H-SPACES WITH TORSION

3

1.1.7. Horn (fiM,N) ^ Hom(M,£N).

This is routine using 1.1.6.

1.1.8. DEFINITION. Let £ = Z/pZ[X] where grade X = +1. In [3] and [20]

a regrading of objects in UFU/G is given such that the X-action of

1.1.3. is subsumed under the usual theory of graded modules over a

graded algebra.

Let X be an object in Uto/G with X-action 1.1.3. For each integer

n 0 set

G2n(X) = ^ 0 ^

P

/Y\ _

Y

2n+1 .

y

„(2np+2)p

G2n+lW " X + Zk0 X

Define a new graded structure on G^(X) by

G2n(X)k = X

2 n

^

k

x2n+1 , k = o

G 2n

+ 1

« =

x

(

2

n p

+

2 ) p -

k 5 1

.

Then with this regrading X is a graded module over the graded

algebra Z/pZ[X] in the usual sense. Let UFtl/X denote the result-

ing category and

A: UIT1/G • Uln/£

the functor which regards objects in UlTl/G as modules over £ via

the regrading process. Then we make the definition, M e obj Uft/G

is A-projective if A(M) is a projective in Urn/£. It is routine to

verify that M is A-projective if and only if the X-action is monic.

In this paper the term "A-projectiveM is used in place of the term

"free X-module" used in [20] and elsewhere.

1.1.9. PROJECTIVES IN UIU/G. Let F(n) = lU G/B(n) .

(b) F(n) is projective in Ulft/G. We write i for the

G-generator

(b) F(n) is A-projective

(c) Any projective in Ulft/G is expressible as a direct sum of

modules of the form F(n), with the decomposition unique up

to order, [20].

(d) fiF(n) = F(n-l), for n 1.

1.1.10. We define F'(n) as the quotient