Folgenräume als universelle Generatoren für Klassen lokalkonvexer Räume

Motivated by the known characterizations of equicontinuity in the dual of a Schwartz space, a nuclear space, or a strongly nuclear space,we introduce the concepts of a -sequence and of a ()-sequence in the dual of an arbitrary lcs [E,], and we investigate the corresponding topologies and ₍₎ on E of uniform convergence on these sequences. Here is a normal sequence space such that . Under favorable enough conditions on , including the nuclearity of its normal topology , [,] acts as a universal generator for those lcs [E,] which satisfy =. Under somewhat weaker assumptions on , [,₍₎] is a universal generator for the lcs [E,] with =₍₎. These results cover e.g. the cases of -nuclear spaces and of nuclear spaces known from the recent literature. As an application we show that every non-trivial ultrabornological lcs is representable as an inductive limit of isomorphic copies of [, ( , )], where is any nuclear power series space of infinite type with stable exponent sequence.     

On the representation type of one point extensions of tame concealed algebras

Letk be an algebraically closed field and a finite dimensionalk-algebra. Letq be the quadratic Tits form associated with . If is tame we show thatq is weakly semipositive. Let be a one-point extension of a tame concealed algebra, then is tame iffq is weakly semipositive.     

Large indecomposables over representation-infinite orders and algebras

Let R be a complete discrete valuation domain with quotient field K, and let be an R-order in a semisimple K-algebra. Butler, Campbell, and Kovács have shown that R-free -modules decompose into -lattices when is representation-finite. Using the theory of ladder functors, we prove the converse by constructing indecomposable R-free -modules of infinite rank if is not representation-finite.Received: 23 March 2004     

Isomorphism of one-place functors Ext

Let be an associative ring with identity. One considers the category of left (unitary) -modules m and also the contravariant and the covariant functors Ext ¹ ( ,A) and Ext ¹ (A, ): M_z M. One proves the following results: (1) If the homomorphism of -modules A B induces an isomorphism Ext ¹ ( ,A)Ext ¹ ( ,B), then there exist injective -modules J₁ and J₂ such that AJ₁BJ₂. (2) Every functorial morphism Ext ¹ ( ,A)Ext ¹ ( ,B) induces a certain homomorphism of -modules AB. One also obtains a dual result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 71–74, 1981.     

A family of noetherian rings with their finite length modules under control

We investigate the category mod of finite length modules over the ring =A _k , where is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module E _j gives rise to a quasiprogenerator P _j = A E _j . By a result of K. Fuller, P _j induces a category equivalence from which we deduce that mod _j mod EndP _j . As a consequence we can(1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring such that modA mod(2) find twisted versions of algebras of wild representation type such that itself is of finite or tame representation type (in mod)(3) describe for certain rings the minimal almost split morphisms in mod and observe that almost all of these maps are not almost split in Mod.     

Une variante à la Coxeter du groupe de Steinberg

LetG _n ()be the semi-direct product of the symmetric groupS _n by the Steinberg groupSt _n ()of a ringWe first prove thatG _n ()has a Coxeter-type presentation. The canonical morphism St _n () GL _n ()extends to a group homo G_n() GL _n ()We next determine the kernel of for n = We also give an expression for the generator of the algebraic K group K ₂(Z)of the integers in terms of permutation matrices.     

Computation of Almost Split Sequences with Applications to Relatively Projective and Prinjective Modules

Let be an Artin algebra, let mod be the category of finitely generated -modules, and let Amod be a contravariantly finite and extension closed subcategory. For an indecomposable and not Ext-projective module CA, we compute the almost split sequence 0ABC0 in A from the almost split sequence 0DTrCEC0 in mod. Since the computation is particularly simple if the minimal right A-approximation of DTrC is indecomposable for all indecomposable and not Ext-projective CA, we manufacture subcategories A with the desired property using orthogonal subcategories. The method of orthogonal subcategories is applied to compute almost split sequences for relatively projective and prinjective modules.     

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

Subnormalizer of net subgroups in the general linear group over a ring

Let be a commutative ring in which the elements of the form ²–1, * generate the unit ideal and assume that a is any D-net of ideals of of order n. It is shown that the normalizerN() of the net subgroup G() (RZhMat, 1977, 2A280) coincides with its subnormalizer in GL(n, ). For noncommutative the corresponding result is obtained under the assumptions: 1) in the elements of the form — 1, where runs through all invertible elements of the center of , generate the unit ideal, and 2) the subgroup G() contains the group of block diagonal matrices with blocks of order 2.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 14–19, 1982.     

