Surjectivity of multiplication and F -regularity of multigraded rings

Let R be a noetherian Z-graded integral domain. Then the subset Σ(R) := {λ ∈ Z | Rλ 6= 0} is a finitely generated subsemigroup of Z. We say that R is surjectively graded if for any λ, μ ∈ Σ(R), the product Rλ⊗R0 Rμ → Rλ+μ is surjective. This is essentially a generalization of the degree-one generation property of N-graded rings. The purpose of this paper is to study this property, mainly for normal domains. After that, we show that surjectively graded normal domains in positive characteristic behaves well with respect to strong F -regularity, utilizing the notion of global F -regularity defined and studied by Smith [17]. This approach gives yet another abstraction of beautiful ring theoretic properties of multicones over G/B, as in [9], [11]. In section 2, we review the definition and some basic properties of global F regularity (with an obvious generalization). This section is essentially a proper subset of [17]. A lemma on multicones will be used later. In section 3, we define and study the first property of surjectively graded rings. As is a multihomogeneous coordinate ring of a closed subscheme of a product of projective spaces, such a ring gives a projective variety. As combinations of the Segre and the Veronese embeddings give various embeddings of the same projective variety, we prove that lines in the ‘middle’ of graded part give homogeneous coordinate rings of the same projective variety. We can prove more for a normal surjectively graded algebra, and we discuss this in section 4. Such rings are given as multicones over the projective schemes given in section 3. As an application, we prove a criterion for strong F -regularity of surjectively graded algebras in section 5. Note that if the ring is N-graded, then much more has long been known [20, (3.4)]. A multicone over G/B is a typical example. Finally, we prove that a normal semigroup scheme in characteristic zero which admits a dominating semigroup homomorphism from a reductive group has at most rational singularities.