Theorem 3

Let K be a commutative ring, A a K-algebra, and P an $(A,A)$-bimodule. Suppose that the $(A\otimes_KA^o)$-module A is projective. We then have $H^n(A,P) = 0$ for every integer $n\geqslant 1$.

We must prove that for every integer $n\geqslant 1$, every element $f$ of $C^n(A,P)$ with $\partial^n(f) = 0$ is of the form $\partial^{n-1}(g)$ for an element $g$ of $C^{n-1}(A,P)$. By Proposition 5, this is an immediate consequence of Proposition 6.