bỏu ú Tỏc gi xin c gi li cm n ti: - Khoa S phm Toỏn hc v Phũng o to Sau i hc, Trng i hc Vinh, - Khoa Toỏn - ng dng, Trng i hc Si Gũn, - Khoa Toỏn hc, Trng i hc ng Thỏp, ó h tr v to mi iu kin thun li tỏc gi hon thnh nhim v ca mt nghiờn cu sinh Cui cựng, tỏc gi xin by t lũng bit n ti gia ỡnh v nhng ngi bn thõn thit ó luụn giỳp v ng viờn tỏc gi sut quỏ trỡnh hc Ch Th Kim Phng Footer Page of 258 Header Page of 258 MC LC Mc lc Mt s ký hiu c dựng lun ỏn Bng thut ng S mi liờn h gia cỏc khỏi nim M u Mt s kin thc chun b 1.1 Phm trự monoidal 16 16 1.2 Nhúm phm trự bn v phm trự Picard 1.3 Nhúm phm trự phõn bc 19 22 1.4 i ng iu ca cỏc -mụun 25 1.5 Nhúm phm trự phõn bc bn v phm trự Picard phõn bc 26 1.6 Kt lun ca Chng 29 H nhõn t cỏc phm trự Picard phõn bc 30 2.1 H nhõn t ly h t phm trự Picard 31 2.2 H nhõn t ly h t phm trự Picard (M, N, h) 35 2.3 M rng -mụun 41 2.4 Kt lun ca Chng 46 Mụun chộo bn v nhúm phm trự cht ch bn 47 3.1 Mụun chộo bn v nhúm phm trự cht ch bn 48 3.2 Mụun chộo aben v phm trự Picard cht ch 55 3.3 M rng aben kiu mụun chộo aben 59 3.4 Kt lun ca Chng 66 -mụun chộo bn v nhúm phm trự phõn bc cht ch bn 67 4.1 -mụun chộo bn v nhúm phm trự phõn bc cht ch bn 68 4.2 M rng -mụun kiu -mụun chộo aben 77 4.3 Kt lun ca Chng 84 Footer Page of 258 Header Page of 258 M rng nhúm ng bin v nhúm phm trự phõn bc cht ch 85 5.1 Nhúm phm trự phõn bc cht ch 5.2 Ht nhõn ng bin 86 87 5.3 Phõn lp cỏc m rng nhúm ng bin l m rng tõm 90 5.4 Hp thnh ca nhúm phm trự phõn bc vi -ng cu 93 5.5 Kt lun ca Chng 96 Kt lun chung 97 Danh mc cụng trỡnh liờn quan trc tip n lun ỏn 98 Ti liu tham kho 99 Ch mc Footer Page of 258 103 Header Page of 258 MT S Kí HIU C DNG TRONG LUN N Ký hiu AbCross BrCross n H,ab Ngha phm trự cỏc mụun chộo aben phm trự cỏc mụun chộo bn phm trự ri rc m rng tớch chộo ca h nhõn t F cỏc lp tng ng cỏc m rng nhúm ca A bi hm t monoidal nhúm phm trự phõn bc nhúm phm trự -phõn bc kiu (M, N, h) cỏc lp ng luõn ca cỏc hm t t C n C cỏc mi tờn t vt X n vt Y nhúm i ng iu aben th n ca nhúm nhúm i ng iu i xng th n ca nhúm nhúm i ng iu aben th n ca cỏc -mụun n H,s nhúm i ng iu i xng th n ca cỏc -mụun idX n Z,ab mi tờn ng nht ca vt X (-)mụun chộo bn (aben) cỏc mi tờn ca phm trự C cỏc vt ca phm trự C nhúm phm trự bn nhúm phm trự bn kiu (M, N, h) nhúm phm trự phõn bc bn cỏc lp vt ng cu ca phm trự C cỏc t mi tờn ca vt n v I phm trự thu gn ca phm trự P phm trự thu gn nhúm cỏc n-i chu trỡnh aben ca nhúm nhúm cỏc n-i chu trỡnh aben ca cỏc -mụun n Z,s