A desingularization theorem of Néron type

Summary LetA→A′ be a regular morphism between two noetherian rings. Then for everyA-algebra of finite typeB, everyA-morphism fromB toA′ can be factorised through aB-algebra of finite typeB′ which is standard (smooth) overA and which is ?as smooth as possible? overB (i.e. except above the singular locus ofB overA). This result is a positive answer to one conjecture due to M. Artin.

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Riassunto SiaA→A′ un omomorfismo regolare di anelli noetheriani. Si dimostra che per ogniA-algebraB di tipo finito, ogniA-omomorfismoB→A′ si può fattorizzare mediante unaB-algebraB′ di tipo finito che è liscia suA ed è ?la più liscia posibile? suB (più precisamente liscia suB meno il luogo singolare diB suA). Questo risultato è una risposta affermativa a una congettura di M. Artin.

     

Some embeddings of infinite-dimensional spaces

It is shown that ifA is a weakly infinite-dimensional subset of a metric spaceR then aG _δ setB ofR exists such thatA⊆B andB is weakly infinite-dimensional. A similar result holds for a set having strong transfinite inductive dimension. As a consequence each weakly infinite-dimensional metric space possesses a weakly infinite-dimensional complete metric extension. A similar result holds also for a space having strong transfinite inductive dimension.     

Balleans of topological groups

A subset S of a topological group G is called bounded if, for every neighborhood U of the identity of G, there exists a finite subset F such that S ⊆ FU, S ⊆ UF. The family of all bounded subsets of G determines two structures on G, namely the left and right balleans B _l (G) and B _r (G), which are counterparts of the left and right uniformities of G. We study the relationships between the uniform and ballean structures on G, describe all topological groups admitting a metric compatible both with uniform and ballean structures, and construct a group analogue of Higson’s compactification of a proper metric space.     

Chaos caused by a strong-mixing measure-preserving transformation

The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m _i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer _i such that lim_i→∞ f ^r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.     

Commutators with finite spectrum II

The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum a（[dx, d′x]） is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then there exists an element a in the socle of A such that [d, d′] is the inner derivation implemented bv a.     

Measures on the Product of Compact Spaces

 If K is an uncountable metrizable compact space, we prove a “factorization” result for a wide variety of vector valued Borel measures μ defined on K ⁿ . This result essentially says that for every such measure μ there exists a measure μ′ defined on K such that the measure μ of a product A ₁ × ⋯ × A _n of Borel sets of K equals the measure μ′ of the intersection A ₁′∩⋯∩A _n ′, where the A _i ′’s are certain transforms of the A _i ’s. Partially supported by DGICYT grant PB97-0240. Received August 23, 2001; in revised form March 21, 2002     

Partitions of the natural numbers into infinitely oscillating bases and nonbases

The setA of nonnegative integers is a basis if every sufficiently large integerx can be written in the formx=a+a′ witha, a′∈A. IfA is not a basis, then it is a nonbasis. We construct a partition of the natural numbers into a basisA and a nonbasisB such that, as random elements are moved one at a time fromA toB, fromB toA, fromA toB, …, the setA oscillates from basis to nonbasis to basis … and the setB oscillates simultaneously from nonbasis to basis to nonbasis…     

Basic Propositional Calculus II. Interpolation

Let ℒ and ? be propositional languages over Basic Propositional Calculus, and ℳ = ℒ∩?. Weprove two different but interrelated interpolation theorems. First, suppose that Π is a sequent theory over ℒ, and Σ∪ {C⇒C′} is a set of sequents over ?, such that Π,Σ⊢C⇒C′. Then there is a sequent theory Φ over ℳ such that Π⊢Φ and Φ, Σ⊢C⇒C′. Second, let A be a formula over ℒ, and C ₁, C ₂ be formulas over ?, such that A∧C ₁⊢C ₂. Then there exists a formula B over ℳ such that A⊢B and B∧C ₁⊢C ₂. Received: 7 January 1998 / Published online: 18 May 2001     

Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.     

Flattening Antichains

A flat antichain is a collection of incomparable subsets of a finite ground set, such that |B|−|C|≤1 for every two members B, C. Using Lieby’s results, we prove the Flat Antichain Conjecture, which says that for any antichain there exists a flat antichain having the same cardinality and average set size.     

The triadjoint of an orthosymmetric bimorphism

Let A and B be two Archimedean vector lattices and let (A′)′ _n and (B′)′ _n be their order continuous order biduals. If Ψ: A × A → B is a positive orthosymmetric bimorphism, then the triadjoint Ψ***: (A′)′ _n × (A′)′ _n → (B′)′ _n of Ψ is inevitably orthosymmetric. This leads to a new and short proof of the commutativity of almost f-algebras.     

