Serre dimension of monoid algebras

Let R be a commutative Noetherian ring of dimension d, M a commutative cancellative torsion-free monoid of rank r and P a finitely generated projective RM]-module of rank t. Assume M is Φ-simplicial seminormal. If \(M\in \mathcal {C}({\Phi })\), then Serre dim RM]≤d. If r≤3, then Serre dim Rint(M)]≤d. If \(M\subset \mathbb {Z}_{+}^{2}\) is a normal monoid of rank 2, then Serre dim RM]≤d. Assume M is c-divisible, d=1 and t≥3. Then P?∧ ^t P⊕RM] ^t?1. Assume R is a uni-branched affine algebra over an algebraically closed field and d=1. Then P?∧ ^t P⊕RM] ^t?1.