Non-Absoluteness of Model Existence at ℵω\aleph_\omega

In [FHK13], the authors considered the question whether model-existence of $L_{\omega_1,\omega}$-sentences is absolute for transitive models of ZFC, in the sense that if $V \subseteq W$ are transitive models of ZFC with the same ordinals, $\varphi\in V$ and $V\models "\varphi \text{ is an } L_{\omega_1,\omega}\text{-sentence}"$, then $V \models "\varphi \text{ has a model of size } \aleph_\alpha"$ if and only if $W \models "\varphi \text{ has a model of size } \aleph_\alpha"$. From [FHK13] we know that the answer is positive for $\alpha=0,1$ and under the negation of CH, the answer is negative for all $\alpha>1$. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each $\alpha>1$, except the case when $\alpha=\omega$ which is an open question in [FHK13]. We answer the open question by providing a negative answer under GCH even for $\alpha=\omega$. Our examples are incomplete sentences. In fact, the same sentences can be used to prove a negative answer under GCH for all $\alpha>1$ assuming the consistency of a Mahlo cardinal. Thus, the large cardinal assumption is relaxed from a supercompact in [FHK13] to a Mahlo cardinal. Finally, we consider the absoluteness question for the $\aleph_\alpha$-amalgamation property of $L_{\omega_1,\omega}$-sentences (under substructure). We prove that assuming GCH, $\aleph_\alpha$-amalgamation is non-absolute for $1<\alpha<\omega$. This answers a question from [SS]. The cases $\alpha=1$ and $\alpha$ infinite remain open. As a corollary we get that it is non-absolute that the amalgamation spectrum of an $L_{\omega_1,\omega}$-sentence is empty

Properties of chains of prime ideals in an amalgamated algebra along an ideal

Let $f:A \to B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we study the amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by ${A\Join^fJ}$), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D'Anna and Fontana in 2007, and other classical constructions (such as the $A+ XB[X]$, the $A+ XB[[X]]$ and the $D+M$ constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension.Comment: J. Pure Appl. Algebra (to appear

Galois-stability for Tame Abstract Elementary Classes

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this context. We assume that \K is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: (1) Galois-stability above the Hanf number implies that \kappa(K) is less than the Hanf number. Where \kappa(K) is the parallel of \kapppa(T) for f.o. T. (2) We use (1) to construct Morley sequences (for non-splitting) improving previous results of Shelah (from Sh394) and Grossberg & Lessmann. (3) We obtain a partial stability-spectrum theorem for classes categorical above the Hanf number.Comment: 23 page

Continuous Family of Invariant Subspaces for R-diagonal Operators

We show that every R-diagonal operator x has a continuous family of invariant subspaces relative to the von Neumann algebra generated by x. This allows us to find the Brown measure of x and to find a new conceptual proof that Voiculescu's S-transform is multiplicative. Our considerations base on a new concept of R-diagonality with amalgamation, for which we give several equivalent characterizations.Comment: 35 page

Upward Stability Transfer for Tame Abstract Elementary Classes

Grossberg and VanDieren have started a program to develop a stability theory for tame classes. We prove, for instance, that for tame abstract elementary classes satisfying the amlagamation property and for large enough cardinals kappa, stability in kappa implies stability in kappa^{+n} for each natural number n

Bi-Amalgamated algebras along ideals

Let $f: A\rightarrow B$ and $g: A\rightarrow C$ be two commutative ring homomorphisms and let $J$ and $J'$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J')$. The \emph{bi-amalgamation} of $A$ with $(B, C)$ along $(J, J')$ with respect to $(f,g)$ is the subring of $B\times C$ given by $$A\bowtie^{f,g}(J,J'):=\big\{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J\times J'\big\}.$$ This paper investigates ring-theoretic properties of \emph{bi-amalgamations} and capitalizes on previous works carried on various settings of pullbacks and amalgamations. In the second and third sections, we provide examples of bi-amalgamations and show how these constructions arise as pullbacks. The fourth section investigates the transfer of some basic ring theoretic properties to bi-amalgamations and the fifth section is devoted to the prime ideal structure of these constructions. All new results agree with recent studies in the literature on D'Anna-Finocchiaro-Fontana's amalgamations and duplications.Comment: 15 page