Serre Dimension of Monoid Algebras

TL;DR

This paper investigates the Serre dimension of monoid algebras over Noetherian rings, establishing bounds and structural results for projective modules under various conditions on the monoid and the base ring.

Contribution

It provides new bounds on the Serre dimension of monoid algebras for specific classes of monoids and rings, extending classical results and offering structural decompositions of projective modules.

Findings

Serre dimension of R[M] is bounded by the dimension d under certain monoid conditions.

Projective modules over R[M] decompose as direct sums involving their top exterior power.

Results apply to monoids with normal, seminormal, or divisible properties, and to specific base rings.

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 $t$. $(1)$ Assume $M$ is $\Phi$-simplicial seminormal. $(i)$ If $M\in \CC(\Phi)$, then {\it Serre dim} $R[M]\leq d$. $(ii)$ If $r\leq 3$, then {\it Serre dim} $R[int(M)]\leq d$. $(2)$ If $M\subset \BZ_+^2$ is a normal monoid of rank $2$, then {\it Serre dim} $R[M]\leq d$. $(3)$ Assume $M$ is $c$-divisible, $d=1$ and $t\geq 3$. Then $P\cong \wedge^t P\op R[M]^{t-1}$. $(4)$ Assume $R$ is a uni-branched affine algebra over an algebraically closed field and $d=1$. Then $P\cong \wedge^t P\op R[M]^{t-1}$.