Conditions for the Yoneda algebra of a local ring to be generated in low degrees

The powers mnmn of the maximal ideal mm of a local Noetherian ring R are known to satisfy certain homological properties for large values of n  . For example, the homomorphism R→R/mnR→R/mn is Golod for n≫0n≫0. We study when such properties hold for small values of n, and we make connections with the structure of the Yoneda Ext algebra, and more precisely with the property that the Yoneda algebra of R is generated in degrees 1 and 2. A complete treatment of these properties is pursued in the case of compressed Gorenstein local rings.