Serre dimension of monoid algebras

Abstract

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 R[M]-module of rank t. Assume M is Φ-simplicial seminormal. If $M\in \mathcal {C}({\Phi })$ , then Serre dim R[M]≤d. If r≤3, then Serre dim R[int(M)]≤d. If $M\subset \mathbb {Z}_{+}^{2}$ is a normal monoid of rank 2, then Serre dim R[M]≤d. Assume M is c-divisible, d=1 and t≥3. Then P≅∧ t P⊕R[M] t−1. Assume R is a uni-branched affine algebra over an algebraically closed field and d=1. Then P≅∧ t P⊕R[M] t−1.