nhúm cỏc n-i chu trỡnh i xng ca cỏc -mụun Zsn nhúm cỏc n-i chu trỡnh i xng ca nhúm kt thỳc chng minh Dis M (F) Ext(, A) (F, F , F ) G (M, N, h) Hom[C, C ] Hom(X, Y ) n Hab Hsn M Mor(C) Ob(C) P (M, N, h) P (C) (C) = Aut(I) P(h) Red N n Zab Footer Page of 258 Header Page of 258 BNG THUT NG Ting Vit cn tr nh lý phõn lp i ng iu i xng -mụun chộo -mụun chộo aben -mụun chộo bn -mụun chộo i xng gi hm t hm t monoidal hm t monoidal i xng ht nhõn ng bin h nhõn t lý thuyt cn tr lý thuyt Schreier mụun chộo mụun chộo aben mụun chộo bn mụun chộo i xng mụun chộo ng bin mụun chộo ng bin aben mụun chộo ng bin bn mụun chộo ng bin i xng m rng -mụun m rng nhúm ng bin m rng tõm nhúm phm trự nhúm phm trự bn nhúm phm trự phõn bc bn nhúm phm trự cht ch nhúm phm trự i xng Footer Page of 258 Ting Anh obstruction classification theorem symmetric cohomology -crossed module abelian -crossed module braided -crossed module symmetric -crossed module pseudofunctor monoidal functor symmetric monoidal functor equivariant kernel factor set obstruction theory Schreier theory crossed module abelian crossed module braided crossed module symmetric crossed module equivariant crossed module abelian equivariant crossed module braided equivariant crossed module symmetric equivariant crossed module -module extension equivariant group extension central extension categorical group braided categorical group braided graded categorical group strict categorical group symmetric categorical group Header Page of 258 nhúm phm trự phõn bc i xng nhúm phm trự phõn bc nhúm phm trự phõn bc cht ch nhúm phm trự phõn bc cht ch bn phm trự monoidal phm trự monoidal i xng phm trự Picard phm trự Picard cht ch phm trự Picard phõn bc phm trự Picard phõn bc cht ch phm trự tenx bn phộp bin i t nhiờn rng buc rng buc n v rng buc giao hoỏn rng buc kt hp tớch chộo tng ng monoidal Footer Page of 258 symmetric graded categorical group graded categorical group strict graded categorical group braided strict graded cate-group monoidal category symmetric monoidal category Picard category strict Picard category graded Picard category strict graded Picard category braided tensor category natural transformation constraint unit constraint commutativity constraint associativity constraint crossed product monoidal equivalence Header Page 10 of 258 S MI LIấN H GIA CC KHI NIM Nhúm phm trự phõn bc bn o o Nhúm phm trự bn Nhúm phm trự bn o O ? o Phm trự Picard ? _ Phm trự Picard phõn bc ?_ ?