These are notes on van den Bergh’s analogue of Poincar´e duality in Hochschild (co)homology
[VdB98]. They are based on survey talks that I gave in 2006 in G¨ottingen, Cambridge
and Warsaw and consist of an elementary explanation of the proof in terms of Ischebeck’s
spectral sequence [Isch69] and a detailed discussion of the commutative case, plus some
motivating background material. The reader is assumed to be familiar with standard homological
algebra, but the commutative algebra and algebraic geometry needed to understand
the commutative case is recalled
