arXiv subjects
Publications and source records attributed to Fred Rohrer.
Fairness and centredness of ideals in commutative rings, i.e., the relations between assassins and weak assassins of a module, its small or large torsion submodule, and the corresponding quotients, are studied. General criteria as well as more specific results about idempotent or nil ideals are given, and several examples are presented.
The behaviour under coarsening functors of simple, entire, or reduced graded rings, of free graded modules over principal graded rings, of superfluous monomorphisms and of homological dimensions of graded modules, as well as adjoints of degree restriction functors, are investigated.
We investigate scalar restriction, scalar extension, and scalar coextension functors for graded modules, including their interplay with coarsening functors, graded tensor products, and graded Hom functors. This leads to several characterisations of epimorphisms of graded rings.
Let $\mathfrak{a}$ be an ideal in a commutative ring $R$. For an $R$-module $M$, we consider the small $\mathfrak{a}$-torsion $\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{a}^n\subseteq(0:_Rx)\}$ and the large $\mathfrak{a}$-torsion $\overline{\Gamma}_{\mathfrak{a}}(M)=\{x\in M\mid\mathfrak{a}\subseteq\sqrt{(0:_Rx)}\}$. This gives rise to two functors $\Gamma_{\mathfrak{a}}$ and $\overline{\Gamma}_{\mathfrak{a}}$ 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 non-noetherian rings.
Let $R$ be a ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. Let $\Gamma_{\mathfrak{a}}$ denote the $\mathfrak{a}$-torsion functor. Conditions are given for the (weakly) associated primes of $\Gamma_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ containing $\mathfrak{a}$, and for the (weakly) associated primes of $M/\Gamma_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ not containing $\mathfrak{a}$.
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. Behaviour of ITI under formation of rings of fractions, tensor products and idealisation is studied. Applications to local cohomology over non-noetherian rings are given.
This is a comprehensive study of the relations between the global, local and pointwise variants of irreducibility and integrity of schemes, including examples and counterexamples, and aimed especially at learners of algebraic geometry.
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 Mustata 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.
A correspondence between quasicoherent sheaves on toric schemes and graded modules over some homogeneous coordinate ring is presented, and the behaviour of several finiteness properties under this correspondence is investigated.
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.
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.
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.
Let X be the toric scheme over a ring R associated with a fan Sigma. It is shown that there are a group B, a B-graded R-algebra S and a graded ideal I of S such that there is an essentially surjective, exact functor ~ from the category of B-graded S-modules to the category of quasicoherent O_X-modules that vanishes on I-torsion modules and that induces for every B-graded S-module F a surjection Xi_F from the set of I-saturated graded sub-S-modules of F onto the set of quasicoherent sub-O_X-modules of ~F. If Sigma is simplicial, the above data can be chosen such that ~ vanishes precisely on I-torsion modules and that Xi_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.
It is investigated how graded variants of integral and complete integral closures behave under coarsening functors and under formation of group algebras.
It is characterized when coarsening functors between categories of graded modules preserve injectivity of objects, and when they commute with graded covariant Hom functors.
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.