Fred Rohrer

Completions of fans

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.

Quasicoherent sheaves on toric schemes

Abstract Let X be the toric scheme over a ring R associated with a fan Σ . It is shown that there are a group B , a B -graded R -algebra S and a graded ideal I ⊆ S such that there is an essentially surjective, exact functor • ˜ from the category of B -graded S -modules to the category of quasicoherent 풪 X -modules that vanishes on I -torsion modules and that induces for every B -graded S -module F a surjection Ξ F from the set of I -saturated graded sub- S -modules of F onto the set of quasicoherent sub- 풪 X -modules of F ˜ . If Σ is simplicial, the above data can be chosen such that • ˜ vanishes precisely on I -torsion modules and that Ξ F is bijective for every F . In case R is noetherian, a toric version of the Serre–Grothendieck correspondence is proven, relating sheaf cohomology on X with B -graded local cohomology with support in I .

Torsion functors with monomial support

The dependence of torsion functors on their supporting ideals is investigated, especially in the case of monomial ideals of certain subrings of polynomial algebras over not necessarily Noetherian rings. As an application it is shown how flatness of quasicoherent sheaves on toric schemes is related to graded local cohomology.

The geometry of toric schemes

Abstract 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.

Injective modules and torsion functors

ABSTRACT A commutative ring is said to have ITI with respect to an ideal 𝔞 if the 𝔞-torsion functor preserves injectivity of modules. Classes of rings with ITI or without ITI with respect to certain sets of ideals are identified. Behavior of ITI under formation of rings of fractions, tensor products, and idealization is studied. Applications to local cohomology over non-noetherian rings are given.

Coarsening of Graded Local Cohomology

Some criteria for graded local cohomology to commute with coarsening functors are proven, and an example is given where graded local cohomology does not commute with coarsening.

Graded integral closures

It is investigated how graded variants of integral and complete integral closures behave under coarsening functors and under formation of group algebras.

Torsion functors, small or large

Let

An avoidance principle with an application to the asymptotic behaviour of graded local cohomology

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