SOME EXACT SEQUENCES FOR THE HOMOTOPY (BI-)MODULE OF A MONOID

For finitely presented monoids the homological finiteness conditions left-[Formula: see text], left-[Formula: see text], right-[Formula: see text] and right-[Formula: see text], the homotopical finiteness conditions of having finite derivation type [Formula: see text] and of being of finite homological type [Formula: see text] are developed and the relationship between these notions is investigated in detail. In particular, a result of Pride [40] and Guba and Sapir [27] on the exactness of a sequence of bimodules for the homotopy module is proved in a completely different, purely combinatorial manner. This proof is then translated into a proof of the corresponding result for the left homotopy module, thus giving new insights into the relationship between the finiteness conditions considered.