H-SPACES WITH TORSION
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  and 
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
Let X be an object in Uto/G with X-action 1.1.3. For each integer
n 0 set
G2n(X) = ^ 0 ^
G2n+lW " X + Zk0 X
Define a new graded structure on G^(X) by
G2n(X)k = X
x2n+1 , k = o
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  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
(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, .
(d) fiF(n) = F(n-l), for n 1.
1.1.10. We define F'(n) as the quotient