  Semi-infinite cohomology is constructed from scratch as the proper
generalization of finite dimensional Lie algebra cohomology. The differential d
and other operators are realized as universal inner deri- vations of a
completed algebra, which acts on any appropriate semi-infinite complex. In
particular, d is shown to be the unique derivation satisfying the "Cartan
identity" and certain natural degree conditions. The proof that d is
square-zero may well be the shortest (arguably, the only) one in print.
