The quotient field as a torsion-free covering module

R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom _R (G, M) there exists μ∈Hom _R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.     

The ring as a torsion-free cover

LetR be an integral domain andI a non-zero ideal ofR. The canonical mapR→R/I is called atorsion-free cover ofR/I if everyR-homomorphism from a torsion-freeR-module intoR/I can be factored throughR. The main result of this paper is thatR→R/I is a torsion-free cover if and only ifR is complete in theR-topology andI is an ideal of injective dimension 1. In this caseI is contained in the Jacobson radical ofR. And if Λ is the endomorphism ring ofI, then Λ is a quasi-local domain. IfI is a flatR-module, thenQ→Q/Λ is a torsion-free cover, whereQ is the quotient field ofR. And thenQ/Λ is an indecomposable injectiveR (and Λ) module. Special results are obtained ifR is a Noetherian domain or a Prüfer domain.     

The homomorphismM*⊗N → Hom (M,N)

In this note we prove two theorems. In theorem 1 we prove that if M andN are two non-zero reflexive modules of finite projective dimensions over a Gorenstein local ring, such that Hom (M, N) is a third module of syzygies, then the natural homomorphismM* ⊗N → Hom (M, N) is an isomorphism. This extends the result in [7]. In theorem 2, we prove that projective dimension of a moduleM over a regular local ringR is less than or equal ton if and only if Ext_R ⁿ (M, R) ⊗M → Ext_R ⁿ (M, M) is surjective; in which case it is actually bijective. This extends the usual criterion for the projectivity of a module.     

大子代数中的极大幂零理想和代数同构

§ 1　Introduction and preliminariesLet Hbe a complex Hilbertspace,B(H) be the setof all linear bounded operators.A set Nofprojections on His called a nestif(1 ) Nistotally ordered by the usual oper-ator order,(2 ) Ncontains the zero operator0 and the identity operator I,and(3 ) givenany subset N0 N,the operators sup{ N:N∈N0 } and inf{ N:N∈ N0 } are both mem-bers of N.For N∈Nwith N>0 ,we define N- =sup{ N′∈N:N′相似文献   

The structure of injective covers of special modules

In this paper, we prove that the injective cover of theR-moduleE(R/B)/R/B for a prime ideal B ofR is the direct sum of copies ofE(R/B) for prime ideals D ⊃ B, and if B is maximal, the injective cover is a finite sum of copies ofE(R/B). For a finitely generatedR-moduleM withn generators andG an injectiveR-module, we argue that the natural mapG ⁿ →G ⁿ/Hom _R (M, G) is an injective precover if Ext _R ¹ (M, R) = 0, and that the converse holds ifG is an injective cogenerator ofR. Consequently, for a maximal ideal R ofR, depth_R R ≧ 2 if and only if the natural mapE(R/R) →E(R/R)/R/R is an injective cover and depth_R R > 0.     

Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

On sperner families in which nok sets have an empty intersection III

LetR be anr-element set and ℱ be a Sperner family of its subsets, that is,X ⊈Y for all differentX, Y ∈ ℱ. The maximum cardinality of ℱ is determined under the conditions 1)c≦|X|≦d for allX ∈ ℱ, (c andd are fixed integers) and 2) nok sets (k≧4, fixed integer) in ℱ have an empty intersection. The result is mainly based on a theorem which is proved by induction, simultaneously with a theorem of Frankl.     

Quasi-localizations of ℤ

Motivated by the categorical notion of localizations applied to the quasi-category of abelian groups, we call a homomorphism α: A → B a quasi-localization of abelian groups if for each ϕ ∈ Hom(A,B) there is an n ∈ ℕ and a unique ψ ∈ End(B) such that nϕ = ψ ∘ α. In this case we call B a quasi-localization of A. In this paper we investigate quasi-localizations of the integers ℤ. While it is well-known that localizations of ℤ are just the E-rings, quasi-localizations of ℤ are much more abundant; an injection α: ℤ → M with M torsion-free, is a quasi-localization if and only if, for R = End(M), one has . We call R the ring of the quasi-localization M. Some old results due to Zassenhaus and Butler show that all rings with free additive groups of finite rank are indeed rings of quasi-localizations of ℤ. We will extend this result and show that there are also rings of infinite rank with this property. While there are many realization results of rings R as endomorphism rings of torsion-free abelian groups M in the literature, the group M is usually not contained in the divisible hull of R ⁺, as is required here. We will use a particular case of a category of left R-modules M with a distinguished family of submodules and thus . We will restrict our discussion to the case M = R such that , and in this case we call the family of left ideals E-forcing, not to be confused with the notion of forcing in set theory. We will provide many examples of quasi-localizations M of ℤ, among them those of infinite rank as well as matrix rings for various rings of finite rank.     

Periodic number for Anosov maps

LetX be a compact metric space and letf: X→X be an Anosov map,i.e., an expansive selfmap with the pseudoorbit tracing property (abbr. POTP) (see Lemma 1). IfNn(f) denotes the number of fixed points off ⁿ which we name here then-periodic number then we prove in the case asn tends to infinity thatn ^M ≤N_n(f)≤Hⁿ, whereM andH are two positive integers.     

