In the last talk, I discussed the minimal free resolution of the Apéry toric ideal. In this fifth talk, I will discuss that a minimal free resolution of Apéry toric ideals can be obtained as the tensor product of a minimal free resolution of defining toric ideals with a Koszul complex. Moreover, if numerical semigroups \(S\) and \(T\) lie in the interior of the same face of the Kunz cone \(C_m\), then their Apéry and defining toric ideals have identical Betti numbers.
