Free resolutions over commutative Koszul algebras

For $R=Q/J$ with $Q$ a commutative $\BN$-graded algebra over a field and $J\ne0$, we relate the slopes of the minimal resolutions of $R$ over $Q$ and of $k=R/R_{+}$ over $R$. When $Q$ and $R$ are Koszul and $J_1=0$ we prove $\Tor iQ{R}k_j=0$ for $j >2i\ge0$, and also for $j=2i$ when $i >\dim Q-\dim R$ and $\pd_QR$ is finite.

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Published 1 January 2010