An improved bound for the de Bruijn–Newman constant

The Riemann hypothesis is equivalent to the conjecture that the de Bruijn–Newman constant satisfies 0. However, so far all the bounds that have been proved for go in the other direction, and provide support for the conjecture of Newman that 0. This paper shows how to improve previous lower bounds and prove that –2.710^–9<. This can be done using a pair of zeros of the Riemann zeta function near zero number 10²⁰ that are unusually close together. The new bound provides yet more evidence that the Riemann hypothesis, if true, is just barely true.     

Exceptional Sequences in Hall Algebras and Quantum Groups

Let be a finite-dimensional hereditary algebra over a finite field k, () and () be, respectively, the Hall algebra and the composition algebra of , be the isomorphism classes of finite dimensional -modules and I the isomorphism classes of simple -modules. We define and , in , to be the right and left derivations of () respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of -mod, we provide an effective algorithm of calculating the root vectors of real Schur roots. This means that we get an inductive method to express u as the combinations of elements u_i in the Hall algebra, where i I and in is any exceptional -module. Because of the canonical isomorphism between the Drinfeld–Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups. In particular, all the root vectors are obtained in this way in the finite type cases.     

Triangular extension identities

It is shown that if the - -bimodule M generates a category of - -bimodules, then the ideal of identities of the triangular extension of the direct sum of algebras and by means of the bimodule M is equal to the product of ideals of identities of the algebras and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 5–11, 1983.     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

Perversité et variation

Let S be the spectrum of a strictly henselian discrete valuation ring with residue characteristic p and =/, where is a prime number p and is an integer 1. For a scheme X of finite type over S and smooth over S along the special fiber X _s outside a closed point x, we study the vanishing cycles complex R() and the tame variation , for in the tame inertia group I _t . In particular, we show that if X is regular, flat over S of relative dimension n1, and is a topological generator of I _t , then R ^q () _x =0 for qn and is an isomorphism. Mathematics Subject Classification (2000):14F20, 14D05, 14D06     

Stabilization theorem for the Milnor K2-functor

Let be an associative ring. For every natural number n there is a canonical homomorphism _n: K_2,n()K₂(), where K₂ is the Milnor functor and K_2,n() the associated unstable K-group. Dennis and Vasershtein have proved that if n is larger than the stable rank of , _n is an epimorphism. It is proved in the article that if n – 1 is greater than the stable rank of , the homomorphism _n is an isomorphism.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 131–152, 1976.     

Division Algebras that Ramify only on a Plane Quartic Curve with Simply Connected Components

A central division algebra over the field of rational functions in two variables with coefficients over an algebraically closed field ramifies along a divisor on P ². If the ramification divisor of is a quartic curve which is the union of simply connected curves, we show that is a symbol algebra and satisfies the index equals exponent equation.     

On nonlinear integral equations and Λ-bounded variation

Summary We discuss the existence or the existence and uniqueness of global and local -bounded variation (BV) solutions as well as continuous BV-solutions of nonlinear Hammerstein and Volterra-Hammerstein integral equations formulated in terms of the Lebesgue integral. Since the space of functions of bounded variation in the sense of Jordan is a proper subspace of functions of -bounded variation and for some class of functions , the space of functions of bounded -variation in the sense of Young is also a proper subspace of the space under consideration, our results extend known results in the literature.     

Differentiation for Orders and Artinian Rings

The method of differentiation for the category -lat of lattices over an order will be extended to integral almost Abelian categories A instead of -lat. In particular, this yields a differentiation for finitely generated left modules over left Artinian rings.     

A Homological Bridge Between Finite and Infinite-Dimensional Representations of Algebras

Given a finite-dimensional algebra , we show that a frequently satisfied finiteness condition for the category -mod) of all finitely generated (left) -modules of finite projective dimension,namely contravariant finiteness of (-mod) in -mod, forces arbitrary modules of finite projective dimension to be direct limits of objects in (-mod). Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimensionconjecture, that is, for the little finitistic dimension of to coincide with the big (this is well known to fail overfinite-dimensional algebras in general).     

Special Fibres and Critical Locus for a Pencil of Plane Curve Singularities

We will analyze the relationships between the special fibres of a pencil of plane curve singularities and the Jacobian curve J of (defined by the zero locus of the Jacobian determinant for any fixed basis ). From the results, we find decompositions of J (and of any special fibre of the pencil) in terms of the minimal resolution of . Using these decompositions and the topological type of any generic pair of curves of , we obtain some topological information about J. More precise decompositions for J can be deduced from the minimal embedded resolution of any pair of fibres (not necessarily generic) or from the minimal embedded resolution of all the special fibres.