Ring Of Global Sections Research Articles

On the geometry of Prüfer intersections of valuation rings

Let $F$ be a field, let $D$ be a subring of $F$ and let $Z$ be an irreducible subspace of the space of all valuation rings between $D$ and $F$ that have quotient field $F$. Then $Z$ is a locally ringed space whose ring of global sections is $A = \bigcap_{V \in Z}V$. All rings between $D$ and $F$ that are integrally closed in $F$ arise in such a way. Motivated by applications in areas such as multiplicative ideal theory and real algebraic geometry, a number of authors have formulated criteria for when $A$ is a Pr\"ufer domain. We give geometric criteria for when $A$ is a Pr\"ufer domain that reduce this issue to questions of prime avoidance. These criteria, which unify and extend a variety of different results in the literature, are framed in terms of morphisms of $Z$ into the projective line ${\mathbb{P}}^1_D$

Rings of differential operators on toric varieties

Let be a finite dimensional toric variety over an algebraically closed field of characteristic zero, k. Let be the sheaf of differential operators on . We show that the ring of global sections, is a finitely generated Noetherian k-algebra and that its generators can be explicitly found. We prove a similar result for the sheaf of differential operators with coefficients in a line bundle.

Cohen-Macaulay seminormalizations of unions of linear subspaces

This paper is concerned with the unions of subspaces of an afIine space regarded as affine varieties. Let R denote the co-ordinate ring of such a union. Then according to a theorem of Orecchia’s [S], the seminormaliza- tion S of R can be represented as the section ring (or pull-back) of the sheaf d of the co-ordinate rings on the intersection poset X of the subspaces. In this paper, we give a homological condition on X and d equivalent to S being Cohen-Macaulay (Theorem 4.3). Two particular cases of this result have been known before. The Reisner Theorem [6] covered the case of co-ordinate subspaces (when R = S is the face ring of a complex) and Geramita and Weibel [2] soIved the problem for planes. The main idea of the proof is to construct a resolution C of S consisting of S-modules of a known depth and then estimate the depth of S by the length of C. This idea was applied in [7] to the section rings of flasque sheaves. Since in this paper the sheaves are not necessarily flasque, the con- struction of C is a bit more complicated and involves the local Tech cohomology of X with coefficients in &. A drawback of the method is that it gives very little information about R itself when it is not seminormal. This paper is organized as follows. In Section 1, the terminology and background of sheaves on posets is given, In Section 2, we study the (Tech) cohomology of a covering of a poset with coefficients in a sheaf. The next section is technical. In it we compare the depth of the rings of sections on certain closed subsets with their depth as modules over the ring of global sections. In Section 4, we prove the main result of the paper. Finally, Section 5 contains examples and some unsolved problems.

Sheaves of Noncommutative Algebras and the Beilinson-Bernstein Equivalence of Categories

Let $X$ be an irreducible algebraic variety defined over a field $k$, let $\mathcal {R}$ be a sheaf of (noncommutative) noetherian $k$-algebras on $X$ containing the sheaf of regular functions $\mathcal {O}$ and let $R$ be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of $R$ obtained from the local sections of $\mathcal {R}$) there is an equivalence between the category of $R$-modules and the category of sheaves of $\mathcal {R}$-modules which are quasicoherent as $\mathcal {O}$-modules. This shows that the equivalence of categories established by Beilinson and Bernstein as the first step in their proof of the KazhdanLusztig conjectures (where $R$ is a primitive factor ring of the enveloping algebra of a complex semisimple Lie algebra, and $\mathcal {R}$ is a sheaf of twisted differential operators on a generalised flag variety) is valid for more fundamental reasons than is apparent from their work.

Sheaves of noncommutative algebras and the Beilinson-Bernstein equivalence of categories

Let $X$ be an irreducible algebraic variety defined over a field $k$, let $\mathcal {R}$ be a sheaf of (noncommutative) noetherian $k$-algebras on $X$ containing the sheaf of regular functions $\mathcal {O}$ and let $R$ be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of $R$ obtained from the local sections of $\mathcal {R}$) there is an equivalence between the category of $R$-modules and the category of sheaves of $\mathcal {R}$-modules which are quasicoherent as $\mathcal {O}$-modules. This shows that the equivalence of categories established by Beilinson and Bernstein as the first step in their proof of the KazhdanLusztig conjectures (where $R$ is a primitive factor ring of the enveloping algebra of a complex semisimple Lie algebra, and $\mathcal {R}$ is a sheaf of twisted differential operators on a generalised flag variety) is valid for more fundamental reasons than is apparent from their work.

Commutative regular rings and Boolean-valued fields

In this paper we present an equivalence between the category of commutative regular rings and the category of Boolean-valued fields, i.e., Boolean-valued sets for which the field axioms are true. The author used this equivalence in [12] to develop a Galois theory for commutative regular rings. Here we apply the equivalence to give an alternative construction of an algebraic closure for any commutative regular ring (the original proof is due to Carson [2]).Boolean-valued sets were developed in 1965 by Scott and Solovay [10] to simplify independence proofs in set theory. They later were applied by Takeuti [13] to obtain results on Hilbert and Banach spaces. Ellentuck [3] and Weispfenning [14] considered Boolean-valued rings which consisted of rings and associated Boolean-valued relations. (Lemma 4.2 shows that their equality relation is the same as the one used in this paper.) To the author's knowledge, the present work is the first to employ the Boolean-valued sets of Scott and Solovay to obtain results in algebra.The idea that commutative regular rings can be studied by examining the properties of related fields is not new. For several years algebraists and logicians have investigated commutative regular rings by representing a commutative regular ring as a subdirect product of fields or as the ring of global sections of a sheaf of fields over a Boolean space (see, for example, [9] and [8]). These representations depend, as does the work presented here, on the fact that the set of central idempotents of any ring with identity forms a Boolean algebra. The advantage of the Boolean-valued set approach is that the axioms of classical logic and set theory are true in the Boolean universe. Therefore, if the axioms for a field are true for a Boolean-valued set, then other properties of the set can be deduced immediately from field theory.

Projective modules over rings with many units

Projective modules over rings with many units