Théorie tannakienne non commutative

Inspired by a recent paper by Deligne [2], we extend the Tannaka-Krein duility results (over a field) to the non-commutative situation. To be precise, we establish a 1-1 corresponde:ice between ‘tensorial autonomous categories’ equipped with a ‘fibre functor’ (i. e. tannakian categories without the commutativity condition on the tensor product), and ‘quantum groupoids’ (as defined by Maltsiniotis, [9]) which are ‘transitive’ (7.1.). When the base field is perfect, a quantum groupoid over Spec B is transitive iff it is projective and faithfully fiat over B? _k B. Moreover, the fibre functor is unique up to ‘quantum isomorphism’ (7.6.). Actually, we show Tannaka-Krein duality results in the more general setting where there is no monoidal structure on the category (and the functor); the algebraic object corresponding to such a category is a ‘semi-transitive’ coalgebroid (5.2. and 5.8.).     

On commutative rings which are strongly prüfer

Abstract

As defined by Nicholson [Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278] an element of a ring R is clean if it is the sum of a unit and an idempotent, and a subset A of R is clean if every element of A is clean. It is shown that a semiprimitive Gelfand ring R is clean if and only if Max(R) is zero-dimensional; if and only if for each M ∈ Max(R), the intersection all prime ideals contained in M is generated by a set of idempotents. We also give several equivalent conditions for clean functional rings. In fact, a functional ring R is clean if and only if the set of clean elements is closed under sum; if and only if every zero-divisor is clean; if and only if; R has a clean prime ideal.     

Erdős-Ginzburg-Ziv theorem for finite commutative semigroups

Let \(\mathcal{S}\) be a finite additively written commutative semigroup, and let \(\exp(\mathcal{S})\) be its exponent which is defined as the least common multiple of all periods of the elements in \(\mathcal{S}\) . For every sequence T of elements in \(\mathcal{S}\) (repetition allowed), let \(\sigma(T) \in\mathcal{S}\) denote the sum of all terms of T. Define the Davenport constant \(\mathsf{D}(\mathcal{S})\) of \(\mathcal{S}\) to be the least positive integer d such that every sequence T over \(\mathcal{S}\) of length at least d contains a proper subsequence T′ with σ(T′)=σ(T), and define \(\mathsf{E}(\mathcal{S})\) to be the least positive integer ? such that every sequence T over \(\mathcal{S}\) of length at least ? contains a subsequence T′ with \(|T|-|T'|= \lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil \exp(\mathcal{S})\) and σ(T′)=σ(T). When \(\mathcal{S}\) is a finite abelian group, it is well known that \(\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil\exp (\mathcal{S})=|\mathcal{S}|\) and \(\mathsf{E}(\mathcal{S})=\mathsf{D}(\mathcal{S})+|\mathcal{S}|-1\) . In this paper we investigate whether \(\mathsf{E}(\mathcal{S})\leq \mathsf{D}(\mathcal{S})+ \lceil\frac{|\mathcal{S}|}{\exp(\mathcal {S})} \rceil \exp(\mathcal{S})-1\) holds true for all finite commutative semigroups \(\mathcal{S}\) . We provide a positive answer to the question above for some classes of finite commutative semigroups, including group-free semigroups, elementary semigroups, and archimedean semigroups with certain constraints.     

The nonexistence of sensitive commutative group actions on graphs

In this paper,we observe a special kind of continuous functions on graphs.By estimating the integrals of these functions,we prove that there are no sensitive commutative group actions on graphs.Furthermore,we consider a 1-dimensional continuum composed of a spiral curve and a circle and show that there exist sensitive homeomorphisms on it,which answers negatively a question proposed by Kato in 1993.     

Dedekind’s linear independence theorem over commutative semirings

In this paper, Dedekind’s theorem on the linear independence of homomorphisms of commutative semirings is studied. Furthermore, this theorem is extended to the case of linear independence of compositions of homomorphisms and powers of a derivation. The results obtained in this paper generalize and develop previous results for fields.     

Direct product decompositions of bounded commutative residuated ℓ-monoids

The notion of bounded commutative residuated ℓ-monoid (BCR ℓ-monoid, in short) generalizes both the notions of MV-algebra and of BL-algebra. Let be a BCR ℓ-monoid; we denote by ℓ( ) the underlying lattice of . In the present paper we show that each direct product decomposition of ℓ( ) determines a direct product decomposition of . This yields that any two direct product decompositions of have isomorphic refinements. We consider also the relations between direct product decompositions of and states on . This work was supported by Slovak Research and Development Agency under the contract No APVV-0071-06. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information (grant I/2/2005).     

Two cancellative commutative congruences and group diagrams

Remmers (Adv. Math. 36:283–296, 1980) uses group diagrams in the Euclidean plane to demonstrate how equality in a semigroup S “mirrors” that inside the group G sharing the same presentation with S, when S satisfies Adyan’s condition—no cycles in the left/right graphs of the semigroup’s presentation. Goldstein and Teymouri (Semigroup Forum 47:299–304, 1993) introduce a conjugacy equivalence relation for semigroups S. By closely examining the geometry of annular group diagrams in the plane, they show how their equivalence relation mirrors conjugacy inside G, for S satisfying Adyan’s. In this article we introduce two cancellative commutative congruences. Following their leads, we examine the geometry of group diagrams on closed surfaces of higher genera to demonstrate how these congruences mirror equality inside two naturally associated Abelian quotient groups G/[G,G] and G/G ², respectively. In these instances we can drop Adyan’s condition.     

