q+v
The Category of H-modules over a Spectrum 25
Let (B,$) be a spectrum, v e 4Z. Define a spectrum (S A B,3')
by setting (S A B) = S A B and by defining
q+v q
6' : S4 A sP+V A B - SP+V A B
A
by
q+v q q+4
2
(S4 A S P + V A b) = 3 P + V A g(S4 A b).
1.19. Proposition. There is an isomorphism T: T T (B) - T T (S A B) .
q q+p
Q+r
Proof. Suppose x e T T (B) is represented by f: S - B . Then T(x)
p+v cr+r p+v
is represented by 1 A f: s^ A s^ - S^ A B . Let a be the least
p+v
non-negative integer such that a+p z 4Z. For x e T T (S A B)
q+p
q+p+v+z p+v -1
represented by g: S - S A B , T (x) is represented by
a q+p+v+z 1 A g a P+v 1 A ft 6
S
A
s^ ^ ^~ S A
SF
A B ^~ —z~+ B
z z+p+v+a
It is easily seen that T and T as defined are homomorphisms and that
T o T and T o T are the identity homomorphisms on T T (S A B)
q+p
and T T (B) , respectively.**
As a notational aid, we will reserve R and V for special use. R
will always denote a k-tuple (r , ,r ) of integers in 4Z, where k _ 0,
and V will always denote a j-tuple (v , »v.) of integers in 4Z, j _ 0.
By (R,q) we will mean (r,,••• ,r ,q) , where, if k = 0, (R,q) = (q) . If
M , •••, M are spaces, we denote M A»»«A M by M ; if k = 0,
ri rk ri rk R
M A x = X. We denote r +•••+ r by ER. The notation M and IV is
similar.
Let S(q) denote the group of permutations of a set containing q
elements. Given a e S(k) and R = (r ,«»«,r ), we denote
(r . .. ,r , _ ) by a(R) and M A---AM by M . . .
a(l) a
k)N
J r r
J
a (R)
a(l) a(k)
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