_ Nhúm phm trự o cht ch O bn ? o ? _ Phm trự Picard Phm trự Picard / Mụun chộo bn O ? / Mụun chộo aben cht ch M rng aben o ? _ M rng aben kiu mụun chộo aben Nhúm phm trự phõn bc bn o O ? Phm trự Picard phõn bc o Nhúm phm trự phõn ? _ bc chtO ch bn o / -mụun chộo bn O ? ? _ Phm trự Picard o ? / -mụun chộo aben phõn bc cht ch M rng -mụun o ? _ M rng -mụun kiu -mụun chộo aben Footer Page 10 of 258 Header Page 94 of 258 90 Do ú ta cú (p h)(x, y, ) + f (x, y) + f (xy, ) = f (x, ) + x f (y, )f (x, y) (5.2.2) iu ny chng t (p h)(x, y, ) = k(x, y, ) Cui cựng, ta da vo tớnh hm t ca H chng minh (p h)(x, , ) = (0,) (0, ) k(x, , ) Vi hp thnh r s t, ta cú H[(0, ) (0, )] (1.3.1) = H[(h(r, , ), )] = H[h(r, , ), 1) (0, )] = H[(h(r, , ), 1)] H[(0, )] = (h(r, , ), 1) (g(r, ), ) (5.1.1) = (g(r, ) + h(r, , ), ) Mt khỏc, (g(r,),) (g(s, ), ) H(0, ) H(0, ) = H(r) H(s) H(t) (5.1.1) = ( g(r,)+g(r, ), ) H(r) H(t) Hn na vỡ h(r, , ) ZG v tớnh hm t ca H nờn h(r, , ) + g(r, ) = g(r, ) + g(r, ), ngha l h(r, , ) = g Vỡ vy ta c (p h)(x, , ) = h(px, , ) = g(px, , ) = f (x, , ) (5.2.3) = k(x, , ) nh lý ó c chng minh 5.3 Phõn lp cỏc m rng nhúm ng bin l m rng tõm Trong mc ny, chỳng tụi phõn lp cỏc m rng nhúm -ng bin A vi A ZE qua cỏc t ng cu ca -nhúm phm trự E (, A, 0) trỡnh by kt qu ny, ta ký hiu Extc (, A) l cỏc lp tng ng ca cỏc m rng nhúm -ng bin A Footer Page 94 of 258 E vi A ZE Header Page 95 of 258 91 5.3.1 nh lý (Lý thuyt Schreier cho cỏc m rng tõm ca cỏc nhúm ng bin) Gi s l -nhúm v A l -mụun -ng bin Khi ú tn ti song ỏnh Extc (, A) Endid (, A, 0) , ú Endid phõn bc (F, F ) t (, A, 0) l cỏc lp ng luõn ca cỏc hm t monoidal (, A, 0) n chớnh nú tha F (x) = x, x , F (b, 1) = (b, 1), b A Chng minh Gi s (F, F ) Endid (, A, 0) Khi ú (F, F ) xỏc nh hm : ( ì ) ( ì ) A vi ((x, y), 1) = Fx,y , (0,) ((x, ), ) = F (x x), ú (x, ) = Tớnh tng thớch ca (F, F ) vi cỏc rng buc kộo theo (x, 1) = = (1, y) v (x, y) + (xy, z) = x((y, z)) + (x, yz) (5.3.1) Tớnh t nhiờn ca F v tớnh hm t ca F ln lt suy (x, y) + (xy, ) = (x, ) + (x)(y, ) + (x, y), (5.3.2) (x, ) = (x, ) + ( x, ), (5.3.3) vi mi x, y, z G, , T hm ta dng c tớch chộo E = A ì vi phộp toỏn (a, x) + (b, y) = (a + xb + (x, y), xy) H thc (5.3.1) v tớnh chun tc ca hm suy cu trỳc nhúm ca E Cỏc h thc (5.3.2) v (5.3.3) m bo rng E l -nhúm vi -tỏc ng (a, x) = (a + (x, ), x) Khi ú tn ti dóy khp i q 0A E Footer Page 95 of 258 Header Page 96 of 258 92 vi i(a) = (a, 1) v q(a, x) = x Hn na, d thy A ZE Ngc li, gi s ta cú m rng ng bin q i 0A E vi