Lower bound on the profile of degree pairs in cross-intersecting set systems

We raise the following problem. For natural numbers m, n ≥ 2, determine pairs d′, d″ (both depending on m and n only) with the property that in every pair of set systems A, B with |A| ≤ m, |B| ≤ n, and A ∩ B ≠ 0 for all A ∈ A, B ∈ B, there exists an element contained in at least d′ |A| members of A and d″ |B| members of B. Generalizing a previous result of Kyureghyan, we prove that all the extremal pairs of (d′, d″) lie on or above the line (n − 1) x + (m − 1) y = 1. Constructions show that the pair (1 + ɛ / 2n − 2, 1 + ɛ / 2m − 2) is infeasible in general, for all m, n ≥ 2 and all ɛ > 0. Moreover, for m = 2, the pair (d′, d″) = (1 / n, 1 / 2) is feasible if and only if 2 ≤ n ≤ 4. The problem originates from Razborov and Vereshchagin’s work on decision tree complexity. Research supported in part by the Hungarian Research Fund under grant OTKA T-032969.     

On Some Rational Triangles

We prove that every set of n ≥ 3 points in \mathbbR²{\mathbb{R}^2} can be slightly perturbed to a set of n points in \mathbbQ²{\mathbb{Q}^2} so that at least 3(n − 2) of mutual distances between those new points are rational numbers. Some special rational triangles that are arbitrarily close to a given triangle are also considered. Given a triangle ABC, we show that for each ε > 0 there is a triangle A′B′C′ with rational sides and at least one rational median such that |AA′|, |BB′|, |CC′| < ε and a Heronian triangle A′′B′′C′′ with three rational internal angle bisectors such that A^￠￠, B^￠￠, C^￠￠ ? \mathbbQ²{A^{\prime\prime}, B^{\prime\prime}, C^{\prime\prime} \in \mathbb{Q}^2} and |AA′′|, |BB′′|, |CC′′| < ε.     

An acyclic extension of the braid group

We relate Artin's braid groupB _℞=lim_→B_n to a certain groupF′ ofpl-homeomorphisms of the interval. Namely, there exists a short exact sequence 1→B _∞→A→F′→1 whereH _kA=0,k≥1.     

Separable algebras over infinite fields are 2-generated and finitely presented

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M ₂(R), under an additional assumption that both pairs generate M ₂(R) as an R-algebra.     

r-Maximal major subsets

The question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (A⊂_rm B) arose in attempts to extend Lachlan’s decision procedure for the αε-theory of ℰ^*, the lattice of r.e. sets modulo finite sets, and Soare’s theorem thatA andB are automorphic if their lattice of supersets ℒ^*(A) and ℒ^*(B) are isomorphic finite Boolean algebras. We characterize the r.e. setsA with someB⊂_rm A as those with a Δ₃ function that for each recursiveR _i specifiesR _i or as infinite on and to be preferred in the construction ofB. There are r.e.A andB with ℒ^*(A) and ℒ^*(B) isomorphic to the atomless Boolean algebra such thatA has anrm subset andB does not. Thus 〈ℰ^*,A〉 and 〈ℰ^*,B〉 are not even elementarily equivalent. In every non-zero r.e. degree there are r.e. sets with and withoutrm subsets. However the classF of degrees of simple sets with norm subsets satisfies . The authors were partially supported by NSF Grants MCS 76-07258, MCS 77-04013 and MCS 77-01965 respectively.     

Hausdorff measures of different dimensions are not Borel isomorphic

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H ^s ) and (ℝ, B, H ^t ) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H ^d is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d. We also prove this statement in a more general form: If A ⊂ ℝⁿ is Borel and ƒ: A → ℝ^m is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A). Partially supported by the Hungarian Scientific Research Fund grant no. T 49786.     

On some infinite dimensional linear groups

Let F be a field, A be a vector space over F, and GL(F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim _F (BFG/B) is finite. A subspace B is called almost G-invariant, if dim _F (B/Core _G (B)) is finite. In the present article we begin the study of subgroups G of GL(F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = charF.      

Structure of locally graded CDN[)-groups

We introduce the notion of CDN[)-groups:G is a CDN[)-group if, for any pair of its subgroupsA andB such thatA is a proper nonmaximum subgroup, ofB, there exists a normal subgroupN which belongs toG and satisfies the inequalitiesA≤N<B. Fifteen types of nilpotent non-Dedekind groups and nine types of nonnilpotent locally graded groups of this kind are obtained. Ukrainian Economic Financial Institute, Irpen. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 383–388, March, 1999.