By Robin Hartshorne, C. Musili

ISBN-10: 3540051848

ISBN-13: 9783540051848

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We shall also find it convenient to call a complex almost positive if A, = 0 for all n < - 1; in this case the map d o : A o+ A - 1 has a special role and is called the augmentation map, designated as E. To unify notation we shall retain do. There is a category Vumfi whose objects are the complexes and whose morphisms f: A + A‘ are Z-tuples (fi) of morphisms f ; : A i + A : for each i, such that the following diagram commutes: d”+ I . f”+. ---Ai+I ... __I, If” 4 + ,A:, d:, ,A : , - , - . - These morphisms f are called chain maps.

If K,(R,) = ([R,]) then K , ( R ) = ([R]). Before presenting the proof of this important result, we should note the same proof shows K , ( R ) % K,(R,). This result will be generalized further in appendix A. We follow Bass [68B] and start by noting various properties 20 Homology and Cohomology R,, an ideal of of graded modules, of independent interest. Let R+ = R/R+. 1. R. 34‘: If M E R-Y#-Mod and R+M = M then M = 0. (Indeed, otherwise, take n minimal such that M, # 0. 6. d. M = t < cc and M is graded then M has a projective resolution of length t in R-%-Aod.

Po fo ----+M-O 44 Proofi do:A, A, such that Homology snd Cobomoiogy -+ A _ , is epic so we can lift gf,: Po + A _ , to a map go: Po -+ = d,g,. , gf, = dogo. +l = 0, so gnfn+ 1 = 7 = d,+ 10 for some Q: P,+ 1 A, + and we are done taking g, + = c. D. -1 P = . 14, then apply T, and finally take the n-th homology map. , independent of the choice of projective resolution. 16: Suppose (A; (d,)) and (A'; (di)) are complexes. 17: Homotopic chain maps induce the same maps on the homology modules. Proofi Suppose g : A' -+ A and h: A' -+ A are homotopic.

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