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0 ! D ! U ! C / ! 0 were equivalent to a sequence 0 ! D ! f / ! C / ! 0 associated with a morphism f W C ! D. 3 it was noted that if 0 ! D ! U ! C / ! 0 splits at the module level then the sequence is equivalent to a mapping cone short exact sequence. R-Mod/, the condition that every morphism f W C ! C; D/ D 0. 5 The Koszul Complex In this section, we let R be a commutative ring and r 2 R. Let K be the complex r with the R’s in the 1st and 0th positions. 0! have the following. 1. K; D/ D 0 if and only if 0 !

C ! C 00 ! R-Mod/. 1. If f g 0 ! C 0 ! C ! C 00 ! R-Mod/, then the following are equivalent: a) f admits a retraction r W C ! C 0 (so r ı f D idC 0 ) b) g admits a section t W C 00 ! C (so g ı t D idC 00 ). g t We note that if 0 ! C 0 ! C ! C 00 ! 0 is split exact with a retraction r W C ! C 0 , then for each n 2 Z, fn gn 0 ! Cn0 ! Cn ! Cn00 ! 0 is split exact with a retraction rn W Cn ! Cn0 . It can happen that each 0 ! Cn0 ! Cn ! Cn00 ! 0 is split exact without 0 ! 0 C ! C ! C 00 ! 0 being split exact.

R-Mod/. 5. R-Mod/, argue that C is exact and that each term Ck of C has projective dimension at most n. 6. R-Mod/ with proj: dim C D C1. 7. Let l: gl: dim R Ä n < C1. R-Mod/ is exact, then proj: dim E Ä n. 8. –7. above for injective dimension of complexes. 9. M; N / for all n. NN // D 0 if k ¤ 0; 1. 10. R-Mod/, a morphism R ! C /. Given such a morphism, compute the pushout of the diagram  R  / R  C Argue that the pushout is a complex of the form d3 d0 ! C3 ! C2 ! C1 ˚ R ! C0 ! C d 1 1 ! C 2 !

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COBE (Explorer 66)


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