Crossed extensions of algebras and Hochschild cohomology (Q1605630)

\textit{J. Huebschmann} [Comment. Math. Helv. 55, 302-313 (1980; Zbl 0443.18019)] has shown that crossed \(n\)-fold extensions over a group \(G\) by a \(G\)-module \(M\) represent elements in the cohomology \(H^{n+1}(G,M)\). Given an algebra \(B\) and a \(B\)-module \(M\), the authors define \(n\)-fold extensions of \(B\) by \(M\) which represent elements in the Hochschild cohomology \(HH^{n+1}(B,M)\) for \(n\geq 2\). In particular, each chain (resp.\ cochain) algebra \(C\) yields a crossed module over the homology (resp.\ cohomology) algebra \(B=HC\) representing a characteristic class \(\langle C\rangle\) in the Hochschild cohomology \(HC\).