Residual Finiteness of Finitely Generated Quasi-commutative Semigroups

A semigroup S is called residually finite if for any pair of distinct elements a,b∈S, there exists a congruence P on S such that S/p is finite and (a,b) P. In1958, Malcev proved the following theorem: Any finitely generated abelian semigroupis residually finite . In this paper,we prove that a finitely generated quasi-commu-tative semigroup is residually finite. It generalizes the above theorem.     

Cohen–Macaulay Modules in Dimension >s and Results on Local Cohomology

Let (R,𝔪) be a local ring and s ≥ ?1. Using the notion of M-sequence in dimension > s, we introduce Cohen–Macaulay modules in dimension > s. Among other things concerning Cohen–Macaulay modules in dimension > s, some finiteness results of the support and the associated primes of local cohomology modules are investigated.     

Cohomology of the Orlik–Solomon Algebras and Local Systems

The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.     

Cycle-Finite Algebras with Finitely Many τ-Rigid Indecomposable Modules

We prove that every cycle-finite artin algebra with finitely many isomorphism classes of τ-rigid indecomposable modules is of finite representation type.     

When Cyclic Modules Have Σ-Injective Hulls

Abstract

A theorem of Cartan-Eilenberg (Cartan, H., Eilenberg, S. (1956). Homological Algebra. Princeton: Princeton University Press, pp. 390.) states that a ring Ris right Noetherian iff every injective right module is Σ-incentive. The purpose of this paper is to study rings with the property, called right CSI, that, all cyclic right R-modules have Σ-injective hulls, i.e., injective hulls of cyclic right R-modules are Σ-injective. In this case, all finitely generated right R-modules have Σ-injective hulls, and this implies that Ris right Noetherian for a lengthy list of rings, most notably, for Rcommutative, or when Rhas at most finitely many simple right R-modules, e.g., when Ris semilocal. Whether all right CSIrings are Noetherian is an open question. However, if in addition, R/rad Ris either right Kasch or von Neuman regular (=VNR), or if all countably generated (sermisimple) right R-modules have Σ-injective hulls then the answer is affirmative. (See Theorem A.) We also prove the dual theorems for Δ-injective modules.     

On the Second Cohomology of K?hler Groups

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact K?hler manifold, then virtually H²(G, \mathbb R) 1 0{H^{2}(\Gamma, {\mathbb R}) \ne 0} . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( \mathbbC{\mathbb{C}} -VHS) on the K?hler manifold. We prove the conjecture under some assumption on the \mathbbC{\mathbb{C}} -VHS. We also study some related geometric/topological properties of period domains associated to such a \mathbbC{\mathbb{C}} -VHS.     

The Properties of Finitely Generated Shift-invariant Subspace in L_(2π)~2

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On Fraïssé’s Theorem for Uncountable Classes of Finitely Generated Structures

We prove a weak version of Fraïssé’s theorem for uncountable classes of finitely generated structures.We also solve Hodges’ problem on failure of the complete analog of this theorem for uncountable classes of finitely generated structures.     

When is the Boston mechanism strategy-proof?

Kumano (2013) is the first to investigate the Boston school choice mechanism (BOSM) under restricted priority domains. This paper strengthens and extends his result and shows that the BOSM is strategy-proof, if and only if it is fair, if and only if it is equivalent to the student-optimal stable mechanism (SOSM), and if and only if the number of total seats at any two schools exceeds the number of students.     

When is the Diagonal Functor Frobenius?

Given a complete, cocomplete category 𝒞, we investigate the problem of describing those small categories I such that the diagonal functor Δ: 𝒞 → Functors(I, 𝒞) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → 𝒞 are naturally isomorphic. We find necessary conditions on I for a certain class of categories 𝒞, and, as an application, we give both necessary and sufficient conditions in the two special cases 𝒞 =Set or _R ?, the category of left modules over a ring R.     

When is the Comaximal Graph Split?

Let R be a commutative ring with non zero unity. Let Ω(R) be a graph with vertices as elements of R whose two distinct vertices x and y are adjacent if and only if Rx + Ry = R. A graph (V, E) is said to be a split graph if V is the disjoint union of two sets K and S where K induces a complete subgraph and S is an independent set. We investigate the properties of R when Ω(R) is split.     

A Krull―Schmidt Theorem for Artinian Modules over Local Rings

The paper contains a generalization of Fitting's lemma for Artinian modules of arbitrary length over some non-Noetherian local rings. A counterexample to a strong form of the KrullSchmidt theorem for Artinian modules over a local ring is constructed. Bibliography: 4 titles.     

Approximations of Complete Modules by Complete Big Cohen–Macaulay Modules over a Cohen–Macaulay Local Ring

The Auslander–Buchweitz theory for finitely generated modules over a Cohen–Macaulay local ring is extended to complete modules, finitely generated or not. This also allows us to extend to complete modules some other known facts concerning finitely generated ones.     

When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals?

Recently, quasi-Monte Carlo algorithms have been successfully used for multivariate integration of high dimensiond, and were significantly more efficient than Monte Carlo algorithms. The existing theory of the worst case error bounds of quasi-Monte Carlo algorithms does not explain this phenomenon. This paper presents a partial answer to why quasi-Monte Carlo algorithms can work well for arbitrarily larged. It is done by identifying classes of functions for which the effect of the dimensiondis negligible. These areweightedclasses in which the behavior in the successive dimensions is moderated by a sequence of weights. We prove that the minimalworst caseerror of quasi-Monte Carlo algorithms does not depend on the dimensiondiff the sum of the weights is finite. We also prove that the minimal number of function values in the worst case setting needed to reduce the initial error by ε is bounded byCε^−p, where the exponentp∈ [1, 2], andCdepends exponentially on the sum of weights. Hence, the relatively small sum of the weights makes some quasi-Monte Carlo algorithms strongly tractable. We show in a nonconstructive way that many quasi-Monte Carlo algorithms are strongly tractable. Even random selection of sample points (done once for the whole weighted class of functions and then the worst case error is established for that particular selection, in contrast to Monte Carlo where random selection of sample points is carried out for a fixed function) leads to strong tractable quasi-Monte Carlo algorithms. In this case the minimal number of function values in theworst casesetting is of order ε^−pwith the exponentp= 2. The deterministic construction of strongly tractable quasi-Monte Carlo algorithms as well as the minimal exponentpis open.