Coherence relative to a weak torsion class

Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules

$$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$

such that F₀,..., F_n?1 are finitely generated free and K_n?1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext ⁿ _R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor ^R _n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.

     

The quantum mechanical virial theorem and the absence of positive energy bound states of Schrödinger operators

For a class of potentialsq which may become as singular asr ^?2 at the origin we show that the eigenfunctionsu of the Friedrichs extensionT _Fof (?Δ+q)?C ₀ ^∞ (R ⁿ /{0}) satisfy the virial theorem 2(?Δu,u)=(u,rq,u). Two proofs of this relation are given. One extends the Fock-Weidmann method of scale transformations, the other makes an argument by Finkelstein rigorous and can be viewed of as a substitute of the “commutator proof” of (*) employed in the physical literature. The virial theorem provides a convenient tool for proving the nonexistence of eigenvalues embedded in the continuum or the total absence of eigenvalues ofT _F.     

On a Field-Theoretic Invariant for Extensions of Commutative Rings

Abstract

If R ? T is an extension of (commutative integral) domains, Λ(T/R) is defined as the supremum of the lengths of chains of intermediate fields in the extension k _R (Q ∩ R) ? k _T (Q), where Q runs over the prime ideals of T. The invariant Λ(T/R) is determined in case R and T are adjacent rings and in case Spec(R) = Spec(T) as sets. It is proved that if R is a domain with integral closure R′, then Λ(T/R) = 0 for all overrings T of R if and only if R′ is a Prüfer domain such that Λ(R′/R) = 0. If R ? T are domains such that the canonical map Spec(T) → Spec(R) is a homeomorphism (in the Zariski topology), then Λ(T/R) is bounded above by the supremum of the lengths of chains of rings intermediate between R and T. Examples are given to illustrate the sharpness of the results.     

Rings of formal power series with homeomorphic prime spectra

LetR?T be domains, not fields, such that Spec(R)=Spec(T) as sets; that is, such that the prime ideals ofT coincide, as sets, with those ofR. It is proved that the canonical map Spec(T[[X]])→Spec(R[[X]]) is a homeomorphism. This generalizes a result of Girolami in caseR is a pseudovaluation domain with the SFT (strong finite type)—property andT is its associated valuation domain. The analogous property for polynomial rings is also characterized: Spec(T[X])→Spec(R[X]) is a homeomorphism if and only ifR/M?T/M is a purely inseparable (algebraic) field extension, whereM is the maximal ideal ofR.     

Adjacent Ideals to Simple Complete Ideals in Two-Dimensional Regular Local Rings

Let R be a two-dimensional regular local ring with infinite residue field, and ? be a simple complete residually rational ideal of R of order r which determines R _h . Let 𝒯 be the set of quadratic transforms T of R _h with [T: R _h ] = 1, and 𝒮 the set of simple complete ideals of R of order r which are adjacent to ? from below. If R _h is free respectively a satellite, then there exist T* ∈ 𝒯 respectively T*, T** ∈ 𝒯 and a bijective map between the set 𝒮 and the set 𝒯?{T*} respectively 𝒯?{T*, T**}.     

Power series extensions of half-factorial domains

An integral domain is said to be a half-factorial domain (HFD) if every non-zero element a that is not a unit may be factored into a finite product of irreducible elements, while any other such factorization of a has the same number of irreducible factors. While it is known that a power series extension of a factorial domain need not be factorial, the corresponding question for HFD has been open. In this paper we show that the answer is also negative. In the process we answer in the negative, for HFD, an open question of Samuel for factorial domains by showing that for certain quadratic domains R, and independent variables, Y and T, R[[Y]][[T]] is not HFD even when R[[Y]] is HFD. The proof hinges on Samuel’s theorem to the effect that a power series, in finitely many variables, over a regular factorial domain is factorial.     

(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

Given an (H,R)-Lie coalgebra Γ, we construct (H,R _T )-Lie coalgebra Γ^T through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L ^T , [, ], Δ_T, R _T ) is an (H,R _T )-Lie bialgebra of an ordinary Lie algebra.     

Jordan isomorphisms of upper triangular matrix rings

Let R be a 2-torsionfree ring with identity 1 and let T_n(R), n ? 2, be the ring of all upper triangular n × n matrices over R. We describe additive Jordan isomorphisms of T_n(R) onto an arbitrary ring and generalize several results on this line.     

∗-Polynomial Identities of a Nonsymmetric ∗-Minimal Algebra

Let F be an infinite field of characteristic ≠?2. We study the ?-polynomial identities of the ?-minimal algebra R?=?UT _?(F?⊕?F, F). We describe the generators of T _?(R) and a linear basis of the relatively free algebra of R. When char.F?=?0, these results allow us to provide a complete list of polynomials generating irreducible GL × GL-modules decomposing the proper part of the relatively free algebra of R. Finally, the ?-codimension sequence of R is explicitly computed.     

