Acyclic ideals and Koszul complexes

Abstract

If A is a differential module, then the computation of its homology may frequently be simplified by finding a large acyclic submodule N, for then H(A)≅H( A N ) as modules, and hopefully A N is more tractable than A. The same idea works if A is a differential algebra, but in that case it is critical to factor out by an acyclic ideal I⊂ A, so that H(A)≅H( A I ) as algebras. This reduction technique in the classical (ungraded) case is used by Rees [7] and Tate [11], for example. I used a graded version in my thesis [8,9] to study the cohomology of two-stage Postnikov systems. Recently this Acyclic Ideal theorem has been used by Mann, May, Milgram and Sigaard [6] and there has also developed a body of work on the Koszul complex by Józefiak [2,3] and others in which this theorem fits naturally.