Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring (R 0 , m 0 ). Then, the nth graded component H i R + (M) n of the ith local cohomology module of M with respect to the irrelevant ideal R + of R is a finitely generated R 0 -module which vanishes for all n 0. In various situations we show that, for an m 0 -primary ideal q 0 ⊆ R 0 , the multiplicityM) n is antipolynomial in n of degree less than i. In particular we consider the following three cases: (a) i < g(M), where g(M) is the so-called cohomological finite length dimension of M;In cases (a) and (b) we express the degree and the leading coefficient of the representing polynomial in terms of local cohomological data of M (e.g. the sheaf induced by M) on Proj(R).We also show that the lengths of the graded components of various graded submodules of H i R + (M) are antipolynomial of degree less than i and prove invariance results on these degrees.
In a finite-dimensional real vector space furnished with a rational structure with respect to a subfield of the field of real numbers, every (simplicial) rational semifan is contained in a complete (simplicial) rational semifan. In this paper this result is proved constructively on use of techniques from polyhedral geometry.Comment: minor correction
Geometric properties of schemes obtained by gluing algebras of monoids, including separation and finiteness properties, irreducibility, normality, catenarity, dimension, and Serre's properties (S_k) and (R_k), are investigated. This is used to show how the geometry of a toric scheme over an arbitrary base is influenced by the geometry of the base.Comment: to appear in J. Pure Appl. Algebra; corrected typos, updated reference
Let a be an ideal in a commutative ring R. For an R-module M , we consider the small a-torsion Γ a (M ) = {x ∈ M | ∃n ∈ AE : a n ⊆ (0 : R x)} and the large a-torsion Γ a (M ) = {x ∈ M | a ⊆ (0 : R x)}. This gives rise to two functors Γ a and Γ a that coincide if R is noetherian, but not in general. In this article, basic properties of as well as the relation between these two functors are studied, and several examples are presented, showing that some well-known properties of torsion functors over noetherian rings do not generalise to nonnoetherian rings.
Studying toric varieties from a scheme-theoretical point of view leads to toric schemes, i.e. "toric varieties over arbitrary base rings". It is shown how the base ring affects the geometry of a toric scheme. Moreover, generalisations of results by Cox and Mustaţǎ allow to describe quasicoherent sheaves on toric schemes in terms of graded modules. Finally, a toric version of the Serre-Grothendieck correspondence relates cohomology of quasicoherent sheaves on toric schemes to local cohomology of graded modules. From toric varieties to toric schemesDuring the last forty years a huge amount of work on toric varieties was and still is published. Their theory was generalised in several directions, and this often lead to a better understanding of classical toric varieties. However, the generalisation that seems to be the most natural and the most important -the study of toric varieties from a scheme-theoretical point of view -was never actually carried out. It is clear that to do this one has to be able to make arbitrary base changes. Hence, instead of considering toric varieties over an algebraically closed field (or, as often done, over the field of complex numbers), one needs to study toric schemes, that is "toric varieties over arbitrary base rings". Special cases of this generalisation were mentioned briefly in [3, §4] (for regular fans and mainly over the ring of integers) and [9, IV.3] (over discrete valuation rings). But unfortunately later authors seemed to ignore this, and hence the knowledge of toric schemes is very small compared to the one of toric varieties.Besides yielding a better understanding of the geometry of toric varieties, there are concrete applications of the above generalisation, as the following remark shows.(0) Let X be the toric variety over an algebraically closed field K associated with a fan Σ. A fundamental question in algebraic geometry is then if the Hilbert functor Hilb X/K of X over K is representable, i.e. if the Hilbert scheme of X exists (cf. [7]). If X is projective, then this is indeed the case and follows from Grothendieck's more general result [7, Théorème 3.1]. However, toric varieties are not necessarily projective, and in general it is not known whether their Hilbert schemes exist. Studying Hilb X/K amounts to studying quasicoherent sheaves on the base change X ⊗ K R for every K-algebra R, and it turns out that X ⊗ K R is the same as the toric scheme over R associated with Σ.
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