Graphs and their parallel groups

Given an immersion of a manifoldf: M→R ^n+k , dimensionM=n, the parallel groupP(f) off is formed by the diffeomorphisms ofM such that the normalk-planes at points of each orbit are parallel. In [3] we studied the parallel group of a plane closed curve. Here we concentrate on immersionsf: R ⁿ →R ⁿ⁺¹, special attention being paid to graphs of smooth maps fromR toR. Graphs of smooth mapsf: S ⁿ →R ^m are also dealt with and we characterise those maps of which the graph has nontrivial parallel group. To end up we find a sufficient condition for the triviality of the tangent group.     

The multiplication map for global sections of line bundles and rank 1 torsion free sheaves on curves

LetX be an integral projective curve andL ∃ Pic^a(X),M ∃ Pic^b (X) with h¹(X, L)= h¹(X, M) = 0 andL, M general. Here we study the rank of the multiplication map μ _L,M :H ⁰(X,L)⊗H ⁰(X,M)→H ⁰(X,L⊗M). We also study the same problem whenL andM are rank 1 torsion free sheaves onX. Most of our results are forX with only nodes as singularities.     

On mappings into ℝ2ℓ

LetM be a compact orientable manifold. We know how to calculateX(M), the Euler characteristic ofM, from a stable mapf: M→R, with information only onS(f), the singular set off. This result was extended to stable maps into the plane by H. Levine [L-2] whenM has dimension 2n, and it is also calculated fromS(f). The purpose of this work is to generalize the above result for maps intoR ^2l, wheren≥l. In this caseS(f) is not a manifold. We use the process of resolution of singularities [L-3] to get a homomorphism having only singularities of codimension 1 and use simmilar technics as in [L-2]. Supported by FAPESP and FINEP.     

ℱ-subnormal and ℱ-subabnormal chains in finite groups

The class ℱ_pc of all finite groupsG is defined such thatX≤G implies that there exists an ℱ-subnormal subgroupS ofG containingX such thatX is ℱ-subabnormal inS, where ℱ is a saturated formation, closed under taken subgroups. Groups in ℱ_pc are characterized by ℱ-projectors and ℱ-covering subgroups. Project supported by the National Natural Science Foundation of China (Grant No. 19760001) and the Natural Science Foundation of Guangxi Autonomous Region.     

Remarks on the generic rank of a CR mapping

We study germs of smooth CR mappings between embedded real hypersurfaces in complex spaces of the same dimension. In particular, we are interested in the generic rank of such mappings. IfH:M →M′ is a CR map between two hypersurfacesM andM′, we prove that ifM′ does not contain any germ of a holomorphic manifold then eitherH is constant or the generic rank ofH is odd. We also prove that if there is no formal holomorphic vector field tangent toM, then eitherH is constant or genericallyH is a local diffeomorphism. It follows, as a special case, that ifM andM′ are of D-finite type (in the sense of D’Angelo) thenH is either constant or is generically a local diffeomorphism. Supported by NSF Grant DMS 8901268.     

Nielsen’s theorem for model aspherical manifolds

The main issue of this paper is the discussion of Nielsen’s realisation-problem for aspherical manifolds arising from (generalised) Seifert fiber space constructions. We present sufficient conditions on such “model” aspherical manifoldsM to have that a finite abstract kernel ψ:G → Out (π₁ (M)) can be (effectively) geometrically realised by a group of fiber preserving homeomorphisms ofM if and only if ψ can be realised by an (admissible) group extension 1 → (π₁ (M)) →E’ →G → 1. Then an algebraic approach to a (partial) study of the symmetry ofM is possible. Our result covers all situations already described in literature and we show with an example that we also deal with other types of Seifert fiber space constructions which were not yet treated before. Research Assistant of the Belgian National Fund for Scientific Research (N.F.W.O.)     

Alcuni risultati sugli ideali massimali di anelli di polinomi

Riassunto In questo lavoro viene studiata la cardinalità minima dei sistemi di generatori di ideali massimaliM dell'anello di polinomiR[X ₁, …,X _n] (R dominio d'integrità) tali cheM⊃R=0. In particolare è dimostrato che sen≥2 edR è unS-dominio di dimensione 1, ogni siffatto ideale massimale può essere generato dan elementi.

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Summary This paper is concerned with estimates of the minimal number of generators for maximal idealsM in the polynomial ringR[X ₁, …,X _n] (R an integral domain) such thatM⊃R=0. In particular it is proved that ifn≥2 andR is a 1-dimensionalS-domain, every such maximal ideal can be generated byn elements.

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Lavoro eseguito nell'ambito dell'attività del G.N.S.A.G.A. del C.N.R.     

One-Parameter Families of Modules for Tame Algebras and Bocses

Let Λ be a finite-dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. Consider the family ℱ(u) of the indecomposables M∈mod Λ such that , where is the subspace of morphisms which factorize through semisimple modules. If P,Q are projectives in mod Λ, ℱ(u)(P,Q) is the family of those modules M∈ℱ(u) such that a minimal projective presentation is of the formf_M: P→Q. We prove that if Λ is of tame representation type then each ℱ(P,Q) has only a finite number of isomorphism classes or is parametrized by μ(u,P,Q) one-parameter families. We give an upper bound for this number in terms of u,P and Q. Then we give some sufficient conditions for tame of polynomial growth type. For the proof we consider similar results for bocses. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G60, 16G70, 16G20.     

Abelian groups as endomorphic modules over their endomorphism ring

Let R be an associative ring with a unit and N be a left R-module. The set M _R(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring E _R(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if M _R(N) = E _R(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 229–233, 2007.     

Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:Z →X which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH ¹(Z, O) →H ¹(X, EndE) is surjective. Dedicated to the memory of Professor K G Ramanathan