The Category of H-modules over a Spectrum

X A Q(f );

•• - T T (N(f.,---,f. )) — - 7T (N(f.,---,f. .)) —

z 1 i-l z+p. 1 i-l

* * z

+ P

.

( i

"

f

i ' - '

f

i " — v i

( N (

v - '

f

i - i - ••• •

6.1. Proposition. Let N be a convergent H-module over M. For

p.+v.

i = 1,2,• • • , represent x. e T T (M), p. 0, by f.:S -*M ,

l p . l — l v .

i l

v . 0. Assume t h a t p . - +°° and v . - +°° a s i - +°°. Then t h e r e i s a

l — l l

convergent H-module N(f°°) over M such that for each positive integer z,

there is a positive integer I(z) and an H-map

Q(I(z)): N(f ,..«,f ) -*N(f°°) which induces a TT^(M)-isomorphism

1 1 \ Z;

Q(I(z))#:

7r r^N f 1'"##'fI(z)))

• Trr(N(f»)), for r^z .

Let M denote any one of the Thorn spectra MO, MSO, MU, MSU, and MSp.

8.1. Proposition. M is a convergent H-module over itself.

Section 1 is purely preliminary. The category of compactly generated

spaces and maps is briefly discussed and some results about mapping spaces

for later use are proved. Spectra in the sense of G. W. Whitehead [9] and

related concepts are defined. To avoid unnecessary complications, we define a

spectrum M to be a sequence of spaces, •••/ M . . , M ., •••, with, for

4

each k e Z, a map a.. : S A M.. • M.. . . Let 4Z denote the set

4 k 4 k 4k+ 4

{...,-4,0,4,8,...}.

Section 2 introduces the category of H-modules over M, a ring

spectrum with pairing w: (M,M) - M. Let (R,q) denote the (k+1)-tuple

(r,,•• •,r, ,q) of integers in 4Z. Let M = M A«••A M ,

I k R rn r,

1 k

ZR = rn +•••+ r,, and a (R) = (r , _ . , • • • ,r . . . ) , where a e S (k) , the

1 k a(l) a(k)

group of permutations of {l,*..,k}. For V = (v ,•••,v .) , M , ZV,