Local Rings of Bounded Cohen–Macaulay Type

Let be a complete local Cohen–Macaulay (CM) ring of dimension one. It is known that R has finite CM type if and only if R is reduced and has bounded CM type. Here we study the one-dimensional rings of bounded but infinite CM type. We will classify these rings up to analytic isomorphism (under the additional hypothesis that the ring contains an infinite field). In the first section we deal with the complete case, and in the second we show that bounded CM type ascends to and descends from the completion. In the third section we study ascent and descent in higher dimensions and prove a Brauer–Thrall theorem for excellent rings. Presented by J. HerzogMathematics Subject Classifications (2000) 13C05, 13C14, 13H10.     

Noetherian types of homogeneous compacta and dyadic compacta

The Noetherian type of a space is the least κ such that it has a base that is κ-like with respect to reverse inclusion. Just as all known homogeneous compacta have cellularity at most c, they satisfy similar upper bounds in terms of Noetherian type and related cardinal functions. We prove these and many other results about these cardinal functions. For example, every homogeneous dyadic compactum has Noetherian type ω. Assuming GCH, every point in a homogeneous compactum X has a local base that is c(X)-like with respect to containment. If every point in a compactum has a well-quasiordered local base, then some point has a countable local π-base.     

A homological approach to representations of algebras I: The wild case

We prove that the pure global dimension of a polynomial ring over an integral domain k in a finite or countable number n?2 of commuting (non-commuting, resp.) variables is t + 1, provided $\text{|k| = ?}{}_{\mathrm{t}}$. As an application, we determine the pure global dimension of wild algebras of quiver type, also (in case k is an algebraically closed field) of the wild local and wild commutative algebras of finite k-dimension.     

Powerful digraphs

We introduce the concept of a powerful digraph and establish that a powerful digraph structure is included into the saturated structure of each nonprincipal powerful type p possessing the global pairwise intersection property and the similarity property for the theories of graph structures of type p and some of its first-order definable restrictions (all powerful types in the available theories with finitely many (> 1) pairwise nonisomorphic countable models have this property). We describe the structures of the transitive closures of the saturated powerful digraphs that occur in the models of theories with nonprincipal powerful 1-types provided that the number of nonprincipal 1-types is finite. We prove that a powerful digraph structure, considered in a model of a simple theory, induces an infinite weight, which implies that the powerful digraphs do not occur in the structures of the available classes of the simple theories (like the supersimple or finitely based theories) that do not contain theories with finitely many (> 1) countable models.     

Automorphism Group of q-Quantum Torus Lie Algebra with q a Root of Unity*

Let α be an automorphism of a ring R. We study the skew Armendariz of Laurent series type rings (α-LA rings), as a generalization of the standard Armendariz condition from polynomials to skew Laurent series. We study on the relationship between the Baerness and p.p. property of a ring R and these of the skew Laurent series ring R[[x, x ^?1; α]], in case R is an α-LA ring. Moreover, we prove that for an α-weakly rigid ring R, R[[x, x ^?1; α]] is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of S _?(R) has a generalized countable join in R. Various types of examples of α-LA rings are provided.     

A FORMALISM FOR SOME CLASS OF FORCING NOTIONS

We introduce a class of forcing notions, called forcing notions of type S, which contains among other Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for this forcing notions, which allows all standard tricks for iterations or products with countable supports of Sacks forcing. On the other hand it does not involve internal combinatorial structure of conditions of iterations or products. We prove that the class of forcing notions of type S is closed under products and certain iterations with countable supports.     

Modules with Few Types over a Commutative Valuation Ring

It is proved that a module M over a countable commutative valuation ring has few models if and only if M is ω-stable. Let T be an arbitrary complete theory of modules over either a countable serial ring or a countable commutative Prüfer ring or a countable hereditary Noetherian prime ring. It is proved that Martin's conjecture is true for T. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.     

