On the quadratic character of quadratic units

Let be a prime. Let a,b∈Z with p?a(a²+b²). In the paper we mainly determine by assuming p=c²+d² or p=Ax²+2Bxy+Cy² with AC−B²=a²+b². As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1).     

Isomorphisms between unital C-algebras

It is shown that every almost linear bijection of a unital C^∗-algebra A onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all unitaries u∈A, all y∈A, and n=0,1,2,…, and that almost linear continuous bijection of a unital C^∗-algebra A of real rank zero onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all , all y∈A, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C^∗-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all x∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C^∗-algebra isomorphisms between unital C^∗-algebras.     

Size bipartite Ramsey numbers

Let B, B₁ and B₂ be bipartite graphs, and let B→(B₁,B₂) signify that any red-blue edge-coloring of B contains either a red B₁ or a blue B₂. The size bipartite Ramsey number is defined as the minimum number of edges of a bipartite graph B such that B→(B₁,B₂). It is shown that is linear on n with m fixed, and is between c₁n²2ⁿ and c₂n³2ⁿ for some positive constants c₁ and c₂.     

Noetherianity and Ext

Let R=R₀⊕R₁⊕R₂⊕? be a graded algebra over a field K such that R₀ is a finite product of copies of K and each R_i is finite dimensional over K. Set J=R₁⊕R₂⊕? and . We study the properties of the categories of graded R-modules and S-modules that relate to the noetherianity of R. We pay particular attention to the case when R is a Koszul algebra and S is the Koszul dual to R.     

Castelnuovo-Mumford regularity for complexes and weakly Koszul modules

Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and the category of finitely generated graded left A-modules. Following Jørgensen, we define the Castelnuovo-Mumford regularity of a complex in terms of the local cohomologies or the minimal projective resolution of M^•. Let A^! be the quadratic dual ring of A. For the Koszul duality functor , we have . Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear modules) over A^!. As an application, refining a result of Herzog and Römer, we show that if J is a monomial ideal of an exterior algebra E=?〈y₁,…,y_d〉, d≥3, then the (d−2)nd syzygy of E/J is weakly Koszul.     

Quadratic algebras with Ext algebras generated in two degrees

Green and Marcos (2005) [2] call a graded k-algebra δ-Koszul if the corresponding Yoneda algebra is finitely generated and there exists a function δ:N→N such that is zero if j≠δ(i). For any integer m≥3 we exhibit a noncommutative quadratic δ-Koszul algebra for which the Yoneda algebra is generated in degrees (1,1) and (m,m+1). These examples answer a question of Green and Marcos. These algebras are not Koszul but m-Koszul (in the sense of Backelin).     

The structure of max-min hyperplanes

In this article, continuing [12,13], further contributions to the theory of max-min convex geometry are given. The max-min semiring is the set endowed with the operations ⊕=max,⊗=min in . A max-min hyperplane (briefly, a hyperplane) is the set of all points satisfying an equation of the form

a₁⊗x₁⊕…⊕a_n⊗x_n⊕a_n+1=b₁⊗x₁⊕…⊕b_n⊗x_n⊕b_n+1,     

Cubic residues and binary quadratic forms

Let p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p?u²−dv² and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=PUn−QUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.     

Codimensions of algebras and growth functions

Let A be an algebra over a field F of characteristic zero and let cn(A), , be its sequence of codimensions. We prove that if cn(A) is exponentially bounded, its exponential growth can be any real number >1. This is achieved by constructing, for any real number α>1, an F-algebra Aα such that exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.     

Torsion theories induced from commutative subalgebras

We begin a study of torsion theories for representations of finitely generated algebras U over a field containing a finitely generated commutative Harish-Chandra subalgebra Γ. This is an important class of associative algebras, which includes all finite W-algebras of type A over an algebraically closed field of characteristic zero, in particular, the universal enveloping algebra of (or ) for all n. We show that any Γ-torsion theory defined by the coheight of the prime ideals of Γ is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in have the same coheight. Hence, the coheight of these associated prime ideals is an invariant of a given simple U-module. This implies the stratification of the category of U-modules controlled by the coheight of the associated prime ideals of Γ. Our approach can be viewed as a generalization of the classical paper by Block (1981) [4]; it allows, in particular, to study representations of beyond the classical category of weight or generalized weight modules.     

Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals

Let M₁,…,M_n be right modules over a ring R. Suppose that the endomorphism ring of each module M_i has at most two maximal right ideals. Is it true that every direct summand of M₁⊕?⊕M_n is a direct sum of modules whose endomorphism rings also have at most two maximal right ideals? We show that the answer is negative in general, but affirmative under further hypotheses. The endomorphism ring of uniserial modules, that is, the modules whose lattice of submodules is linearly ordered under inclusion, always has at most two maximal right ideals, and Pavel P?íhoda showed in 2004 that the answer to our question is affirmative for direct sums of finitely many uniserial modules.     

Rudin's Dowker space, strong base-normality and base-strong-zero-dimensionality

It is well known that every pair of disjoint closed subsets F₀,F₁ of a normal T₁-space X admits a star-finite open cover U of X such that, for every U∈U, either or holds. We define a T₁-space X to be strongly base-normal if there is a base B for X with |B|=w(X) satisfying that every pair of disjoint closed subsets F₀,F₁ of X admits a star-finite cover B^′ of X by members of B such that, for every B∈B^′, either or holds. We prove that there is a base-normal space which is not strongly base-normal. Moreover, we show that Rudin's Dowker space is strongly base-(collectionwise)normal. Strong zero-dimensionality on base-normal spaces are also studied.     

Toeplitz operators associated to commuting row contractions

Let H be a complex Hilbert space and let {Tn}_n?1 be a sequence of commuting bounded operators on H such that . Let denote the space of all operators X in B(H) for which and suppose that . We will show that there exists a triple {K,Γ,{Un}_n?1} where K is a Hilbert space, Γ:K→H is a bounded operator and {Un}_n?1⊂B(K) is a sequence of commuting normal operators with such that TnΓ=ΓUn for n?1, and for which the mapping Y?ΓYΓ^∗ is a complete isometry from the commutant of {Un}_n?1 onto the space . Moreover we show that the inverse of this mapping can be extended to a ^∗-homomorphism

     

On bimodules determining stable equivalences

In the paper one shows that for two indecomposable non-simple self-injective algebras over a field K we have that if the functor induces a stable equivalence then the bimodule _AN_B is contained in the frame of a connected component in the Auslander-Reiten quiver Γ_A⊗KB^op.     

Chem classes and Lie-Rinehart algebras

Let A be a F-algebra where F is a field, and let W be an A-module of finite presentation. We use the linear Lie-Rinehart algebra V_W of W to define the first Chern-class c₁(W) in , where U in Spec(A) is the open subset where W is locally free. We compute explicitly algebraic V_W-connections on maximal Cohen-Macaulay modules W on the hypersurface-singularities B_mn2 = x^m + yⁿ + z², and show that these connections are integrable, hence the first Chern-class c₁(W) vanishes. We also look at indecomposable maximal Cohen-Macaulay modules on quotient-singularities in dimension 2, and prove that their first Chern-class vanish.     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .