Frank DeMeyer

On generic polynomials

Abstract We examine over arbitrary fields the possible implications among the concepts due to D. Saltman of generic Galois extension, retract rational extension, the lifting property for Galois extensions, the notions due to G. Smith of generic polynomial, and of descent generic polynomial due to F. DeMeyer.

Computing the Brauer-Long group of Z/2 dimodule algebras

The Brauer-Long group of Z/2 dimodule algebras over a connected commutative ring R in which 2 is invertible is computed in terms of the cohomology of R. Consequences of this calculation are given.

On the Brauer group of toric varieties

We compute the cohomological Brauer group of a normal toric variety whose singular locus has codimension less than or equal to 2 everywhere

Homomorphisms of progenerator modules

Abstract Homomorphisms defined on progenerator modules over a commutative ring are studied through the introduction of a relation called homotopy, which is coarser than the usual notion of equivalence. The set of homotopy classes of homomorphisms forms a commutative monoid with operation induced by tensor product. For a given commutative ring, classification of homomorphisms is carried out by determining the algebraic structure of the monoid of homotopy classes and then giving a representing homomorphism for each class. This classification is carried out explicitly for Dedekind domains.

Invariants of group rings

Abstract The Schur group, uniform (distribution) group, Schur index, and Schur exponent are examined in the context of separable group rings of finite groups over commutative rings.

TWISTED FORMS OF FINITE ÉTALE EXTENSIONS AND SEPARABLE POLYNOMIALS

Examples of twisted forms of finite etale extensions and separable polynomials are calculated using Mayer-Vietoris sequences for non-abelian cohomology.

Local to global equivalence for homomorphisms

his paper studies the connection between sheaf cohomology and local vs. global equivalence for homomorphisms of finitely presented modules over commutative rings. The connection between cohomology and the local/global problem is worked out, basic cohomology sequences are derived and connections with the known interpretation of local/global equivalence in terms of modules and some calculations are given.

GENERALIZED EQUIVALENCE OF MATRICES OVER PREFER DOMAINS

Two m x n matrices A,B over a commutative ring R are equivalent i,.ve,-tible nmtrices P,O over R with B PAQ. While any m x n matrix over a principle ideal dota.i, ca, be diagonalized, the same is not true for Dedekind domains. The first author and T..I. Ford ittroduced a coarser equivalence relation on matrices called homotopy and showed any x mtrix over a. Dedekind domain is homotopic to a direct stun of x 2 matrices. In this article wc giw, necessary and sufficient conditions on a Prefer domain that any m x n matrix be homotolfic to a.

Etale cohomology of toric varieties defined by infinite fans

Abstract Toric varieties are a special class of normal rational varieties defined by means of a combinatorial object called a fan. The fan defining the variety gives a designated open cover and a dictionary describing how the open sets in the cover intersect. When all the open sets in the designated cover and their intersections have trivial cohomology, the cohomology of the whole variety is determined by the fan. In this article we calculate the low degree cohomology on the Zariski and Etale sites of toric varieties over an algebraically closed field of characteristic = 0. These computations extend (with different proofs) most of the results in DeMeyer et al., (1993) where the fan was assumed to be finite. As a result, we show any countable direct product of finite cyclic groups is isomorphic to the cohomological Brauer group of some toric variety. All the definitions and basic facts about toric varieties which we use can be found in Oda (1988).

The Brauer Long group of Z/2 dimodule algebras

Beginning with the papers of G. Azumaya [2] and M. Auslander and O. Goldman [1] introducing the Brauer group B(R) of a commutative ring R, there have followed a series of extensions of the Brauer group of division algebras over a field to satisfy various needs. A. Grothendeick viewed the Brauer group of a commutative ring as the local part of a Brauer group of schemes and related the Brauer group to the second etale cohomology group of the scheme with values in the units scheaf [11]. At about the same time, C. T. C. Wall introduced in [14] a Brauer group of equivalence classes of Z/2 graded algebras over a field with multiplication induced by a twisted tensor product to study the Witt ring of quadratic forms. Wall’s construction was extended to commutative rings R by H. Bass and C. Small [13], and this group is now called the Brauer Wall group of R and denoted BW(R). A Brauer group of algebras over a field graded by an arbitrary finite abelian group was introduced by M. Knus [9], and extended to arbitrary commutative rings (with a twisted multiplication) by L. Childs, G. Garfinkel and M. Orzech [5]. The Childs-Garfinkel-Orzech construction contained the Brauer Wall group as a special case. An “equivariant Brauer group” of algebras on which a fixed group acted as a group of automorphisms was constructed by O. Frolich and C. T. C. Wall. In his thesis [10], F. W. Long introduced a Brauer group of dimodule algebras on which a grading group G acted as a group of automorphisms which included the affine versions of all the previous extensions of the Brauer group as subgroups. Long’s group is now called the Brauer Long group of R and is denoted BD(R, G). After Long introduced his group, a steady stream of papers have considered the properties and calculations of BD(R, G) and its siblings. Some of these are listed among the references at the end of this report.