The Category of H-modules over a Spectrum 11

d' = min{d(x_ ) , • • • ,d(x ,t ) }. 1,1t_

n n

Denote [p (f,g,a)]~ (U) by V, and let T T : X x i - x be the

projection. For x e T T (K), suppose (x,a) e V. Then there are an open

set A(x) ex, x e A(x), and d(x) 0 such that

[A(x)x(a-d(x),a+d(x) ) ] n (xxi) = v. If (x,a) £ V, then (x,a) ft YL, and

so there are an open set A(x) ex, x e A(x), and d(x) 0 such that

[A(x)x(a-d(x),a+d(x))] n K = $, the empty set. In either case, denote

A(x) x (a-d(x),a+d(x) ) by C(x). Since T T (K) X

a

c u C(x), there

xeTTx(K)

is a finite subcovering {C(x, ) , • • • ,C(x )} of TT, (K) X a. Let

1 m X

0 d" min{d (x ) , • • • ,d(x )}. Now suppose that x e T T (Kn[a-d",a+d"] ) .

Then (x,a) e T T (K) X a, and there is a C (x. ) , 1 _ i _ _ m, such that

(x,a) e C(x.). There exists z, a-d" _ z ^ a+d", such that (x,z) e K,

and so (x,z) e C(x.), and C(x.) n K ^ $. Hence

(xx[a-d",a+d"3) n (Xxi) c v.

Define a map k: xxi - Xxi by

k(x,t) = (x,0), for 0_t_a;

= (x, (t-a)/(l-a)), for a _ t _ 1.

For (x,t) e Kn(Xx[a,l]), g(k(x,t)) e U. There are an open set A(x,t) c x,

x e A(x,t), and d(x,t) 0 such that

[A(x,t) x (t-2d(x,t),t+2d(x,t)) ] n (Xxi) ck" (g~ (U)) . Denote

A(x,t) x (t-d(x,t),t+d(x,t) ) by D(x,t). Since K n (Xx[a,l]) is compact,

there is a finite set {(x_ft_),•••,(x ,t )} such that

1 1 p p

K n (Xx[a,l]) c D(x1-t ) u«--u D(xp,tp). Let

1

d = min{d(x ,t ) , • • • ,d(x ,t )}.