Homology of the Shafarevich complex and noncommutative complete intersections

General properties of the Shafarevich complex construction are studied. They are used to provide a proof of the theorem which characterizes non-commutative complete intersections in terms of the homology algebras of Shafarevich complexes. This theorem is a non-commutative analogue of (a generalized version of) the Tate–Assmus theorem on commutative complete intersections.