By J. F. Adams (auth.)
Read or Download Category Theory, Homology Theory and Their Applications III: Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, June 24–July, 19, 1968 Volume Three PDF
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Extra resources for Category Theory, Homology Theory and Their Applications III: Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, June 24–July, 19, 1968 Volume Three
W X 1 V " X 1 / ! II WI V WI X' 2 V X" . 2 "" - 35 We c a n n o w (i,i). ) setwo But comparison iso for r. It r e m a i n s sequences. a direct a "telescope" has ("To" of h o m o l o g i c a l X resolution comparison comparison finite struct 21 to the m a p a third sequences. these theorem all Lemma We obtain quence both apply - to d i s c u s s Given limit the p r o p e r t y convergence a resolution X~ of or i t e r a t e d the the of of t h e s e X, objects we Xr mapping-cylinder). (X r) = 0 .
N o v i k o v case s h o w s E approximate p < n, run any s y s t e m of f i n i t e H p (MU;MU* (S 0) ) i,: is E 17 . We h a v e to let limit 17 are s a t i s f i e d Therefore Again, to let ~ Hsn(i)K(Z2) i H* (MU;MU* (S O ) ) is trivia l. = sn; of P r o p o s i t i o n whose known run o v e r MO limit (iv) p E E directly. conditions MO ~ V sn(i)K(Z2) i Take and of c o u r s e A s s u m p - It is s u f f i c i e n t o v e r any s y s t e m of f i n i t e (iii) satisfied, verified E = K(Zp). spectrum.
E) > is projective this involves remarking E,(E) over which is known to be trivial In the cases spaces BU, BSp B_~O,(BSp) (vi), (vii)). sequence > MSp* (MSp) (see A s s u m p t i o n 20, Example (v)). we apply this argument to the to show that the modules are projective tive modules t E = BU, BO in the case that the spectral sequence is trivial, by duality with the spectral H* (MSp;MSp* (S O ) ) E, (S0) ; BU, (BU), (compare A s s u m p t i o n 20, examples We then remark that a direct limit of projecis flat.