We construct free resolutions of compressed Artinian graded algebra quotients of polynomial rings and give a method to reduce them to a minimal resolutions. Our result generalizes results of El Khoury and Kustin for Gorenstein algebras of even socle degree with a different proof.

Then we use this to show current progress towards constructing dg-algebra structures in the Gorenstein case. For this we will discuss two general homological tools less known in the commutative algebra world, namely of transferring A∞ structures (and dg-algebra structures in nice situations) along homotopy equivalences and a tool for creating new homotopy equivalences from old ones.

This is joint work with Hamid Rahmati.