Differential algebras on semigroup algebras

Critical Modules of the Ring of Differential Operators of an Affine Semigroup Algebra

“…In this section, for a prime ideal p of F [s], we define a preorder [p] and an equivalence relation ∼ [p] , which are generalizations of those considered in (Musson, Van den Bergh, 1998;Saito, 2001;Saito, Traves, 2001;Saito, 2007) for maximal ideals (Proposition 3.3 and Remark 3.4). This equivalence relation plays a central role in this paper.…”

“…In (Saito, Traves, 2001, Definition 4.1.1) a preorder a b was defined by E τ (a) ⊆ E τ (b) for all faces τ of the cone R ≥0 A generated by A, where…”

“…Here ZC stands for taking Zariski closure in F d , and the last equivalence follows from (Saito, Traves, 2001, Lemma 4.1.4).…”

“…Let R A denote the semigroup algebra over F of the monoid generated by A, and D(R A ) the ring of differential operators of R A ; both of them have natural Z dgradings. Musson, Van den Bergh (1998) (Saito, Traves, 2001). ) In this paper, considering the Krull dimension of finitely generated Z d -graded D(R A )-modules, we characterize critical Z d -graded D(R A )-modules.…”

“…In (Saito, Traves, 2001), a preorder a b was defined; the equivalence relation induced from was used for the classifications of Z d -graded simple modules and A-hypergeometric systems (Musson, Van den Bergh, 1998;Saito, 2001;Saito, Traves, 2001;Saito, 2007). This preorder is for maximal ideals of the polynomial ring…”