A ZE Khi ú vi mi x , ta chn phn t i din ux E vi u1 = H i din {ux } cm sinh hm chun tc , nhn giỏ tr A cho ux + uy = (x, y) + uxy , ux = (x, ) + ux Cỏc h thc ux + (uy + uz ) = (ux + uy ) + uz , (ux + uy ) = (ux ) + (uy ), (ux ) = ux ln lt kộo theo cỏc h thc (5.3.1)-(5.3.3) Do ú ta cú th xỏc nh hm t monoidal phõn bc (F, F ) Endid F x = x, (, A, 0) nh sau: F (b, 1) = (b, 1), (0,) F (x x) = ((x, ), ), Fx,y = ((x, y), 1) Hn na, mi m rng tõm E tng ng vi mt m rng tớch chộo E liờn kt vi hm t monoidal phõn bc (F, F ) bi ng cu : a + ux (a, x) Tip theo ta s chng minh hai m rng thuc Extc (, A) tng ng v ch hai hm t monoidal phõn bc tng ng ng luõn Gi s F, F : (, A, 0) (, A, 0) l hai hm t monoidal phõn bc ng luõn bi ng luõn : F F c xỏc nh (e(x),1) x = (x x), x vi e : A Vỡ l mt ng luõn v cỏc h thc (1.3.1), (1.3.2) nờn e(1) = 0, (5.3.4) (x, ) + e(x) = e(x) + (x, ), (5.3.5) (x, y) + e(xy) = e(x) + xe(y) + (x, y) (5.3.6) vi x, y v Khi ú tng ng : E E (a, x) (a + e(x), x) Footer Page 96 of 258 Header Page 97 of 258 93 l mt tng ng gia hai m rng tõm nu v ch nu cỏc iu kin (5.3.4)(5.3.6) c tha Mt khỏc, nu : E E l mt ng cu thỡ (a, x) = (a + e(x), x), ú e : A l mt hm tha e(1) = Do vy, t cỏc lp lun nh trờn ta cú x = (e(x), 1) l mt ng luõn ca F v F 5.4 Hp thnh ca nhúm phm trự phõn bc vi -ng cu Theo S MacLane [26, tr 113], vi ng cu nhúm : v m rng i q E:0A B 1, ú A l nhúm aben thỡ tn ti m rng E ca A bi cho E = E Trong nh lý sau õy, chỳng tụi xột bi toỏn tng t cho trng hp nhúm phm trự phõn bc cht ch 5.4.1 nh lý Gi s H l mt nhúm phm trự phõn bc cht ch vi cỏc bt bin , C , h v p : l mt ng cu nhúm ng bin Khi ú tn ti nhúm phm trự phõn bc cht ch G tng ng vi ( , C, h ), ú C l -mụun vi tỏc ng xc = p(x)c vi x , c C v h = p h Chng minh Ta xõy dng nhúm phm trự phõn bc cht ch G nh sau Tp vt ca G l Ob(G) = {(x, X)| x , X p(x)} Mt -mi tờn (x, X) (x, Y ) l mt b ba (x, u, ), ú u : X Y l (x,u,) (x,v, ) -mi tờn H Hp thnh ca hai mi tờn (x, X) (x, Y ) (x, Z) l (x, v, ) (x, u, ) = (x, v u, ) Tớch tenx trờn cỏc vt v cỏc mi tờn ca G c xỏc nh (x, X) (y, Y ) = (xy, X Y ), Footer Page 97 of 258 Header Page 98 of 258 94 (x, u, ) (y, v, ) = (xy, u v, ) i vi mi tờn (x, u, ) G , ta cú (x, u, )1 = (x, u1 , ) Hai hm t phõn bc gr : G v I : G ln lt cho bi (x, u, ) , (1,idI ,) ((1, I) (1, I)) Vt n v ca G l (1, I) vi I l vt n v ca H Cỏc rng buc kt hp v rng buc n v l cỏc ng nht Gi s H cú h nhõn t chớnh qui {FH , } Khi ú G cú h nhõn t chớnh qui {FG , } vi FG (x, X) = (x, FH X), FG (x, u, ) = (x, FH (u), ) Vỡ vy G l mt nhúm phm trự phõn bc cht ch Tip theo ta s chng minh G tng ng vi nhúm phm trự phõn bc cht ch ( , C, h ) Ta xỏc nh cp ỏnh x (, f ), ú : (G) , [(x, X)] x, f : (G) (H) = C (1, c, ) (c, ) Khi ú l mt ng cu ca cỏc -nhúm v f l mt ng cu -mụun -ng bin, ú C cú cu trỳc -mụun bi xc = p(x)c, x , c C Hm t monoidal phõn bc (F, F ) : G H c cho bi F (x, X) = X, F (x, u, ) = (u, ), F = id Khi ú (F, F ) cm sinh hm t monoidal phõn bc (, ) : G(hG ) H(h) vi H(h) = (, C, h) v ú [(x, X)] = F0 [(x, X)] = [F (x, X)] = [X] = p(x) = p[(x, X)], (1, c, ) = F1 (1, c, ) = F1(1,I) F (1, c, ) = I1 (c, ) = (c, ) = f (1, c, ), ú (c, ) l mt mi tờn H(h) Vỡ vy (, ) l mt hm t kiu (p, f ) Footer Page 98 of 258 Header Page 99 of 258 95 Gi s hG Z3 (0 (G), (G)) Theo [11, nh lý 3.2], cn tr ca cp (p, f ) phi trit tiờu H (0 G, H) = H (0 G, C), ngha l (p) h = f hG + Nu ta t h = f hG thỡ cp J = (, f ) v J = id lp thnh mt hm t monoidal phõn bc t G(hG ) n J = ( , C, h ) Khi ú hp thnh (G,G) (J,J) G G(hG ) J l mt tng ng t G n J Cui cựng, ta chng minh rng h thuc cựng lp i ng iu vi p h Tht vy, gi s K = (1 , f ) : J G(hG ) Khi ú K cựng vi K = id l mt hm t monoidal phõn bc v hp thnh (, ) (K, K) : J H(h) l mt hm t monoidal phõn bc lm cho biu sau giao hoỏn G(hG ) H(h) K K J Rừ rng K l mt hm t monoidal phõn bc kiu (p, id) v vỡ vy cn tr ca K trit tiờu Do ú ta cú p h h = g, ngha l h = p h 5.4.2 Nhn xột (i) Trong trng hp c bit = 1, t nh lý 5.4.1 ta thu c Mnh 14 [35] (ii) Nhúm phm trự phõn bc G c gi l nhúm phm trự phõn bc hp thnh ca nhúm phm trự phõn bc H vi -ng cu p v ký hiu l G = H p Theo nh lý 5.4.1, G id0 G = G, G (p p ) = (G p) p nờn G l mt hm t phn bin theo bin (G) c nh (G) Footer Page 99 of 258 Header Page 100 of 258 96 5.5 Kt lun ca Chng Trong chng ny, lun ỏn ó gii quyt c nhng sau: - Ch mi liờn h gia bt bin th ba ca nhúm phm trự phõn bc cht ch Hol G vi cn tr ca ht nhõn ng bin; - Trỡnh by lý thuyt Schreier cho cỏc m rng nhúm ng bin ca A bi l m rng tõm nh vo cỏc t hm t monoidal phõn bc ca nhúm phm trự phõn bc cht ch (, A, 0); - Xõy dng nhúm phm trự phõn bc cht ch t mt nhúm phm trự phõn bc cht ch vi mt -ng cu Footer Page 100 of 258 Header Page 101 of 258 97 KT LUN CHUNG Lun ỏn ó thu c cỏc kt qu chớnh sau õy: - a mt cỏch tip cn mi i vi cỏc phm trự Picard phõn bc thụng qua khỏi nim gi hm t (h nhõn t) gii thớch nhúm i ng iu i xng th v th ca cỏc -mụun T ú, thu li c kt qu v s phõn lp cỏc phm trự Picard phõn bc ca A M Cegarra v E Khmaladze, v thu