On simplicial commutative algebras with vanishing André-Quillen homology

In this paper, we study the André-Quillen homology of simplicial commutative ℓ-algebras, ℓ a field, having certain vanishing properties. When ℓ has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an application, indicate how the algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for algebras of finite type over Noetherian rings, having non-zero characteristic. Oblatum 3-III-1999 & 3-V-2000?Published online: 11 October 2000     

An Assmus–Mattson theorem for codes over commutative association schemes

We prove an Assmus–Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with s classes). This in particular generalizes the Assmus–Mattson-type theorems for \(\mathbb {Z}_4\)-linear codes due to Tanabe (Des Codes Cryptogr 30:169–185, 2003) and Shin et al. (Des Codes Cryptogr 31:75–92, 2004), as well as the original theorem by Assmus and Mattson (J Comb Theory 6:122–151, 1969). The weights of a code are s-tuples of non-negative integers in this case, and the conditions in our theorem for obtaining t-designs from the code involve concepts from polynomial interpolation in s variables. The Terwilliger algebra is the main tool to establish our results.     

Leading coefficients of Kazhdan–Lusztig polynomials and fully commutative elements

Let W be a Coxeter group of type . We show that the leading coefficient, μ(x,w), of the Kazhdan–Lusztig polynomial P _x,w is always equal to 0 or 1 if x is fully commutative (and w is arbitrary).     

Vaught’s conjecture for modules over a commutative Prüfer ring

We prove that Vaught’s conjecture is true for modules over a commutative Prüfer ring. It is shown that a positive solution to Vaught’s conjecture for modules over 1-dimensional Noetherian domains would imply the same for modules over finitely presented algebras. This article was written during the visit of the second author to the University of Manchester supported by EPSRC grant GR/L68827. She would like to thank the University for hospitality. Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 419–435, July–August, 1999.     

The spectral theory of commutative C∗-algebras: The constructive Gelfand-Mazur theorem

It is shown, for a commutative C^?-algebra in any Grothendieck topos E, that the locale MFn A of multiplicative linear functionals on A is isomorphic to the locale Max A of maximal ideals of A, extending the classical result that the space of C^?-algebra homomorphisms from A to the field of complex numbers is isomorphic to the maximal ideal space of A, that is, the Gelfand-Mazur theorem, to the constructive context of any Grothendieck topos. The technique is to present Max A, in analogy with our earlier definition of MFn A, by means of a propositional theory which expresses one's natural intuition of the notion involved, and then to establish various properties, leading up to the final result, by formal reasoning within these theories.     

Auslander’s defect formula and a commutative triangle in an exact category

We prove Auslander’s defect formula in an exact category,and obtain a commutative triangle involving the Auslander bijections and the generalized Auslander Reiten duality.     

Gr?bner bases in difference-differential modules with coefficients in a commutative ring

Insa and Pauer presented a basic theory of Grbner bases for differential operators with coefficients in a commutative ring and an improved version of this result was given by Ma et al.In this paper,we present an algorithmic approach for computing Grbner bases in difference-differential modules with coefficients in a commutative ring.We combine the generalized term order method of Zhou and Winkler with SPoly method of Insa and Pauer to deal with the problem.Our result is a generalization of theories of Insa and Pauer,Ma et al.,Zhou and Winkler and includes them as special cases.     

On thick subgroups of uncountable σ-compact locally compact commutative groups

We show that, for any uncountable commutative group (G,+), there exists a countable covering where each Gj is a subgroup of G satisfying the equality card(G/Gj)=card(G). This purely algebraic fact is used in certain constructions of thick and nonmeasurable subgroups of an uncountable σ-compact locally compact commutative group equipped with the completion of its Haar measure.     

Universal problem for Kähler differentials in A-modules: Non-commutative and commutative cases

Let A be an associative and unital K-algebra sheaf, where K is a commutative ring sheaf, and ε an (A, A)-bimodule, that is, a sheaf of (A, A)-bimodules. We construct an (A, A)-bimodulc which is K-isomorphic with the K-module D _K (A, ε) of germs of K-derivations. A similar isomorphism is obtained, this time around with respect to A, between the K-module D _K (A, ε) with the A-module Hom _A (Ω _K (A), ε). where A, in addition of being associative and unital, is assumed to be commutative, and Ω _K (A) denotes the A-module of germs of Kähler differentials. Finally, we expound on functoriality of Kähler differentials.     

Stability of a simple Levi–Civitá functional equation on non-unital commutative semigroups

In this paper, we study the Hyers–Ulam stability of a simple Levi–Civitá functional equation f(x+y)=f(x)h(y)+f(y) and its pexiderization f(x+y)= g(x) h(y)+k(y) on non-unital commutative semigroups by investigating the functional inequalities |f(x+y)?f(x)h(y)?f(y)|≤?? and |f(x+y)?g(x)h(y)?k(y)|≤??, respectively. We also study the bounded solutions of the simple Levi–Civitá functional inequality.     

Normalizer of a group of “diagonal” automorphisms in algebras over a commutative ring

Let S be a faithful algebra over commutative ring R. It is assumed that S is additively generated by its invertible elements. It is shown that the nomalizer of subgroup Aut(S_s) of group Aut(S_R) coincides with the semidirect product Aut(S_S) Aut(S/R),where the second factor is the group of all ring automorphisms of ring S identical on R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 5–8, 1991.