分担四个或五个小函数的亚纯函数

本文证明了两个非常数亚约函数在具有五个ＩＭ分担小函数的前提下附加一定条件,则这两个函数必恒等,并研究了两个具有四个ＣＭ分担小函数的非常数亚纯函数之间的关系,推广了Ｎｅｖａｎｌｉｎｎａ四值定理．     

The finite-dimensional decomposition property in non-Archimedean Banach spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property（OFDDP） if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck＇s approximation theory,where an open problem is the following： Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E.Does D have the OFDDP？ In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP.Next we prove that,however,for certain classes of Banach spaces of countable type,the OFDDP is preserved by taking finite-codimensional subspaces.     

Pure injectivity of n-cotilting modules: the Prüfer and the countable case

We will prove that if a ring R is either countable or a Prüfer domain, then n-cotilting R-modules are pure injective. We can apply and modify a very nice argument due to Jensen and Lenzing [14] showing that algebraic compactness follows from splitting of canonical direct sums in cartesian products.Received: 3 February 2004     

On Ext-indices of ring extensions

In this paper, we investigate the finiteness of Ext-indices for certain ring extensions. In this direction, we introduce some conjectures and discuss the relationships among them. We also prove these conjectures in some special cases. Furthermore, we prove that the trivial extension of an Artinian local ring by its residue class field is always of finite Ext-index, and prove a generalization of the Auslander-Reiten conjecture for this type of ring.     

Automorphisms of Chevalley Groups of Type G 2 Over Local Rings Without 1/2

In this paper, we prove that every automorphism of a Chevalley group of type G ₂ over a commutative local ring without 1/2 is the composition of a ring automorphism and a conjugation by some matrix.     

M-gap conjecture and m-normal theories

We find a small weakly minimal theory with an isolated weakly minimal type ofM-rank ∞ and an isolated weakly minimal type of arbitrarily large finiteM-rank. These examples lead to the notion of an m-normal theory. We prove theM-gap conjecture for m-normalT. In superstable theories with few countable models we characterize traces of complete types as traces of some formulas. We prove that a 1-based theory with few countable models is m-normal. We investigate generic subgroups of small superstable groups. We compare the notions of independence induced by measure (μ-independence) and category (m-independence). Research supported by KBN grant 2 P03A 006 09.     

Local variational principle concerning entropy of a sofic group action

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group.     

Topological simplicity,commensurator super-rigidity and nonlinearities of Kac-Moody groups (appendix by P. Bonvin)

We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator super-rigidity theorem proved in the Appendix by P. Bonvin.     

A note on countable models of 1-based theories

 We prove that the existence of a nonisolated type having a finite domain and which is orthogonal to ?in a 1-based theory implies that it has a continuum nonisomorphic countable models. Received: 6 March 2000 Published online: 12 July 2002     

Koszul Modules and Gorenstein Algebras

I first define Koszul modules, which are a generalization to arbitrary rank of complete intersections. After a study of some of their properties, it is proved that Gorenstein algebras of codimension one or two over a local or graded CM ring are Koszul modules, thus generalizing a well known statement for rank one modules. The general techniques used to describe Koszul modules are then used to obtain a structure theorem for Gorenstein algebras in codimension one and two, over a local or graded CM ring.     

Products with linear and countable type factors

The basic theorem presented shows that the product of a linearly ordered space and a countable (regular) space is normal. We prove that the countable space can be replaced by any of a rather large class of countably tight spaces. Examples are given to prove that monotone normality cannot replace linearly ordered in the base theorem. However, it is shown that the product of a monotonically normal space and a monotonically normal countable space is normal.

     

Hypersurface sections and S2-modules

We prove a hypersection section type criterion for a module over a noetherian local ring to satisfy Serre' condition S₂.     

Normalizers of Chevalley Groups of Type G 2 Over Local Rings Without 1/2

In this paper, we prove that every element of the linear group GL₁₄(R) normalizing the Chevalley group of type G ₂ over a commutative local ring R without 1/2 belongs to this group up to some multiplier. This allows us to improve our classification of automorphisms of these Chevalley groups showing that an automorphism-conjugation can be replaced by an inner automorphism. Therefore, it is proved that every automorphism of a Chevalley group of type G ₂ over a local ring without 1/2 is a composition of a ring and an inner automorphisms.