\(\mathcal K_2\) factors of Koszul algebras and applications to face rings. (Q1945892)

Let \(A\) be an \(\mathbb N\)-graded connected locally finite dimensional algebra over a field \(k\) generated by elements of degree 1. \(A\) is a \(\mathcal K_2\)-algebra if its Yoneda algebra \(\text{Ext}_A(k,k)\) is generated as \(k\)-algebra by homogeneous components of degree 1 and 2. Similarly one can define a graded \(\mathcal K_2\)-module over \(A\). Suppose that \(A\) is a Koszul algebra with a homogeneous ideal \(I\) and \(B=A/I\) acts trivially on \(\text{Ext}_A(B,k)\). Let \(I\) be \(\mathcal K_2\) as \(A\)-module. Then \(B\) is a \(\mathcal K_2\)-algebra. Suppose that \(\Delta\) is a simplicial complex and \(\Delta^*\) its Auslander dual. The algebra \(k[\Delta]\) is a \(\mathcal K_2\)-algebra if \(\Delta^*\) is (sequentially) Cohen-Macaulay.