Homological properties of rings defined by $n+1$ general quadrics in $n$ variables

TL;DR

This paper investigates the homological properties of rings defined by $n+1$ general quadrics in $n$ variables, revealing their rational Poincaré series, minimal rate, and Betti number bounds, extending properties of Koszul rings.

Contribution

It provides new insights into the homological behavior of these almost complete intersection rings, including explicit formulas and bounds for Poincaré series and Betti numbers.

Findings

Finitely generated modules have rational Poincaré series.

The ring $A$ has minimal rate and its Ext algebra is generated in degrees 1 and 2.

Bounds and specific values for Betti numbers of $R$ and $A$ are established.

Abstract

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 Poincar\'e series and we give concrete formulas for the Poincar\'e 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.