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The powers ${\mathfrak m}^n$ of the maximal ideal $\mathfrak m$ of a localNoetherian ring $R$ are known to satisfy certain homological properties forlarge values of $n$. For example, the homomorphism $R\to R/{\mathfrak m}^n$ isGolod for $n\gg 0$. We study when such properties hold for small values of $n$,and we make connections with the structure of the Yoneda Ext algebra, and moreprecisely with the property that the Yoneda algebra of $R$ is generated indegrees $1$ and $2$. A complete treatment of these properties is pursued in thecase of compressed Gorenstein local rings.

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