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BIBLIOGRAPHIC INFORMATION LOCAL ID AUTHOR ARTICLE AUTHOR Ravenel, Douglas
TITLE Conference on Homotopy Theory: Evanston, Ill., ARTICLE TITLE Dieudonne modules for abelian Hopf algebras 1974: [papers]/ IMPRINT Mexico, D.F. : Sociedad Matematica Mexicana, FORMAT Book 1975. EDITION ISBN VOLUME NUMBER DATE 1975 SERIES NOTE Notas de matematica y simposia ; nr. 1. PAGES 177-183
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NOTAS DE MATEMATICAS Y SIMPOSIA
NUMERO 1
COMITE EDITORIAL CONFERENCE ON HOMOTOPY THEORY
IGNACIO CANALS N. SAMUEL GITLER H.
FRANCISCO GONZALU ACUNA LUIS G. GOROSTIZA
Evanston, Illinois, 1974
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NOTAS DE MATEMATICAS Y SIMPOSIA Editado por Donald M. Davis
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SOCIEDAD MATEMATICA MEXICANA Mexico, D. F., 1975 ;.. V '1 ' "" A. 612 .7 C67 CON'rENIOO 1974
Lista de Participantes . r.:/,TH VII Lista de Colaboradores. VIII DAVIS, D.M.: Introducci6~ IX DAVIS, D.M.: Problemas . 1
CONFERENC IAS
BROWN, E.H., Jr. and PETERSON, F.P.: H* (MO) as an algebra over the Steenrod algebra .... 11 , . . f 8l+7 DAVIS, D.M.: T h e immersion conJecture or JR 21 GLOVER, H.H. and MISLIN, G.: V~ctor fields on 2-equivalent manifolds 29 JOHNSON, D.C., MILLER, H.R., WILSON, W.S. and ZAHLER,R.S.: Boundary homomorphisms in the generalized Adams spectral sequence and the nontriviality of infinitely many yt in stable homotopy .... 47 KAHN, D.S.: Homology of the Barratt-Eccles decomposition maps 60 LIULEVICIUS, A.: On the birth and death of elements in complex cobordism .•...... 78 MA1'H· MAHOWALD, M.: On the unstable J-homomorphism 99 STA'T, \.\?,RARY MAY, J.P.: Problems in infinite loop space theory 106 MILGRAM, R.J.: The Steenrod algebra and its dual for connective K-theory 127 1 MILLER, H.R. and WILSON, W.S.: On Novikov's Ext modulo an invariant prime ideal .. 159 MOORE, J.C.: An algebraic version of the incompressibility theorem of S. Weingram 167 RAVENEL, D.C.: Dieudonne modules for abelian Hopf algebras 177
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DIEUDONNE MODULES FOR ABELIAN HOPF ALGEBRAS Preliminary Report in Honor of SAMUEL EILENBERG
(*) Douglas c. Ravenel
By Abelian Hopf algebra we mean graded connected biassocia tive strictly bi-commutative Hopf algebra of finite type over a perfect field k of characteristic p. Let A denote the cate gory of such objects. ~ is known to be abelian ([12]) and our purpose here is to show that it is isomorphic to a certain cate gory of modules. An analogous theorem for the nongraded case was proved long ago by Dieudonne, and the modules that he used have been studied extensively (see [l], Chapter v, and [4]). I am grateful to Bill Singer for first bringing this work to my attention and suggesting the problem of carrying it over to the graded case.
The ring D in question is a noncommutative power series over W(k) (the Witt ring of k) in two variables F and V subject to the relations
FV = VF = p
Fw = w ( *) Research partially supported by N.S.F. 178 179 7 for w E W(k), where wcr denotes the action of the Frobenius established by showing that the endomorphism ring of H*(BU;k) automorphism of k lifted to W(k). acts canonically on any abelian Hopf algebra. Part (b) follows In our case we will obtain modules over a commutative graded from the fact that a Hopf algebra map sends primitives to primi ring E = W(k) [[F,V]]/(FV-p) where dim F = 1, dim V = -1.. F tives. Part (c) is trivial. will be seen to correspond to the Frobenius endomorphism of a We now construct a set of projective generators for TA. 1 Hopf algebra A which sends x EA to xP, while V corresponds Let be k[b ,b , ,bn1 with dim b, = i and coproduct Bn E ~ 1 2 ... l. to the dual of F, commonly known as the Verschiebung. .pb, = L b s®bt where bo = 1. Let w be the type 1 factor l. s+t=i n The relation between abelian Hopf algebras and E-modules of B It is a polynomial algebra k[w ,w , ... ,w J with pn O 1 n will be described in Theorem 3" below, which is our main result. dim w, = pi. The coproduct is obtained lifting to W(k) and l. Our first result is a decomposition theorem. m , m-i defining the Witt polynomials f (w) = p 1.wf o '.S_m '.S, n, to m L i=O Definition. Let n be an integer prime to p. An Abelian be primitive. Hopf algebra is of !Y£§_ n if each of its primitives and gener- i Theorem 2. w is a projective object in A and its dual some Let T Ac A denote the n ators has dimension np