Conditions of multiplicity and applications for almost Gorenstein graded rings (Q6588204)

The authors consider Cohen-Macaulay local/graded rings \(A, B\) and \(R\) whose multiplicity and Cohen-Macaulay type satisfy the following conditions \N\[\Ne(\text{coker}(\varphi_R)) \ge e(B)\cdot e(\text{coker}(\varphi_A)) + e(A)\cdot e(\text{coker}(\varphi_B))\ \text{and}\ r(R) \le r(A)\cdot r(B),\N\]\Nhere \(\varphi_{*}\) is the monomorphism of degree 0 with a shift of their canonical module. They prove that if \(R\) is almost Gorenstein, then \(A, B\) and \(R\) are Gorenstein (see Theorem 3.2 for the local case and Theorem 3.3 for the graded case).\N\NApplications to tensor products of semi-standard graded rings and certain classes of affine semigroup rings -- specifically, numerical semigroup rings, edge rings, and stable set rings -- are discussed in Section~4.