Homological properties of rings defined by $n+1$ general quadrics in $n$ variables
We study the almost complete intersection ring $R$ defined by $n+1$ general quadrics in a polynomial ring in $n$ variables over a field $\sf{k}$ and a corresponding linked Gorenstein ring $A$. The overarching theme is that, while not Koszul (except for some small values of $n$), these rings have homological properties that extend those of Koszul rings. We establish that finitely generated modules over these rings have rational Poincaré series and we give concrete formulas for the Poincaré series of $\sf{k}$ over both $A$ and $R$. We also show that $A$ has minimal rate and its Yoneda algebra $\text{Ext}_A(\sf{k},\sf{k})$ is generated by its elements of degrees $1$ and $2$. While the graded Betti numbers of $R$ and $A$ over the polynomial ring are not known when $n$ is odd, our approach provides bounds and yields values for two of these Betti numbers, showing in particular that $R$ is level.