Koszul almost complete intersections (Q1703220)

Let \(R = S/I\) be a quotient of a standard graded polynomial ring over a field. We assume that \(I\) is generated by quadrics. A paper of Avramov, Conca and Iyengar [\textit{L. L. Avramov} et al., Math. Res. Lett. 17, No. 2, 197--210 (2010; Zbl 1231.13012)] asked the following. If \(R\) is Koszul and \(I\) is minimally generated by \(m\) elements, do the Betti numbers satisfy \(\beta_i^S (R) \leq \binom{m}{i}\)? In particular, is \(\text{pd}_S R \leq m\)? This is known to hold in some special cases, but remains open in general. The current paper proves that it holds for Koszul \textit{almost complete intersections} (i.e. when \(I\) is generated by \(\text{ht } I +1\) elements) generated by any number of quadrics. This is a consequence of a stronger result, namely a complete classification of the structure of Koszul almost complete intersections.