c kt qu mi v phõn lp cỏc m rng -mụun; - Xõy dng tng ng phm trự gia phm trự cỏc mụun chộo bn vi phm trự cỏc nhúm phm trự cht ch bn; Xõy dng tng ng phm trự cho phm trự cỏc mụun chộo aben v phm trự cỏc phm trự Picard cht ch; - a nh ngha nhúm phm trự phõn bc cht ch bn v xõy dng tng ng phm trự gia phm trự cỏc nhúm phm trự phõn bc cht ch bn vi phm trự cỏc -mụun chộo bn; - Phỏt biu v gii bi toỏn m rng aben kiu mụun chộo aben v bi toỏn m rng -mụun kiu -mụun chộo aben; - ng dng nhúm phm trự phõn bc cht ch vo bi toỏn phõn lp cỏc m rng nhúm ng bin l m rng tõm Footer Page 101 of 258 Header Page 102 of 258 98 DANH MC CễNG TRèNH LIấN QUAN TRC TIP N LUN N N T Quang, P T Cuc and C T K Phung (2013), Factor sets in graded Picard categories, Universal Journal of Mathematics and Mathematical 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Page 107 of 258 CH MC Zs2 (Coker d, Ker d ), 57 (, A, h), 24 (F, F , F ), 17 P, 19, 20 (C, gr), 23 P(h), 20 BrCross, 55 E , 60 BrGr , 55 F , 31 n (M, N ), 25 H,ab M, 48 F , 33 (P), 20 Dis M , 41 (P), 20 Extc (, A), 90 h, 64 ExtZ (M, N ), 41 gr, 23 , 23 AbCross, 58 -mụun chộo, 87 BrCross, 53 i xng, 68 BrGr , 54 aben, 77 PiGr , 55 bn, 68 Picstr, 58 -phõn bc, 23 BrCross, n nh, 23 BrGr , 75 76 Hom(,f ) [S, S ], 29 Hom(,f ) [S, S ], 22 Hom,s [Dis M, Red N ], 42 ng cu mụun chộo Obs(p), 87 ng bin bn, 72 Red N , 42 bn, 51 AbCross , 59 Picstr , 59 SymCross, 55 PiGr , bn, 19 cn tr 77 SymCross, ng luõn, 18 ca hm t, 22 77 ca ht nhõn ng bin, 87 (, A, 0) , 90 ca mt hm t phõn bc, 28 103 Footer Page 107 of 258 Header Page 108 of 258 104 hm t kiu (, f ), 21 cht ch bn liờn kt, 49 hm t monoidal, 17 cht ch i xng, 49 i xng, 19 cht ch bn, 49 phõn bc, 23 phõn bc, 24 phõn bc i xng, 27 thu gn, 25 ht nhõn ng bin, 87 phõn bc bn thu gn, 28 h i din, 21 phõn bc cht ch, 86 h nhõn t i xng, 31 phõn bc cht ch bn, 69 chớnh qui, 69 khỏ cht ch, 32 hai m rng tng tng, 41 m rng -mụun, 41 kiu -mụun chộo aben, 78 m rng aben kiu mụun chộo aben, 60 phm trự monoidal, 16 phõn bc bn, 27 bn, 19 phõn bc, 23 phõn bc i xng, 27 phm trự phõn bc, 23 phm trự Picard, 19 m rng phõn bc, 27 cht ch, 56 m rng tớch chộo cht ch liờn kt, 56 liờn kt, 61 mụun chộo, 48 ng bin i xng, 68 ng bin bn, 68 i xng, 49 ng bin, 87 aben, 56 aben liờn kt, 56 bn, 49 bn liờn kt, 50 nhúm phm trự, 18 phõn bc bn, 27 i xng, 19 bn, 19 bn thu gn, 20 Footer Page 108 of 258 phõn bc, 27 phõn bc thu gn, 28 thu gn, 21 rng buc n v, 17 giao hoỏn, 19 kt hp, 17 tng ng monoidal, 18 phõn bc, 24