Generalized intersection patterns and two-symbol balanced arrays

Let m be a fixed integer, m = {0,1,?,m ? 1}; let C be a family of nonvoid subsets of m, and let R be a hereditary subfamily of C. Given finite sets A_m,…,A_m?1 such that ∩ _i?BA_i = Ø for all B ? m, B ? C, the vector of |∩_i?RA_i| (R_j?R) is called a C-supported R-intersection pattern. The characterization of the Y_RC of such patterns is a difficult combinatorial problem even for m=5 and simple families R and C. We study the algebraic structure of the convex cone Y_RC and its dual, and an integer linear-programming aspect of the problem; in particular we introduce the notion of content and pseudocontent. A relaxation leads to quadratic and higher forms over certain subsets of reals. As an application we study the natural link between highly symmetric patterns and two-symbol balanced arrays.     

On universally catenarian pairs

If 1≤n<∞ and R⊆S are integral domains, then (R,S) is called an n-catenarian pair if for each intermediate ring T (that is each ring T such that R⊆T⊆S) the polynomial ring in n indeterminates, T[n] is catenarian. This implies that (R,S) is m-catenarian for all m<n. The main purpose of this paper is to prove that 1-catenarian and universally catenarian pairs are equivalent in several cases. An example of a 1-catenarian pair which is not 2-catenarian is given.     

Weighted sums of squares in local rings and their completions, I

Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let ${\widehat{T}}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if ${{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if Ext¹_R (M, T) = 0{{\rm Ext}^{1}_{R}\,(M, T)\,=\,0} for all torsion modules T, and M is Mittag-Leffler in case the canonical map M?_R ?_{i ? I}Q_i? ?_{i ? I}(M?_RQ_i){M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)} is injective where {Q_i}_{i ? I}{\{Q_i\}_{i\in I}} are arbitrary left R-modules. We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag-Leffler modules.     

On galois groups and their maximal 2-subgroups

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL ₁ of ℚ(T) with Galois groupG. LetL be the subfield ofL ₁ fixed byH. We make the hypothesis thatL ₁ admits a quadratic extensionL ₂ which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL ₁ then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL ₂. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL ₁ admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

Representations of matroids and free resolutions for multigraded modules

Let k be a field, let R=k[x₁,…,xm] be a polynomial ring with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, and let be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T_•(Φ,S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation ? over k of a matroid M a canonical finite complex of finite dimensional k-vector spaces T_•(?) that is a resolution of Ker?. We also show that the length of T_•(?) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map ?.     

Harmonic measure on simply connected domains of fixed inradius

LetD?C be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. ForR>0, letω _D (R) denote the harmonic measure at 0 of the set {z:|z|?R}??D. Then it is shown thatthere exist β>0and C>0such that for each such D,ω _D (R)≤Ce ^?βR ,for every R>0. Thus a natural question is: What is the supremum of all β′s , call it β₀, for which the above inequality holds for every suchD? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β₀ is found. Upper bounds for β₀ can be obtained by constructing examples of domainsD. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.     

On deformation rings of residually reducible Galois representations and R = T theorems

We introduce a new method of proof for R = T theorems in the residually reducible case. We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation ρ ₀ of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents ρ ₁ and ρ ₂. Under some assumptions on Selmer groups associated with ρ ₁ and ρ ₂ we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaïche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R = T-theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic field and a 4-dimensional representation over Q.     

Symmetries of PDE systems and correspondences between jet spaces

Let M be a manifold. A PDE system can be prolonged to another one R^?⊆T^?M (Jiménez et al. (2005) [10]). In analogy with the higher-order symmetries, symmetries of R^? will be called higher-dimensional symmetries of R. For a broad class of PDE systems we prove that every (infinitesimal or finite) symmetry of R comes from another one of R^?. We show that R^? does not have internal (infinitesimal) symmetries (modulo trivial symmetries). This fact allows us, in the infinitesimal case, to compute the internal symmetries of R as external symmetries of R^?. We also give an algorithmic method to obtain solutions of R invariant by a given internal symmetry.     

Simultaneous orderings in function fields

For every Dedekind domain R, Bhargava defined the factorials of a subset S of R by introducing the notion of p-ordering of S, for every maximal ideal p of R. We study the existence of simultaneous ordering in the case S=R=O_K, where O_K is the ring of integers of a function field K over a finite field F_q. We show, that when O_K is the ring of integers of an imaginary quadratic extension K of F_q(T), K=F_q(T)/(Y²-D(T)), then there exists a simultaneous ordering if and only if degD?1.     

Étude asymptotique d'un modèle simple de flammes prémélangées dans un domaine extérieur de RN

We consider a semilinear elliptic equation ?ΔT_ε+u·?T_ε=f_ε(T_ε)(1?T_ε) in outer domains of $\text{R}{}^{\mathrm{N}}$ with Dirichlet's boundary conditions. This Note deals with the questions of existence, uniqueness and the asymptotic behavior of solutions T_ε as ε tends to 0 and the reaction term behaves as a Dirac distribution. Such problems arise in the modelling of premixed flames in the limit of high activations energies. To cite this article: G. Sagon, C. R. Acad. Sci. Paris, Ser. I 336 (2003).