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 no...

For t in Nn, E.Miller has defined a category of t-determined modules over the polynomial ring S in n variables. We consider the Auslander-Reiten translate, Nt, on the (derived) category of such modules. A monomial ideal I is t-determined if every generator x has a ≤ t. We compute the multigraded cohomologyand betti spaces of N k t (S/I) for every iterate k, and also the Smodule structure of the...

Abstract In this paper, we survey the theory of Cartwright–Sturmfels ideals. These are $\mathbb {Z}^{n}$ ℤ n -graded ideals, whose multigraded generic initial ideal is radical. ideals have surprising properties, mostly stemming from fact that thei...

We investigate the minimal number of generators μ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely divisorial ideals I such that μ(I) ≤ C. It follows that there...

Recent work has discovered perhaps unexpectedly rich structures for base loci and asymptotic invariants on the cone of big divisors of smooth complex projective varieties. One may ask what sorts of cones and functions can occur. In general this question is not currently well-understood. However, a key feature of this work (implicit in [7] and explicit in [3]) is that the structures depend on al...