By Samuel; Moore, J.C. Eilenber

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**Extra resources for Foundations of Relative Homological Algebra**

**Sample text**

This yields the conclusion. We thus find that the so·called "double resolutions" of a complex considered in [4. Chapter XVII] are precisely the strongly projective resolutions. 4. Subcategories of Let Cf be an abelian category. Given - 00 cCf $ p and q $ 00 we consider the full subcategory Cq(j p of C(f determined by the complexes A with An n < p and for n > q. We usually omit the symbol p if P = - 00 and the symbol q if q = = 0 for 00 • Complexes A in which the differentiation dn : An - An - 1 is zero for all n € Z, determine a full subcategory cCf of cCf.

If E € CC(f) Cn (E), Zn (E), Z ~ (E), Bn (E), Hn (E) are exact for all n € A. Proof. The exact sequence o -> Zn (E) -> Cn (E) -> B n (E) -> 0 implies that Bn (E) is exact for every n € Z. The exact sequences o---. B n (E) -> Zn (E) -> Hn (E) -> 0, 0-+ Bn(E) -> Cn(E) -> Z~(E) -> 0 now yield the same conclusion for Hand Z'11 . 5. If the abelian category a is projectively perfect then the strong Ca. exact sequences form a projective class fi,s in Further, for any object A in C(f the following properties are equivalent.

2. 1. Let (f be an abelian category and ~ a full subcategory of C(1. Let M and N be subsets of Z such that 32 SAMUEL EILENBERG AND J. C. MOORE (I) If A € et, m € M and N € N, then Qm(A) and Rn(A) are in (11) If IACTl is a locally finite family of objects in Cet is in :D. N where nm C-m 1 &,] n [ nn Z-n I&,] :D. Further, for any object A in (i) A is :D :D :D. N -projective. :)RnHn(A]]. projective objects Em ,F n € (j, m € M, n € N such that Proof. 5) imply Qm --j Rn Cm --j Zn :(:D, et), :(:D, (1), m €M, n € N.