Daniel Herden

The Existence of Large E(R)-Algebras That Are Sharply Transitive Modules

Let R be any principal ideal domain of cardinality < 2ℵ0 , but not a field. We will construct arbitrarily large E(R)-algebras A (of size ≥ 2ℵ0 ) which are at the same time principal ideal domains over R. It follows that the automorphism group of the R-module R A acts sharply transitive on the pure elements of R A. In answering a question of Emmanuel Dror Farjoun, the existence of such large uniquely transitive (UT-modules) for R = ℤ was shown in Göbel and Shelah (2004). The new method, passing first through ring theory, simplifies the arguments; this idea, using localizations, comes from Herden (2005). We applied it recently in Göbel and Herden (2007b) to find such E(R)-algebras of size ≤ 2ℵ0 .

An upper cardinal bound on absolute E-rings

We show that for every abelian group A of cardinality ≥ κ(ω) there exists a generic extension of the universe, where A is countable with 2 N 0 injective endomorphisms. As an immediate consequence of this result there are no absolute E-rings of cardinality > κ(ω). This paper does not require any specific prior knowledge of forcing or model theory and can be considered accessible also for graduate students.

Constructing sharply transitive R-modules of rank ⩽

Abstract In this note we will give an elementary proof of the existence of sharply transitive R-modules M over principal ideal domains R. An R-module is sharply transitive (or a UT-module) if its R-automorphism group acts sharply transitively on the pure elements of M. We will assume that M is torsion-free; thus pure elements are simply those elements divisible only by units of R in M. We provide examples of UT-modules of rank ⩽ , while the existence of UT-modules of rank ⩾ was shown recently in Göbel and Shelah [R. Göbel and S. Shelah. Uniquely transitive torsion-free abelian groups. In Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Applied Math. 236 (Marcel Dekker, 2004), pp. 271–290.] using the more complicated machinery of prediction principles. The existence of countable abelian UT-groups, which follows from this note, was left open in earlier works. Here we require and exploit the existence of algebraically independent elements over the base ring R. (Thus we will need |R| < .) First we will convert the UT problem on modules (as suggested in Herden [D. Herden. Uniquely transitive R-modules. Ph.D. thesis. University of Duisburg-Essen, Campus Essen (2005).]) into a problem on suitable R-algebras. This reduces its solution to a few simple steps and makes the proofs more transparent, requiring only basic results in module theory.

The Hilbert series and

Abstract Let V be a finite-dimensional representation of the complex circle C × determined by a weight vector a ∈ Z n . We study the Hilbert series Hilb a ( t ) of the graded algebra C [ V ] C a × of polynomial C × -invariants in terms of the weight vector a of the C × -action. In particular, we give explicit formulas for Hilb a ( t ) as well as the first four coefficients of the Laurent expansion of Hilb a ( t ) at t = 1 . The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomials that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of C [ V ] C a × in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.

a

We will prove the following theorem: Let D be the ring of algebraic integers of a finite Galois field extension F of \(\mathbb{Q}\) and E a D-algebra such that E is a locally free D-module of countable rank and all elements of E are algebraic over F. Then there exists a left D-submodule M ⊇ E of FE = E ⊗ D F such that the left multiplications by elements of E are the only D-linear endomorphisms of M.

-invariant of circle invariants

Abstract A group homomorphism e : H → G {e:H\to G} is a cellular cover of G if for every homomorphism φ : H → G {\varphi:H\to G} there is a unique homomorphism φ ¯ : H → H {\bar{\varphi}:H\to H} such that φ ¯ ⁢ e = φ {\bar{\varphi}e=\varphi} . Group localizations are defined dually. The main purpose of this paper is to establish 2 ℵ 0 {2^{\aleph_{0}}} varieties of groups which are not closed under taking cellular covers. This will use the existence of a special Burnside group of exponent p for a sufficiently large prime p as a key witness. This answers a question raised by Göbel in [12]. Moreover, by using a similar witness argument, we can prove the existence of 2 ℵ 0 {2^{\aleph_{0}}} varieties not closed under localizations. Finally, the existence of 2 ℵ 0 {2^{\aleph_{0}}} varieties of groups neither closed under cellular covers nor under localizations is presented as well.

An Extension of M. C. R. Butler’s Theorem on Endomorphism Rings

An

Group varieties not closed under cellular covers and localizations

R

Separable

-module

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M