Ms = 1) (for each 2) m-1 :::: m: Y -> 3) For each s, with the following properties. s. ) Y• the following diagram is homotopy commutative. J,. - > SM s-l If we have such a realization of a ladder, then we shall obtain induced maps of all our exact sequences, and hence a map of sp ctral sequences. duced by the maps AS. On El If we assume that we started with two resolutions, then the induced map on the induced map of Ext.

2. so that II Set up an exact sequence (S o r' So) ~ (So M) r+l' n 1T r > 0 43 for t - s > 0 Condition III is satisfied by. Settil~ and s > o. M. up the spectral sequence Supnose given an object Y and a sequence of order 2, Cs fS are free modules over A. suppose that it is a resolution. ,... ,... induces the map 8: C - > H" (Y). o In general, I don't assert that a sequence of order 2 has a realization: but if the sequence is a resolution, then it does. Viz. We can choose an Eilenberg-MacLane object H~:' (K ) = C o 0 8: Co - > H~:' (Y) • that and a map f o : Y -> K 0 Ko such inducing This follows from the last theorem of 44 las t lec tnre • ~"Je \ve form the "quotient" look at the exact sequence <- Ht (y) <-,- Ht (K) <--,:- Ht 01 f .