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 G 2n(X) = ^ 0 ^ P /Y\ _ Y 2n+1 . y „(2np+2)p G 2n+lW " X + Z k0 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
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