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so our task is to analyze Co Colonel of I star n minus 1 yeah this is the task we kept in front of us so to analyze it we first have to discuss free resolution so what is a free resolution of an abelian group so this is nothing but an exact sequence yeah so you form an exact sequence somehow of the following form yeah so this is this is an exact sequence of kernel is equal to image and thus the homology groups of the sequence are trivial here so this is the exact sequence so if you can form such an exact sequence and the construction of this exact sequence is given on page 194 of Hatcher in one of the proofs so if you have such an exact sequence this exact sequence ending an edge is called the free resolution of the abelian group edge and here the important thing is that each of these f is that is f 2 f 1 f 0 and so on are all of them are free now we do lies it yeah by applying the home so obviously we will have to reverse the arrows and what we get as a coach in complex now extremely