for i. n W* is therefore injective. full subcategory -,f l:yp2 n Abelian Hopf algebras. n Proof. Let S be the simple object k1x 7/xP, dim x r. Theorem 1.. There is a canonical categorical splitting r " r- r r Any Abelian Hopf algebra can be built up out of these simple A~ TT TA, n i.e. (n,p)=l- objects by multiple extensions, so it suffices to show a) Every Abelian Hopf algebra is canonically a direct 1 ExtA (w = O which is a simple calculation. n ,S) r \Jr, product of type n Abelian Hopf algebras. Now let Yi c T A denote the full subcategory whose objects b) There are no nontrivial maps between a type n Hopf 1 are the W. Let FW denote the category of contravariant algebra and a type m Hopf algebra for m ~ n. n - funccors from w to the category of finite W(k) modules. c) Moreover, T A T A n 1 n V This category is abelian. We define a functor Such a decomposition is well-known for the Hopf algebra D: T A-FW H*(BU;k) (see [3] for example) The general decomposition is - 1 - 180 181 by to the dual endomorphism, i.e. the Verschiebung. _!2(A) (Wn) Hom (W ,A) To make this more concise let E+ denote the where A is A n __Q 0 projective and A is polynomial. A 1 (If is not finitely gen- Now we can state our main result: erated, one can still construct and A and they need 0 Al but Theorem 3. The functor _!2 defined above is an equivalence not be of finite type). of abelian categories. This is a consequence of 2 The proof is analogous to that of Theorem V, §1,4.3 of [l]. Theorem 6. ExtA(B,A) O for all A iff B is polyno- Theorem 3 can be described in a more useful way by analyzing mial. the structure of W. Let V : w l<--w be the inclusion and n n- n We will conclude by identifying some well-known Hopf alge- let F W --w be defined by F (w.) WP Note that n n+l n n l i-1 bra functors wit~ standard functors from homological algebra. V F = F V p. Then we have n n-1 n n+l + It is convenient at this point to embed EO in ~, the full Lemma 4. The endomorphism ring of w is W(k) /pn+l and n category of graded E-modules and maps of all degrees. Hence for these endomorphisms along with the F and V generate all of M , N E E , Hom ( M, N) n n - E is also an E-module. Moreover, if N is the morphisms of W. nonnegative and M does not have any generators in positive dimensions then HomE(M,N) will also be nonnegatively graded. Hence Theorem 3 can be paraphrased as Define modules P = E/VE, R = E/FE. Theorem 3'. A type 1 Abelian Hopf algebra is characterized Theorem 7. Let AET A. Then Hom (P,C(A)) is isomorphic by a sequence of W(k) modules W (A) = Hom(W ,A) and maps 1 E n n to the abelian restricted Lie algebra of primitives of A (where F : W (A)--W (A) and V : W (A)-W (A) where n n n+ 1 n n n- 1 VnFn-1 F corresponds to the restriction), and Ext~(T,C(A)) is isomer F nv n+l p. phic to the abelian restrict Lie coalgebra (with v correspond If we identify f E W (A) with the element f(w ) EA, we decomposable A. n n ing to the corestriction) of elements of have (F f) (w ) = f (w ) p EA, i.e. F corresponds to the n n+ 1 n n 1 The functors ExtE ( P, C ( A) ) and HomE ( R, C ( A) ) are the Frobenius endomorphism of A, while V corresponds similarly n A A functors P and Q respectively defined in [6] and also in [SJ 182 183 §3. Hence an extension in T A induces six term exact 1963. 1 sequences relating these functors as was shown in [6]. (Note [5] J. Moore and L. Smith, "Hopf algebras and multiplicative 2 that ExtE (P, -) =Ext 2 (R,-) = 0). the con- fibrations", II, Amer. J. Math., 90(1968), 1113-1150. E It is evident that necting homomorphisms of these sequences must be E-module maps, [6] G.C. Wraith, "Abelian Hopf algebras", Journal of Algebra, i.e. they must preserve the restriction and corestriction res- §.(1967), 135-156. pectively. Hence the argument of 4.10 of [6] (which leads to contradictions of Theorems 2 and 4) is incorrect. COLUMBIA UNIVERSITY N.B. These results were also obtained by C. Schoeller, "Etude de la categorie des Algebres de Hopf commutatives Connexes sur un Corps", Manuscripta Math. l( 1970) , 133-155. References [11 M. Demazure and P. Gabriel, "Groupes Algebriques", North Holland, 1970. [2: V.K.A.M. Gugenheim, "Extensions of algebras, coalgebras, Hopf algebras", I and II, Amer. J. Math., 84 ( 1962), 349- 382. [31 D. Husemoller, "The structure of the Hopf algebra H*(BU) over a Z( Amer. Math., 93(1971), 329-349. - p )-algebra", J. - C4J Y.I. Manin, "The theory of commutative formal groups over fields of finite characteristic", Russian Math. Surveys,