Reports of the Midwest Category Seminar V

261 Pages·1971·6.59 MB·English

by  M. André, M. Barr, M. Bunge, A. Frei, J. W. Gray, P. A. Grillet, P. Leroux, F. E. J. Linton, J. MacDonald, P. Palmquist, P. B. Shay, F. Ulmer (auth.), J. W. Gray (eds.)

Lecture Notes in
Mathematics
A collection of informal reports and seminars
Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich
195
M. Andr6, M. Barr, M. Bunge, A. Frei,
J. W. Gray, P. A. Grillet, P. Leroux,
F. E. J. Linton, J. MacDonald,
P. Palmquist, P. B. Shay, F. Ulmer
Reports of the
Midwest Category Seminar V
Edited by
J. W. Gray, University of Illinois at Urbana-Champaign and
Forschungsinstitut for Mathematik, ETH Z0rich and
S. Mac Lane, University of Chicago
Springer-Verlag
Berlin. Heidelberg New York 19 71
AMS Subject Classifications (1970): 18 A xx, 18 C 15, 18 D 10, 18 E xx , 18 H 05
I S B N 3-540-05442-1 Spr inge r -Ver l ag Ber l in • H e i d e l b e r g • N e w York
I S B N 0-387-05442-1 Spr inge r -Ver l ag N e w Y o r k • H e i d e l b e r g - Ber l in
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TABLE OF CONTENTS
Michel
HOPF AND EILENBERG-MACLANE ALGEBRAS
Michel Andr6
Received June, 1970
The purpose of this note is to give a short proof of Cartan's
structure theorem on Eilenberg-MaeLane algebras, in using a structure
theorem on Hopf algebras with divided powers. The proof appearing
here follows Cartan's ideas with two improvements. On the one hand,
we do not use very mueh the multiplicative structures in the induc-
tive proof; on the other hand we can introduce the Eilenberg-MacLane
simplicial sets into the homological machinery more or less in any
form.
In characteristic 0, according to Milnor-Moore, a connected co-
commutative Hopf algebra is the enveloping algebra of a graded Lie
algebra. Dually a connected commutative Hopf algebra is the enveloping
ooalgebra of a graded Lie coalgebra. In characteristic p, that result
does not hold in general, but it does hold if the Hopf algebra has
divided powers (the comultiplication being a homomorphism of algebras
with divided powers).
Now let us consider a field K, an Eilenberg-MacLane space
K(~,n) and its singular homology H(~,n,K) . Actually H(~,n,K) is
a Hopf algebra with divided powers ; consequently H(~,n,K) is the
enveloping eoalgebra of a graded Lie coalgebra. Since the Hopf algebra
is oooommutative, the Lie coalgebra is abelian, in other words it is
a graded vector space. It remains to compute this graded vector space
depending on w, n and K .
The ground field K is fixed. Its characteristic is p ~ 0,2
For the case of characteristic 0 or 2, see the end of this note.
I. HOPF ALGEBRAS WITH DIVIDED POWERS
A Hopf algebra with divided powers is both an algebra with di-
vided powers and a Hopf algebra, the eomdltiplieation being a homo-
morphism of algebras with divided powers. For more details see [3]
for algebras with divided powers, [4] for Hopf algebras and [2] for
Hopf algebras with divided powers. The notion of a graded Lie coalge-
bra is dual to the notion of a graded Lie algebra. To a graded Lie
coalgebra L there corresponds an enveloping coalgebra U(L) which
is actually a Hopf algebra with divided powers.
Theorem i. Let H be a connected Hopf algebra with divided powers.
Then there is one and only one graded Lie coalgebra L (up to an
isomorphism) which appears in an isomorphism H ~ U(L) of Hopf alge-
bras with divided powers.
For the proof see [2] . That result can be rephrased in the
following way.
Theorem 2. Let H be the category of connected Hopf algebras with
divided powers and ~ the category of positively graded Lie coalge-
bras. Then the categories H and ~ are equivalent through U .
Actually we only need the abelian case of that result.
Theorem 3. Let C be a connected eocommutative Hopf algebra with
divided powers. Then there is one and only one graded vector space V
(up to an isomorphism) which appears in an isomorphism C ~ U(V) of
Hopf algebras with divided powers.
Theorem 4. Let C be the category of connected cocommutative Hopf
algebras with divided powers and ~ the category of positively
graded vector spaces. Then the categories ! and ~ are equivalent
through U .
In the abelian case there is an explicit description of the
Hopf algebras with divided powers U(V) On the one hand we define
E (x,2q-l) ~ U(V)
P
where the graded vector space V has exactly one generator x ,
appearing in degree 2q-i . We have
Ep(x,2q-l) : K.I + K.x
where x belongs to U2q_I(V) ; the multiplication maps x ® x onto
0 and the comultiplication maps x onto x ® i + i ® x . On the
other hand we define
P (y,2q) ~ U(V)
P
where the graded vector space V has exactly one generator y
appearing in degree 2q . We have
Pp(y,2q) = ~ K.y k
kZ0
where Yk belongs to U2kq(V) ; the multiplication maps Yi ® Yj
onto (i,j)yi+ J~ and the eomultiplication maps Yk onto E Yi ®
Y j;
i+j=k
the k-th divided power of Ym is equal to (m,m-l)(2m,m-l) ....
(m(k-l),m-l)Ymk
Proposition 5. Let V be a positively graded vector space with the
generators x i in degree 2qi-i (i ~ I) and the generators yj in
degree 2qj (j e J). Then there is a natural isomorphism of Hopf alge-
bras with divided powers U(V) = [ ® Ep(Xi,2qi-l) ]® [ ® Pp(yj,2qj)].
iEl j~J
Of the Hopf algebra with divided powers P (y,2q) we shall use
P
later essentially the algebra structure. Let us define the following
graded algebra
Qp(z,2q) : ~ K.z k
0Nk<p
where zk appears in degree 2kq ; the multiplication maps z i ® zj
onto zi+ j if i+j < p and onto 0 otherwise.
Lemma 6. There is a natural isomorphism of graded algebras
Pp(y,2q) ~ ® Qp (Zk,2pkq).
k~0
The element zk corresponds to the k+l-st divided power of y •
Let us study some functors of the category A of abelian groups.
Let
i:ZZ÷ ZZ /pZZ
be the homomorphism mapping i onto I mod p and let
j :Zg + ~ /pZ~ ,Zg /pnzz
n
be the homomorphism mapping i onto i mod pn and i mod p onto
n-i pn
p mod . By means of the structure theorem of the finitely gene-
rated abelian groups, the following result can be proved.
Lemma 7. Let F be a functor from the category of abelian groups to
the category of vector spaces over the field K of characteristic
p > 0 . Let us suppose that the functor satisfies the following con-
ditions :
i) the functor F is additive
F(~) + F(~') ~ F(~ + ~')
2) the functor F is union preserving
lim F(~.) ~ F(Uw.)
÷ l l
3) the homomorphism F(i) is a monomorphism
4) the homomorphism F(Jn) is an epimorphism for any n_>l